C A R B O N 5 0 ( 2 0 1 2 ) 4 9 0 3 – 4 9 1 0

.sc ienced i rec t .com

Avai lab le a t www

journal homepage: www.elsevier .com/ locate /carbon

Elastic properties of a macroscopic graphene samplefrom phonon dispersion measurements

Antonio Politano a, Antonio Raimondo Marino a, Davide Campi b,Daniel Farıas c,d, Rodolfo Miranda c,d,e, Gennaro Chiarello a,f,*

a Dipartimento di Fisica, Universita degli Studi della Calabria, 87036 Rende (Cs), Italyb Dipartimento di Scienza dei Materiali, Universita Milano-Bicocca, 20125 Milano, Italyc Departamento de Fısica, Universidad Autonoma de Madrid, Campus Universitario de Cantoblanco, 28049 Madrid, Spaind Instituto de Ciencia de Materiales ‘‘Nicolas Cabrera’’, Universidad Autonoma de Madrid, Cantoblanco, 28049 Madrid, Spaine Instituto Madrileno de Estudios Avanzados (IMDEA) en Nanociencia, Cantoblanco, 28049 Madrid, Spainf Consorzio Nazionale Interuniversitario di Scienze Fisiche della Materia, via della Vasca Navale 84, 00146 Roma, Italy

A R T I C L E I N F O

Article history:

Received 18 March 2012

Accepted 9 June 2012

Available online 17 June 2012

0008-6223/$ - see front matter � 2012 Elsevihttp://dx.doi.org/10.1016/j.carbon.2012.06.019

* Corresponding author at: Dipartimento di FE-mail address: gennaro.chiarello@fis.uni

A B S T R A C T

The average elastic properties and the interatomic force constants of a quasi-freestanding

graphene sample have been evaluated by analyzing the phonon dispersion measured by

means of angle-resolved energy loss spectroscopy. We found a Poisson’s ratio of 0.19 and

a Young’s modulus of 342 N/m. Such findings are in excellent agreement with calculations

performed for a free-standing graphene membrane. Present results indicate that the high

crystalline quality of graphene grown on metal substrates leads to macroscopic samples

of high tensile strength and bending flexibility to be used for technological applications

such as electromechanical devices and carbon-fiber reinforcements.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Graphene membranes are attracting much interest for their

wide possible technological applications [1,2]. Understanding

the elastic properties is mandatory in order to tailor graph-

ene’s mechanical properties. The intrinsic strength of graph-

ene, which is higher with respect to other materials [3], opens

the way to several applications such as actuators [4] and

nano-electromechanical devices [5,6] and, moreover, as car-

bon-fiber reinforcement in polymeric nanocomposites [7].

Atomic force microscopy (AFM) is widely used to measure

elastic properties [8–10]. The main advantages of AFM for

such experiments are the nanoscale lateral resolution and

the imaging capability. However, the tip shape and the con-

tact geometry between tip and sample, which are not known,

represent a remarkable hurdle to extensive and precise

measurements.

er Ltd. All rights reserved

isica, Universita degli Stucal.it (G. Chiarello).

Herein, we present an alternative in situ method for eval-

uating the average elastic properties (Young’s modulus and

the Poisson’s ratio) of a quasi-freestanding graphene sample

grown on a metal substrate, based on the investigation of

the phonon dispersion.

To date, experimental investigations on elastic properties

of graphene are complicated by the lack of macroscopic sam-

ples with good crystalline quality. On the other hand, the

preparation of highly-ordered monolayer graphene (MLG)

could be extended up to the millimeter scale for epitaxial

graphene grown on metal substrates [11,12].

Among graphene systems, the epitaxial growth of MLG on

Pt(111) is characterized by the weakest graphene-metal inter-

action [13–17]. Moreover, in contrast with other MLG/metal

interfaces [18–20], MLG/Pt(111) is nearly-flat. MLG on Pt(111)

behaves as nearly-flat, free-standing graphene membrane

[13]. Thus, such system represents an ideal playground for

.

di della Calabria, 87036 Rende (Cs), Italy. Fax: +39 0984494401.

4904 C A R B O N 5 0 ( 2 0 1 2 ) 4 9 0 3 – 4 9 1 0

investigating phonon modes and elastic properties without

the influence of the overlayer corrugation and, moreover, of

the substrate.

High-resolution electron energy loss spectroscopy

(HREELS) is a powerful tool for investigating the surface pho-

non dispersion [21]. Its main advantages are the high surface

sensitivity, the excellent resolution in both the energy and

momentum domains, and the wide energy and momentum

windows [22,23]. Present HREELS measurements consent us

to directly evaluate the longitudinal and transverse sound

velocities of acoustic waves. From these data, we estimate

the Poisson’s ratio (0.19) and Young’s modulus (342 N/m) of

the graphene sheet. Despite the macroscopic size of our

graphene sample which usually reduces the tensile strength

for the presence of defects and grain boundaries, the above

parameters well agree with results reported for suspended

graphene membranes [10] with diameter of 1.0–1.5 lm. Hence,

our results demonstrate that high-quality and macroscopic

samples of epitaxial graphene on metal substrates exhibit

the tensile strength predicted by theory. Moreover, inter-

atomic force constants (FC) for MLG/Pt(111) have been calcu-

lated and used for fitting the experimental phonon dispersion.

