probing young’s modulus and poisson’s ratio in · probing young’s modulus and poisson’s...

Nano Res

1

Probing Young’s modulus and Poisson’s ratio in

graphene/metal interfaces and graphite: a comparative

study

Antonio Politano1 () and Gennaro Chiarello1,2

Nano Res., Just Accepted Manuscript • DOI: 10.1007/s12274-014-0691-9

http://www.thenanoresearch.com on December 16 2014

© Tsinghua University Press 2014

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DOI 10.1007/s12274-014-0691-9

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study

The Young’s modulus and the Poisson’s ratio in various

graphene/metal interfaces and in graphite has been studied by

phonon-dispersion experiments.

Provide the authors’ webside if possible.

Antonio Politano, https://www.researchgate.net/profile/Antonio_Politano?ev=hdr_xprf

https://www.researchgate.net/profile/Antonio_Politano?ev=hdr_xprf



study

Antonio Politano1 () and Gennaro Chiarello1,2

Received: day month year

Revised: day month year

Accepted: day month year

(automatically inserted by

the publisher)

© Tsinghua University Press

and Springer-Verlag Berlin

Heidelberg 2014

KEYWORDS

Young’s modulus,

Poisson’s ration, elastic

properties, graphene

ABSTRACT

By analyzing phonon dispersion, we have evaluated the average Young’s

modulus and Poisson’s ratio in graphene grown on Ru(0001), Pt(111), Ir(111),

Ni(111), BC3/NbB2(0001) and, moreover, in graphite. In both flat and

corrugated graphene sheets and graphite, we find a Poisson’s ratio of 0.19

and a Young’s modulus of 342 N/m. The unique exception is graphene/Ni(111),

for which we find different values (0.36 and 310 N/m, respectively) due to the

stretching of C-C bonds occurring in this commensurate overstructure. Such

findings are in excellent agreement with calculations performed for a

free-standing graphene membrane. 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.

1 Introduction

The elastic moduli of single-layer graphene sheets

have attracted considerable interest in recent years.

In fact, the extraordinary intrinsic strength of

graphene[1] makes graphene a suitable material for

applications such as actuators[2] and

nano-electromechanical devices[3, 4] and, moreover,

as carbon-fiber reinforcement in polymeric

nanocomposites[5].

Graphene can be formed by graphite exfoliation [6],

thermal decomposition of SiC [7] and by epitaxial

growth on metal surfaces[8]. The preparation of

highly ordered monolayer graphene could be

extended up to the millimeter scale when graphene is

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2 Nano Res.

epitaxially grown on transition-metal substrates. [9]

Furthermore, the possibility of transfer of the

graphene sheet onto insulating substrates may be a

promising route toward large scale production of

graphene devices [10]. Thus, it is important to know

the interaction strength between the graphene layer

and the metallic substrate in order to discern

between physisorption and chemisorption of

graphene and, moreover, to appraise the quality of

the contacts between metallic electrodes and

graphene devices[11-13].

Graphene has been grown on different transition

metal substrates: Pt(111)[14], Ni(111)[15],

Ru(0001)[16], Ir(111)[17], Rh(111)[18], Pd(111)[19],

Re(0001)[20], Cu(111)[21], and Co(0001)[22]. Among

the above-mentioned graphene systems, three

general classes may be distinguished.

Firstly, for the class of Ni and Co substrates, the

mismatch in the lattice parameter is negligible, thus

the graphene unit cell may be directly matched with

the substrate unit cell by slightly quenching or

stretching the bonds between carbon atoms of the

graphene lattice. In this case, a strong hybridization

between the substrate d bands and the π bands of

graphene occurs[11]. Thus, graphene is chemisorbed

onto these substrates with a small graphene-substrate

distance (2.1 Å for Ni [23] and 1.5-2.2 Å for Co [22]).

Whenever the mismatch in the lattice parameter

approaches 5-10%, a Moiré pattern appears. In this

case, the graphene sheet may be weakly (Pt, Ir) or

strongly bonded (Ru, Re, Pd, Rh) to the substrate.

The strong interaction occurring for graphene on

Ru(0001) [24] and Re(0001) [20] causes a strong

corrugation of the graphene sheet.

