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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 different 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 efficient 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 different heterostructures such as graphene-MoS2, graphene-hBN, graphene-stanene and stanene-MoS2. The proposed formulae will enable efficient characterization of mechanical properties in developing a wide range of application-specific nano-heterostructures.
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SCiEnTifiC RepoRts | 7: 15818 | DOI:10.1038/s41598-017-15664-3
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Eective 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 dierent 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 ecient 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 dierent heterostructures such as
graphene-MoS2, graphene-hBN, graphene-stanene and stanene-MoS2. The proposed formulae will
enable ecient characterization of mechanical properties in developing a wide range of application-
specic nano-heterostructures.
A generalized analytical approach is presented to derive closed-form formulae for the eective 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 scientic community for investigating more prospective two-dimensional
and quasi-two-dimensional materials that could possess interesting electronic, optical, thermal, chemical
and mechanical characteristics24. 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 dierent class of materials that possess hexagonal
monoplanar nanostructure with dierent 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 dierent 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 dierent 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 dierent, as discussed in the preceding paragraph.
Subsequently, these materials exhibit signicantly dierent 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 dierent two dimensional hexagonal nano-structural forms, recently the
development of novel application-specic heterostructures has started receiving considerable attention from the
scientic community due to the tremendous prospect of combining dierent single layer materials in intelligent
and intuitive ways to achieve several such desired physical and chemical properties2026.
e hexagonal nano-heterostructures can be broadly classied into three categories based on structural cong-
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 dierent 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 dierent 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 dierent forms of multi-layer heterostructures have started receiving immense atten-
tion from the scientic 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, eective 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.7,8,3133. is approach of mechanical property characterization for single-layer nanostructures is com-
putationally very ecient, 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 ecient 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 ecient 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-specic 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 dierent 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 dierent 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 scientic literature31,34,35. Besides that the mechanics of mono-planar hexagonal
honeycomb-like structure is found to be widely investigated across dierent length scales3640. erefore, the
main contribution of this article lies in proposing computationally ecient 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 dierent nano-heterostructures
belonging to the three dierent 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, eect 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 eective 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 Poissons 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
(Equations36) 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 eect of
inter-layer stiness 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 dierent atoms).
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Validation and analytical predictions for the elastic moduli of heterostructures. Results are pre-
sented for the eective 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 dierent forms of heterostructures is very scarce in scien-
tic literature. We have considered four dierent 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 scientic com-
munity for dierent 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 Poissons 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 Poissons ratios, new results are provided for the other three considered heterostructures accounting for the
eect 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 eective 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 Tables15, 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 Poissons ratios, the reported values in scientic literature for
graphene and hBN show wide range of variability, while the reference values of Poissons 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.
Table1 presents the value of two Young’s moduli obtained from the proposed analytical formulae for
nano-heterostructures considering dierent stacking sequences of graphene and MoS2. e results are compared
with the numerical values reported in scientic literature. It can be noted that the dierence 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 dierent for multiplanar single-layer nanostructural forms (such as stanene and MoS2). A
similar trend has been reported before by Li41 for MoS2. us the eective Young’s moduli of the heterostructures
with at least one layer of multiplanar structural form is expected to exhibit dierent E1 and E2 values. In Table1
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 dierent 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 Poissons ratios, as Poissons ratios have
not been reported for graphene–MoS2 heterostructures in literature. e analytical predictions of Poissons 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 graphene-MoS2 heterostructure, the two in-plane Poisson’s ratios (ν12 and ν21)
are found to have dierent 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 Equations36.
Table2 provides the results for elastic moduli of graphene-hBN heterostructure considering dierent 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.
Conguration
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 dierent stacking sequences (e results obtained using the
proposed formulae are compared with the existing results from literature, as available. However, as the Poissons
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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Table3 presents the results for elastic moduli of graphene-stanene heterostructure considering dierent stacking
sequences. As stanene has a multi-planar structural form, the two Young’s moduli and two in-plane Poissons
ratios show dierent values (i.e. E1 E2 and ν12 ν21) when at least one of the constituent layers of the hetero-
structure is stanene. Table4 presents the results for elastic moduli of stanene-MoS2 heterostructure considering
dierent 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 dierent values (i.e. E1 E2 and ν12 ν21). e results of
dierent 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 dierent 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. Figures3, 4, 5, 6 present
the variation of dierent elastic moduli with number of layers of the constituent materials considering the four
dierent heterostructures belonging from the three dierent categories, as described in the preceding section. It
is observed that the trend of variation for two Young’s moduli and two in-plane Poissons ratios are similar for
graphene-MoS2 and graphene-stanene heterostructures with little dierence 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
Conguration 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 Poissons ratios (ν12 and
ν21) of graphene-hBN (G–H) heterostructure with dierent stacking sequences (e thickness of single layer of
graphene and hBN are considered as 0.34 nm and 0.33 nm, respectively).