Fig. 1 – LEED pattern of graphene on Pt(111), recorded at

Ep = 74 eV and for a sample temperature of 100 K.

2. Experimental

Experiments were carried out in an ultra-high vacuum (UHV)

chamber operating at a base pressure of 5 · 10�9 Pa. The sam-

ple was a single crystal of Pt(111). The substrate was cleaned

by repeated cycles of ion sputtering and annealing at 1300 K.

Surface cleanliness and order were checked using Auger elec-

tron spectroscopy (AES) and low-energy electron diffraction

(LEED) measurements, respectively.

Graphene was obtained by dosing ethylene onto the clean

Pt(111) substrate held at 1150 K. The saturation of the first

layer was reached upon an exposure of 3 · 10�8 mbar for

10 min (24 L, 1 L = 1.33 · 10�6 mbar s). After removing the

C2H4 gas from the chamber the temperature was held at

1150 K for further 60 s. Only MLG have been found in the whole

sample, as confirmed by our own ex-situ Raman spectroscopy

and scanning electron microscopy (SEM) experiments, cur-

rently under way [24]. Similar conclusions have been reported

for other investigations on the same system [15,25].

The inspection of the LEED pattern clearly shows the pres-

ence of well-resolved spots which are fingerprint of the order

of the MLG over-structure.

The arcs in the LEED pattern indicate the existence of dif-

ferent rotational domains. Nonetheless, preferred orienta-

tions aligned with the substrate (R0) and rotated by thirty

degrees (R30) are clearly distinguished. Despite the presence

of other domains, the predominance of R0- and R30-oriented

domains in the whole sample has been clearly inferred by Ra-

man, SEM and LEEM measurements [24].

HREEL experiments were performed by using an electron

energy loss spectrometer (Delta 0.5, SPECS). The energy reso-

lution of the spectrometer was degraded to 4 meV so as to in-

crease the signal-to-noise ratio of loss peaks. Dispersion of

the loss peaks, i.e., Eloss(q||), was measured by moving the ana-

lyzer while keeping the sample and the monochromator in a

fixed position. To measure the dispersion relation, values for

the parameters Ep, impinging energy and hi, the incident an-

gle, were chosen so as to obtain the highest signal-to-noise

ratio. The primary beam energy used for the dispersion,

Ep = 20 eV, provided, in fact, the best compromise among sur-

face sensitivity, the highest cross-section for mode excitation

and q|| resolution.

As

�h qjj!¼ �hðki

!sin hi � ks

!sin hsÞ;

the parallel momentum transfer, q|| depends on Ep, Eloss, hi and

hs according to:

qjj ¼ffiffiffiffiffiffiffiffiffiffiffiffi2mEp

p�h

sin rhi �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� Eloss

Ep

ssin hs

!

where Eloss is the energy loss and hs is the electron scattering

angle [26].Accordingly, the integration window in reciprocal

space [23] is

Dqjj �ffiffiffiffiffiffiffiffiffiffiffiffi2mEp

p�h

coshi þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� Eloss

Ep

scoshs

!a

where a is the angular acceptance of the apparatus (±0.5� in

our case). For the investigated range of q||, the indeterminacy

has been found to range from 0.005 (near �C) to 0.022 A�1 (for

higher momenta). To obtain the energies of loss peaks, a poly-

nomial background was subtracted from each spectrum. The

resulting spectra were fitted by a Gaussian line shape (not

shown herein).

All measurements were made at room temperature.

3. Results and discussion

The phonon dispersion was measured with the sample

aligned along the �C� �M direction of the Pt(111) substrate,

which corresponds to the �C� �M direction of R0 domains

and the �C� �K direction of R30 domains. Loss measurements

Fig. 3 – Dispersion relation for phonon modes in MLG/

Pt(111). Experimental data (empty circles). The dotted lines

represent the calculated dispersion relations (along �C� �K

and �C� �M) according to the model by Aizawa et al. [41] using

the best fit parameters reported in Table 3. Comparing

experimental and theoretical dispersion curves we can infer

that the main contribution to the phonon dispersion curves

come from the �C� �M direction (R0 domains).

Fig. 2 – HREEL spectra for MLG/Pt(111) as a function of the

scattering angle. The incidence angle is 80.0� and the

impinging energy is 20 eV.

C A R B O N 5 0 ( 2 0 1 2 ) 4 9 0 3 – 4 9 1 0 4905

of MLG/Pt(111) recorded at Ep = 20 eV as a function of the scat-

tering angle hs are reported in Fig. 2, while the dispersion rela-

tion Eloss(q||) is shown in Fig. 3.

