Abstract
Twodimensional and quasitwodimensional materials are important nanostructures because of their exciting electronic, optical, thermal, chemical and mechanical properties. However, a singlelayer nanomaterial may not possess a particular property adequately, or multiple desired properties simultaneously. Recently a new trend has emerged to develop nanoheterostructures by assembling multiple monolayers of different nanostructures to achieve various tunable desired properties simultaneously. For example, transition metal dichalcogenides such as MoS_{2} show promising electronic and piezoelectric properties, but their low mechanical strength is a constraint for practical applications. This barrier can be mitigated by considering grapheneMoS_{2} heterostructure, as graphene possesses strong mechanical properties. We have developed efficient closedform expressions for the equivalent elastic properties of such multilayer hexagonal nanohetrostructures. Based on these physicsbased analytical formulae, mechanical properties are investigated for different heterostructures such as grapheneMoS_{2}, graphenehBN, graphenestanene and staneneMoS_{2}. The proposed formulae will enable efficient characterization of mechanical properties in developing a wide range of applicationspecific nanoheterostructures.
Introduction
A generalized analytical approach is presented to derive closedform formulae for the effective inplane elastic moduli of hexagonal multiplanar nanostructures and heterostructures. Hexagonal nanostructural forms are common in various twodimensional and quasitwodimensional materials. The fascinating properties of graphene^{1}, a twodimensional allotrope of carbon with hexagonal nanostructure, has led to an enormous interest and enthusiasm among the concerned scientific community for investigating more prospective twodimensional and quasitwodimensional materials that could possess interesting electronic, optical, thermal, chemical and mechanical characteristics^{2,3,4}. The interest in such hexagonal twodimensional materials has expanded over the last decade from hBN, BCN, graphene oxides to Chalcogenides like MoS_{2}, MoSe_{2} and other forms of twodimensional materials like stanene, silicene, sermanene, phosphorene, borophene etc.^{5,6}. Among these twodimensional materials, hexagonal honeycomblike nanostructure is a prevalent structural form^{3}. Four different classes of singlelayer materials with hexagonal nanostructure exist from a geometrical point of view, as shown in Fig. 1(a–d). For example, graphene^{7} consists of a single type of atom (carbon) to form a honeycomblike hexagonal lattice structure in a single plane, while there is a different class of materials that possess hexagonal monoplanar nanostructure with different constituent atoms such as hBN^{8}, BCN^{9} etc. Unlike these monoplanar hexagonal nanostructures, there are plenty of other materials that have the atoms placed in multiple planes to form a hexagonal top view. Such multiplanar hexagonal nanostructures may be consisted of either a single type of atom (such as stanene^{10,11}, silicene^{11,12}, germanene^{11,12}, phosphorene^{13}, borophene^{14} etc.), or different atoms (such as MoS_{2} ^{15}, WS_{2} ^{16}, MoSe_{2} ^{17}, WSe_{2} ^{16}, MoTe_{2} ^{18} etc.). Even though these twodimensional materials show promising electronic, optical, thermal, chemical and mechanical characteristics for exciting future applications, a single nanomaterial may not possess a particular property adequately, or multiple desired properties simultaneously. To mitigate this lacuna, recently a new trend has emerged to develop nanoheterostructures by assembling multiple monolayers of different nanostructures for achieving various tunable desired properties simultaneously.
Although the singlelayer of twodimensional materials have hexagonal lattice nanostructure (topview) in common, their outofplane lattice characteristics are quite different, as discussed in the preceding paragraph. Subsequently, these materials exhibit significantly different mechanical and electronic properties. For example, transition metal dichalcogenides such as MoS_{2} show exciting electronic and piezoelectric properties, but their low inplane mechanical strength is a constraint for any practical application. In contrast, graphene possesses strong inplane mechanical properties. Moreover, graphene is extremely soft in the outofplane direction with a very low bending modulus, whereas the bending modulus of MoS_{2} is comparatively much higher, depending on their respective singlelayer thickness^{19}. Having noticed that graphene and MoS_{2} possess such complementary physical properties, it is a quite rational attempt to combine these two materials in the form of a grapheneMoS_{2} heterostructure, which could exhibit the desired level of electronic properties and inplane as well as outofplane strengths. Besides intense research on different two dimensional hexagonal nanostructural forms, recently the development of novel applicationspecific heterostructures has started receiving considerable attention from the scientific community due to the tremendous prospect of combining different single layer materials in intelligent and intuitive ways to achieve several such desired physical and chemical properties^{20,21,22,23,24,25,26}.
