Size-dependent elastic modulus of single-layer MoS2

nano-sheets

Hongwei Bao1, Yuhong Huang2, Fei Ma1,3,*, Zhi Yang1, Yaping Miao1,3, Kewei Xu1,4,*, and Paul K. Chu3,*

1State Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, China2College of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, Shaanxi, China3Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China4Department of Physics and Opt-electronic Engineering, Xi’an University of Arts and Science, Xi’an 710065, Shaanxi, China

Received: 30 January 2016

Accepted: 11 April 2016

Published online:

20 April 2016

� Springer Science+Business

Media New York 2016

ABSTRACT

The physical properties of transition metal dichalcogenides (TMDs) with a few

layers can be tailored by strain engineering because of the good elasticity with a

strain limit of more than 10 %. In this work, elastic deformation under uniaxial

tension in single-layer MoS2 nano-sheets is investigated theoretically by

molecular dynamics simulation. A size-dependent elastic modulus is observed

especially from sheets with a width of less than 10 nm. The elastic modulus of

the nano-sheets with zigzag edges decreases as the width becomes narrower but

that of the nano-sheets with armchair edges increases anomalously. The ani-

sotropic behavior is ascribed to the opposite variation of the edge stress induced

by the atomic configurations at the edges. Based on the continuum theory, a

composite model composed of edge and core regions is established and the

parameter, a, is adopted to describe the edge stiffness. The value of a is negative

and positive for the zigzag and armchair edges, respectively, but the fitted edge

width is 0.63 nm that is independent of the edge chirality.

Introduction

Since the discovery of graphene, many types of two-

dimensional (2D) materials have been fabricated [1].

In principle, nearly all bulkmaterials with a laminated

structure and bonded by van der Waals force can be

exfoliated into graphene-like layers including transi-

tion metal dichalcogenides (TMDs) [2], transition

metal oxides [3], boron nitride [4], black phosphorus

[5], and even artificially produced honeycomb struc-

tures [6]. In the family of TMDs, there aremore than 40

compounds and their electronic structures depend on

the coordination around the transition metals as well

as the electron density in the non-bonding d bands [7].

Consequently, TMDs possess physical properties

varying from insulating to superconducting having

many potential applications in nano-electronics [8],

photovoltaics [9], photo-catalysis [10], photo-detec-

tion [11], lithium ion battery [12], and nano-elec-

tromechanical systems (NEMS) [13]. Experimental

and theoretical investigations indicate that 2D mate-

rials can sustain an elastic strain of over 10 % that is

Address correspondence to E-mail: mafei@mail.xjtu.edu.cn; kwxu@mail.xjtu.edu.cn; paul.chu@cityu.edu.hk

DOI 10.1007/s10853-016-9972-x

J Mater Sci (2016) 51:6850–6859

one order of magnitude larger than that of conven-

tional materials and more importantly, the physical

properties can be tuned by strain [14]. Guinea et al.

[15] demonstrated quantized Landau-like electronic

levels in graphene under strain under a zero magnetic

field and Mitchell et al. [16] found that an in-plane

biaxial strain could induce polarization perpendicular

to the sheets. Based on the density functional theory

(DFT) calculation, Scalise et al. [17] showed that under

tensile strain, the band gap of single-layer molybde-

num disulfide (SLMoS2) changed from direct to indi-

rect and also decreased. SLMoS2 would transform

from being semiconducting to metallic at a tensile

strain of about 8 % and He et al. [18] observed

reduction in the direct band gap for a strain rate of

about 70 meV per percent from SLMoS2. Ding et al.

[19] conducted molecular dynamics (MD) simulation

to study the phonon spectra in multilayeredMoS2 and

found that compressive/tensile strain led to blue/red

shift in addition to decreased/increased thermal

conductivity along the in-plane direction. Compres-

sive strain of 10 % produced 10 times increase in the

thermal conductivity, while tensile strain of 5 %

reduced the thermal conductivity by 90 %. Chen et al.

[20] demonstrated that a tensile strain enhanced the

ferromagnetic stability of monolayer CrSiTe3 and

compressive strain induced a ferromagnetic to anti-

ferromagnetic transition. In brief, elastic strain engi-

neering produces remarkable effects on the properties

of 2D materials.

Atomic force microscopy (AFM) and in situ tensile

tests within transmission electron microscopy (TEM)

have been employed to measure the elastic modulus

of 2D materials. Bertolazzi et al. [21] measured the

effective elastic modulus of monolayer MoS2 by

AFM-based nanoindentation and obtained a value of

270 ± 100 GPa that was slightly larger than 210 GPa

predicted by density functional theory (DFT) [22].