HREEL spectra, recorded as a function of the scattering an-

gle, show several dispersing features all assigned to phonon

excitations. In graphene, two kinds of phonons exist: lattice

vibrations in the plane of the sheet giving rise to transverse

and longitudinal acoustic (TA and LA) and optical (TO and

LO) branches, and lattice vibrations out of the plane of the

layer which give rise to the so-called flexural phonons (ZA

and ZO). Modes classified with ‘‘T’’ are shear in-plane phonon

excitations; ‘‘L’’ modes are longitudinal in-plane vibrations;

while ‘‘Z’’ indicates out-of-plane polarization. In graphite

and graphene, the ZO mode is significantly softened with re-

spect to the other two optical modes, i.e. TO and LO. This is

due to the higher freedom for atom motion perpendicular to

the plane with respect to the in-plane motion.

Measurements were repeated for several preparations of

the MLG over-structure also by using different impinging

energies and incident angles. All these experiments provided

the same phonon dispersion. The sharpness of phonon

modes observed in Fig. 2 indicates an excellent crystalline or-

der in the graphene sample, in spite of the presence of several

rotational domains. This confirms the predominance of only

R0 and R30 domains (spots in the LEED pattern in Fig. 1) over

the other ones (arcs in the same Figure).

Modes associated with vibrations of carbon atoms in the

direction of the r bonds, i.e. the LA and LO phonons, are sim-

ilar to those of graphite (except a stiffening of the LO mode at�C by 5 meV). Moreover, a careful comparative analysis with re-

spect to graphite (Fig. S1 of the Supporting Information)

showed a softening of the TA mode (with a maximum differ-

ence of 18 meV). On the other hand, by comparing the pho-

non dispersions of MLG/Pt(111) (our data) and MLG/Ni(111)

[27,28] (Supporting Information, Fig. S1), we note a significant

difference for out-of-plane modes (ZA and ZO), which are re-

lated to perpendicular vibrations of carbon atoms with re-

spect to the surface. In particular, in MLG/Ni(111) the ZA

phonon is stiffened while the ZO mode is softened by 10–

15 meV at the �C point with respect to MLG/Pt(111). This is

caused by of the orbital mixing of the p-states of the MLG with

Ni d-bands [29].

A particular attention should be devoted to the ZA phonon

(Supporting Information, Fig. S2), as it has been recently

found that such mode carries most of the heat [30] in sus-

pended graphene. It is a bending mode in which the two

atoms in the unit cell are involved in an in-phase motion in

the out-of-plane direction.

In principle, EELS planar scattering from an isolated graph-

ene sheet does not allow the observation of the TA branch

(unlike graphite, where there is no vertical mirror plane along�K). In fact, its intensity is very reduced and only after long

acquisition time it can be detected.

While the TA and the LA phonons are characterized by a

linear dispersion, the ZA mode has a quadratic dispersion

near the �C point (Fig. 2), as also in layered crystals [31]. Its dis-

persion depends on the bending rigidity s, which is an impor-

tant parameter for mechanical properties of membranes:

4906 C A R B O N 5 0 ( 2 0 1 2 ) 4 9 0 3 – 4 9 1 0

xZAðqjjÞ ¼ffiffiffiffiffiffiffis

q2D

rj~qjjj2

where q2D ¼ 4mC=ð3ffiffiffi3p

a2Þ is the two-dimensional (2D) mass

density (mC is the atomic mass of carbon atoms and a is the

in-plane lattice parameter).

We found that the bending rigidity s could be estimated to

be about 2 eV, in agreement with results in Ref. [32]. For the

sake of completeness, calculations in Ref. [33] for MLG at

T = 0 reported a value of 1.4 eV which is close to ab initio pre-

dictions of 1.46 eV [34] or 1.6 eV [35,36]. However, it should be

considered that the lattice mismatch between graphene and

Pt(111) is accommodated by moire structures with different

rotational variants [13,15,37]. Moire reconstruction in do-

mains of graphene/Pt(111) is expected to increase the bend-

ing rigidity at long wavelengths, as found for ripples in free-

standing, suspended graphene [38].

The degeneracy of LA and LO phonons at higher momenta

was observed, even if the energy of such degenerate mode at

higher momenta is blue-shifted by 25 meV with respect to the

calculated value in Refs. [39] and [40].

Although both the �C� �M direction of the R0 domains and

the �C� �K direction of the R30 domains contribute to the mea-

sured phonon dispersion curves, by comparing the experi-

mental dispersion curve with theoretical calculations for

both symmetry directions we can infer that the main contri-

butions come from the �C� �M direction of the R0 domains.

However, sound velocities were extracted from the experi-

mental slope of the acoustic branches in the low-q|| limit,

for which TA and LA phonons along the �C� �K and �C� �M

directions coincide.

We obtain 14.0 and 22.0 km/s for the TA and the LA

branches, respectively. These results agree well with the

sound velocities for single-crystalline graphite that can be

determined using inelastic X-ray scattering in Ref. [42], i.e.

14.7 and 22.2 km/s. Previous EELS investigation in Ref. [43]

for the graphite (0001) surface phonons found sound veloci-

ties of 14 and 24 km/s for TA and LA modes, respectively.