To date, a comparative investigation of elastic moduli

of graphene/metal interfaces is hitherto missing. Such

a comparative analysis could clarify whether the

elastic properties of periodically rippled graphene on

Ru(0001)[8, 25, 26] are different with respect to

systems in which the graphene overstructure is

nearly flat. Moreover, it would be interesting to

understand the influence of the stretching of C-C

bonds occurring in graphene/Ni(111) on the elastic

properties of the graphene sheet.

Herein, we estimate the average elastic properties

(Young’s modulus and the Poisson’s ratio) in

graphene epitaxially grown on Ru(0001), Pt(111),

Ir(111), Ni(111), BC3/NbB2(0001) and graphite, based

on the investigation of the phonon dispersion.

In most cases, we estimate the same values of the

Poisson’s ratio (0.19) and the Young’s modulus

(342 N/m) of the graphene sheet. The unique

exception is represented by graphene/Ni(111), for

which theirs values are 0.36 and 310 N/m,

respectively. 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[27] with diameter of 1.0-1.5 μm.

Hence, our results demonstrate that high-quality

and macroscopic samples of epitaxial graphene on

metal substrates exhibit the tensile strength

predicted by theory. Moreover, we have

demonstrated that surface corrugation and the

graphene-substrate interaction do not play any

peculiar role on the elastic moduli of the graphene

sheet.

2 Experimental

Experiments were carried out in an ultra-high

vacuum (UHV) chamber operating at a base

pressure of 5∙10-9 Pa. The samples were single

crystals delivered from MaTecK GmbH.

Substrates have been cleaned by repeated cycles

of ion sputtering and annealing at 1300 K.

Surface cleanliness and order were checked using

Auger electron spectroscopy (AES) and

low-energy electron diffraction (LEED)

measurements, respectively. Graphene was obtained by dosing ethylene onto the

clean substrate held at 1150 K, with the exception of

graphene on Ni(111), for which a lower sample

temperature was used (800 K) [28]. The presence of a

single sheet of graphene in the whole sample has

been confirmed by ex-situ Raman spectroscopy[29].

Similar conclusions have been reported in other

works on the same systems in the same experimental

conditions [14, 30].

The inspection of the LEED pattern clearly shows the

presence of well-resolved spots which are

fingerprint of the order of the graphene

overstructure.

Graphene grows on Ru(0001) with a single

www.theNanoResearch.com∣www.Springer.com/journal/12274 | Nano Research

3 Nano Res.

)sinE

E1(sin

mE2q S

p

lossi

p

macroscopic domain which extend up to millimeter

scale [9]. Similar results have been obtained for

graphene on Ni(111) [31]. By contrast, micrometric

graphene domains grow on Ir(111) [32] and Pt(111)

[33, 34] with two and three rotational orientations,

respectively.

High-resolution electron energy loss spectroscopy

(HREELS) experiments were performed by using an

electron energy loss spectrometer (Delta 0.5, SPECS).

The energy resolution of the spectrometer was

degraded to 4 meV so as to increase the

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

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

the analyzer while keeping the sample and the

monochromator in a fixed position. To measure the

dispersion relation, values for the parameters Ep,

impinging energy and θi , the incident angle, 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 surface sensitivity, the highest

cross-section for phonon excitation and q||

resolution.

As

the parallel momentum transfer, q|| depends on Ep,

Eloss, θi and θs according to:

where Eloss is the energy loss and θs is the electron

scattering angle [35].

Accordingly, the integration window in reciprocal

space [36] is

where α 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 ) to 0.022 Å -1 (for higher momenta).

The phonon dispersion for all systems has been

measured with the sample aligned along the

M . To obtain the energies of loss peaks, a

polynomial 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

HREEL spectra, recorded as a function of the

scattering angle, show several dispersing features,

all assigned to phonon excitations. As a selected

case, we show in Figure 1 measurements of

graphene/Ru(0001) recorded at Ep=20 eV as a

function of the parallel momentum transfer q||.

Phonon modes are excited in electron scattering by

the impact mechanism[37]. Thus, the intensity of

phonon modes notably increases with q|| (Figure 1),

even if they are noticeable also at small momenta

just by increasing the acquisition time for improving

the signal-to-noise ratio (not shown).

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 respect 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. Figure 2 shows a

selected HREELS spectrum showing the above-mentioned

six phonon modes of the graphene lattice. The sharpness

of phonon modes observed in Figure 2 indicates an

excellent crystalline order in the graphene sample.

