Thickness Dependent Adhesion Force and Its
Correlation to Surface Roughness in Multilayered
Graphene
Hoorad Pourzand, Pradeep Pai and Massood Tabib-Azar
Department of Electrical and Computer Engineering
University of Utah
Salt Lake City, USA
[email protected]
Abstract— In this paper adhesion force of multi-layered
graphene films with different thickness is experimentally
determined and studied for the first time. In thin graphene
layers, the adhesion force increases with decreasing thickness
and approaches that of substrate which is correlated to Van Der
Waals force transparency of mono-layer graphene. In very thick
graphene layers, a rise in adhesion force is seen which we
correlate to surface roughness changes of the substrate. Simple
models for measuring adhesion force for a flat surface with subnanometer roughness and tip-radius scale roughness are
proposed. Based on these models we understand that small
surface roughness decreases adhesion force while large
roughness will increase the adhesion force. This deduction is
corroborated by the experimental data obtained from adhesion
force measurements. It is also shown that roughness in the scale
of AFM tip-radius can double the adhesion force easily and
surface roughness plays an important role in the adhesion force
measurements.
I.
INTRODUCTION
Since graphene was introduced for the first time [1],
different groups have been trying to investigate possible
applications of this new material in different fields. For
example, graphene shows a very high electron mobility that is
133 times mobility of silicon [2], and many groups have been
trying to incorporate that in designing high speed transistors
and other electronic components for post-Moore era [3].
Graphene also has thermal conductivity which is 33 times that
of silicon [4] and an elastic modulus which is 6 times greater
than silicon [5]. These two facts make graphene a suitable
candidate for a switch that experiences large heat gradients at
the contacts. The high thermal conductivity facilitates rapid
dissipation of heat from the hot spots and the high elasticity
modulus maintains rigidity of the structure while allowing
ample flexure. In our earlier work on contact resistance studies
of different metals we identified surface pitting, fatigue,
asperity generation, atom migration, stiction and micro
welding to be the potential problems in switches at micro and
nanoscales [6]. Graphene is expected to work well as a switch
978-1-4673-4642-9/13/$31.00 ©2013 IEEE
because of the close-pack structure of carbon atoms which is
resistant to introducing defects by not letting the defect atom
to replace original atoms and the fact that it can flake off some
graphene layers to renew the contact layer. In designing
switches out of graphene, adhesion force becomes very
important at nanoscale. Adhesion force can be used to design
non-volatile memory devices in which cantilevers stick in
place by adhesion force until a new writing voltage is applied.
In order to design these switches, a good understanding of
adhesion force of graphene versus thickness is needed.
Friction force of graphene layers has been investigated before
[7], but there is no article on adhesion force of graphene. In
this paper we focus our studies on adhesion force of graphene
layers and correlate it with surface roughness of the layers.
II.
SAMPLE PREPARATION
In order to generate graphene with different thickness we
used mechanical exfoliation of natural graphite using scotch
tape. Highly ordered pyrolytic graphite (HOPG) can also be
used as a source if natural graphite is not available. A silicon
die with 300 nm thick thermally grown oxide was used as the
substrate. The thickness was chosen based on reports from
earlier work where it was found to accentuate the presence of
graphene[1]. Scotch tape was used to detach some graphite
from the natural graphite. The graphite film on the scotch tape
was then successively exfoliated by fresh tape every time until
a thin lusterless gray film was left behind. This layer was then
transferred to the substrate by sticking the graphite embedded
tape to the substrate and peeling the tape off. The graphene
layers are stuck to the substrate through surface forces.
III.
MEASUREMENT SETUP
The adhesion force of graphene pieces with different
thicknesses was measured using the Veeco Multimode AFM.
To measure the thickness of each piece, the step height was
measured from the scanned topology and adhesion forces were
extracted from the force curves. The samples were mounted
IV.
RESULTS AND DISCUSSION
The mean value and distribution of the measurement data
are shown in the figure 2. The most important observation in
the plot is the manner in which the adhesion force approaches
that of the substrate for thin layers of graphene.
Graphene pieces
300nm oxide
AFM cantilever
AFM disc
Silicon substrate
Figure 1. Measurement setup, 300 nm of oxide is grown on the silicon to
highlight the graphene pieces and they are consequently scanned with an
AFM tip for surface roughness measurement.
Adhesion Force (nN)
on AFM discs using conducting silver paste and were placed
over the magnetic holder of the Piezo stage. To successfully
obtain a force curve, first, the voltage to displacement
sensitivity of the Position Sensitive Detector (PSD) sensor was
calculated by touching down the flexible probe on a hard
sample (such as a steel disc); second, the stiffness of the
cantilever was calculated using the thermal-tune option of the
AFM or external nano-indenter. Using these two conversion
factors the voltage-displacement curve was converted to
deflection-displacement and, consequently, force displacement
curve. In this manner, the force was obtained from the voltage
difference generated by the laser spot movement on the PSD
sensor. The horizontal axis represents the vertical
displacement of the sample piezo stage. Prior to making
contact with the sample surface, the AFM tip is not deflected.