Data on the dispersion of acoustic phonons of graphene

can provide information on its elastic properties. According

to the procedure illustrated in Ref. [44], the sound velocities

of the TA and LA branches could be used for calculating the

Table 1 – Poisson’s ratio m, as reported in different experimenta

Experimental, MLG/Pt(111)Experimental, basal plane of graphite, Refs. [45,46]Atomistic Monte Carlo, Ref. potential, Ref. [34]Continuum plate theory, Ref. [50]Density functional theory, Ref. [51]First-principles total-energy calculations, combined to continuuAb initio, Ref. [53]Ab initio, Ref. [54]Valence force model, Ref. [32]Molecular dynamics, Ref. [48]Empirical force-constant calculations, Ref. [55]Continuum elasticity theory and tight-binding atomistic simulaAb initio, Ref. [57]Brenner’s potential, Ref. [58]

in-plane stiffness j (the 2D analogous of the bulk modulus)

and the shear modulus l of the graphene sheet, respectively:

vL ¼ffiffiffiffiffiffijþlq2D

qvT ¼

ffiffiffiffiffiffil

q2D

qThus, we estimate j and l to be 211 and 144 N/m, respec-

tively. It is worth noticing that graphene is a true 2D material,

therefore its elastic behavior is properly described by 2D prop-

erties with units of force/length.

On the other hand, the 2D shear and bulk moduli are also

defined as a function of the Poisson’s ratio t and the Young’s

modulus for 2D samples E2D:

j ¼ E2D

2ð1�mÞ l ¼ E2D

2ð1þmÞ

Hence, from j and l it is possible to estimate the Poisson’s

ratio, i.e. the ratio of transverse contraction strain to longitu-

dinal extension strain in the direction of the stretching force:

t ¼jl� 1jlþ 1

� 0:19

The obtained value, i.e. 0.19, agrees well with results for

graphite in the basal plane (0.165) [45,46] while it is 0.28 in car-

bon nanotubes [47]. It represents an intermediate value with

respect to those reported by calculations for graphene, as

shown in Table 1.

It is worth noticing that, according to molecular dynamics

calculations [48], the Poisson’s ratio increases with the size of

the graphene sample up to reach a saturation value and it

also depends on temperature. This opens the possibility to

tailor the mechanical properties of graphene for engineering

applications.

The Poisson’s ratio could be used as a powerful test among

the various existing calculations on phonon dispersion in

graphene. As an example, the calculated LA and TA modes

in Ref. [59] would lead to a clearly underestimated value of

the Poisson’s ratio (�0.05).

It is also possible to estimate the Young’s modulus E2D,

which is a measure of the stiffness of an isotropic elastic

material. It is defined as the ratio of the uniaxial stress over

the uniaxial strain.

l and theoretical investigations.

Poisson’s ratio m

0.190.1650.1490.160.162

m elasticity, Ref. [52] 0.1690.1730.1860.200.220.227

tions, Ref. [56] 0.310.320.397

Table 2 – 2D Young’s modulus E2D, expressed in N/m, as reported in different experimental and theoretical investigations.

Young’s modulus E2D (N/m)

Experimental (HREELS), MLG/Pt(111) 342Experimental (AFM) on graphene membranes, Ref. [10] 340Tersoff-Brenner potential, Ref. [60] 235Energetic model, Ref. [64] 307Continuum elasticity theory and tight-binding atomistic simulations, Ref. [56] 312Brenner’s potential, Ref. [58] 336First-principles total-energy calculations, combined to continuum elasticity, Ref. [52] 344Tersoff–Brenner potential, Ref. [34] 345Ab initio, Ref. [54] 350Atomistic Monte Carlo, Ref. [49] 353Density functional theory, Ref. [51] 356Empirical force-constant calculations, Ref. [55] 384Experimental (AFM) on MLG/copper foils, Ref. [65] 55

C A R B O N 5 0 ( 2 0 1 2 ) 4 9 0 3 – 4 9 1 0 4907

As reported in Table 2, many theoretical investigations

found Young’s moduli ranging from 307 to 356 N/m. The ob-

tained value of E2D for MLG/Pt(111), i.e. 342 N/m, agrees well

with most theoretical results (Table 2), a part from calcula-

tions in Ref. [60] (underestimated E2D). In particular, a good

agreement exists between present results and first-principles

total-energy calculations, combined to continuum elasticity,

reported in Ref. [52].

It will be helpful to compare present results with the case

of three-dimensional (3D) materials and, in particular, with

bulk graphite. To obtain the corresponding 3D parameter,

the value of E2D should be divided by the distance between

the MLG and the underlying Pt(111) substrate (3.31 A)

[14,61]. Thus, E2D as obtained by vibrational measurements

corresponds to a 3D Young’s modulus E = 1.03 TPa. This is in

fair agreement with experiments on bulk graphite yielding

1.02 TPa for the in-plane Young’s modulus [45]. For the sake

of completeness, the Young’s modulus obtained for single-

walled carbon nanotubes ranges from 0.45 and 1.47 TPa [62],

while for multi-walled carbon nanotubes it was found to

range from 0.27 to 0.95 TPa [63].

