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SCiEnTifiC RepoRts | 7: 15818 | DOI:10.1038/s41598-017-15664-3

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Eective mechanical properties of

multilayer nano-heterostructures

T. Mukhopadhyay1, A. Mahata2, S. Adhikari

3 & M. Asle Zaeem

2

Two-dimensional and quasi-two-dimensional materials are important nanostructures because of

their exciting electronic, optical, thermal, chemical and mechanical properties. However, a single-

layer nanomaterial may not possess a particular property adequately, or multiple desired properties

simultaneously. Recently a new trend has emerged to develop nano-heterostructures by assembling

multiple monolayers of dierent nanostructures to achieve various tunable desired properties

simultaneously. For example, transition metal dichalcogenides such as MoS2 show promising electronic

and piezoelectric properties, but their low mechanical strength is a constraint for practical applications.

This barrier can be mitigated by considering graphene-MoS2 heterostructure, as graphene possesses

strong mechanical properties. We have developed ecient closed-form expressions for the equivalent

elastic properties of such multi-layer hexagonal nano-hetrostructures. Based on these physics-based

analytical formulae, mechanical properties are investigated for dierent heterostructures such as

graphene-MoS2, graphene-hBN, graphene-stanene and stanene-MoS2. The proposed formulae will

enable ecient characterization of mechanical properties in developing a wide range of application-

specic nano-heterostructures.

A generalized analytical approach is presented to derive closed-form formulae for the eective in-plane elas-

tic moduli of hexagonal multiplanar nano-structures and heterostructures. Hexagonal nano-structural forms

are common in various two-dimensional and quasi-two-dimensional materials. e fascinating properties of

graphene1, a two-dimensional allotrope of carbon with hexagonal nanostructure, has led to an enormous interest

and enthusiasm among the concerned scientic community for investigating more prospective two-dimensional

and quasi-two-dimensional materials that could possess interesting electronic, optical, thermal, chemical

and mechanical characteristics2–4. e interest in such hexagonal two-dimensional materials has expanded

over the last decade from hBN, BCN, graphene oxides to Chalcogenides like MoS2, MoSe2 and other forms of

two-dimensional materials like stanene, silicene, sermanene, phosphorene, borophene etc.5,6. Among these

two-dimensional materials, hexagonal honeycomb-like nano-structure is a prevalent structural form3. Four dif-

ferent classes of single-layer materials with hexagonal nano-structure exist from a geometrical point of view, as

shown in Fig.1(a–d). For example, graphene7 consists of a single type of atom (carbon) to form a honeycomb-like

hexagonal lattice structure in a single plane, while there is a dierent class of materials that possess hexagonal

monoplanar nanostructure with dierent constituent atoms such as hBN8, BCN9 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 stanene10,11, silicene11,12, germanene11,12, phosphorene13, borophene14 etc.), or dierent atoms (such

as MoS215, WS216, MoSe217, WSe216, MoTe218 etc.). Even though these two-dimensional 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 nano-heterostructures by assembling multiple

monolayers of dierent nanostructures for achieving various tunable desired properties simultaneously.

Although the single-layer of two-dimensional materials have hexagonal lattice nano-structure (top-view) in

common, their out-of-plane lattice characteristics are quite dierent, as discussed in the preceding paragraph.

Subsequently, these materials exhibit signicantly dierent mechanical and electronic properties. For example,

transition metal dichalcogenides such as MoS2 show exciting electronic and piezoelectric properties, but their

low in-plane mechanical strength is a constraint for any practical application. In contrast, graphene possesses

1Department of Engineering Science, University of Oxford, Oxford, UK. 2Department of Materials Science and

Engineering, Missouri University of Science and Technology, Rolla, USA. 3College of Engineering, Swansea

University, Swansea, UK. Correspondence and requests for materials should be addressed to T.M. (email: tanmoy.

mukhopadhyay@eng.ox.ac.uk) or S.A. (email: s.adhikari@swansea.ac.uk)

Received: 6 June 2017

Accepted: 27 October 2017

Published: xx xx xxxx

OPEN

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SCiEnTifiC RepoRts | 7: 15818 | DOI:10.1038/s41598-017-15664-3

strong in-plane mechanical properties. Moreover, graphene is extremely so in the out-of-plane direction with a

very low bending modulus, whereas the bending modulus of MoS2 is comparatively much higher, depending on

their respective single-layer thickness19. Having noticed that graphene and MoS2 possess such complementary

physical properties, it is a quite rational attempt to combine these two materials in the form of a graphene-MoS2

heterostructure, which could exhibit the desired level of electronic properties and in-plane as well as out-of-plane

strengths. Besides intense research on dierent two dimensional hexagonal nano-structural forms, recently the

development of novel application-specic heterostructures has started receiving considerable attention from the

scientic community due to the tremendous prospect of combining dierent single layer materials in intelligent

and intuitive ways to achieve several such desired physical and chemical properties20–26.