Hence, it is of fundamental and technological

importance to study the size-dependent deformation

in 2D materials even though experimental measure-

ment is quite difficult. Although DFT method has

been widely applied to study the mechanical prop-

erties of nanomaterials and can deal with a system

with thousands of atoms, it becomes difficult to deal

with larger system and extremely long computation

time is required. In this respect, molecular dynamics

(MD) simulation which describes the interaction

between atoms by semi-empirical potentials can bet-

ter handle a large system with millions of atoms and

is more suitable for studying the size dependence of

the mechanical properties of 2D materials [23–26].

In this work, SLMoS2 as an example of 2D models

is studied. The 2D nano-sheets with size between

3 9 3 nm and 50 9 50 nm are established and the

deformation process in 2D materials and size

dependence under tensile loading are assessed. Our

results show that the elastic modulus of nano-sheets

with zigzag edges decreases as the width is reduced

but that of the nano-sheets with armchair edges

increases. It can be ascribed to the distinct edge stress

at different edges and a composite model based on

the continuum theory is established to elucidate the

size-dependent elastic moduli.

Simulation methods

In SLMoS2, the Momonolayer is sandwiched by two S

monolayers and Mo and S atoms are covalently bon-

ded with each other. Based on a lattice constant of 2H-

MoS2 (a = 3.16 A), the initial model of SLMoS2 is

established. Figure 1a schematically displays a

hexagonal unit cell of SLMoS2 with the bond length

between S andMo being 2.40 A [27]. The in-plane and

out-of-plane S–Mo–S angles are 80.5� and 84.5�,respectively [8–19]. Figure 1b presents the top view of

the atomic configuration in SLMoS2 and Fig. 1c and d

displays the cross-sectional images of SLMoS2 along

the zigzag and armchair edges, respectively. The

SLMoS2 nano-sheets are cut out of the ideal infinite

sheets and the size varies from 3 9 3 nm to

50 9 50 nm. The initial SLMoS2 nano-sheets are

Figure 1 Schematic model of SLMoS2: a hexagonal unit cell;

b top view of a SLMoS2 nano-sheet; c and d side view of the

armchair and zigzag edges.

J Mater Sci (2016) 51:6850–6859 6851

relaxed in theNPT (the number of atomsN, pressureP,

and temperature T are constant) ensemble at T = 10 K

and P = 0 Pa for 2 ns and the nano-sheets are ther-

mally equilibrated using theNose-Hoover thermostat.

In the NPT ensemble adopted in theMD simulation in

this work, periodic boundary conditions should be

applied along all three directions. Hence, a vacuum

region 30 A inwidth is addedalong the threedirections

in order to avoid the interaction between atoms near

the opposite edges of the nano-ribbons [23–26]. The

two edges perpendicular to the loading direction are

clamped, and the other two edges along the loading

direction are indeed free. In this way, nano-ribbons are

constructed, although periodic boundary conditions

are applied. All the simulations are done at a very low

temperature of 10 K and the boundaries are fixed in

relaxation process to suppress wrinkling. After full

relaxation, uniaxial tensile loading is simulated.

An open source package LAMMPS (Large-scale

Atomic/Molecular Massively Parallel Simulator) is

employed in MD simulation [28]. Based on the

second-generation reactive empirical bond-order

(REBO) formalism, Liang et al. [29] have developed

an inter-atomic potential for the Mo–S system and

Jiang et al. [30, 31] have proposed a new Stillinger–

Webber (SW) potential to describe the inter-atomic

interaction within SLMoS2. The SW potential is

obtained from fitting the phonon spectrum in which

two-body and three-body interactions are consid-

ered and the additional cut-off for the three-body

interaction is embedded. Bond stretching and angle

bending are described by the following

relationships:

V2 ¼ Ae½q=r�rmax�ðB�r4 � 1Þ and ð1Þ

V3 ¼ Ae½q1=ðr12�rmax12Þþq2=ðr13�rmax13Þ�ðcos h� cos h0Þ2: ð2Þ

The simulated elastic properties, deformation

behavior, thermal conductivity, and phonon spec-

trum of SLMoS2 based on the SW potential show

good agreement with the DFT and experimental

results [21, 22]. Therefore, the SW potential is chosen

to describe the interaction between atoms in this

work. The atomic stress is computed according to the

virial theorem in the following form [32]

rij ¼1

V

1

2

XN

a¼1

XN

b 6¼a

U0ðcabDxabi Dxabj

cab�XN

a¼1

ma _xai _x

bj Þ

!