We want to point out that the experimental measurement

of elastic properties of graphene in Ref. [10] have been carried

out for graphene flakes with a diameter of 1.0–1.5 lm. On the

other hand, our measurements are related to the average

elastic properties of a 9 · 9 mm sample. Graphene epitaxy

Table 3 – Best-fit values of FC for MLG/Pt(111) for the �C� �M (ourMLG/Ni(111) [68].

MLG/Pt(111) �C� �M(our calculations and Ref. [67])

a1 (N/m) 397394 [67]

a2 (N/m) 54.553.7 [67]

c1 (10�19 J) 7.98.3 [67]

c2 (10�19 J) 3.13.21 [67]

d (10�19 J) 3.01.9 [67]

as (N/m) 00 [67]

on metal substrates produces initially sparse graphene do-

mains which then extend up to reach macroscopic dimen-

sions in correspondence of the completion of the MLG.

Epitaxial graphene grown on metal templates promises also

to be used as carbon-fiber reinforcements.

Recently, nanoindentation AFM measurements [65] have

been performed on MLG grown by chemical vapor deposition

(CVD) on copper foils and successively transferred onto sili-

con nitride grids with arrays of pre-patterned holes. These

experiments have revealed a notably reduced E2D (55 N/m)

with respect to the present finding (342 N/m). Such extremely

low value of E2D could be a consequence of the modification of

the membrane structure induced by the transfer process (see

Ref. [65] for more details). On the other hand, MLG on Pt(111)

is nearly-flat and quasi-freestanding on Pt and thus present

measurements are not influenced by such effects.

In addition, in the linear elastic regime, it is possible to

estimate the elastic constants C11 and C12, from E2D and m:

E2D ¼ C211�C2

12C11

m ¼ C12C11

We found C11 = 422 N/m and C12 = 80 N/m, which are in

good agreement with values reported by Cadelano et al. [52]

(354 and 60 N/m).

Their corresponding 3D values are 1.27 and 0.24, respec-

tively, which agree well with experimental findings for graph-

ite reported in Ref. [66] (1.11 and 0.18 TPa).

calculations and Ref. [67]); for pristine graphite [41]; and for

Graphite �C� �M(from Ref. [41])

MLG/Ni(111) �C� �M(data taken from Ref. [68])

364 339

61.9 45.0

8.3 7.24

3.38 2.12

3.17 1.46

0 11.0

4908 C A R B O N 5 0 ( 2 0 1 2 ) 4 9 0 3 – 4 9 1 0

The interatomic FC can be extracted from the dispersion

relation of phonon modes, by using the model proposed by

Aizawa et al. in Ref. [41] which is based on six phenomenolog-

ical FC parameters determined from experimental data. A

least-squares procedure was used to fit the parameters. In

Table 3 we compare our calculations for MLG/Pt(111) for the�C� �M direction, which has a major spectral contribution in

our HREEL spectra (predominance of R0 domains), with

parameters reported for the �C� �M direction of the same sys-

tem by Aizawa et al. [67]. In addition, values calculated for

graphite [41] and for MLG/Ni(111) [68] are also reported. We

note that the FC between nearest neighbors (a1) for MLG/

Pt(111) and graphite is similar. The stretching FC between

second-nearest neighbors (a2) exhibits a slightly decrease for

MLG/Pt(111) as compared with graphite [41]. The three-body

in-plane angle-bending FC (c1) is slightly reduced (5%) in our

calculations with respect to previous result [67] and with re-

spect to pristine graphite [41].

The FC c2 represents a four-body out-of-plane angle-bend-

ing. This parameter is almost the same for pristine graphite

and MLG/Pt(111) while it is considerably reduced (up to

40%) in MLG/Ni(111) (for both directions).

The twisting FC d in our calculation is appreciably different

from that calculated in Ref. [67] and more similar to that ob-

tained for pristine graphite in Ref. [41].

The FC related to the vertical interaction with the sub-

strate aS is different from zero only in MLG/Ni(111) [27,68].

The inspection of results in Table 3 confirms that the inter-

action between MLG and Pt(111) behaves as the weak van der

Waals interlayer interaction between the topmost sheets in

pristine graphite. The FC related to vertical vibrations (c2

and as) are different with respect to MLG/Ni(111), for which

a strong hybridization between p states of graphene and Ni

d states exists [69].

Finally, we point out that the elastic moduli are substan-

tially independent of the direction due to the intrinsic

mechanical basal isotropy of the hexagonal structure [70].

Thus, the results on the elastic moduli and their discussion

remain perfectly valid regardless of the existence of R0 and

R30 domains in the graphene sample.

4. Conclusions

The average elastic properties of a macroscopic graphene

sample grown on Pt(111) were accurately evaluated by pho-

non dispersion experiments performed by using the HREELS

technique. We found a Poisson’s ratio of 0.19 and a Young’s

modulus of 342 N/m. Interestingly, these values are signifi-

cantly higher than those reported for MLG grown on copper

foils. Very likely, such dissimilarities reflect the different sam-

ple preparations. MLG grown on copper foils exhibit a large

surface roughness (3–6 nm) while MLG/Pt(111) is nearly-flat.