e hexagonal nano-heterostructures can be broadly classied into three categories based on structural cong-

uration, as shown in Fig.1: heterostructure containing only mono-planar nanostructures (such as graphene-hBN

Figure 1. (a) Top view and side views of single-layer hexagonal nanostructures where all the constituent atoms

are same and they are in a single plane (e.g. graphene). (b) Top view and side views of single-layer hexagonal

nanostructures where the constituent atoms are not same but they are in a single plane (e.g. hBN, BCN). (c)

Top view and side views of single-layer hexagonal nanostructures where the constituent atoms are same but

they are in two dierent planes (e.g. silicene, germanene, phosphorene, stanene, borophene). (d) Top view and

side views of single-layer hexagonal nanostructures where the constituent atoms are not same and they are

in two dierent planes (e.g. MoS2, WS2, MoSe2, WSe2, MoTe2). (e) ree dimensional view and side views of

heterostructures consisted of only monoplanar layers of materials (such as graphene-hBN heterostructures).

(f) ree dimensional view and side views of heterostructures consisted of only multiplanar layers of materials

(such as stanene-MoS2 heterostructures). (g,h) ree dimensional view and side views of heterostructures

consisted of both monoplanar and multiplanar layers of materials (such as graphene-MoS2 and graphene-

stanene heterostructures).

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SCiEnTifiC RepoRts | 7: 15818 | DOI:10.1038/s41598-017-15664-3

heterostructure)22,23,27, heterostructure containing both mono-planar and multi-planar nanostructures (such as

graphene-MoS2 heterostructure19,21, graphene-stanene heterostructure24, phosphorene-graphene heterostruc-

ture28, phosphorene-hBN heterostructure28, multi-layer graphene-hBN-TMDC heterostructure26) and hetero-

structure containing only multi-planar nanostructures (such as stanene-MoS2 heterostructure25, MoS2-WS2

heterostructure20). Recently dierent forms of multi-layer heterostructures have started receiving immense atten-

tion from the scientic community for showing interesting chemical, thermal, optical, electronic and transport

properties24,25,29,30. Even though the heterostructures show various exciting physical and chemical characteris-

tics, eective mechanical properties such as Young’s moduli and Poisson’s ratios are of utmost importance for

accessing the viability in application of such nano-heterostructures in various nanoelectromechanical systems.

e research in this eld is still in a very nascent stage and investigations on elastic properties of these built-up

structural forms are very scarce to nd in literature20,21.

e common practises to investigate these nanostructures are rst principle studies/ab-initio 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 nano-heterostructures, which are consisted of mul-

tiple materials. e accuracy of molecular dynamics simulation depends on the interatomic potentials and the

situation can become worse in case of nano-heterostructures due to the possibility of having lesser accuracy for

built-up structural forms. Molecular mechanics based analytical closed form formulae are presented by many

researchers for materials having hexagonal nano-structures in a single layer such as graphene, hBN, stanene,

MoS2 etc.7,8,31–33. is approach of mechanical property characterization for single-layer nanostructures is com-

putationally very ecient, yet accurate and physically insightful. However, the analytical models concerning

two-dimesional hexagonal nano-structures developed so far are limited to single-layer structural forms; devel-

opment of ecient analytical approaches has not been attempted yet for nano-heterostructures. Considering the

future prospect of research in this eld, it is essential to develop computationally ecient closed-form formulae

for the elastic moduli of nano-hetrostructures 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.

is will accelerate the process of novel material development based on the application-specic 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 closed-form and high delity expressions for characterizing the mechanical properties of a wide

range of hexagonal nano-heterostructures. Elastic properties of four dierent heterostructures (graphene-hBN,

graphene-MoS2, graphene-stanene and stanene-MoS2), belonging to all the three classes as discussed in the

preceding paragraphs, are investigated considering various stacking sequences. e analytical formulae for elas-

tic moduli of heterostructures are applicable to any number of dierent constituent single-layer materials with

multi-planar or mono-planar hexagonal nanostructures.

Results

Closed-form analytical formulae for the elastic moduli of heterostructures. In this section, the

closed-form analytical expressions of elastic moduli for generalized multiplaner hexagonal nano-heterostructures

are presented. e molecular mechanics based approach for obtaining the equivalent elastic properties of atomic

bonds is well-documented in scientic literature31,34,35. Besides that the mechanics of mono-planar hexagonal

honeycomb-like structure is found to be widely investigated across dierent length scales36–40. erefore, the

main contribution of this article lies in proposing computationally ecient and generalized analytical formulae

for nano-heterostructures (having constituent single-layer materials with monoplanar and multiplanar struc-

tural form) and thereby presenting new results for various stacking sequence of dierent nano-heterostructures

belonging to the three dierent classes as described in the preceding section (graphene-MoS2, graphene-hBN,

graphene-stanene and stanene-MoS2).