The hexagonal nanoheterostructures can be broadly classified into three categories based on structural configuration, as shown in Fig. 1: heterostructure containing only monoplanar nanostructures (such as graphenehBN heterostructure)^{22,23,27}, heterostructure containing both monoplanar and multiplanar nanostructures (such as grapheneMoS_{2} heterostructure^{19,21}, graphenestanene heterostructure^{24}, phosphorenegraphene heterostructure^{28}, phosphorenehBN heterostructure^{28}, multilayer graphenehBNTMDC heterostructure^{26}) and heterostructure containing only multiplanar nanostructures (such as staneneMoS_{2} heterostructure^{25}, MoS_{2}WS_{2} heterostructure^{20}). Recently different forms of multilayer heterostructures have started receiving immense attention from the scientific community for showing interesting chemical, thermal, optical, electronic and transport properties^{24,25,29,30}. Even though the heterostructures show various exciting physical and chemical characteristics, effective mechanical properties such as Young’s moduli and Poisson’s ratios are of utmost importance for accessing the viability in application of such nanoheterostructures in various nanoelectromechanical systems. The research in this field is still in a very nascent stage and investigations on elastic properties of these builtup structural forms are very scarce to find in literature^{20,21}.
The common practises to investigate these nanostructures are first principle studies/abinitio and molecular dynamics, which can reproduce the results of experimental analysis with the cost of computationally expensive and time consuming supercomputing facilities. Moreover, availability of interatomic potentials can be a practical barrier in carrying out molecular dynamics simulation for nanoheterostructures, which are consisted of multiple materials. The accuracy of molecular dynamics simulation depends on the interatomic potentials and the situation can become worse in case of nanoheterostructures due to the possibility of having lesser accuracy for builtup structural forms. Molecular mechanics based analytical closed form formulae are presented by many researchers for materials having hexagonal nanostructures in a single layer such as graphene, hBN, stanene, MoS_{2} etc.^{7,8,31,32,33}. This approach of mechanical property characterization for singlelayer nanostructures is computationally very efficient, yet accurate and physically insightful. However, the analytical models concerning twodimesional hexagonal nanostructures developed so far are limited to singlelayer structural forms; development of efficient analytical approaches has not been attempted yet for nanoheterostructures. Considering the future prospect of research in this field, it is essential to develop computationally efficient closedform formulae for the elastic moduli of nanohetrostructures that can serve as a ready reference for the researchers without the need of conducting expensive and time consuming molecular dynamics simulations or laboratory experiments. This will accelerate the process of novel material development based on the applicationspecific need of achieving multiple tunable properties simultaneously to a desirable extent.
In this article, we aim to address the strong rationale for developing a generalized compact analytical model leading to closedform and high fidelity expressions for characterizing the mechanical properties of a wide range of hexagonal nanoheterostructures. Elastic properties of four different heterostructures (graphenehBN, grapheneMoS_{2}, graphenestanene and staneneMoS_{2}), belonging to all the three classes as discussed in the preceding paragraphs, are investigated considering various stacking sequences. The analytical formulae for elastic moduli of heterostructures are applicable to any number of different constituent singlelayer materials with multiplanar or monoplanar hexagonal nanostructures.
Results
Closedform analytical formulae for the elastic moduli of heterostructures
In this section, the closedform analytical expressions of elastic moduli for generalized multiplaner hexagonal nanoheterostructures are presented. The molecular mechanics based approach for obtaining the equivalent elastic properties of atomic bonds is welldocumented in scientific literature^{31,34,35}. Besides that the mechanics of monoplanar hexagonal honeycomblike structure is found to be widely investigated across different length scales^{36,37,38,39,40}. Therefore, the main contribution of this article lies in proposing computationally efficient and generalized analytical formulae for nanoheterostructures (having constituent singlelayer materials with monoplanar and multiplanar structural form) and thereby presenting new results for various stacking sequence of different nanoheterostructures belonging to the three different classes as described in the preceding section (grapheneMoS_{2}, graphenehBN, graphenestanene and staneneMoS_{2}).
