Actuation of a suspended nano-graphene sheet by impact with an argon cluster.
ABSTRACT Using a molecular dynamics simulation, we examine the actuation of nanodrums consisting of a single graphene sheet. The membrane of the nanodrum, which contains 190 carbon atoms, is bent by collision with a cluster consisting of 10 argon atoms. The choice of an appropriate cluster velocity enables nanometre deformation of the membrane in sub-picosecond time without rupturing the graphene sheet. Theoretical results predict that, if an adsorbed molecule exists on the graphene sheet, the quick deformation due to the impact with the cluster can break the weak bonding between the adsorbed molecule and the graphene sheet and release the molecule from the surface; this suggests that this system has attractive potential applications for purposes of molecular ejection.
- SourceAvailable from: Kuniyasu Saitoh[Show abstract] [Hide abstract]
ABSTRACT: Nanocluster impact on a free-standing graphene is performed by the molecular dynamics simulation, and the dynamical motion of the free-standing graphene is investigated. The graphene is bended by the incident nanocluster, and a transverse deflection wave isotropically propagated in the graphene is observed. We find that the time evolution of the deflection is semiquantitatively described by the linear theory of elasticity. We also analyze the time evolution of the temperature profile of the graphene, and the analysis based on the least dissipation principle reproduces the result in the early stage of impact.Physical review. B, Condensed matter 01/2010; 81(11). · 3.66 Impact Factor
- [Show abstract] [Hide abstract]
ABSTRACT: RATIONALECollisions of clusters with solids have become important, especially in the fields of thin film growth or surface processing such as etching or topography smoothing. However, it is not clear how much of the theory or model used in macroscopic collisions is appropriate for the consideration of microscopic collisions.METHODS We considered a cluster ion consisting of thousands of argon atoms as a continuum and examined the possibility that classical mechanics could analyze its collision with metals. A mass spectrometric analysis of the dissociated ions of argon cluster ions (Ar+1500) in collision with five different metals was performed.RESULTSIn the mass spectra at an incident kinetic energy per atom of less than 10 eV, no monatomic argon ions (Ar+) were observed regardless of the prominence of Ar2+ or Ar3+. The branching ratio for the ion yield Ar2+/∑Arn+ (n ≥ 2), representing the dissociation rate, was found to be significantly different for each metal. The relationship between the branching ratio and the impulsive stress caused by the collision of the cluster ion with metal was investigated. The impulsive stress was calculated based on the Young's modulus and density of the clusters and metal, under the assumption that the collision was initially elastic. As a result, the magnitude correlation in the branching ratio corresponded well with that in the impulsive stress.CONCLUSIONS This result is important in that it indicates that collision of nano-sized clusters with solids at low energies can be modeled using elastic theory. Furthermore, the result suggests a new method for evaluating a physical property of a material such as its Young's modulus. Copyright © 2014 John Wiley & Sons, Ltd.Rapid Communications in Mass Spectrometry 10/2014; 28(19). · 2.51 Impact Factor
- [Show abstract] [Hide abstract]
ABSTRACT: By a molecular dynamics method and using different incident energy and particle density, we calculated the argon-atom bombardment on a graphene sheet. The results show that, the damage of the bombardment on the graphene sheet depends not only on the incident energy but also on the particle density of argon atoms. To compare and analyze the effect of the incident energy and the particle density in the argon-atom bombardment, we defined the impact factor on graphene sheet of the incident energy and the particle density by analyzing the structural Lindeman- index and calculating the broken-hole area of the sheet, respectively. The results indicate that, there is a critical incident energy and particle density for destroying the graphene sheet, and there is an exponential accumulated-damage for the impact of both the incident energy and the particle density in argon-atom bombardment on a graphene sheet. Our results supply some valuable mechanics parameters for fabrication of potential graphene-based electronic devices with high particle radiation.01/2010;
Actuation of a suspended nano-graphene sheet by impact with an argon cluster
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
2008 Nanotechnology 19 505501
IP Address: 126.96.36.199
The article was downloaded on 31/10/2011 at 14:01
Please note that terms and conditions apply.