For atomic level behaviour of nano-scale materials, the total interatomic potential energy can be expressed

as the sum of various individual energy terms related to bonding and non-bonding interactions34. Total strain

energy (E) is expressed as the sum of energy contributions from bending of bonds (Eb), bond stretching (Es), tor-

sion of bonds (Et) and energies associated with non-bonded terms (Enb) such as the van der Waals attraction, the

core repulsions and the coulombic energy (refer to Fig.2).

=+++EE EEE(1)

sbtnb

However, among all the energy components, eect of bending and stretching are predominant in case of small

deformation31,35. For the multiplanar hexagonal nano-structures (such as stanene and MoS2), the strain energy

caused by bending consists of two components, in-plane component (EbI) and out-of-plane component (EbO). e

out-of-plane component becomes zero for monoplanar nanostructures such as graphane and hBN. us the total

interatomic potential energy (E) can be expressed as

θα

=+ +

=∆+∆+∆

θθ

EEEE

kl kk

1

2

() 1

2

() 1

2

()

(2)

sbIbO

r

22 2

where Δl, Δθ and Δα denote the change in bond length, in-plane and out-of-plane angle respectively. e quan-

tities kr and kθ represents the force constants for bond stretching and bending respectively. The molecular

mechanics parameters (kr and kθ) and structural mechanics parameters (EA and EI) of a uniform circular beam

with cross-sectional area A, length l, Young’s modulus E, and second moment of area I, are related as: =

KrEA

l

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SCiEnTifiC RepoRts | 7: 15818 | DOI:10.1038/s41598-017-15664-3

and =

θ

kEI

l

31,34,35. Based on this relationship, the closed form expressions for the eective elastic moduli of mul-

tilayer hexagonal nano-heterostructures are derived following a multi-stage idealization scheme using force equi-

librium and deformation compatibility conditions. e closed form expressions for the two in-plane Young’s

moduli of nano-heterostructures are derived as

∑

ψ

ψψψα

=

+

++

ψα

=

θ

(3)

E

t

1cos

(1 sin) (sin cossin )

i

n

i

i

l

kii

ik

1

1

12

222coscos

i

i

ii

ri

222

∑

ψ

ψψψα αψ

=

+

++++

α

=

θ

(4)

E

t

11sin

cos(cossin sin2sin) (sin 2)

i

n

i

i

l

kii

ii

ki

2

1

12

2222

cos2

i

i

i

ri

22

e subscript i in the above expressions indicates the molecular mechanics and geometrical properties (as

depicted in Fig.2(a,b)) corresponding to ith layer of the heterostructure. e overall thickness of the hetero-

structure is denoted by t. n represents the total number of layers in the heterostructure. Expressions for the two

in-plane Poisson’s ratios are derived as

ν=

∑

∑

ψ

ψψψα

ψψα

=+

++

=

θ

ψα

θ

(5)

i

n

i

nk

l

12

1

cos

(1 sin) (sin cossin )

1

12

sincos cos

i

i

li

kiiii

ii

kri

i

ii ii

2

12

222cos2cos2

22

ν=

∑

∑

ψ

ψψψα αψ

ψψα

=

+

++++

=

θ

α

θ

(6)

i

n

i

nk

l

21

1

1sin

cos(cossin sin2sin) (sin 2)

112

sincos cos

i

ili

kiiiii

i

kri i

i

ii ii

2

12 2222

cos22

22

Here ν12 and ν21 represent the in-plane Poisson’s ratios for loading directions 1 and 2 respectively. us the

elastic moduli of a hexagonal nano-heterostructure can be obtained using the closed-form analytical formulae

(Equations3–6) from molecular mechanics parameters (kr and kθ), bond length (l), in-plane bond angle (ψ) and

out-of-plane angle (α), which are well-documented in the molecular mechanics literature. e analytical for-

mulae are valid for small deformation of the structure (i.e. the linear region of stress-strain curve). e eect of

inter-layer stiness contribution due Lennard-Jones potentials are found to be negligible for the in-plane elastic

moduli considered in this study and therefore, neglected in the analytical derivation (refer to section 7 of the

supplementary material).

Figure 2. (a,b) Top view and side view of a generalized form of multiplanar hexagonal nano-structure. (e in-

plane angles θ and ψ are indicated in Fig.2(a), wherein it is evident that

ψ=−

θ

90 2

. e out-of-plane angle α

is indicated in Fig.2(b)). (c) Energy components associated with the in-plane (1–2 plane) and out-of-plane (1–3

plane) deformation mechanisms (Direction 1 and 2 are indicated in the gure. Direction 3 is perpendicular to

the 1–2 plane. Here A and B indicate two dierent atoms).