Hence, MLG/Pt(111) offers the unique opportunity to perform

measurements for ripples-free and quasi-freestanding graph-

ene samples. The excellent crystalline quality of graphene

grown on metal substrates (with a reduced number of defects

and grain boundaries) leads to macroscopic samples of high

bending flexibility and tensile strength, which could be used

for applications in advanced nanocomposites. Due to its ther-

mal stability up to 1200 K, chemical stability and robustness,

epitaxial graphene represents a promising candidate for

application in nano-electromechanical devices.

Acknowledgments

We thank prof. L. Colombo for helpful discussions. Work sup-

ported by the Ministerio de Educacion y Ciencia through pro-

jects CONSOLIDER-INGENIO 2010 on Molecular Nanoscience

(CSD 2007-00010), FIS2010-18847, and by Comunidad de

Madrid through the program NANOBIOMAGNET (S2009/MAT-

1726).

Appendix A. Supplementary material

Supplementary data associated with this article can be found,

in the online version, at http://dx.doi.org/10.1016/j.carbon.

2012.06.019.

R E F E R E N C E S

[1] Castro Neto AH. Graphene: Phonons behaving badly. NatMater 2007;6(3):176–7.

[2] Politano A, Marino AR, Formoso V, Chiarello G. Evidence ofKohn anomalies in quasi-freestanding graphene on Pt(111).Carbon 2012;50(2):734–6.

[3] Zhang YY, Wang CM, Cheng Y, Xiang Y. Mechanical propertiesof bilayer graphene sheets coupled by sp3 bonding. Carbon2011;49(13):4511–7.

[4] Park S, An J, Suk JW, Ruoff RS. Graphene-based actuators.Small 2010;6(2):210–2.

[5] Wang Y, Yang R, Shi Z, Zhang L, Shi D, Wang E, et al. Super-elastic graphene ripples for flexible strain sensors. ACS Nano2011;5(5):3645–50.

[6] Scharfenberg S, Rocklin DZ, Chialvo C, Weaver RL, GoldbartPM, Mason N. Probing the mechanical properties of grapheneusing a corrugated elastic substrate. Appl Phys Lett2011;98(9):091908.

[7] Alzari V, Nuvoli D, Sanna R, Scognamillo S, Piccinini M, KennyJM, et al. In situ production of high filler content graphene-based polymer nanocomposites by reactive processing. JMater Chem 2011;21(41):16544–9.

[8] Tomasetti E, Legras R, Nysten B. Quantitative approachtowards the measurement of polypropylene/(ethylene–propylene) copolymer blends surface elastic properties byAFM. Nanotechnology 1998;9(4):305.

[9] Reynaud C, Sommer F, Quet C, El Bounia N, Duc TM.Quantitative determination of Young’s modulus on a biphasepolymer system using atomic force microscopy. Surf InterfAnal 2000;30(1):185–9.

[10] Lee C, Wei X, Kysar JW, Hone J. Measurement of the elasticproperties and intrinsic strength of monolayer graphene.Science 2008;321(5887):385–8.

[11] Pan Y, Zhang H, Shi D, Sun J, Du S, Liu F, et al. Highly ordered,millimeter-scale, continuous, single-crystalline graphenemonolayer formed on Ru (0001). Adv Mater2009;21(27):2777–80.

[12] Coraux J, N‘Diaye AT, Busse C, Michely T. Structuralcoherency of graphene on Ir(1 1 1). Nano Lett2008;8(2):565–70.

[13] Sutter P, Sadowski JT, Sutter E. Graphene on Pt(111): growthand substrate interaction. Phys. Rev. B 2009;80(24):245411.

C A R B O N 5 0 ( 2 0 1 2 ) 4 9 0 3 – 4 9 1 0 4909

[14] Gao M, Pan Y, Zhang C, Hu H, Yang R, Lu H, et al. Tunableinterfacial properties of epitaxial graphene on metalsubstrates. Appl Phys Lett 2010;96(5):053109.

[15] Gao M, Pan Y, Huang L, Hu H, Zhang LZ, Guo HM, et al.Epitaxial growth and structural property of graphene onPt(111). Appl Phys Lett 2011;98(3):033101.

[16] Politano A, Marino AR, Formoso V, Chiarello G. Hydrogenbonding at the water/quasi-freestanding graphene interface.Carbon 2011;49(15):5180–4.

[17] Politano A, Marino AR, Formoso V, Farıas D, Miranda R,Chiarello G. Quadratic dispersion and damping processes of pplasmon in monolayer graphene on Pt(111). Plasmonics;2012. http://dx.doi.org/10.1007/s11468-11011-19317-11461.

[18] Preobrajenski AB, Ng ML, Vinogradov AS, Martensson N.Controlling graphene corrugation on lattice-mismatchedsubstrates. Phys Rev B 2008;78(7):073401.

[19] Borca B, Barja S, Garnica M, Minniti M, Politano A, Rodriguez-Garcıa JM, et al. Electronic and geometric corrugation ofperiodically rippled, self-nanostructured grapheneepitaxially grown on Ru(0001). New J Phys 2010;12(9):093018.