For atomic level behaviour of nanoscale materials, the total interatomic potential energy can be expressed as the sum of various individual energy terms related to bonding and nonbonding interactions^{34}. Total strain energy (E) is expressed as the sum of energy contributions from bending of bonds (E _{ b }), bond stretching (E _{ s }), torsion of bonds (E _{ t }) and energies associated with nonbonded terms (E _{ nb }) such as the van der Waals attraction, the core repulsions and the coulombic energy (refer to Fig. 2).
However, among all the energy components, effect of bending and stretching are predominant in case of small deformation^{31,35}. For the multiplanar hexagonal nanostructures (such as stanene and MoS_{2}), the strain energy caused by bending consists of two components, inplane component (E _{ bI }) and outofplane component (E _{ bO }). The outofplane component becomes zero for monoplanar nanostructures such as graphane and hBN. Thus the total interatomic potential energy (E) can be expressed as
where Δl, Δθ and Δα denote the change in bond length, inplane and outofplane angle respectively. The quantities k _{ r } and k _{ θ } represents the force constants for bond stretching and bending respectively. The molecular mechanics parameters (k _{ r } and k _{ θ }) and structural mechanics parameters (EA and EI) of a uniform circular beam with crosssectional area A, length l, Young’s modulus E, and second moment of area I, are related as: \({K}_{r}=\frac{EA}{l}\) and \({k}_{\theta }=\frac{EI}{l}\) ^{31,34,35}. Based on this relationship, the closed form expressions for the effective elastic moduli of multilayer hexagonal nanoheterostructures are derived following a multistage idealization scheme using force equilibrium and deformation compatibility conditions. The closed form expressions for the two inplane Young’s moduli of nanoheterostructures are derived as
The subscript i in the above expressions indicates the molecular mechanics and geometrical properties (as depicted in Fig. 2(a,b)) corresponding to i ^{th} layer of the heterostructure. The overall thickness of the heterostructure is denoted by t. n represents the total number of layers in the heterostructure. Expressions for the two inplane Poisson’s ratios are derived as
Here ν _{12} and ν _{21} represent the inplane Poisson’s ratios for loading directions 1 and 2 respectively. Thus the elastic moduli of a hexagonal nanoheterostructure can be obtained using the closedform analytical formulae (Equations 3–6) from molecular mechanics parameters (k _{ r } and k _{ θ }), bond length (l), inplane bond angle (ψ) and outofplane angle (α), which are welldocumented in the molecular mechanics literature. The analytical formulae are valid for small deformation of the structure (i.e. the linear region of stressstrain curve). The effect of interlayer stiffness contribution due LennardJones potentials are found to be negligible for the inplane elastic moduli considered in this study and therefore, neglected in the analytical derivation (refer to section 7 of the supplementary material).
Validation and analytical predictions for the elastic moduli of heterostructures
Results are presented for the effective elastic moduli of hexagonal multilayer nanoheterostructures based on the formulae proposed in the preceding section. As investigations on nanoheterostructures is a new and emerging field of research, the results available for the elastic moduli of different forms of heterostructures is very scarce in scientific literature. We have considered four different nanoheterostructures to present the results: grapheneMoS_{2}, graphenehBN, graphenestanene and staneneMoS_{2} (belonging to the three categories as depicted in the introduction section). Though all these four heterostructures have received attention from the concerned scientific community for different physical and chemical properties recently, only the grapheneMoS_{2} heterostructure has been investigated using molecular dynamics simulation for the Young’s modulus among all other elastic moduli^{20,21}. Thus we have validated the proposed analytical formulae for Young’s moduli of grapheneMoS_{2} heterostructure with available results from literature. New results are presented for the two inplane Poisson’s ratios of grapheneMoS_{2} heterostructure using the analytical formulae, which are validated by carrying out separate molecular dynamics simulations. Having the developed analytical formulae validated for the two Young’s moduli and Poisson’s ratios, new results are provided for the other three considered heterostructures accounting for the effect of stacking sequence. Moreover, it can be noted that for single layer of the heterostructure (i.e. for n = 1), the proposed analytical formulae can be used to predict the effective elastic moduli of monoplanar (i.e. α = 0) and multiplanar (i.e. α ≠ 0) materials. The analytical predictions for the Young’s moduli and Poisson’s ratios of such singlelayer materials are further validated with reference results from literature, as available.