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SCiEnTifiC RepoRts | 7: 15818 | DOI:10.1038/s41598-017-15664-3

Validation and analytical predictions for the elastic moduli of heterostructures. Results are pre-

sented for the eective elastic moduli of hexagonal multi-layer nano-heterostructures based on the formulae

proposed in the preceding section. As investigations on nano-heterostructures is a new and emerging eld of

research, the results available for the elastic moduli of dierent forms of heterostructures is very scarce in scien-

tic literature. We have considered four dierent nano-heterostructures to present the results: graphene-MoS2,

graphene-hBN, graphene-stanene and stanene-MoS2 (belonging to the three categories as depicted in the introduc-

tion section). ough all these four heterostructures have received attention from the concerned scientic com-

munity for dierent physical and chemical properties recently, only the graphene-MoS2 heterostructure has been

investigated using molecular dynamics simulation for the Young’s modulus among all other elastic moduli20,21.

us we have validated the proposed analytical formulae for Young’s moduli of graphene-MoS2 heterostruc-

ture with available results from literature. New results are presented for the two in-plane Poisson’s ratios of

graphene-MoS2 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

eect 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 eective elastic moduli of monoplanar (i.e. α = 0) and

multiplanar (i.e. α ≠ 0) materials. e analytical predictions for the Young’s moduli and Poisson’s ratios of such

single-layer materials are further validated with reference results from literature, as available.

As shown in Tables1–5, in the case of single-layer hexagonal nanostructures (n = 1) belonging to all the four

classes as described in the preceding section (graphene, hBN, stanene and MoS2), the in-plane Young’s moduli

obtained using the proposed analytical formulae are in good agreement with reported values in literature for

graphene, hBN, stanene and MoS2. ese observations corroborate the validity of the proposed analytical for-

mulae in case of a single-layer. However, in case of Poisson’s ratios, the reported values in scientic literature for

graphene and hBN show wide range of variability, while the reference values of Poisson’s ratios for stanene and

MoS2 are very scarce in available literature. e results predicted by the proposed formulae agree well with most

of the reported values for Poisson’s ratios.

Table1 presents the value of two Young’s moduli obtained from the proposed analytical formulae for

nano-heterostructures considering dierent stacking sequences of graphene and MoS2. e results are compared

with the numerical values reported in scientic literature. It can be noted that the dierence between E1 and E2 is

not recognized in most of the previous investigations and the results presented as E1 = E2. e Young’s moduli E1

and E2 are found to be dierent for multiplanar single-layer nanostructural forms (such as stanene and MoS2). A

similar trend has been reported before by Li41 for MoS2. us the eective Young’s moduli of the heterostructures

with at least one layer of multiplanar structural form is expected to exhibit dierent E1 and E2 values. In Table1

it can be observed that for single and bi-layer of graphene E1 = E2, while for single and bi-layer of MoS2 E1 ≠ E2.

In case of heterostructures consisting of both graphene and MoS2 the value of E2 is observed to be higher than E1.

However, the numerical values of E1 for dierent stacking sequences are found to be in good agreement with the

values of Young’s modulus reported in literature (presumably obtained for direction-1) corroborating the validity

of the developed closed-form expressions. We have carried out separate molecular dynamics simulations for

graphene – MoS2 heterostructures to validate the analytical predictions of Poisson’s ratios, as Poisson’s ratios have

not been reported for graphene–MoS2 heterostructures in literature. e analytical predictions of Poisson’s ratios

reported in Table1 are found to be in good agreement with the results of molecular dynamics simulations. Similar

to the results of Young’s moduli for graphene-MoS2 heterostructure, the two in-plane Poisson’s ratios (ν12 and ν21)

are found to have dierent values when at least one multi-planar structural form is present in the heterostructure.

us having the analytical formulae for all the elastic moduli validated, we have provided new results for three

other nano-heterostructures in the following paragraphs based on Equations3–6.

Table2 provides the results for elastic moduli of graphene-hBN heterostructure considering dierent stacking

sequences. It is observed that the two Young’s moduli and two in-plane Poisson’s ratios are equal (i.e. E1 = E2 and

ν12 = ν21) in case of graphene-hBN heterostructure as these are consisted of only mono-planar structural forms.