[20] Politano A, Borca B, Minniti M, Hinarejos JJ, Vazquez de PargaAL, Farıas D, et al. Helium reflectivity and Debye temperatureof graphene grown epitaxially on Ru(0001). Phys Rev B2011;84(3):035450.

[21] Politano A, Marino AR, Chiarello G. Phonon dispersion ofquasi-freestanding graphene on Pt(1 1 1). J. Phys: CondensMatter 2012;24(10):104025.

[22] Politano A, Chiarello G. Vibrational investigation of catalystsurfaces: change of the adsorption site of CO molecules uponcoadsorption. J Phys Chem C 2011;115(28):13541–53.

[23] Politano A, Formoso V, Colavita E, Chiarello G. Probingcollective electronic excitations in as-deposited and modifiedAg thin films grown on Cu(111). Phys Rev B 2009;79(4):045426.

[24] Cazzanelli E, Caruso T, Castriota M, Marino AR, Politano A,Chiarello G, et al.; in preparation.

[25] Politano A, Marino AR, Formoso V, Farıas D, Miranda R,Chiarello G. Evidence for acoustic-like plasmons on epitaxialgraphene on Pt(111). Phys Rev B 2011;84(3):033401.

[26] Rocca M. Low-energy EELS investigation of surface electronicexcitations on metals. Surf Sci Rep 1995;22(1–2):1–71.

[27] Farıas D, Rieder KH, Shikin AM, Adamchuk VK, Tanaka T,Oshima C. Modification of the surface phonon dispersion of agraphite monolayer adsorbed on Ni(111) caused byintercalation of Yb, Cu and Ag. Surf Sci 2000;454(1–2):437–41.

[28] Shikin AM, Farıas D, Adamchuk VK, Rieder KH. Surfacephonon dispersion of a graphite monolayer adsorbed onNi(111) and its modification caused by intercalation of Yb, Laand Cu layers. Surf Sci 1999;424(1):155–67.

[29] Wintterlin J, Bocquet ML. Graphene on metal surfaces. SurfSci 2009;603(10–12):1841–52.

[30] Seol JH, Jo I, Moore AL, Lindsay L, Aitken ZH, Pettes MT, et al.Two-dimensional phonon transport in supported graphene.Science 2010;328(5975):213–6.

[31] Hartmut Z. Phonons in layered compounds. J Phys: CondensMatter 2001;13(34):7679.

[32] Perebeinos V, Tersoff J. Valence force model for phonons ingraphene and carbon nanotubes. Phys Rev B2009;79(24):241409.

[33] Lindahl N, Midtvedt D, Svensson J, Nerushev OA, Lindvall N,Isacsson A, et al. Determination of the bending rigidity ofgraphene via electrostatic actuation of buckled. Membranes;2012. doi:arXiv:1203.4397v1201.

[34] Kudin KN, Scuseria GE, Yakobson BI. C2F, BN, and C nanoshellelasticity from ab initio computations. Phys Rev B2001;64(23):235406.

[35] Zhang DB, Akatyeva E, Dumitrica T. Bending ultrathingraphene at the margins of continuum mechanics. Phys RevLett 2011;106(25):255503.

[36] Garcia-Sanchez D, van der Zande AM, Paulo AS, Lassagne B,McEuen PL, Bachtold A. Imaging mechanical vibrations insuspended graphene sheets. Nano Lett 2008;8(5):1399–403.

[37] Merino P, Svec M, Pinardi AL, Otero G, Martin-Gago JA. Strain-driven moire superstructures of epitaxial graphene ontransition metal surfaces. ACS Nano 2011;5(7):5627–34.

[38] Fasolino A, Los JH, Katsnelson MI. Intrinsic ripples ingraphene. Nat Mater 2007;6(11):858–61.

[39] Viola Kusminskiy S, Campbell DK, Castro Neto AH. Lenosky’senergy and the phonon dispersion of graphene. Phys Rev B2009;80(3):035401.

[40] Karssemeijer LJ, Fasolino A. Phonons of graphene andgraphitic materials derived from the empirical potentialLCBOPII. Surf Sci 2011;605(17–18):1611–5.

[41] Aizawa T, Souda R, Otani S, Ishizawa Y, Oshima C. Bondsoftening in monolayer graphite formed on transition-metalcarbide surfaces. Phys Rev B 1990;42(18):11469.

[42] Bosak A, Krisch M, Mohr M, Maultzsch J, Thomsen C.Elasticity of single-crystalline graphite: inelastic x-rayscattering study. Phys Rev B 2007;75(15):153408.

[43] Oshima C, Aizawa T, Souda R, Ishizawa Y, Sumiyoshi Y.Surface phonon dispersion curves of graphite (0001) over theentire energy region. Solid State Commun 1988;65(12):1601–4.

[44] Sanchez-Portal D, Artacho E, Soler JM, Rubio A, Ordejon P. Abinitio structural, elastic, and vibrational properties of carbonnanotubes. Phys Rev B 1999;59(19):12678.