ð3Þ

where V is the total volume of the SLMoS2 nano-

sheets. In order to calculate the volume, half of the

lattice constant along the c axis in bulk MoS2 is usu-

ally taken as the thickness of monolayer sheets. Based

on the experimental [21] and DFT calculation results

[22], the value of 0.61 nm is adopted. N is the total

number of atoms, _xai is the ith component of the

velocity of atom a, Dxabj = _xaj � _xbj , ma is the mass of

atom a, cab is the distance between atoms a and b and

U0 is the potential energy function. Essentially, the

stress is calculated by summing the contributions

from each atom in a given region, such as, at edges,

Figure 2 Stress–strain curves of SLMoS2 nano-sheets under tensile loading: a nano-sheets with zigzag edges and b Armchair edges. The

stress–strain curves in the strain range of 0–2 % are shown in the insets.

6852 J Mater Sci (2016) 51:6850–6859

and then is averaged by the volume of the given

region.

Results and discussion

Figure 2a and b shows the stress–strain curves of the

SLMoS2 nano-sheets with zigzag and armchair edges,

respectively, under tensile loading. The stress

increases almost linearly with strain up to 10 % and

the elastic modulus is calculated by fitting the stress–

strain curves in a small strain range of 0–2 % as

shown by the insets. Figure 3 summarizes the elastic

moduli of the nano-sheets with zigzag and armchair

edges. When the sheet width is larger than 10 nm, the

elastic moduli become constant at 117 GPa and 122

GPa for the zigzag and armchair edges, respectively.

The values are very close to those predicted by Jiang

et al. [30], Xiong et al. [33], and Zhao et al. [34], but it

is lower than the experimental values [21]. The dis-

crepancy between experimental and simulation

results might be due to the SW potential adopted in

describing the interaction between atoms. However,

this does not affect the tendency of size-dependent

elastic modulus revealed in this work. Although the

elastic moduli are size independent and isotropic for

large nano-sheets, they are highly size-dependent

and anisotropic for small nano-sheets. When the

sheet width is reduced from 10 to 3 nm, the elastic

moduli of the nano-sheets with zigzag edges decrease

gradually from 117 to 92 GPa by 21.4 %, but those of

the nano-sheets with armchair edges increase from

122 to 173 GPa by 41.8 %. The elastic moduli of the

SLMoS2 nano-sheets with zigzag and armchair edges

exhibit opposite tendency with width similar to the

observation from graphene nano-ribbons and h-BN

nano-ribbons [35–37]. Griffith predicted that the ideal

fracture stress of a defect-free material is about 10 %

of the Young’s modulus [14]. This has been proved in

2D materials by nanoindentation and tensile loading

[13, 18]. Hence, it is reasonable to believe that the

fracture stress of SLMoS2 nano-sheets with zigzag

edges might be lower than that with armchair edges

referencing to the elastic modulus. Essentially, the

area ratio of edge region with respect to the whole

sheet dominates the size-dependent elastic modulus

of 2D materials, but the influence of sheet thickness is

negligible. Therefore, as predicted in this work, sin-

gle-layer MoS2 sheets have almost the same critical

width as that of graphene nano-ribbons, although

three atomic layers are involved and the S–Mo bond

length (2.40 A) is much longer than C–C bond length

(1.4 A). It is universal for other 2D materials, such as,

germanene, hexagonal boron nitride, and phospho-

rene nano-ribbons. Furthermore, the close critical

sizes were also found in nanowires and nanotubes, as

predicted by MD simulations [38–46].

To understand the size-dependent elastic proper-

ties of nanomaterials, many theoretical models such

as the stick-spiral model [38], truss-spring model [39],

core-shell model [40], core-surface model [41], beam

model [42], and nano-scale continuum models based

on the Cauchy–Born rule have been proposed [43].

Yao et al. [47] have modified the core-shell (MC-S)

model based on the continuum theory to investigate

the elastic properties of nanowires. The surface elas-

ticity is believed to vary with the depth exponentially

and a reasonable thickness in the surface region and

surface factor are given to describe the change in the

surface elasticity. The calculated elastic moduli of

ZnO and Si nanowires agree with both the experi-

mental and MD results. Herein, a composite model

based on the continuum theory is proposed to qual-

itatively elucidate the edge effect in SLMoS2 nano-

sheets. Figure 4a schematically shows the composite

model of SLMoS2 nano-sheets in which two free

(edge) boundaries and an inner (core) region are

included. The edge width is denoted by Ledge and the

core width is indicated by L0. It is assumed that the

elastic modulus at the edge region (Eedge) changes

exponentially but that at the core region (E0) is

Figure 3 Effective elastic modulus as a function of sheet width

with the solid lines highlighting the fitted results.