Conguration

Present results

Reference (E1 = E2)

Present results

Reference (ν12 = ν21)E1E2ν12 ν21

G 1.0419 1.0419 1.0519, 1 ± 0.167 0.2942 0.2942 0.3468, 0.19569

G/G 1.0419 1.0419 1.0619, 1.04 ± 0.170 0.2942 0.2942 0.2798 [MD]

M 0.1778 0.3549 0.1619, 0.27 ± 0.171 0.0690 0.1376 0.1019 [MD], 0.2172

M/M 0.1778 0.3549 0.2719, 0.2 ± 0.171 0.0690 0.1376 0.1018 [MD]

G/M 0.4893 0.6025 0.5319, 0.49 ± 0.0520 0.1672 0.2059 0.2153 [MD]

G/M/G 0.6357 0.7189 0.6819, 0.5621 0.2058 0.2328 0.1805 [MD]

M/G/M 0.3678 0.5059 0.4519 0.1318 0.1813 0.1859 [MD]

Table 1. Results for two Young’s moduli (E1 and E2, in TPa) and two in-plane Poisson’s ratios (ν12 and ν21)

of graphene-MoS2 (G–M) heterostructure with dierent stacking sequences (e results obtained using the

proposed formulae are compared with the existing results from literature, as available. However, as the Poisson’s

ratios for the heterostructures are not available in literature, we have conducted molecular dynamics (MD)

simulations for the Poisson’s ratios. e thickness of single layer of graphene and MoS2 are considered as

0.34 nm and 0.6033 nm, respectively).

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SCiEnTifiC RepoRts | 7: 15818 | DOI:10.1038/s41598-017-15664-3

Table3 presents the results for elastic moduli of graphene-stanene heterostructure considering dierent stacking

sequences. As stanene has a multi-planar structural form, the two Young’s moduli and two in-plane Poisson’s

ratios show dierent values (i.e. E1 ≠ E2 and ν12 ≠ ν21) when at least one of the constituent layers of the hetero-

structure is stanene. Table4 presents the results for elastic moduli of stanene-MoS2 heterostructure considering

dierent stacking sequences. As stanene and MoS2 both have multi-planar structural form, the two Young’s mod-

uli and two in-plane Poisson’s ratios show considerably dierent values (i.e. E1 ≠ E2 and ν12 ≠ ν21). e results of

dierent 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 nano-heterostructures presented in this article are capable of

obtaining the elastic moduli corresponding to any stacking sequence of the constituent layer of nano-materials.

However, from the expressions it can be discerned that the numerical values of elastic moduli actually depend

on the number of layers of dierent constituent materials rather than their exact stacking sequences. From a

mechanics view-point, this is because of the fact that the in-plane properties are not a function of the distance

of individual constituent layers from the neutral plane of the entire heterostructure. Figures3, 4, 5, 6 present

the variation of dierent elastic moduli with number of layers of the constituent materials considering the four

dierent heterostructures belonging from the three dierent categories, as described in the preceding section. It

is observed that the trend of variation for two Young’s moduli and two in-plane Poisson’s ratios are similar for

graphene-MoS2 and graphene-stanene heterostructures with little dierence in the actual numerical values. e

variation of elastic moduli for graphene-hBN heterostructure are presented for E1 and ν12 as the numerical values

are exactly same for the two Young’s moduli and two in-plane Poisson’s ratios, respectively. e plots furnished in

this section can readily provide an idea about the trend of variation of elastic moduli with stacking sequence of

Conguration E1E2ν12 ν21

G 1.049 1.049 0.2942 0.2942

G/G 1.049 1.049 0.2942 0.2942

H 0.8056 0.8056 0.2901 0.2901

H/H 0.8056 0.8056 0.2901 0.2901

G/H 0.9255 0.9255 0.2925 0.2925

G/H/G 0.9647 0.9647 0.2931 0.2931

H/G/H 0.8859 0.8859 0.2918 0.2918

Table 2. Results for two in-plane Young’s moduli (E1 and E2, in TPa) and two in-plane Poisson’s ratios (ν12 and

ν21) of graphene-hBN (G–H) heterostructure with dierent stacking sequences (e thickness of single layer of

graphene and hBN are considered as 0.34 nm and 0.33 nm, respectively).

Conguration E1E2ν12 ν21

G 1.049 1.049 0.2942 0.2942

G/G 1.049 1.049 0.2942 0.2942

S 0.3166 0.3736 0.1394 0.1645

S/S 0.3166 0.3736 0.1394 0.1645

G/S 0.7982 0.8174 0.2563 0.2625

G/S/G 0.8955 0.9070 0.2726 0.2761

S/G/S 0.6771 0.7058 0.2333 0.2432

Table 3. Results for two in-plane Young’s moduli (E1 and E2, in TPa) and two in-plane Poisson’s ratios (ν12 and

ν21) of graphene-stanene (G–S) heterostructure with dierent stacking sequences (e thickness of single layer

of graphene and stanene are considered as 0.34 nm and 0.172 nm, respectively).