View the table of contents for this issue, or go to the journal homepage for more
HomeSearch CollectionsJournalsAboutContact usMy IOPscience
Nanotechnology 19 (2008) 505501 (7pp)
Actuation of a suspended nano-graphene
sheet by impact with an argon cluster
Norio Inui, Kozo Mochiji and Kousuke Moritani
Graduate School of Engineering, University of Hyogo, 2167, Shosha, Himeji,
Hyogo 671-2280, Japan
Received 29 August 2008, in final form 14 October 2008
Published 24 November 2008
Online at stacks.iop.org/Nano/19/505501
Using a molecular dynamics simulation, we examine the actuation of nanodrums consisting of a
single graphene sheet. The membrane of the nanodrum, which contains 190 carbon atoms, is
bent by collision with a cluster consisting of 10 argon atoms. The choice of an appropriate
cluster velocity enables nanometre deformation of the membrane in sub-picosecond time
without rupturing the graphene sheet. Theoretical results predict that, if an adsorbed molecule
exists on the graphene sheet, the quick deformation due to the impact with the cluster can break
the weak bonding between the adsorbed molecule and the graphene sheet and release the
molecule from the surface; this suggests that this system has attractive potential applications for
purposes of molecular ejection.
(Some figures in this article are in colour only in the electronic version)
Graphene, a single layer of graphite, has recently attracted
attention not only for its unique electrical properties [1, 2] but
alsofor itsnovelmechanical properties. Asuspendedgraphene
sheet, for example, is a promising candidate for membranes
of nanodrums, which may be used in mechanical resonators
designed for mass detection .
The actuation of the suspended graphene sheet  is a
new technical subject.Bunch et al actuated a suspended
graphene sheet by applying a time-varying radio-frequency
to it .We propose another method using a gas cluster
ion beam, wherein the graphene sheet is actuated by collision
with an argon cluster. The advantage of this method is that
high pressure is generated locally in a very short time without
the use of electrical circuits, although temporal control of
the displacement is difficult. By using a cluster instead of a
monomeras the source of kinetic energy, the kineticenergy per
atom is decreased, and therefore the damage to the graphene
sheet can be decreased.
The main purpose of this paper is to investigate the
time evolution of the displacement of a graphene sheet after
collision with an argon cluster.
the relation between the maximum displacement and several
important parameters: the velocity of the cluster, the size
of the cluster and the size of the graphene sheet.
In particular, we consider
application of the actuation of the graphene sheet, we examine
the ejection of an adsorbed molecule from the graphene sheet.
If the graphene is forced to move rapidly, the interaction
between the adsorbed molecule and the graphene is broken,
and the adsorbed molecule is released from the surface of the
graphene.Thus, this technique allows the realization of a
molecular injector, which may be used for manipulation in a
nanochemical reactor or a mass spectrometer.
This paper is organized as follows.
explain the method of the molecular dynamics simulation used
to simulate the collision of the argon cluster with the graphene
sheet. In section 3, we estimate the bending rigidity from the
fundamental frequency of the graphene sheet using continuum
mechanics. In section 4, we show the time dependence
of the displacement of the suspended graphene after impact
with argon for different impact speeds.
the dependence of the deformation of the graphene sheet on
the argon cluster size. In section 5, we show an application
of our sheet method to molecular ejection.
we summarize the obtained results and discuss some of the
problems encountered in the molecular dynamics simulation.