J Mater Sci (2016) 51:6850–6859 6853

constant. In the model, the edge region is divided

into innumerable infinitesimal regions of the same

width Dl and ‘‘i’’ in Fig. 4a is used to indicate the

region number, that is, ‘‘i’’ changes from 1 to N. The

elastic modulus changes from one region to another,

and the elastic modulus in region i is denoted as Ei.

The gradual variation in the elastic modulus at the

edge region can avoid mismatch between the edge

and core regions. Hence, Eedge can be expressed as

Eedge ¼ E0eaðl�1

2L0Þ;1

2L0 � l� 1

2L0 þ Ledge; ð4Þ

where a is an edge factor related to the stiffness in the

edge region determined by the width and atom con-

figuration of the edge region. Specifically, a positive ameans that the edge region is stiffer than the core

region and Eedge is larger than E0. On the contrary, a

negative a indicates that the core region is stiffer than

the edge region andEedge is smaller thanE0. According

to the continuum model, the effective elastic modulus

of the edge region, Eedge is taken as the weighted

average of Ei. The total area of the edge region is

DAedge ¼Z Lx

0

Z Lyþdl

Ly

dLxdLy ¼ Ldl: ð5Þ

Therefore, the effective elastic modulus of the

SLMoS2 nano-sheets can be calculated by the fol-

lowing relationship:

EeffAeff ¼ E0A0 þ 2Xn

i

EedgeðiÞDAedgeðiÞ

¼ E0A0 þ 2

Z

edge

EedgeAedge; ð6Þ

where Aeff is the total effective area of the SLMoS2nano-sheets and A0 is the area of the core region.

Combining Eq. (5) and (6), we have

Eeff ¼ E0ðL� LedgeÞ þ2

L

Z

edge

E0eaðl�1

2L0Þdl

¼ E0ðL� LedgeÞ þ2

L

Z Ledge

0

E0e12aðlþLedgeÞdl; ð7Þ

in which Ledge is the edge width, L is the total width,

E0 is the elastic modulus at the core region, and a is

the edge factor. For the SLMoS2 nano-sheet of a given

width, L, the effective elastic modulus Eeff can be

obtained from the simulated stress–strain curve.

Then Ledge, E0 and a can be fitted from a series of Eeff

and L according to Eq. (7).

The solid lines in Fig. 3 show the fitted curves of

the zigzag and armchair configurations as a function

of sheet width. The theoretical results are in good

agreement with those derived by MD simulation thus

validating the proposed composite model. With

regard to the nano-sheets with zigzag edges, the fit-

ted parameters are as follows: E0 = 117 GPa, a = - 1

and Ledge = 0.63 nm and for the armchair edges, the

fitted parameters are E0 = 122 GPa, a = 1 and

Ledge = 0.63 nm. The elastic modulus of the core

region (E0) is indeed the value in bulk and almost

isotropic, while that of the edge region is not a con-

stant and dominates the effective elastic modulus of

SLMoS2 nano-sheets with a width narrower than

10 nm. For zigzag edged nano-sheets, the core region

is stiffer than the edges, indicating that the elastic

modulus decreases gradually from the core region to

edges, the value of a is negative. On the contrary, for

Figure 4 a Schematic representation of the composite model of the SLMoS2 nano-sheet composed of edge and core regions and b Ratio

of the edge area with respect to the total area of the nano-sheets as a function of sheet width.

6854 J Mater Sci (2016) 51:6850–6859

armchair edged nano-sheets, the edge region is stiffer

than the core region, and the elastic modulus

increases gradually from core region to edges, the

value of a is positive.

Based on the theoretical analysis, the fitted edge

width is 0.63 nm and is independent of the edge chi-

rality. Figure 4b shows the ratio of the edge area to the

total area in the SLMoS2 nano-sheets with different

widths. It can be fitted as g = 68 exp (-W/

4.46) ? 4.84, where g is the edge area ratio andW is the

width of the SLMoS2 nano-sheets. The solid line in

Fig. 4b indicates the fitted results. The edge area ratio

increases exponentially from 2.52 % in the nano-

sheets 50 9 50 nm in size to 42 % in the nano-sheets

3 9 3 nm in size. Hence, nearly half of the atoms are in

the edge region of the 3 9 3 nm SLMoS2 nano-sheet.