In principle, EELS planar scattering from an isolated

graphene sheet does not allow the observation of the

TA branch for selection rules inhibiting the

observation of odd phonons under reflection

symmetries[37]. However, the presence of an

underlying substrate acts as a symmetry breaking

and this allows to record a weak signal also from the

TA mode. Such a reduced intensity implies that long

acquisition time is required for detecting TA with a

sufficient signal-to-noise ratio.

Figures 3 and 4 report the dispersion of the TA and

LA phonons, respectively, for different systems, that

is graphene epitaxially grown on Ni(111), Pt(111),

Ir(111), Ru(0001), BC3/NbB2(0001) and, moreover,

graphite.


4 Nano Res.

Sound velocities have been extracted from the

experimental slope of the acoustic branches in the

low-q|| limit, for which TA and LA phonons along

the K and M directions coincide. In

particular, we define vL, longitudinal sound velocity,

and vT, transverse sound velocity, as

and where and are the

frequencies of longitudinal and transverse acoustic

phonons, respectively.

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

branches, respectively. The only exception is the

transverse sound velocity for graphene/Ni(111),

which is found to be lower by 11% (12.4 km/s).

The difference for graphene/Ni(111) arises from the

fact that the C−C bonds of the graphene layer are

stretched by 1.48% to form a 1 × 1 structure. The

energetically most favorable configuration is that

with one carbon atom is on top of a Ni atom and the

other carbon atom on a hollow site[38].

Data on the dispersion of acoustic phonons of a

graphene can provide information on its elastic

properties. According to the procedure illustrated in

Ref. [39], the sound velocities of the TA and LA

branches could be used for calculating the in-plane

stiffness (the 2D analogous of the bulk modulus)

and the shear modulus of the graphene sheet,

respectively:

D

T

D

L

v

v

2

2

Thus, we estimate and to be 211 and 144 N/m for

all systems with the exception of graphene/Ni(111),

for which their values are 244 and 114 N/m,

respectively. It is worth noticing that graphene is a

true 2D material, therefore its elastic behavior is

properly described by 2D properties 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 and

the Young’s modulus for 2D samples E2D:

)1(2

)1(22

2

D

D

E

E

Hence, from and it is possible to estimate the

Poisson’s ratio, i.e. the ratio of transverse contraction

strain to longitudinal extension strain in the direction

of the stretching force:

19.0

1

1

The obtained value for most systems, i.e. 0.19, agrees

well with results for graphite in the basal plane (0.165)

[40, 41] while it is 0.28 in carbon nanotubes [42]. It

represents an intermediate value with respect to

those reported by calculations for graphene, as

shown in Table I. Instead, for graphene/Ni(111) its

value is 0.36.

It is worth noticing that, according to molecular

dynamics calculations[43], 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. [44] 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.

Table I. Poisson’s ratio ν, as reported in different

experimental and theoretical works. Poisson’s

ratio ν

Experimental (HREELS), graphene on Pt(111),

Ru(0001), Ir(111), BC3/NbB2(0001), graphite

0.19

Experimental, basal plane of graphite, Refs. [40, 41] 0.165

Experimental (HREELS), graphene/Ni(111) 0.36

Atomistic Monte Carlo, Ref. [45] 0.12

Tersoff-Brenner potential, Ref. [46] 0.149

Continuum plate theory, Ref. [47] 0.16

http://en.wikipedia.org/wiki/Stiffness

http://en.wikipedia.org/wiki/Isotropic

http://en.wikipedia.org/wiki/Stress_(physics)

http://en.wikipedia.org/wiki/Strain_(materials_science)


5 Nano Res.

Density functional theory, Ref. [48] 0.162

First-principles total-energy calculations, combined

to continuum elasticity, Ref. [49]

0.169

Ab initio, Ref. [50] 0.173


DFT, Ref. [52] 0.18



Valence force model, Ref. [55] 0.20

LDA, Ref. [56] 0.20

Cellular material mechanics theory, Ref. [57] 0.21

Molecular dynamics, Ref. [43] 0.22


Empirical force-constant calculations, Ref. [59] 0.227

Brenner’s potential, Ref. [52] 0.27

continuum elasticity theory and tight-binding

atomistic simulations, Ref. [60]

0.31



Brenner’s potential, Ref. [63] 0.397

Multiple component correlation model, Ref. [64] 0.4


As reported in Table II, many theoretical works

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

The obtained value of E2D for most graphene/metal

interfaces, i.e. 342 N/m, agrees well with most

theoretical results (Table II), a part from calculations

in Ref. [66] (underestimated E2D). In particular, a

good agreement exists between present results and

first-principles total-energy calculations, combined to

continuum elasticity, reported in Ref. [49].