In section 2, we
We also consider
In section 6,
2. Simulation method
The collision of an argon cluster with a graphene sheet is
simulated using a molecular dynamics (MD) method . We
© 2008 IOP Publishing LtdPrinted in the UK
Nanotechnology 19 (2008) 505501N Inui et al
Figure 1. Schematic of actuation of the graphene sheet by collision
with an argon atom. The graphene sheet is placed on a substrate
having a square hole. The cluster collides with the graphene sheet
assume that a single layer of graphene is placed on a substrate
with a square hole at the centre, as shown in figure 1. The
argon cluster collides perpendicularly with the graphene sheet
through the hole. To design this set-up, we referred to the
recent experiment carried out by Poot and Zant . They used
a silicon wafer as the substrate, on which circular holes were
etched with buffered hydrofluoric acid using resist masks.
in the graphene sheet are modelled by the Tersoff–Brenner
empirical potential energy function [8–10]. The interactions
between the argon atoms in the cluster and those between
the carbon atoms and the argon atoms are modelled by the
Lennard-Jones potential. The Tersoff–Brenner potential Eb
is expressed as a summation of the binding energy over the
atomic sites i:
Thus, the force acting on atomic site i is given by
where ∇i = (∂/∂xi,∂/∂yi,∂/∂zi). The energy associated
with atomic site i is expressed as the summation over the
interaction potential between atom i and atom j:
where rij is the distance between atom i and atom j. The
functions VR(r) and VA(r) in equation(3) represent the Morse-
typerepulsiveandattractive interactions, respectively, givenby
VR(r) = f (rij)
S − 1e−β√2S(r−Re),
S − 1e−β√2/S(r−Re),
VA(r) = f (rij)DeS
Table 1. C–C potential parameters.
β (˚ A
Table 2. Lennard-Jones potential parameters.
where f (r) is a smooth cutoff function:
The function¯Bij≡ (Bij+Bji)/2in equation(3) represents the
effect of a many-body coupling between bonds. The parameter
Bijis given by
where θijkis the angle between bonds i–j and j–k, and the
function G is given by
The set of parameters is shown in table 1.
formula, a conjugate-compensation term exists; however, on
the basis of the results of the previous study, it is ignored in our
The Lennard-Jones potential is given by
The parameters for ? and σ used in our simulation are shown
in table 2 .
The position of the atom is updated by the velocity Verlet
algorithm  with a time step of less than ?t = 0.3 fs.
To reduce the calculation time, we introduced a cutoff length.
The interaction between argon atoms by a distance 3.5σAr−Ar
or greater is neglected.The same cutoff is applied to the
calculation of the van der Waals interaction between argon
atom and carbon atom.
We assume that the argon cluster and the graphene sheet
are in a low-temperature environment, and any thermostat is
not used in our simulation.
f (r) =
r < R1,
1 + cos
?r − R1
R1< r < R2,
r > R2.
G(θijk) f (rik)
G(θ) = a0
0+ (1 + cosθ)2
φ(r) = 4?
3. Mechanical properties of graphene
The mechanical properties of graphene are important in
studying the collision process between the argon cluster and
Nanotechnology 19 (2008) 505501 N Inui et al
Figure 2. Illustration of the graphene sheet used in the simulation.
Solid circles denote the fixed atoms.
the graphene sheet. Although the prediction of mechanical
properties is not the aim of this paper, the bending rigidity D
and fundamental frequency νfare particularly important values
when considering the dynamics of the graphene sheet in the
following sections. Therefore, we calculate the fundamental
frequency of the suspended nano-graphene shown in figure 2.
This graphene sheet is nine six-membered carbon rings long
and nine wide, and it contains a total of 190 atoms. We apply
a boundary condition that the atoms shown by the solid circles
in figure 2 are fixed in our simulation.
The fundamental frequency calculated by the MD method
for small-amplitude oscillations is νf = 4 × 1011Hz. This
value is higher than the experimentally measured resonance
frequency , because the size of the graphene sheet
considered here is very small in comparison to that used in
the experiments.In the experiment carried out by Bunch
et al, the graphene sheets are suspended over a 5 μm trench
and the fundamental frequency varies from 1 to 170 MHz.
