Elastic Properties of Chemically Derived
Single Graphene Sheets
Cristina Gómez-Navarro,*,†Marko Burghard,†and Klaus Kern†,‡
Max-Planck-Institut fu ¨r Festko ¨rperforschung, Heisenbergstrasse 1,
70569 Stuttgart, Germany, and Ecole Polytechnique Federale de Lausanne,
CH-1015 Lausanne, Switzerland
Received May 14, 2008
Theelasticmodulusof freelysuspendedgraphenemonolayers, obtainedviachemical reductionof grapheneoxide, wasdeterminedthrough
that of pristinegraphene, aswell asahighflexibilitywhichenablesthemtobendeasilyintheir elasticregime. Built-intensionsarefoundto
be significantly lower compared to mechanically exfoliated graphene. The high resilience of the sheets is demonstrated by their unaltered
electrical conductivityaftermultipledeformations. Theelectrical conductivityof thesheetsscalesinverselywiththeelasticmodulus, pointing
toward a 2-fold role of the oxygen bridges, that is, to impart a bond reinforcement while at the same time impeding the charge transport.
Recent experiments have revealed the thermodynamic stabil-
ity of graphene under ambient conditions,1–3which strongly
revived the interest in the electrical and mechanical properties
of this fascinating carbon nanostructure. Two major methods
have been established for the fabrication of graphene
monolayers, namely, mechanical exfoliation of graphite2and
vacuum graphitization of silicon carbide.3,4More recently,
chemical reduction of graphene oxide (GO) has been reported
as an alternative, solution-based route, for obtaining graphene-
like sheets which offers the advantages of being cheap and
up-scalable.5–8Even though GO is a good insulator, deoxy-
genation has been shown to substantially enhance its electri-
cal conductivity, albeit the obtained values of ∼1 S/cm
remain 2-3 orders of magnitude below that of pristine
graphene.6,7Due to the limited efficiency of the reduction
process, the obtained sheets still contain residual oxygenated
functional groups of the starting material. Microscopic studies
of the reduced GO also indicate the coexistence of graphitic
regions with defect clusters6,9in agreement with proposed
models.9–11In addition to its interesting electrical character-
istics, this 2D material is expected to have also unique
mechanical properties. Recent studies have witnessed its
suitability for the fabrication of composites12,13and paper-
like materials14with excellent mechanical properties. How-
ever, in order to promote its application in nanotechnology,
the mechanical characterization of single sheets is of upmost
Here we present a study of the mechanical and electrical
properties of suspended, chemically reduced GO single
sheets. Graphite oxide prepared via Hummers method15was
dispersed in water, deposited onto a Si/SiO2substrate, and
subsequently reduced by hydrogen plasma treatment. The
obtained samples exhibited a high yield of monolayers with
lateral dimensions of 0.1-5 µm. Preselected layers were then
contacted by standard e-beam lithography with Ti/Au
electrodes. In order to freely suspend the layers (Figure 1),
the samples were wet etched by buffered hydrofluoric acid,
followed by critical point drying to prevent adhesion of the
sheets to the substrate. Figure 2a,b displays two atomic force
* Corresponding author, firstname.lastname@example.org.
†Max-Planck-Institut fu ¨r Festko ¨rperforschung.
‡Ecole Polytechnique Federale de Lausanne.
Figure 1. Device geometry: (a) Schematic depiction of the sample
geometry (lateral view, not to scale). The highly doped Si wafer
serves as a back gate in the electrical measurements. (b) Scanning
electron microscopy image of a suspended reduced graphene oxide
monolayer, for which the electrode undercut can be clearly
discerned (scale bar corresponds to 1 µm).
Vol. 8, No. 7
10.1021/nl801384y CCC: $40.75
Published on Web 06/10/2008
2008 American Chemical Society
microscopy (AFM) topographic images of the same GO
monolayer before and after etching. AFM height profiles of
the etched sample taken along the width and length of the
channel (Figure 2c,d) confirm the presence of a free-standing
structure. From these profiles, a channel depth of 75-80 nm
is determined. Due to the different etching rates in the lateral
and vertical directions, this depth is sufficient to ensure
complete suspension of the layer. In the present work, we
were able to realize suspended GO sheets with sizes of up
to 1 µm2. Larger sheets were found to collapse onto the
bottom of the trenches most likely due to residual tensions
surface during the drying process. A few percent of the
smaller layers appeared to be buckled toward the substrate.