Conguration E1E2ν12 ν21

S 0.3166 0.3736 0.1394 0.1645

S/S 0.3166 0.3736 0.1394 0.1645

M 0.1778 0.3549 0.0690 0.1376

M/M 0.1778 0.3549 0.0690 0.1376

S/M 0.2086 0.3591 0.0831 0.1430

S/M/S 0.2282 0.3617 0.0925 0.1466

M/S/M 0.1951 0.3573 0.0768 0.1406

Table 4. Results for two in-plane Young’s moduli (E1 and E2, in TPa) and two in-plane Poisson’s ratios (ν12 and

ν21) of stanene-MoS2 (S–M) heterostructure with dierent stacking sequences (e thickness of single layer of

stanene and MoS2 are considered as 0.172 nm and 0.6033 nm, respectively).

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SCiEnTifiC RepoRts | 7: 15818 | DOI:10.1038/s41598-017-15664-3

multi-layer nano-heterostructures in a comprehensive manner; exact values of the elastic moduli corresponding

to various stacking sequences can be easily obtained using the proposed computationally ecient closed-form

formulae.

Discussion

We have presented computationally ecient analytical closed-form expressions for the eective elastic moduli of

multi-layer nano-heterostructures, wherein individual layers may have multiplanar (i.e. α ≠ 0) or monoplanar

(i.e. α = 0) congurations. It is interesting to notice that the generalized analytical formulae developed for the

Young’s moduli of heterostructures can be reduced to the closed-form expressions provided by Shokrieh and

Raee31 for graphene considering single-layer (i.e. n = 1), α = 0 and ψ = 30°.

==

+

θ

θ

EE

kk

tk

43

9

(7)

r

kl

12

4

r

2

Material Present Results Reference results from literature (E1 = E2 and ν12 = ν21)

Graphene

E1 = 1.0419 1.00 ± 0.1 TPa67, 1.05 TPa73,74, 1.041 TPa31

E2 = 1.0419 1.00 ± 0.1 TPa67, 1.05 TPa73,74, 1.041 TPa31

ν12 = 0.2942 0.3468, 0.1774, 0.4175, 0.19569, 0.653–0.8487

ν21 = 0.2942 0.3468, 0.1774, 0.4175, 0.19569, 0.653–0.8487

hBN

E1 = 0.8056 0.76 ± 0.04576, 0.82177, 0.84278, 0.81579

E2 = 0.8056 0.76 ± 0.04576, 0.82177, 0.84278, 0.81579

ν12 = 0.2901 0.2–0.380, 0.2–0.2480, 0.384–0.3898, 0.384–0.3898, 0.2118, 0.2–0.481

ν21 = 0.2901 0.2–0.380, 0.2–0.2480, 0.384–0.3898, 0.384–0.3898, 0.2118, 0.2–0.481

Stanene

E1 = 0.3166 0.30756

E2 = 0.3736 0.30756

ν12 = 0.1394 —

ν21 = 0.1645 —

MoS2

E1 = 0.1778 0.27 ± 0.099 TPa71, 0.233 TPa82, 0.248 TPa83

E2 = 0.3549 0.27 ± 0.099 TPa71, 0.233 TPa82, 0.248 TPa83

ν12 = 0.0609 0.2172, 0.2963

ν21 = 0.1376 0.2172, 0.2963

Table 5. Results for Young’s moduli (TPa) and Poisson’s ratios of single-layer hexagonal nanostructures).

Figure 3. (a,b) Variation of in-plane Young’s moduli (E1 and E2) with number of layers in a graphene-MoS2

heterostructure. (c,d) Variation of the in-plane Poisson’s ratios (ν12 and ν21) with number of layers in a

graphene-MoS2 heterostructure.

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It can be noted from the presented results that the single-layer materials having regular monoplanar hexagonal

nano-structures (such as graphene and hBN) have equal value of elastic modulus in two perpendicular direc-

tions (i.e. E1 = E2 and ν12 = ν21). However, for single-layer materials with multiplanar nanostructure, the elastic

modulus for direction-2 is more than that of direction-1, even though the dierence is not signicant. Similar

observation is found to be reported in literature41. For single-layer of materials, the formulae of elastic moduli

deduced from Equations3–6 by replacing n = 1, perfectly obeys the Reciprocal theorem (i.e. E1ν21 = E2ν12)42. In

case of nano-heterostructures, the Young’s moduli and Poisson’s ratios possess dierent values if at least any one

of the layers have a material with multiplanar hexagonal nano-structure (i.e. E1 ≠ E2 and ν12 ≠ ν21). An advantage

of the proposed bottom-up approach of considering layer-wise equivalent material property is that it allows us

to neglect the eect of lattice mismatch in evaluating the eective elastic moduli for multi-layer heterostruc-

tures consisting of dierent materials. In the derivation for eective elastic moduli of such heterostructues, the

deformation compatibility conditions of the adjacent layers are satised. is is expected to give rise to some

strain energy locally at the interfaces, which is noted in previous studies21. From the derived expressions it can

be discerned that the numerical values of elastic moduli actually depend on the number of layers of dierent

constituent materials rather than their stacking sequences. In case of multi-layer nanostructures constituted of

the layers of same material (i.e. bulk material), it can be expected from Equations3 and 4 that the Young’s moduli

would reduce due to the presence of inter-layer distances, which, in turn, increase the value of overall thickness t.