Since the fundamental frequency of a rectangular membrane
is in inverse proportion to the square of the side length, the
fundamental frequency greatly increases by decreasing the size
of the graphene sheet.
To estimate the bending rigidity of the graphene sheet
using continuum mechanics, we assume that the graphene
sheet is a square continuous membrane whose centre is the
origin and that displacement w(x, y) satisfies the following
where a and b are the length and width of the membrane,
respectively. According to the theory of elasticity , the
fundamental eigenfrequency is given by
where M is the mass of the graphene and γ = b/a denotes
the aspect ratio. Using this formula, we estimate the bending
rigidity from the eigenfrequency to be 0.2 nN nm.
Let us consider the thickness of the membrane. Since
the thickness of the graphene sheet is approximately that of
a single carbon atom, it is difficult to define the thickness of
the membrane. Therefore, the effective thickness is often used.
The bending rigidity is represented using the thickness of the
membrane h, the in-plane Young’s modulus E and the in-plane
Poisson’s ratio ν as
12(1 − ν2).
Poot and Zant measured a value for the bending rigidity of
several-layers-thick graphene, which closely agrees with that
yielded by equation (12) in which the Young’s modulus and
Poisson’s ratio assume the values of bulk graphite, E =
0.92 TPa and ν = 0.16 . Very recently, Lee et al reported
that the two-dimensional second-order elastic stiffness of
monolayer graphene is 340 N m−1.
interlayer spacing in graphite (0.335 nm), it yields an effective
Young’s modulus of E = 1.0 TPa. They note that graphene is
the strongest material ever measured. If the effective thickness
is defined as the thickness that satisfies equation (12) and
Young’s modulus is 1.0 TPa, then it is 1.3 ˚ A. This calculated
value agrees with the thickness reported by Tserpes .
ratio of the single graphene layer are the same as those of bulk
graphite. Young’s modulus extracted by Frank et al, E =
0.5 TPa, is indeed different from that of bulk graphite .
Therefore, further experimentation is required to determine
the accurate mechanical properties of single-layer graphene;
however, recent experiments have revealed that the graphene
sheet becomes remarkably soft under a perpendicular force as
the number of layers is decreased . Thus, a single graphene
sheet is an excellent candidate to act as a very soft and strong
By dividing the
4. Simulation results
4.1. Collision with an argon atom
We now consider actuation of the graphene sheet occurring
upon being struck by argon atoms.
parameters are the kinetic energy of the argon Ekand the size
of the cluster. If the kinetic energy of the argon is significantly
greater than the binding energy between the carbon atoms, the
graphene sheet is ruptured by the impact. The threshold of
kinetic energy required to rupture the graphene sheet depends
on the impact position. Therefore, we randomly repositioned
the point of impact inside a square region with sides of σAr−Ar
and observed whether or not the graphene was ruptured. If
the kinetic energy of the cluster was less than 30 eV, which is
almostfour timestheinteraction energy per atom, thegraphene
did not rupture within a series of 100 trials. The graphene
ruptured when the kinetic energy of the argon was 40 eV.
Thus, the threshold of the kinetic energy is probably between
30 and 40 eV. When the graphene sheet ruptured, a small
The most important
Nanotechnology 19 (2008) 505501N Inui et al
hole was initially generated by the impact with the argon,
which was later enlarged by the interaction between the carbon
atoms. The process of rupturing is interesting: however, we
concentrate our attention on the bending of the graphene sheet
that does not result in its rupture.
The argon atom is initially located at z = −2σAr−Ar
independent of the kinetic energy of the cluster, and starts
to move toward the graphene sheet with velocity vz
√2Ek/mArwhere the argon mass mAr= 6.6335 × 10−26kg.
Figure 3(a) illustrates the change in the average maximum
displacement zmaxover 100 trials for Ek= 10, 20 and 30 eV.