For the experiments described in the following, only fully
suspended monolayers with no observable slack were se-
Mechanical characterization of the free-standing reduced
GO monolayers was performed by indenting an AFM tip at
the center of the suspended area. Before indentation, the
sheets were localized by AFM imaging in noncontact
dynamic mode in order to avoid damage of the samples. Four
different force vs z piezo displacement curves are displayed
in Figure 3a. Curves acquired on the Si/SiO2substrate were
used as a reference for calculating the applied force (F) and
the resulting deflection of the layers (δ). The obtained F(δ)
curves are linear for small deflections but assume a nonlinear
dependence for deflections exceeding ∼10 nm. The transition
to a nonlinear regime is predicted by elasticity theory and
can be ascribed to the onset of sizable local deformations
originating from the finite size and shape of the indenter.16–18
Importantly, even for large deflections of 50 nm (corre-
sponding to strains of ∼1%), the loading and unloading
curves were observed to overlap, indicating fully elastic
deformations under these conditions. Further evidence for
the excellent resilience of the sheets derives from the fact
that the slope of the F(δ) curves does not change after
repeated deformations, which proves the absence of perma-
nent damage. Dynamic complications such as viscosity
effects can be ruled out since the curves do not depend on
the approaching or retracting speed of the tip (in the range
The F(δ) curves provide access to the stiffness (Keff) of
the GO sheets. According to continuum mechanics, the
effective force constant of a double-clamped beam depends
on both the mechanical properties of the material and the
geometry of the beam. In the pure bending regime (i.e., small
indentations) and for point loading at the beam center, Keff
is related to the elastic modulus, E, of the material by the
where t, w, and l are the thickness, width, and length of the
beam, respectively, and T is the tension in the beam. In the
limit of T f 0, a linear scaling of Keffwith w(t/l)3would
thus be expected. By contrast, the corresponding plot of the
experimental data of Keff(Figure 4a) exhibits a pronounced
scatter, indicating the presence of sizable built-in tensions
(T). The magnitude of this tension can be derived from the
measured Keffvalues and the sheet dimensions: eq 1 can be
rewritten as E ) 32(l/t)3Keff/w - 17Tl2/32wt3where the first
Figure 2. Atomic force microscopy images of a reduced GO monolayer before and after etching. The AFM topographic images show the
same GO sheet before (a) and after (b) rendering it free-standing. The color scales are different for the two images. Panels c and d display
AFM height profiles along the lines marked in color in the AFM images. The profiles indicate a channel depth of ∼75 nm (c), and the
absence of slack along the length of the suspended layer (d).
Nano Lett., Vol. 8, No. 7, 2008
term is obtainable from the experimental data according to
Eeff ) 32(l/t)3Keff/w. This effectiVe modulus (Eeff) depends
on the geometry of the samples as Eeff) E + 17Tl2/32wt3.
Thus, plotting of Eeffvs l2/w for a fixed thickness (only single
layers were selected), as illustrated in Figure 4b, yields a
linear dependence from which an approximately constant
value of ∼4 nN for all the sheets can be inferred. Tensions
introduced during the fabrication process represent a general
hurdle in the development of nanoelectromechanical resona-
tors.19,20For the present samples, the tension is estimated to
be a few nanonewtons, which is 2 orders of magnitude lower
than that reported for mechanically exfoliated graphene.21–24
The presence of such large tensions in the latter case has
been attributed to macroscopic uncontrolled forces applied
during the deposition process. By contrast, the solution-based
approach used in this work leads to significantly lower
tensions, allowing easy access to the intrinsic properties of
these ultrathin membranes.
According to eq 1 and based upon the determined tension
value, a mean elastic modulus of 0.25 TPa is calculated for
the reduced GO monolayers, with a standard deviation of
0.15 TPa (Figure 4c). The reproducibility of the results with
different tips and the linear dependence observed in Figure
4b (indication of experimental accuracy in determination of
geometry and forces) suggest that the scattering present in
the Young modulus is most likely to arise from a difference
Figure 3. Deformation response of suspended GO monolayers. (a)
Plot of force as a function of vertical piezo displacement of the
AFM scanner for a hard substrate (black curve) and three different
suspended layers (colored curves) with different geometries. The
deformation depth is calculated as the difference in Z for a fixed
force. (b) Force vs deformation depth curve corresponding to the
red curve in panel a. The dashed red line indicates the linear
dependence for small deformations, which is the basis for the
calculation of the elastic modulus. (c) Force vs strain curve extracted
from the plot in panel b. The strain has been calculated as the
relative change in length, assuming a point load and a triangular
Figure 4. Elastic modulus for a range of reduced GO monolayers
with different geometries. (a) Measured effective force constant
vs t3wl-3for different GO monolayer samples, with t as the
thickness, w as width, and l as the length of the sheets (note that t
) 1 nm is fixed since only monolayers were selected). The red
and green dashed lines indicate the dependence of Keffexpected
for E ) 1.8 and 0.08 TPa, respectively (in the absence of built-in
tension). (b) Plot of Eeffas a function of l2/w for different monolayer
samples. From the linear fit an averaged tension of 4 nN can be
extracted. (c) Histogram of the values of the elastic modulus
obtained from the experimental data in panel a using eq 1 and the
estimated value of T from panel b.