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 for the selected case of

graphene/Pt(111), the value of E2D should be divided

by the distance between the graphene and the

underlying Pt(111) substrate (3.31 Å )[67, 68]. 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[40]. For the sake of completeness, the

Young’s modulus obtained for single-walled carbon

nanotubes ranges from 0.45 and 1.47 TPa [69], while

for multi-walled carbon nanotubes it was found to

range from 0.27 to 0.95 TPa [70]. In graphene/Ni(111)

E2D is instead 310 N/m.

Table II. 2D Young’s modulus E2D, expressed in N/m, as

reported in different experimental and theoretical works. Young's modulus E2D (N/m)

Experimental (HREELS),

graphene on Pt(111), Ru(0001),

Ir(111), BC3/NbB2(0001),

graphite

342

Experimental (HREELS),

graphene/Ni(111)

310

Experimental (AFM) on

graphene/copper foils, Ref. [71]

339±17

Experimental (AFM) on graphene

membranes, Ref. [27]

340±50

Experimental (AFM) on graphene

membranes, Ref. [72]

350±50

Tersoff-Brenner potential, Ref.

[66]

235

Energetic model, Ref. [73] 307

continuum elasticity theory and

tight-binding atomistic

simulations, Ref. [60]

312

DFT, Ref. [52] 330

Brenner’s potential, Ref. [63] 336

First-principles total-energy

calculations, combined to

continuum elasticity, Ref. [49]

344

Tersoff-Brenner potential, Ref.

[46]

345

Ab initio, Ref. [53] 350

Atomistic Monte Carlo, Ref. [45] 353

Density functional theory, Ref.

[48]

356

Empirical force-constant

calculations, Ref. [59]

384

Experimental (AFM) on

graphene/copper foils, Ref. [74]

55

Recently, nanoindentation AFM measurements [74]

have been performed on graphene grown by CVD on

copper foils and successively transferred onto silicon

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. [74]

for more details).

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

to estimate the elastic constants C11 and C12 , from E2D

and ν:

11

12

11

2

12

2

112

C

C

C

CCE D

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

good agreement with values reported by Cadelano et


6 Nano Res.

al.[49] (354 and 60 N/m).

Their corresponding 3D values are 1.27 and 0.24,

respectively, which agree well with experimental

findings for graphite reported in Ref. [75] (1.11 and

0.18 TPa).

Conclusions

We have demonstrated that the elastic properties in

graphene/metal interfaces are the same recorded in

graphite and free-standing graphene, with the

exception of graphene/Ni(111), where C-C bonds are

stretched by 1.48%. This implies a variation of the 2D

Young’s modulus by 9% (310 N/m versus 342 N/m in

the other systems).

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 thermal

stability up to 1200 K, chemical stability and

robustness, epitaxial graphene represents a

promising candidate for application in

nano-electromechanical devices.

Acknowledgements

We thank Davide Campi and Fernando de Juan for

helpful discussions. References

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Figure 1: HREELS spectra recorded as a function of the

parallel momentum transfer q|| for monolayer graphene on

Ru(0001). The green arrow indicates the weak TA mode.

Figure 2: HREELS spectrum reporting the six phonon

modes of graphene on Ru(0001) for a selected value of the

parallel momentum transfer q|| (~0.95 Å-1). Measurements have

been carried out with a primary electron beam energy of 20 eV,

with a fixed incidence angle of 80° with respect to the sample

normal (grazing incidence). The loss energy is measured in

meV in the bottom panel, while in the top panel the

corresponding value in cm-1 is reported.


10 Nano Res.

Figure 3: Dispersion of the TA mode in graphene

epitaxially grown on BC3/NbB2(0001) (data taken from Ref.

[76]), Ir(111), Pt(111), Ru(0001), Ni(111) and, moreover, in

graphite.

Figure 4: Dispersion of the LA mode in graphene

epitaxially grown on BC3/NbB2(0001) (data taken from Ref.

[76]), Ni(111) Ir(111), Pt(111), Ru(0001), and, moreover, in

graphite.


Nano Res.

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