The edge effect is enhanced as the width is reduced,

especially when the sheet width is smaller than 10 nm.

As mentioned above, the width of the edge region,

Ledge, is size and chirality independent. The value of

0.63 nm is about two times that of the lattice constant

of SLMoS2. That is, the edge region contains two

hexagonal lattices and the atoms in this region are

strongly affected. The variation in the atomic config-

uration and potential energy in this region may be the

reason for the opposite a values along the zigzag and

armchair edges. Figure 5a and b shows the average

potential energy per atom in the edge and core

regions. Using an edge width of 0.63 nm, the potential

energy per atom in the edge region increases as the

sheets become narrower for both the zigzag and

armchair edges and can be fitted with an exponential

function. However, the potential energy per atom in

the core region is size independent and about

-4.25 eV that is close to that of the entire nano-sheets

wider than 10 nm.When the width is reduced from 50

to 3 nm, the average potential energy per atom at the

zigzag edges increases from -3.9376 to -3.7924 eV

and that at the armchair edges increases from-3.8760

to -3.81027 eV. The average potential energy per

atom at zigzag edges is lower than that at armchair

edges because of the different bonding environments

at edges in SLMoS2 nano-sheets. There are four dan-

gling bonds for each hexagonal ring at zigzag edge,

but six at armchair edge. Hence, the zigzag edges are

slightly more stable than the armchair ones and the

results are consistent with other MD and DFT results

[48, 49]. The average potential energy per atom at the

edges is always higher than that at the core regions

indicative of unstable atomic configurations and

dangling bonds at the edges and possibly changes in

the atomic configuration at the edges.

Experimental characterization and DFT calculation

demonstrated that tensile strain could tune the elec-

tronic band structures of 2D materials as well as the

physical properties significantly. This highlights a

great promise for future applications, such as, flexible

electronics, nanomechanical systems, and high-fre-

quency resonators [14–20], and it is indispensible to

understand the elastic deformation behavior in 2D

materials. The edge stress after relaxation is another

physical parameter influencing the elastic modulus of

2D nano-sheets. As shown in Fig. 2, the stress is not

zero at zero strain in the sheet with a width narrower

than 10 nm. Figure 6a and b shows the edge stress at

the zigzag and armchair edges under a strain of 0 and

Figure 5 Average potential energy per atom at the edges and in the core region: a SLMoS2 nano-sheets with zigzag edges and b Armchair

edges.

J Mater Sci (2016) 51:6850–6859 6855

2 %, respectively. The stress at the armchair andzigzag

edges changes with width oppositely. As shown in

Fig. 6a, at a strain of 0 %, the edge stress in the nano-

sheets wider than 10 nm is close to zero for both the

zigzag and armchair edges but becomes non-zero

when the nano-sheets are narrower than 10 nm. The

compressive stress at the zigzag edge increases grad-

ually up to-2.54GPawhen the sheetwidth is reduced

to 3 nm, while the tensile stress at the armchair edge

increases gradually to 1.90GPa.DFT calculations show

that the edge stress of single-layer h-BN nano-ribbons

is positive at zigzag edges but negative at armchair

ones [46]. Moreover, it was reported that hydrogen

passivation and other edge termination could alter the

edge stress in graphene nano-ribbons [30, 47]. For

example, Reddy et al. [30] showed that the edge stress

Figure 6 Edge stress of SLMoS2 nano-sheets with zigzag and

armchair edges as a function of sheet width: a strain of 0 % and

b Strain of 2 %.

Figure 7 a Bond length distribution at the zigzag edges of a

SLMoS2 nano-sheet 3 nm wide; b average bond length along the

atom chains near the zigzag edges; c bond length distribution at the

armchair edges of a SLMoS2 nano-sheet 3 nm wide; d average

bond length along the atom chains near the armchair edges.

6856 J Mater Sci (2016) 51:6850–6859

at a zigzag edge dropped by almost 90 % upon

hydrogen passivation but increased by nearly 50 % for

an armchair edge.Hence, it is reasonable to believe that

the elastic modulus in SLMoS2 nano-sheets might

change if hydrogen passivation and edge reconstruc-

tion occur. For SLMoS2 nano-sheets, the tensile loading

will offset or increase the edge stress at zigzag or

armchair edges. In the strain range of 0–2 %, the edge

stress at the zigzag edges changes from compressive to

tensile but the curves of the edge stress with respect to

sheet widths exhibit the same tendency as zero strain.