Figure 4. (a) Variation of in-plane Young’s modulus (E1) with number of layers in a graphene-hBN

heterostructure (Variation of E2 with number of layers in a graphene-hBN heterostructure is same as E1).

(b) Variation of the in-plane Poisson’s ratio (ν12) with number of layers in a graphene-hBN heterostructure

(Variation of ν21 with number of layers in a graphene-hBN heterostructure is same as ν12).

Figure 5. (a,b) Variation of in-plane Young’s moduli (E1 and E2) with number of layers in a graphene-stanene

heterostructure. (c,d) Variation of the in-plane Poisson’s ratios (ν12 and ν21) with number of layers in a

graphene-stanene heterostructure.

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Eective mechanical properties such as Young’s moduli and Poisson’s ratios are of utmost importance to

access the viability for the use of nano-heterostructures in various nanoelectromechanical applications. e

major contribution of this work is to develop the generalized closed-form analytical formulae for multi-layer

nano-heterostructures. ses formulae are also applicable to single-layer of materials with monoplanar as well

as multiplanar nanostructures. us the developed analytical formulae for elastic moduli can be used as an e-

cient reference for the entire spectrum of materials with lattice-like structural form and the heterostructures

obtained by combining multiple layers of dierent such materials with any stacking sequence. Such generalization

in the derived formulae, with the advantage of being computationally ecient and easy to implement, opens up

a tremendous potential scope in the eld of novel application-specic 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 graphene-MoS2 het-

erostructure, respectively. In-depth new results are presented for the Young’s moduli and Poisson’s ratios of three

other nano-heterostructures (graphene-hBN, graphene-stanene and stanene-MoS2). Even though the results are

presented in this article considering only two dierent constituent materials in a single heterostructure (such

as graphene-MoS2, graphene-hBN, graphene-stanene and stanene-MoS2), the proposed formulae can be used

for heterostructures containing any number of dierent materials26. e physics-based analytical formulae are

capable of providing a comprehensive in-depth insight on the behaviour of such multilayer heterostructures.

Noteworthy feature of the present analytical approach is the computational eciency and cost-eectiveness com-

pared to conducting nano-scale experiments or molecular dynamics simulations. us, besides deterministic

analysis of elastic moduli, as presented in this paper, the ecient closed-form formulae could be an attractive

option for carrying out uncertainty analysis43–50 based on a Monte Carlo simulation based approach (refer to

section 8 of the supplementary material). e bottom-up approach based concept to develop expressions for

hexagonal nano-heterostructures can be extended to other forms of nanostrcutures in future.

Aer several years of intensive investigation, research concerning graphene has logically reached to a rather

mature stage. us investigation of other two dimensional and quasi-two dimensional materials have started

receiving the due attention recently. However, the possibility of combining single layers of dierent two dimen-

sional materials (heterostructures) has expanded this eld of research dramatically; well beyond the scope of con-

sidering a simple single layer graphene or other 2D material. e interest in such heterostructures is growing very

rapidly with the advancement of synthesizing such materials in laboratory22,23, as the interest in graphene did few

years ago. e attentiveness is expected to expand further in coming years with the possibility to consider dier-

ent tunable nanoelectromechanical properties of the prospective combination (single and multi-layer structures

with dierent stacking sequences) of so many two dimensional materials. is, in turn introduces the possibility

of opening a new dimension of application-specic material development that is analogous to metamaterials51,52

in nano-scale. e present article can contribute signicantly in this exciting endeavour.

In summary, we have developed computationally ecient physics-based analytical expressions for predicting

the equivalent elastic moduli of multi-layer nano-heterostructures. e proposed expressions are validated for

graphene–MoS2 heterostructures by carrying out separate molecular dynamics simulations and available results

from literature. New results are presented for graphene–hBN, graphene–stanene and stanene–MoS2 heterostruc-

tures using the developed analytical framework. As the proposed closed-form formulae are general in nature and

Figure 6. (a,b) Variation of in-plane Young’s moduli (E1 and E2) with number of layers in a stanene-MoS2

heterostructure. (c,d) Variation of the in-plane Poisson’s ratios (ν12 and ν21) with number of layers in a stanene-

MoS2 heterostructure.