The initial velocities are vz = 6.95 km s−1, 9.83 km s−1and
12.0 km s−1for Ek = 10, 20 and 30 eV, respectively. The
graphene sheet is deformed by the impact in a very short time,
and the displacements of the atoms near the centre reach their
increases with kinetic energy. The maximum forces generated
by the impact are approximately 4, 8 and 10 nN for Ek= 10,
20 and 30 eV, respectively. The argon atom loses its kinetic
energy quickly, in less than 2 fs. If the velocity decreases to
zero within the interval ?t = 2 fs, the mean forces acting
on the argon, roughly estimated by mArvz/?t, are 4.6, 6.5
and 8 nN for Ek = 10, 20 and 30 eV, respectively. The
time required for the argon to transfer its kinetic energy to
the graphene sheet is significantly smaller than the eigenperiod
of the graphene sheet. Accordingly, the force acting on the
graphene sheet is regarded as an impulsive force.
In addition to the maximum displacement, the mean of
the displacement ¯ z that is averaged over all atoms is important.
conditions used for figure 3(a). Note that the averaged mean
displacement reaches its maximum value after the appearance
of the first peak of the averaged maximum displacement. This
is because the deformation spreads from the centre to the edge
of the graphene sheet.
Figure 3(c) shows the kinetic energy of the graphene
per carbon atom Ek. Although the graphene sheet oscillates
vertically, the fluctuation of the kinetic energy of the graphene
sheet is small. This is because the contribution concerning the
vertical velocity to the total kinetic energy is small. In other
words, the kinetic energy of the argon cluster is transferred
mainly to the horizontal oscillations of the carbon atoms,
4.2. Collision with an argon cluster
Gas cluster ion beams have many potential uses in the field
of surface processing, such as in dry etching, surface cleaning
and surface smoothing . The gas cluster is a good carrier
of kinetic energy. In contrast to a monomer ion beam, the
kinetic energy per atom in the cluster can be decreased without
decreasing the total kinetic energy, simply by increasing the
number of atoms in the cluster. This enables low-damage,
atomic-scale surface smoothing and shallow implantation.
An argon cluster consisting of 10 atoms was made to
collide vertically with the graphene sheet. The argon cluster
initially had the lowest potential  and its centre of mass
was located at (x, y,z) = (0,0,−2σAr−Ar). The mean radius
Figure 3. Plot of the temporal changes in the displacement of the
graphene sheet caused by a single argon atom: (a) average maximum
displacement, (b) average mean displacement and (c) kinetic energy
of the graphene sheet per carbon atom for three different kinetic
energy values of the argon atom (10, 20 and 30 eV).
of the cluster was 0.33 nm. The displacement of the graphene
sheet after collision was dependent on the configuration of
atomsinthecluster. Therefore, we calculated theaverage value
of various displacements of the graphene sheet that occurred
through collisions with different configurations of atoms in the
The values of the averaged maximum displacement and
averaged mean displacement of the graphene sheet recorded
for collisions for over 100 different configurations of the argon
cluster are illustrated in figures 4(a) and (b), respectively, for
Ek = 50, 100 and 150 eV. Both figures clearly show that
collision with an argon cluster resulted in greater deformation
of thegraphene sheetthan didcollisionwitha singleatom. The
Nanotechnology 19 (2008) 505501N Inui et al
Figure 4. Plot of the temporal changes in the displacement of the
graphene sheet caused by an argon cluster containing 10 atoms:
(a) averaged maximum displacement, (b) averaged mean
displacement and (c) kinetic energy of the graphene sheet per carbon
atom for three different kinetic energy values of the argon cluster
(50, 100 and 150 eV).
displacement caused by the collision with the cluster is almost
three times larger than that by a monomer. The total kinetic
energy of the cluster is greater than that of a single argon atom;
however, the kinetic energy per atom is close to that used in the
monomer simulation; the risk of rupturing the graphene sheet
does not increase much.