Conguration 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 Poissons ratios (ν12 and
ν21) of graphene-stanene (G–S) heterostructure with dierent stacking sequences (e thickness of single layer
of graphene and stanene are considered as 0.34 nm and 0.172 nm, respectively).
Conguration 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 Poissons ratios (ν12 and
ν21) of stanene-MoS2 (S–M) heterostructure with dierent 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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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 ecient closed-form
formulae.
Discussion
We have presented computationally ecient analytical closed-form expressions for the eective elastic moduli of
multi-layer nano-heterostructures, wherein individual layers may have multiplanar (i.e. α 0) or monoplanar
(i.e. α = 0) congurations. 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
Raee31 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 dierence is not signicant. Similar
observation is found to be reported in literature41. For single-layer of materials, the formulae of elastic moduli
deduced from Equations36 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 dierent 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 eect of lattice mismatch in evaluating the eective elastic moduli for multi-layer heterostruc-
tures consisting of dierent materials. In the derivation for eective elastic moduli of such heterostructues, the
deformation compatibility conditions of the adjacent layers are satised. 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 dierent
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 Equations3 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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Eective 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 dierent such materials with any stacking sequence. Such generalization
in the derived formulae, with the advantage of being computationally ecient and easy to implement, opens up
a tremendous potential scope in the eld of novel application-specic 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 Poissons ratios of graphene-MoS2 het-
erostructure, respectively. In-depth new results are presented for the Young’s moduli and Poissons 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 dierent 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 dierent 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 eciency and cost-eectiveness com-
pared to conducting nano-scale experiments or molecular dynamics simulations. us, besides deterministic
analysis of elastic moduli, as presented in this paper, the ecient closed-form formulae could be an attractive
option for carrying out uncertainty analysis4350 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.
Aer 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 dierent 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 dier-
ent tunable nanoelectromechanical properties of the prospective combination (single and multi-layer structures
with dierent stacking sequences) of so many two dimensional materials. is, in turn introduces the possibility
of opening a new dimension of application-specic material development that is analogous to metamaterials51,52
in nano-scale. e present article can contribute signicantly in this exciting endeavour.
In summary, we have developed computationally ecient 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 eective
elastic moduli of each individual layer are determined based on a continuum based approach. is is equivalent to
the eective elastic properties of a single-layer nanostructure. e multi-layer heterostructure can be idealized as
a layered plate-like composite structural element with respective eective 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 eective 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 dierent materials, which are
used to obtain numerical results based on Equations36, are provided in the next paragraph.
e molecular mechanics parameters and geometric properties of the bonds are well-documented in scien-
tic 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 mol1 nm2 = 6.52 × 107 Nnm1 and kθ = 126 kcal mol1 rad2
= 8.76 × 1010 Nnm rad2. 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 × 107 Nnm1 and kθ = 6.952 × 1010 Nnm rad2 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 × 107 Nnm1 and kθ =
1.121 × 109 Nnm rad2 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 respectively5659. In case of MoS2, the molecular mechanics parameters kr and kθ can be obtained
from literature as kr = 1.646 × 107 Nnm1 and kθ = 1.677 × 109 Nnm rad2, 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 ively15,6062.
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.
Content courtesy of Springer Nature, terms of use apply. Rights reserved
www.nature.com/scientificreports/
13
SCiEnTifiC RepoRts | 7: 15818 | DOI:10.1038/s41598-017-15664-3
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 signicantly
throughout the entire period by providing necessary scientic inputs.
Additional Information
Supplementary information accompanies this paper at https://doi.org/10.1038/s41598-017-15664-3.
Competing Interests: e authors declare that they have no competing interests.
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