Nano Lett., Vol. 8, No. 7, 20082047
in microscopic structure of the sheets and not from experi-
mental errors. The E value of 0.25 TPa is remarkably high
for a 2D membrane and approaches that predicted for pristine
graphene.25At the same time, this value provides a boundary
for the maximum mechanical performance that could be
reached via optimization of the interlayer coupling within
bulk, paper-like materials based on (reduced) GO.14
From Figure 5a, it can be seen that the conductivity of
the reduced GO sheets scales inversely with the elastic
modulus. On the basis of theoretical predictions of local
bonding reinforcement, and consequently stiffening of the
sheet by oxygen bridges introduced between the graphene
carbons,26the sheets with higher elastic modulus and lower
conductivity could be assigned to those of comparatively high
oxygen content. However, unreduced samples with maximum
oxygen content and a 2-3 orders of magnitude lower
conductivity did not display a further enhanced elastic
modulus (data not shown), which evidence a second factor
governing the mechanical behavior. It is plausible to assume
that this contribution arises from the contained structural
defects, that is, holes or vacancies that constitute weak links
limiting the overall elasticity of the sheets. The existence of
such defects has been experimentally confirmed6and pre-
dicted to significantly impart the mechanical strength of
graphene.27In contrast to the oxygen-containing functional
groups, they cannot be removed upon chemical reduction
and thus remain within the sheets.
A further reflection of the outstanding mechanical perfor-
properties against deformation. In fact, electrical measurements
prior to and after repeated deformation showed that the
resistance of the monolayers remained essentially unchanged,
provided that the indentation depth did not exceed 10 nm
(Figure 5b). The combined mechanical and electrical stability
demonstrates that the electrical contacts, which are expected
to be extremely sensitive against changes in the contact area
and the GO-metal distance, are not affected during the
deformation. For larger indentations (δ ≈ 50 nm), some
devices exhibited a resistance increase by approximately
20%, a finding that might be explained by a reduced physical
contact between the GO sheet and the metal or the local shear
strain introduced by the tip under these conditions.
Finally, it is noteworthy that reduced GO sheets comprised
of more than three layers showed a markedly different
behavior. In particular, they displayed 1 order of magnitude
lower values of the elastic modulus, as compared to those
of the single and double layers. A similar reduction of E
has been observed upon bundling of single-walled carbon
nanotubes.28This behavior is characteristic of stiffened
structures and can be understood on the basis of competing
bending and shear forces. For stiffer structures like the thicker
and shorter multilayers, bending deflections are almost
negligible and shear strains play a dominant role. Thus, their
mechanical response is out of the bending dominated limit,
such that the above performed analysis is not applicable
anymore. Further support for the relevance of shear strain
stems from the observation that the electrical resistance of
the multilayers was consistently found to be significantly
increased after the deformations.
In summary, our study of suspended, chemically derived
graphene sheets reveals an elastic modulus closely approach-
ing the value predicted for pristine graphene. Furthermore,
the chemically reduced graphene oxide combines useful
electrical conductivity with excellent mechanical properties
including high bending flexibility and tensile strength. Due
to their stability, robustness, and reversible response to
mechanical stress, these ultrathin membranes represent
promising candidates for application in nanoelectromechani-
cal devices, for example, resonators with resonance frequen-
cies in the microwave range. Moreover, their high robustness,
low mass, and large surface area render such graphene-based
membranes ideally suited as components of force, mass, or
chemical sensors. Future experimental and theoretical studies
addressing the influence of defects and oxygen functionalities
contained in the graphene layers will be needed to achieve
a balance of their mechanical and electrical properties.
Acknowledgment. We thank J. Weber for his help with
the critical point drier and J. Go ´mez-Herrero for helpful
discussions. C. Go ´mez-Navarro acknowledges support from
the Alexander von Humboldt Foundation.
Figure 5. Correlation between electrical and mechanical properties.
(a) Room temperature conductivity of different monolayers as a
function of the elastic modulus. The experimental data have been
fitted (red curve) by the expression σ ) A + B/E (with A, B fitting
constants) as a guide to the eye. (b) Room temperature electrical
resistance of the suspended GO layer shown in Figure 1, measured
as a function of back gate voltage before (green curve) and after
(red curve) repeated deformation by AFM tip indentation. The
relatively weak gate dependence originates from the reduced
capacitance between the layer and the conductive silicon substrate,
due to the removal of the SiO2 dielectric. The inset shows
corresponding I-V curves recorded at zero gate potential before
and after repeated indentations.
Nano Lett., Vol. 8, No. 7, 2008
Supporting Information Available: Detailed description
of the sample preparation and experimental methods. This
material is available free of charge via the Internet at http://
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