[45] Blakslee OL, Proctor DG, Seldin EJ, Spence GB, Weng T. Elasticconstants of compression-annealed pyrolytic graphite. J ApplPhys 1970;41(8):3373–82.

[46] Seldin EJ, Nezbeda CW. Elastic constants and electron-microscope observations of neutron-irradiated compression-annealed pyrolytic and single-crystal graphite. J Appl Phys1970;41(8):3389–400.

[47] Lu JP. Elastic properties of carbon nanotubes and nanoropes.Phys Rev Lett 1997;79(7):1297.

[48] Jiang J-W, Wang J-S, Li B. Young’s modulus of graphene: amolecular dynamics study. Phys Rev B 2009;80(11):113405.

[49] Zakharchenko KV, Katsnelson MI, Fasolino A. Finitetemperature lattice properties of graphene beyond thequasiharmonic approximation. Phys Rev Lett2009;102(4):046808.

[50] Arghavan S, Singh AV. Free vibration of single layer graphenesheets: lattice structure versus continuum plate theories. JNanotechnol Eng Med 2011;2(3):031005–6.

[51] Bera S, Arnold A, Evers F, Narayanan R, Wolfle P. Elasticproperties of graphene flakes: boundary effects and latticevibrations. Phys Rev B 2010;82(19):195445.

[52] Cadelano E, Palla PL, Giordano S, Colombo L. Elasticproperties of hydrogenated graphene. Phys Rev B2010;82(23):235414.

[53] Gui G, Li J, Zhong J. Band structure engineering of grapheneby strain: first-principles calculations. Phys Rev B2008;78(7):075435.

[54] Liu F, Ming P, Li J. Ab initio calculation of ideal strength andphonon instability of graphene under tension. Phys Rev B2007;76(6):064120.

[55] Michel KH, Verberck B. Theory of the elastic constants ofgraphite and graphene. Phys Status Solidi B2008;245(10):2177–80.

[56] Cadelano E, Palla PL, Giordano S, Colombo L. Nonlinearelasticity of monolayer graphene. Phys Rev Lett2009;102(23):235502.

[57] Zhou G, Duan W, Gu B. First-principles study on morphologyand mechanical properties of single-walled carbon nanotube.Chem Phys Lett 2001;333(5):344–9.

[58] Arroyo M, Belytschko T. Finite crystal elasticity of carbonnanotubes based on the exponential Cauchy–Born rule. PhysRev B 2004;69(11):115415.

4910 C A R B O N 5 0 ( 2 0 1 2 ) 4 9 0 3 – 4 9 1 0

[59] Adamyan V, Zavalniuk V. Phonons in graphene with pointdefects. J. Phys.: Condens. Matt. 2011;23(1):015402.

[60] Zhou J, Huang R. Internal lattice relaxation of single-layergraphene under in-plane deformation. J Mech Phys Solids2008;56(4):1609–23.

[61] Otero G, Gonzalez C, Pinardi AL, Merino P, Gardonio S, LizzitS, et al. Ordered vacancy network induced by the growth ofepitaxial graphene on Pt(111). Phys Rev Lett2010;105(21):216102.

[62] Yu M-F, Files BS, Arepalli S, Ruoff RS. Tensile loading of ropesof single wall carbon nanotubes and their mechanicalproperties. Phys Rev Lett 2000;84(24):5552.

[63] Yu M-F, Lourie O, Dyer MJ, Moloni K, Kelly TF, Ruoff RS.Strength and breaking mechanism of multiwalled carbonnanotubes under tensile load. Science 2000;287(5453):637–40.

[64] Reddy CD, Ramasubramaniam A, Shenoy VB, Zhang Y-W.Edge elastic properties of defect-free single-layer graphenesheets. Appl Phys Lett 2009;94(10):101904.

[65] Ruiz-Vargas CS, Zhuang HL, Huang PY, van der Zande AM,Garg S, McEuen PL, et al. Softened elastic response and

unzipping in chemical vapor deposition graphenemembranes. Nano Lett 2011;11(6):2259–63.

[66] Klintenberg M, Lebegue S, Ortiz C, Sanyal B, Fransson J,Eriksson O. Evolving properties of two-dimensionalmaterials: from graphene to graphite. J Phys: Condens Matter2009;21(33):335502.

[67] Aizawa T, Hwang Y, Hayami W, Souda R, Otani S, Ishizawa Y.Phonon dispersion of monolayer graphite on Pt(111) and NbCsurfaces: bond softening and interface structures. Surf Sci1992;260(1–3):311–8.

[68] Aizawa T, Souda R, Ishizawa Y, Hirano H, Yamada T, TanakaK-I, et al. Phonon dispersion in monolayer graphite formedon Ni(111) and Ni(001). Surf Sci 1990;237(1–3):194–202.

[69] Bertoni G, Calmels L, Altibelli A, Serin V. First-principlescalculation of the electronic structure and EELS spectra at thegraphene/Ni(111) interface. Phys Rev B 2005;71(7):075402.

[70] Benedek G, Onida G. Bulk and surface dynamics of graphitewith the bond charge model. Phys Rev B 1993;47(24):16471–6.