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applicable to wide range of materials and their combinations with hexagonal nano-structures, the present article

can serve as a ready reference for characterizing the material properties in future nano-materials development.

Methods

Analytical framework for equivalent elastic moduli of nano-heterostructures. A concise descrip-

tion of the basic philosophy behind the developed analytical framework is explained in this section (detail der-

ivations are provided as supplementary material with this manuscript). A multi-stage bottom-up idealization

scheme is adopted for deriving the closed-form expressions, as depicted in Fig.7. In the rst stage, the eective

elastic moduli of each individual layer are determined based on a continuum based approach. is is equivalent to

the eective elastic properties of a single-layer nanostructure. e multi-layer heterostructure can be idealized as

a layered plate-like composite structural element with respective eective elastic properties and geometric dimen-

sions (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 eective deformation in a particular direction. e equivalent elastic property

of the entire heterostructure is determined based on force equilibrium and deformation compatibility conditions.

e molecular mechanics parameters (kr and kθ), bond length and bond angles for dierent materials, which are

used to obtain numerical results based on Equations3–6, are provided in the next paragraph.

e molecular mechanics parameters and geometric properties of the bonds are well-documented in scien-

tic literature. In case of graphene, the molecular mechanics parameters kr and kθ can be obtained from literature

using AMBER force led53 as kr = 938 kcal mol−1 nm−2 = 6.52 × 10−7 Nnm−1 and kθ = 126 kcal mol−1 rad−2

= 8.76 × 10−10 Nnm rad−2. e out-of-plane angle for graphene is α = 0 and the bond angle is θ = 120° (i.e. ψ

= 30°), while bond length and thickness of single-layer graphene can be obtained from literature as 0.142 nm

and 0.34 nm respectively7. In case of hBN, the molecular mechanics parameters kr and kθ can be obtained from

literature using DREIDING force model54 as kr = 4.865 × 10−7 Nnm−1 and kθ = 6.952 × 10−10 Nnm rad−2 55. e

out-of-plane angle for hBN is α = 0 and the bond angle is θ = 120° (i.e. ψ = 30°), while bond length and thickness

of single-layer hBN can be obtained from literature as 0.145 nm and 0.098 nm respectively8. In case of stanene, the

molecular mechanics parameters kr and kθ can be obtained from literature as kr = 0.85 × 10−7 Nnm−1 and kθ =

1.121 × 10−9 Nnm rad−2 56,57. e out-of-plane 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 respectively56–59. In case of MoS2, the molecular mechanics parameters kr and kθ can be obtained

from literature as kr = 1.646 × 10−7 Nnm−1 and kθ = 1.677 × 10−9 Nnm rad−2, while the out-of-plane angle, bond

angle, bond length and thickness of single layer MoS2 are α = 48.15°, θ = 82.92° (i.e. ψ = 48.54°), 0.242 nm and

0.6033 nm respect ively15,60–62.

Molecular dynamics simulation for Poisson’s ratios of graphene–MoS2 heterostructures. We

have followed a similar method as reported in literature21,63,64 for calculating the Poisson’s ratios of graphene–

MoS2 bilayers and heterostructures through molecular dynamics simulation. e interatomic potential used for

carbon-carbon, molybdenum-sulfur interactions are the second-generation Brenner interatomic potential65,66.

Figure 7. (a) Idealization scheme for the analysis of a three-layer nano-heterostructure. (b) Idealization scheme

for the analysis of a two-layer nano-heterostructure.

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We have stabilized the heterostructures following the same method as described in literature21. The MoS2

and graphene layers of the heterostructures are coupled by van der Waals interactions, as described by the

Lennard-Jones potential. e adopted cut-o is 10.0A° for M/G/M and 5.0A° for G/M/G heterostructures. ese

cut-o values are determined by stabilizing and minimizing the M/G/M and G/M/G heterostructures21.

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Acknowledgements

T.M. acknowledges the nancial 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. NSF-

CMMI 1537170. e authors are also grateful for computer time allocation provided by the Extreme Science and

Engineering Discovery Environment (XSEDE) under award number TG-DMR140008.

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Author Contributions

T.M. and A.M. primarily conceived the idea of developing analytical expressions for nano-heterostructures.

T.M. derived the analytical formulae and prepared the manuscript. A.M. carried out the molecular dynamics

simulations for the Poisson’s ratios of graphene-MoS2 heterostructure. S.A. and M.A.Z. contributed signicantly

throughout the entire period by providing necessary scientic inputs.

Additional Information

Supplementary information accompanies this paper at https://doi.org/10.1038/s41598-017-15664-3.

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