Specifically, the edge stress of the nano-sheets with the

zigzag edges decreases gradually from 7.95 to 4.8 GPa

by 40 % in the width range from 50 to 3 nm, but that of

the nano-sheets with the armchair edges increases

from 6.19 to 8.46 GPa by 52.8 %. Because of the initial

compressive stress at the zigzag edges, the tensile

stress during loading is compensated to some degree

resulting in a smaller elasticmodulus. On the contrary,

the initial tensile stress at the armchair edges produces

a larger elasticmodulus. Compressive and tensile edge

stress leads to the elastic distortions of nano-sheets,

and this leads to the different size dependences of the

elastic modulus of the SLMoS2 nano-sheets with the

zigzag and armchair edges.

The edge width of the SLMoS2 nano-sheets is

about 0.63 nm and three S–Mo atom chains are

included in the edge region. Figure 7a and c shows

the bond length distributions along the three atomic

chains near the zigzag and armchair edges of the

3 9 3 nm SLMoS2 nano-sheets, respectively. The

atomic configurations are displayed in the insets in

Fig. 7a and c. Figure 7b and d shows the average

bond lengths along different atomic chains near the

zigzag and armchair edges, respectively. The S–Mo

bond length of 2.40 A in the bulk is indicated by the

red dashed lines. For the nano-sheets with zigzag

edges, the bond length in the edge regions becomes

shorter. The bond length in the outer atomic chain is

reduced to 2.38 A by 0.8 % and some bond lengths

decrease to 2.36 A. However, for the nano-sheets

with the armchair edges, the bond length in the

outer atomic chain is elongated to 2.41 A by 0.4 %

and some bond lengths increase to 2.43 A. The

opposite trend should be related to the different

edge stresses. In fact, the smaller the sheet width,

the more substantial is the edge effect. Our results

indicate that the edge plays a dominant role in the

structural stability and mechanical deformation in

2D materials.

Conclusion

MD simulation is conducted to study elastic defor-

mation in SLMoS2 nano-sheets. The elastic modulus is

size and chirality dependent when the sheet is nar-

rower than 10 nm. The elastic modulus of the nano-

sheets with zigzag edges decreases as they become

narrower but that of the nano-sheets with armchair

edges increases abnormally. Based on continuum

theory, a composite model composed of edge and core

regions is established to elucidate the size-dependent

behavior. Exponentially changing elastic moduli are

adopted in the edge region to suppress the influence of

the mismatch between the edge and core regions. The

edge width fitted by the composite model is 0.63 nm

and independent of the edge chirality. The average

potential energy per atom at the edges increases

exponentially as the sheets become narrower thus

affecting the structural stability. The nano-sheets with

the zigzag and armchair edges have compressive and

tensile edge stress in the initial stage giving rise to

smaller bond length at the zigzag edges and larger one

at the armchair edges. The different variation ten-

dencies in the atom configurations in conjunctionwith

the stress at the edges produce opposite changes in the

elastic moduli of SLMoS2 nano-sheets with the zigzag

and armchair edges. Consequently, the edges play an

important role on the structural stability and

mechanical properties of 2D materials.

Acknowledgements

This work was jointly supported by the National

Natural Science Foundation of China (Grant Nos.

51271139, 51471130, 51302162, and 51171145), Natural

Science Foundation of Shaanxi Province (Grant Nos.

2013JM6002 and 2014JQ1016), Fundamental Research

Funds for the Central Universities, and City Univer-

sity of Hong Kong Applied Research, Grant (ARG)

No. 9667104.

References

[1] Xu M, Liang T, Shi M, Chen H (2013) Graphene-like two-

dimensional materials. Chem Rev 113:3766–3798

[2] Ataca C, Sahin H, Ciraci S (2012) Stable, Single-layer MX2

transition-metal oxides and dichalcogenides in a honey-

comb-like structure. J Phys Chem C 116:8983–8999

J Mater Sci (2016) 51:6850–6859 6857

[3] Li LF, Wang YL, Xie SY, Li XB, Wang YQ, Wu RT, Sun

HB, Zhang SB, Gao HJ (2013) Two-dimensional transition

metal honeycomb realized: Hf on Ir (111). Nano Lett

13:4671–4674

[4] Jin C, Lin F, Suenaga K, Iijima S (2009) Fabrication of a

freestanding boron nitride single layer and its defect

assignments. Phys Rev Lett 102:195505

[5] Brent JR, Savjani N, Lewis EA, Haigh SJ, Lewis DJ,

O’Brien P (2014) Fabrication of a freestanding boron nitride

single layer and its defect assignments. Chem Commun

50:13338–13341

[6] Meng L, Wang YL, Zhang LZ, Du SX, Wu RT, Li LF, Zhang

Y, Li G, Zhou HT, Hofer WA (2013) Buckled silicene for-

mation on Ir (111). Nano Lett 613:685–690

[7] Huang X, Zeng ZY, Zhang H (2013) Metal dichalcogenide

nanosheets: preparation, properties and applications. Chem

Soc Rev 42:1934–1946

[8] Mak KF, Lee C, Hone J, Shan J, Heinz TF (2010) Seeing

many-body effects in single- and few-layer graphene:

observation of 2-dimensional saddle point excitons. Phy Rev

Lett 05:136805

[9] Sundaram RS, Engel M, Lombardo A, Krupke R, Ferrari

AC, Avouris P, Steiner M (2013) Electroluminescence in

single layer MoS2. Nano Lett 13:1416–1421

[10] Wu Z, Fang B, Wang Z, Wang C, Liu Z, Liu F, Wang W,

Alfantazi A, Wang D, Wilkinson DP (2013) MoS2 nanosh-

eets: a designed structure with high active site density for the

hydrogen evolution reaction. ACS Catal 3:2101–2107

[11] Fontana M, Deppe T, Boyd AK, Rinzan M, Liu AY,

Paranjape M, Barbara P (2013) Electron-hole transport and

photovoltaic effect in gated MoS2 schottky junctions. Sci

Rep 3:1634

[12] Shanmugam M, Durcan CA, Yu B (2012) Layered semi-

conductor molybdenum disulfide nanomembrane based

schottky-barrier solar cells. Nanoscale 4:7399–7405

[13] Mak KF, McGill KL, Park J, McEuen PL (2014) The valley

hall effect in MoS2 transistors. Science 344:1489–1492

[14] Feng J, Qian X, Huang CW, Li J (2012) Strain-engineered

artificial atom as a broad-spectrum solar energy funnel. Nat

Phot 6:866–872

[15] Guinea F, Katsnelson MI, Geim AK (2010) Energy gaps and

a zero-field quantum hall effect in graphene by strain engi-

neering. Nat Phys 6:30–33

[16] Ong MT, Reed EJ (2011) Engineered piezoelectricity in

graphene. ACS Nano 6:1387–1394

[17] ScaliseE,HoussaM,PourtoisG,Afanas’evV,StesmansA(2012)

Strain-induced semiconductor to metal transition in the two-di-

mensional honeycomb structure of MoS2. Nano Res 5:43–48

[18] He K, Poole C, Mak KF, Shan J (2013) Experimental

demonstration of contiuous electric structure tuning via

strain in atomically thin MoS2. Nano Lett 13:2931–2936

[19] Ding ZW, Jiang JW, Pei QX, Zhang YW (2015) In-plane and

cross-plane thermal conductivities of molybdenum disulfide.

Nanotechnol 26:065703

[20] Chen XF, Qia JS, Shi DN (2015) Strain-engineering of

magnetic coupling in two-dimensional magnetic semicon-

ductor CrSiTe3: competition of direct exchange interaction

and super exchange interaction. Phys Lett A 379:60–63

[21] Bertolazzi S, Brivio J, Kis A (2011) Stretching and breaking

of ultrathin MoS2. ACS Nano 5:9703–9709

[22] Cooper RC, Lee C, Marianetti CA, Wei XD, Hone J, Kysar

JW (2013) Nonlinear elastic behavior of two-dimensional

molybdenum disulfide. Phys Rev B 87:035423

[23] Wang Z, Devel M (2011) Periodic ripples in suspended

graphene. Phys Rev B 83:125422

[24] Feruz Y, Mordehai D, Towards A (2016) Universal size-

dependent strength of face-centered cubic nanoparticles.

Acta Mater 103:433–441

[25] Singh SK, Neek-Amal M, Costamagna S, Peeters FM (2015)

Rippling, buckling, and melting of single-and multilayer

MoS2. Phys Rev B 91:014101

[26] Ma F, Sun YJ, Ma DY, Xu KW, Chu PK (2011) Reversible

phase transformation in graphene nano-ribbons: lattice

shearing based mechanism. Acta Mater 59:6783–6789

[27] Kadantsev ES, Hawrylak P (2012) Electronic structure of a

single MoS2 monolayer. Solid State Commun 152:909–913

[28] Lammps, http://www.cs.sandia.gov/*sjplimp/lammps.html

[29] Liang T, Phillpot SR, Sinnott SB (2009) Parametrization of a

reactive many-body potential for Mo–S systems. Phys Rev B

79:245110

[30] Jiang JW, Park HS, Rabczuk T (2013) Molecular dynamics

simulations of single-layer molybdenum disulphide (MoS2):