As shown in Tables 1–5, in the case of singlelayer hexagonal nanostructures (n = 1) belonging to all the four classes as described in the preceding section (graphene, hBN, stanene and MoS_{2}), the inplane Young’s moduli obtained using the proposed analytical formulae are in good agreement with reported values in literature for graphene, hBN, stanene and MoS_{2}. These observations corroborate the validity of the proposed analytical formulae in case of a singlelayer. However, in case of Poisson’s ratios, the reported values in scientific literature for graphene and hBN show wide range of variability, while the reference values of Poisson’s ratios for stanene and MoS_{2} are very scarce in available literature. The results predicted by the proposed formulae agree well with most of the reported values for Poisson’s ratios.
Table 1 presents the value of two Young’s moduli obtained from the proposed analytical formulae for nanoheterostructures considering different stacking sequences of graphene and MoS_{2}. The results are compared with the numerical values reported in scientific literature. It can be noted that the difference between E _{1} and E _{2} is not recognized in most of the previous investigations and the results presented as E _{1} = E _{2}. The Young’s moduli E _{1} and E _{2} are found to be different for multiplanar singlelayer nanostructural forms (such as stanene and MoS_{2}). A similar trend has been reported before by Li^{41} for MoS_{2}. Thus the effective Young’s moduli of the heterostructures with at least one layer of multiplanar structural form is expected to exhibit different E _{1} and E _{2} values. In Table 1 it can be observed that for single and bilayer of graphene E _{1} = E _{2}, while for single and bilayer of MoS_{2} E _{1} ≠ E _{2}. In case of heterostructures consisting of both graphene and MoS_{2} the value of E _{2} is observed to be higher than E _{1}. However, the numerical values of E _{1} for different stacking sequences are found to be in good agreement with the values of Young’s modulus reported in literature (presumably obtained for direction1) corroborating the validity of the developed closedform expressions. We have carried out separate molecular dynamics simulations for graphene – MoS_{2} heterostructures to validate the analytical predictions of Poisson’s ratios, as Poisson’s ratios have not been reported for graphene–MoS_{2} heterostructures in literature. The analytical predictions of Poisson’s ratios reported in Table 1 are found to be in good agreement with the results of molecular dynamics simulations. Similar to the results of Young’s moduli for grapheneMoS_{2} heterostructure, the two inplane Poisson’s ratios (ν _{12} and ν _{21}) are found to have different values when at least one multiplanar structural form is present in the heterostructure. Thus having the analytical formulae for all the elastic moduli validated, we have provided new results for three other nanoheterostructures in the following paragraphs based on Equations 3–6.
Table 2 provides the results for elastic moduli of graphenehBN heterostructure considering different stacking sequences. It is observed that the two Young’s moduli and two inplane Poisson’s ratios are equal (i.e. E _{1} = E _{2} and ν _{12} = ν _{21}) in case of graphenehBN heterostructure as these are consisted of only monoplanar structural forms. Table 3 presents the results for elastic moduli of graphenestanene heterostructure considering different stacking sequences. As stanene has a multiplanar structural form, the two Young’s moduli and two inplane Poisson’s ratios show different values (i.e. E _{1} ≠ E _{2} and ν _{12} ≠ ν _{21}) when at least one of the constituent layers of the heterostructure is stanene. Table 4 presents the results for elastic moduli of staneneMoS_{2} heterostructure considering different stacking sequences. As stanene and MoS_{2} both have multiplanar structural form, the two Young’s moduli and two inplane Poisson’s ratios show considerably different values (i.e. E _{1} ≠ E _{2} and ν _{12} ≠ ν _{21}). The results of different elastic moduli corresponding to various stacking sequences are noticed to have an intermediate value between the respective elastic modulus for single layer of the constituent materials, as expected on a logical basis.