We expect the extent of deformation to increase with the
size of the graphene sheet and of the cluster. As an example,
we simulate the collision of an argon cluster containing 1000
atoms with a graphene sheet consisting of 51×45 six-member
carbon rings. The length and width of the graphene sheet
are 11.0 nm and 10.9 nm, respectively. Figure 5 shows the
temporal change in the displacement of the graphene sheet
Figure 5. Deformation of a graphene sheet containing 4732 carbon
atoms during impact with a cluster containing 1000 argon atoms. The
kinetic energy of the argon cluster is 5 keV.
caused by collisionwiththe argon cluster whose kinetic energy
is5keV. Thedome-shaped graphene sheet isfoundat5 ps. The
maximum height and volume of the dome are 4 nm and 3 nm3,
respectively. It is noteworthy that no argon atoms penetrate the
Figure 4(c) shows the kinetic energy of graphene per
carbon atom after the collision of the argon cluster.
comparison with figure 3(c), both the absolute values and
fluctuations of the kinetic energy of graphene shown in
figure 4(c) are larger than those in figure 3(c). This means that
the larger amount of energy can be transferred to the graphene
sheet by using argon clusters.
Nanotechnology 19 (2008) 505501 N Inui et al
Figure 6. Illustration of molecular ejection by rapid movement of a
six-member carbon ring.
5. Application of the actuator to molecular ejection
When the argon cluster collides with the graphene sheet, it
causes the sheet to bend quickly. As an application of this
actuation, we consider the ejection of a molecule that bonds
weakly with the graphene. If the binding energy between the
atoms in the molecule ?M−Mis significantly greater than that
between the molecule and the graphene ?M−G, the molecule
is released by providing energy between ?M−Mand ?M−Gto
the molecule without causing it to break.
may be used in mass spectrometry, nanosurface chemistry and
Figure 6 shows a demonstration of molecular ejection by
the impact of an argon cluster with a graphene sheet. An argon
cluster contains 10 atoms and its kinetic energy is 100 eV.
A six-member carbon ring is located on the graphene sheet
with a separation of 3.4 ˚ A, which corresponds to the gap
between graphene sheets found in graphite. We assume that
the interaction between the graphene sheet and the six-member
carbon ring is modelled by the Lennard-Jones potential with
parameters ?C−G= 0.00188 eV and σC−G= 3.33264˚ A .
This binding energy ?C−G is significantly smaller than that
between carbon atoms in the six-member carbon ring.
Two forces act on the six-member carbon ring when the
collision between the graphene sheet and the argon cluster
occurs. The firstforce istheattractiveforce between thecarbon
atoms in the six-member carbon ring and the argon atoms. The
Figure 7. Plot of the trajectories of the ejected six-member carbon
ring for three impact positions. The gap between the centre of the
six-member carbon ring and the centre of mass of the argon cluster at
t = 0 is denoted by ?x.
second force is the repulsive force between the carbon atoms
in the six-member carbon ring and the carbon atoms in the
graphene sheet. The repulsive force between the carbon atoms
rapidly increases as the graphene sheet approaches the six-
member carbon ring. This repulsive force is greater than the
attractive force, causing it to be released from the surface of
the graphene sheet. If the kinetic energy is insufficient to cut
the binding between the graphene sheet and the six-member
carbon ring, the six-member carbon ring stays in its position
relative to the graphene sheet and does not leave the graphene
The trajectory of the carbon ring depends on the position
where the argon cluster collides with the graphene sheet. We
set the initial position of the centre of the argon cluster at
(?x, 0, −2σAr−Ar) and consider the influence of the gap ?x
on the trajectory of the carbon ring. The other parameters are
the same as those used in the previous simulation for ejection.
Figure 7 shows the trajectories of the carbon rings which are
averaged over 100 different configurations of the argon cluster
for three different gaps. The horizontal axis denotes sgn (¯ x)¯ r,
where ¯ x denotes the mean position along the x axis and ¯ r
denotes the mean distance from the centre of mass of the
carbon ring before the collision. To indicate the directionof the
ejection, the function sgn (x) that denotes the sign of x is used.