Stillinger-Weber parametrization, mechanical properties, and

thermal conductivity. J Appl Phys 114:064307

[31] Jiang JW (2015) Parametrization of Stillinger-Weber poten-

tial based on valence force field model: application to single-

layer MoS2 and black phosphorus. Nanotechnol 26:315706

[32] Subramaniyan AK, Sun CT (2008) Continuum interpretation

of virial stress in molecular simulation. Int J Solids Struct

45:4340–4346

[33] Xiong S, Cao GX (2015) Molecular dynamics simulations of

mechanical properties of monolayer MoS2. Nanotechnol

26:185705

[34] Zhao JH, Kou LZ, Jiang JW, Rabczuk T (2014) Tension-

induced phase transition of single layer molybdenum sisul-

phide (MoS2) at low temperatures. Nanotechnol. 25:295701

6858 J Mater Sci (2016) 51:6850–6859

[35] Reddy CD, Ramasubramaniam A, Shenoy VB, Zhang YW

(2009) Edge elastic properties of defect-free single-layer

graphene sheets. Appl Phys Lett 94:101904

[36] Le MQ (2015) Prediction of Young’s modulus of hexagonal

monolayer sheets based on molecular mechanics. J Mech Sci

Tech 11:15–24

[37] Lu Q, Gao W, Huang R (2013) Atomistic simulation and

continuum modeling of graphene nanoribbons under uniax-

ial tension. Modell Simul Mater Sci Eng 19:054006

[38] Jing YH, Sun Y, Niu HW, Shen J (2013) Atomistic simu-

lations on the mechanical properties of silicene nanoribbons

under uniaxial tension. Phys Status Solidi B 8:1505–1509

[39] Liu X, Yang QS, He XQ, Liew KM (2012) Size- and shape-

dependent effective properties of single-walled super carbon

nanotubes via a generalized molecular structure mechanics

method. Comput Mater Sci 61:27–33

[40] Sainath G, Choudhary BK (2015) Molecular dynamics

simulations on size dependent tensile deforma- tion beha-

viour of [110] oriented body centred cubic iron nanowires.

Mater Sci Eng, A 640:98–105

[41] Zhao H, Min K, Aluru NR (2009) Size and chirality

dependent elastic properties of graphene nanoribbons under

uniaxial tension. Nano Lett 9:3012–3015

[42] Liu WT, Hsiao C, Hsua WD (2014) Role of surface on the

size-dependent mechanical properties of copper nano-wire

under tensile load: a molecular dynamics simulation. Appl

Surf Sci 289:47–52

[43] Chen CQ, Shi Y, Zhu J, Yan YJ (2006) Size dependence of

Young’s modulus in ZnO nanowires. Phys Rev Lett

96:075505

[44] Zheng YG, Zhao YT, Ye HF, Zhang HW (2014) Size-de-

pendent elastic moduli and vibrational properties of fivefold

twinned copper nanowires. Nanotechno 25:315701

[45] Qiao L, Zheng XJ (2013) Effect of surface stress on the

stiffness of micro/nanocantilevers: nanowire elastic modulus

measured by nano-scale tensile and vibrational techniques.

J Appl Phys 113:013508

[46] Zhang TY, Wang ZJ, Chan WK (2010) Eigenstress model for

surface stress of solids. Phys Rev B 81:195427

[47] Yao HY, Yun GH, Bai N, Li JG (2012) Surface elasticity

effect on the size-dependent elastic property of nanowires.

J Appl Phys 111:083506

[48] Dattaa D, Nadimpalli SPV, Li YF, Shenoyd VB (2015)

Effect of crack length and orientation on the mixed-mode

fracture behavior of graphene. Extreme Mechanics Letters

5:10–17

[49] Pan H, Zhang YW (2012) Edge-dependent structural, elec-

tronic and magnetic properties of MoS2 nanoribbons.

J Mater Chem 22:7280

[50] Gan CK, Srolovitz DJ (2010) First-principles study of gra-

phene edge properties and flake shapes. Phys Rev B

81:125445

[51] Jun S, Li XB, Meng FC (2011) Elastic properties of edges in

BN and SiC nanoribbons and of boundaries in C-BN

superlattices: a density functional theory study. Phys Rev B

83:153407

J Mater Sci (2016) 51:6850–6859 6859