The physics based analytical formulae for nanoheterostructures presented in this article are capable of obtaining the elastic moduli corresponding to any stacking sequence of the constituent layer of nanomaterials. However, from the expressions it can be discerned that the numerical values of elastic moduli actually depend on the number of layers of different constituent materials rather than their exact stacking sequences. From a mechanics viewpoint, this is because of the fact that the inplane properties are not a function of the distance of individual constituent layers from the neutral plane of the entire heterostructure. Figures 3, 4, 5, 6 present the variation of different elastic moduli with number of layers of the constituent materials considering the four different heterostructures belonging from the three different categories, as described in the preceding section. It is observed that the trend of variation for two Young’s moduli and two inplane Poisson’s ratios are similar for grapheneMoS_{2} and graphenestanene heterostructures with little difference in the actual numerical values. The variation of elastic moduli for graphenehBN heterostructure are presented for E _{1} and ν _{12} as the numerical values are exactly same for the two Young’s moduli and two inplane Poisson’s ratios, respectively. The plots furnished in this section can readily provide an idea about the trend of variation of elastic moduli with stacking sequence of multilayer nanoheterostructures in a comprehensive manner; exact values of the elastic moduli corresponding to various stacking sequences can be easily obtained using the proposed computationally efficient closedform formulae.
Discussion
We have presented computationally efficient analytical closedform expressions for the effective elastic moduli of multilayer nanoheterostructures, wherein individual layers may have multiplanar (i.e. α ≠ 0) or monoplanar (i.e. α = 0) configurations. It is interesting to notice that the generalized analytical formulae developed for the Young’s moduli of heterostructures can be reduced to the closedform expressions provided by Shokrieh and Rafiee^{31} for graphene considering singlelayer (i.e. n = 1), α = 0 and ψ = 30°.
It can be noted from the presented results that the singlelayer materials having regular monoplanar hexagonal nanostructures (such as graphene and hBN) have equal value of elastic modulus in two perpendicular directions (i.e. E _{1} = E _{2} and ν _{12} = ν _{21}). However, for singlelayer materials with multiplanar nanostructure, the elastic modulus for direction2 is more than that of direction1, even though the difference is not significant. Similar observation is found to be reported in literature^{41}. For singlelayer of materials, the formulae of elastic moduli deduced from Equations 3–6 by replacing n = 1, perfectly obeys the Reciprocal theorem (i.e. E _{1} ν _{21} = E _{2} ν _{12})^{42}. In case of nanoheterostructures, the Young’s moduli and Poisson’s ratios possess different values if at least any one of the layers have a material with multiplanar hexagonal nanostructure (i.e. E _{1} ≠ E _{2} and ν _{12} ≠ ν _{21}). An advantage of the proposed bottomup approach of considering layerwise equivalent material property is that it allows us to neglect the effect of lattice mismatch in evaluating the effective elastic moduli for multilayer heterostructures consisting of different materials. In the derivation for effective elastic moduli of such heterostructues, the deformation compatibility conditions of the adjacent layers are satisfied. This is expected to give rise to some strain energy locally at the interfaces, which is noted in previous studies^{21}. From the derived expressions it can be discerned that the numerical values of elastic moduli actually depend on the number of layers of different constituent materials rather than their stacking sequences. In case of multilayer nanostructures constituted of the layers of same material (i.e. bulk material), it can be expected from Equations 3 and 4 that the Young’s moduli would reduce due to the presence of interlayer distances, which, in turn, increase the value of overall thickness t.