The vertical axis denotes the mean position along the z axis.
The horizontal and vertical bars attached to the points in the
figure denote the standard deviation of the position. Figure 7
illustratesthat, iftheargon clustercollidesonthe left-handside
of the carbon ring, the carbon ring is ejected to the right-hand
side. Conversely, if the argon cluster collides on the right-hand
side of the carbon ring, the carbon ring is ejected to the left.
Therefore, the direction of ejection can be roughly controlled
by changing the axis of the argon cluster beam.
We have proposed a simple method of actuating a graphene
sheet. The recent remarkable progress of gas cluster ion beam
technology has enabled us to obtain argon clusters whose size
Nanotechnology 19 (2008) 505501 N Inui et al
and velocity are controlled very precisely. Therefore, it is
possible to realize our proposal by using currently available
Although MD simulation often requires a long calculation
time, it is used in this study to examine the actuation of a
graphene sheet because it is difficult to otherwise analyse the
collision of the argon cluster with the graphene sheet.
particular, the advantage of MD is that the effect when the
argon cluster is broken apart is correctly simulated.
A more precise prediction of the dynamicsof the graphene
sheet calls for some modifications to the simulation. First,
the boundary conditions must be changed. In our simulation,
the atoms on the edge of the graphene sheet were fixed;
however, in reality, the graphene sheet is peeled from the
substrate by the impact. Second, the lattice vibration of the
substrate must be taken into account when future simulations
As an application of our proposal, we considered
molecular ejection from the graphene sheet.
desorption is essential in molecular manipulation, it is more
difficult than adsorption of the molecule onto the graphene
surface. Using our method, the molecule is desorbed without
being directly touched, unlike manipulation using atomic force
microscopy. In addition, the cluster beams can actuate many
suspended graphene sheets simultaneously.
The authors would like to thank Michihiro Hashinokuchi and
Nobuyasu Ito for helpful discussions.
 Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y,
Dubonos S V, Grigorieva I V and Firsov A A 2004 Science
 Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y,
Dubonos S V, Grigorieva I V and Firsov A A 2005 Nature
 Schedin F, Geim A K, Morozov S V, Hill E W, Blake P,
Katsnelson M I and Novoselov K S 2007 Nat. Mater. 6 652
 Meyer J C, Geim A K, Katsnelson M I, Novoselov K S,
Booth T J and Roth S 2007 Nature 446 60
 Bunch L S, van der Zande A M, Verbridge S S, Frank I W,
Tanenbaum D M, Parpia J M, Craighead H G and
McEuen P L 2007 Science 315 490
 Heermann D W 1986 Computer Simulation Methods
 Poot M and van der Zant H S J 2008 Appl. Phys. Lett.
 Brenner D W 1990 Phys. Rev. B 42 9458
 Brenner D W, Shenderova O A, Harrison J A, Stuart S J,
Ni B and Sinnott S B 2002 J. Phys.: Condens. Matter
 Zhou J and Huang R 2008 J. Mech. Phys. Solids 56 1609
 Yamaguchi Y and Gspann J 2001 Eur. Phys. J. D 16 103
 Verlet L 1967 Phys. Rev. 98 159
 Landau L D and Lifshitz E M 1986 Theory of Elasticity
 Lee C, Wei X, Kysr J W and Hone J 2008 Science 231 385
 Tserpes K I and Papanikos P 2005 Composites B 36 468
 Frank I W, Tanenbaum D M, van der Zande A M and
McEuen P L 2007 J. Vac. Sci. Technol. B 25 2558
 Yamada I, Matsuo J, Toyoda N and Kirkpatrick A 2001
Mater. Sci. Rep. 34 231
 Wales D J and Doye J P K 1997 J. Phys. Chem. A 101 5111
 Wang Y, Scheerschmidt K and G¨ osele U 2000 Phys. Rev. B