Effective mechanical properties such as Young’s moduli and Poisson’s ratios are of utmost importance to access the viability for the use of nanoheterostructures in various nanoelectromechanical applications. The major contribution of this work is to develop the generalized closedform analytical formulae for multilayer nanoheterostructures. Thses formulae are also applicable to singlelayer of materials with monoplanar as well as multiplanar nanostructures. Thus the developed analytical formulae for elastic moduli can be used as an efficient reference for the entire spectrum of materials with latticelike structural form and the heterostructures obtained by combining multiple layers of different such materials with any stacking sequence. Such generalization in the derived formulae, with the advantage of being computationally efficient and easy to implement, opens up a tremendous potential scope in the field of novel applicationspecific heterostructure development. We have validated the proposed expressions considering multiple stacking sequences with existing results of literature and separate molecular dynamics simulations for the Young’s moduli and Poisson’s ratios of grapheneMoS_{2} heterostructure, respectively. Indepth new results are presented for the Young’s moduli and Poisson’s ratios of three other nanoheterostructures (graphenehBN, graphenestanene and staneneMoS_{2}). Even though the results are presented in this article considering only two different constituent materials in a single heterostructure (such as grapheneMoS_{2}, graphenehBN, graphenestanene and staneneMoS_{2}), the proposed formulae can be used for heterostructures containing any number of different materials^{26}. The physicsbased analytical formulae are capable of providing a comprehensive indepth insight on the behaviour of such multilayer heterostructures. Noteworthy feature of the present analytical approach is the computational efficiency and costeffectiveness compared to conducting nanoscale experiments or molecular dynamics simulations. Thus, besides deterministic analysis of elastic moduli, as presented in this paper, the efficient closedform formulae could be an attractive option for carrying out uncertainty analysis^{43,44,45,46,47,48,49,50} based on a Monte Carlo simulation based approach (refer to section 8 of the supplementary material). The bottomup approach based concept to develop expressions for hexagonal nanoheterostructures can be extended to other forms of nanostrcutures in future.
After several years of intensive investigation, research concerning graphene has logically reached to a rather mature stage. Thus investigation of other two dimensional and quasitwo dimensional materials have started receiving the due attention recently. However, the possibility of combining single layers of different two dimensional materials (heterostructures) has expanded this field of research dramatically; well beyond the scope of considering a simple single layer graphene or other 2D material. The interest in such heterostructures is growing very rapidly with the advancement of synthesizing such materials in laboratory^{22,23}, as the interest in graphene did few years ago. The attentiveness is expected to expand further in coming years with the possibility to consider different tunable nanoelectromechanical properties of the prospective combination (single and multilayer structures with different stacking sequences) of so many two dimensional materials. This, in turn introduces the possibility of opening a new dimension of applicationspecific material development that is analogous to metamaterials^{51,52} in nanoscale. The present article can contribute significantly in this exciting endeavour.
In summary, we have developed computationally efficient physicsbased analytical expressions for predicting the equivalent elastic moduli of multilayer nanoheterostructures. The proposed expressions are validated for graphene–MoS_{2} heterostructures by carrying out separate molecular dynamics simulations and available results from literature. New results are presented for graphene–hBN, graphene–stanene and stanene–MoS_{2} heterostructures using the developed analytical framework. As the proposed closedform formulae are general in nature and applicable to wide range of materials and their combinations with hexagonal nanostructures, the present article can serve as a ready reference for characterizing the material properties in future nanomaterials development.
Methods
Analytical framework for equivalent elastic moduli of nanoheterostructures
A concise description of the basic philosophy behind the developed analytical framework is explained in this section (detail derivations are provided as supplementary material with this manuscript). A multistage bottomup idealization scheme is adopted for deriving the closedform expressions, as depicted in Fig. 7. In the first stage, the effective elastic moduli of each individual layer are determined based on a continuum based approach. This is equivalent to the effective elastic properties of a singlelayer nanostructure. The multilayer heterostructure can be idealized as a layered platelike composite structural element with respective effective elastic properties and geometric dimensions (such as thickness) of each layer. To ensure the consistency in deformation of the adjacent layers, each of the layers are considered to have equal effective deformation in a particular direction. The equivalent elastic property of the entire heterostructure is determined based on force equilibrium and deformation compatibility conditions. The molecular mechanics parameters (k _{ r } and k _{ θ }), bond length and bond angles for different materials, which are used to obtain numerical results based on Equations 3–6, are provided in the next paragraph.
The molecular mechanics parameters and geometric properties of the bonds are welldocumented in scientific literature. In case of graphene, the molecular mechanics parameters k _{ r } and k _{ θ } can be obtained from literature using AMBER force filed^{53} as k _{ r } = 938 kcal mol^{−1} nm^{−2} = 6.52 × 10^{−7} Nnm^{−1} and k _{ θ } = 126 kcal mol^{−1} rad^{−2} = 8.76 × 10^{−10} Nnm rad^{−2}. The outofplane angle for graphene is α = 0 and the bond angle is θ = 120° (i.e. ψ = 30°), while bond length and thickness of singlelayer graphene can be obtained from literature as 0.142 nm and 0.34 nm respectively^{7}. In case of hBN, the molecular mechanics parameters k _{ r } and k _{ θ } can be obtained from literature using DREIDING force model^{54} as k _{ r } = 4.865 × 10^{−7} Nnm^{−1} and k _{ θ } = 6.952 × 10^{−10} Nnm rad^{−2} ^{55}. The outofplane angle for hBN is α = 0 and the bond angle is θ = 120° (i.e. ψ = 30°), while bond length and thickness of singlelayer hBN can be obtained from literature as 0.145 nm and 0.098 nm respectively^{8}. In case of stanene, the molecular mechanics parameters k _{ r } and k _{ θ } can be obtained from literature as k _{ r } = 0.85 × 10^{−7} Nnm^{−1} and k _{ θ } = 1.121 × 10^{−9} Nnm rad^{−2} ^{56,57}. The outofplane angle for stanene is α = 17.5° and the bond angle is θ = 109° (i.e. ψ = 35.5°), while bond length and thickness of single layer stanene can be obtained from literature as 0.283 nm and 0.172 nm respectively^{56,57,58,59}. In case of MoS_{2}, the molecular mechanics parameters k _{ r } and k _{ θ } can be obtained from literature as k _{ r } = 1.646 × 10^{−7} Nnm^{−1} and k _{ θ } = 1.677 × 10^{−9} Nnm rad^{−2}, while the outofplane angle, bond angle, bond length and thickness of single layer MoS_{2} are α = 48.15°, θ = 82.92° (i.e. ψ = 48.54°), 0.242 nm and 0.6033 nm respectively^{15,60,61,62}.
Molecular dynamics simulation for Poisson’s ratios of graphene–MoS_{2} heterostructures
We have followed a similar method as reported in literature^{21,63,64} for calculating the Poisson’s ratios of graphene–MoS_{2} bilayers and heterostructures through molecular dynamics simulation. The interatomic potential used for carboncarbon, molybdenumsulfur interactions are the secondgeneration Brenner interatomic potential^{65,66}. We have stabilized the heterostructures following the same method as described in literature^{21}. The MoS_{2} and graphene layers of the heterostructures are coupled by van der Waals interactions, as described by the LennardJones potential. The adopted cutoff is 10.0A° for M/G/M and 5.0A° for G/M/G heterostructures. These cutoff values are determined by stabilizing and minimizing the M/G/M and G/M/G heterostructures^{21}.
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Acknowledgements
T.M. acknowledges the financial support from Swansea University through the award of Zienkiewicz Scholarship. S.A. acknowledges the support of the ‘Engineering Nonlinearity’ program grant (EP/K003836/1) funded by the EPSRC. M.A.Z. acknowledges the funding support from the National Science Foundation under Grant No. NSFCMMI 1537170. The authors are also grateful for computer time allocation provided by the Extreme Science and Engineering Discovery Environment (XSEDE) under award number TGDMR140008.
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T.M. and A.M. primarily conceived the idea of developing analytical expressions for nanoheterostructures. T.M. derived the analytical formulae and prepared the manuscript. A.M. carried out the molecular dynamics simulations for the Poisson’s ratios of grapheneMoS_{2} heterostructure. S.A. and M.A.Z. contributed significantly throughout the entire period by providing necessary scientific inputs.
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Mukhopadhyay, T., Mahata, A., Adhikari, S. et al. Effective mechanical properties of multilayer nanoheterostructures. Sci Rep 7, 15818 (2017). https://doi.org/10.1038/s41598017156643
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