This was also the case during the tip-approach phase, hence,
the force curve was horizontal. Upon making contact with the
surface the gradient was formed and this was due to the
compliance of the probe and the contact regions of the sample
and the tip. The retracting curve was almost the same, apart
from the hysteresis behavior generated because of the
adhesion between the probe and sample materials. To
calculate the adhesion forces, the vertical length of the
triangular hysteresis section of the force curve was measured
accurately, by extracting F-Z data points of the force curve
from the AFM software. Each force curve for each graphene
piece was plotted at least 10 times from different parts of the
sample in order to capture the mean value and standard
deviations of measured values for that thickness. These data
are included in the results section below. It is known that a
thin layer of native oxide covers the silicon tip so that the
graph is actually indicative of the adhesion forces between
silicon oxide and mentioned graphene layers. The surface
roughness was also measured for different thicknesses in
terms of Ra and Rq values. Ra is the mean value of the
roughness integrated over the area while Rq is the root mean
square of the roughness integrated over the area.
50
45
40
35
30
25
20
15
10
5
0
Figure 2. Adhesion force versus thickness of different samples. It can be seen
that there are two adhesion force rises as we approach very thin graphene
pieces and very thick ones.
To analyze this, it is necessary to understand the adhesive
forces involved in the interaction of two solid bodies. Each
body is comprised of molecules which interact with the
molecules of the other body. Between each of these
molecules, we have the standard form of the Lennard-Jones
potential for the pairwise interactions, given as:
1
Where C and C are constants for attractive and repulsive
interaction. We assume at first that the presence of molecules
doesn’t affect the potential between other atoms and the total
potential is the sum of each pairwise potential between
molecules. We can assume only the first term in formula 1 for
simplicity and derive a set of equations for the adhesion force
between a molecule and a surface, sphere and a surface,
sphere and sphere and so on. Sometimes a constant named
Hamaker constant is used to simplify equations:
2
Where C is the constant in the potential formula between
molecules and ρ and ρ are number of atoms per unit volume
in each of the bodies.
Adhesion force between the AFM tip and the sample is
basically comprised of the adhesion force between the tip and
graphene layer and adhesion force of the tip and the substrate.
When the thickness of graphene is decreased the first term will
decrease but the second term will increase and the value
approaches the state where there is no graphene present. we
assumed the additivity of pairwise potential to calculate the
total potential which is not correct because in reality atoms
will affect the field of other atoms. This effect is considered in
the Lifshitz theory, but from our experimental results, we can
conclude that thin graphene is transparent to adhesion forces
and doesn’t affect them which has been concluded in other
researches as well [8].
The second observation in the plot is the increase in
adhesion force with thickness for relatively thicker layers of
graphene. We think the reason for increase of the adhesion
force in thicker pieces is surface roughness. Surface roughness
has been shown to affect adhesion force considerably [9].
Humidity cannot be the issue because all measurements were
carried out nearly at the same time. Glue residues cannot be
the reason of increase in adhesion force, unless we transfer
graphene pieces multiple times in each step. Fig 3 plots the
surface roughness versus multilayer graphene thickness.
Surface roughness affects the adhesion force in two ways.
At sub nanometer surface roughness the surface roughness
decreases the adhesion force by separating the tip and the
surface a little bit (Fig 4). However at higher surface
roughness comparable to tip radius the adhesion force will
increase because the contact area will somehow increase by
the surrounding asperities (Fig 5). The surface roughness
graph can be compared with the adhesion force graph. In the
very thin layers of graphene we have substrate effect which
gives rise to an increase in adhesion force. As the thickness
increases to 10 nm we have a slight increase in surface
roughness in sub nanometer range which should decrease the
adhesion force and it does. As we proceed to thicker samples
the roughness will be comparable to tip radius and this will
increase the adhesion force. The only problem is that the
transition in increase in adhesion force is about 15 nm while
for the surface roughness it is 100 nm. This can be due to the
fact that surface roughness is a partly sample dependent and in
each sample there are local asperities that affect the surface
roughness considerably, and also in a fairly smooth surface
there might be moderate trenches that doesn’t change the
surface roughness considerably but affect the adhesion force.
We can quantitatively compute the adhesion forces of
both cases by considering the simple models in figures 5 and
6 and we can relate the adhesion force to surface roughness in
this way. For sub-nanometer surface roughness we can
consider a sphere near a surface and the adhesion force of this
system has been reported to be [10]:
4
6
Where A is the Hamaker constant, R is the tip radius and
D is the displacement between the sphere and the surface
which can be considered as the surface roughness.
RR
b)
a)
DD
Measured surface
Measured surface
Real surface
Real surface
Figure 4. a) Sub-nanometer surface roughness and its effect. b) Simple
model of tip-sample for the adhesion force measurement. Increasing surface
roughness in this regime increases the mean sample-tip distance and
decreases adhesion force between them.
The simple model for larger surface roughness can be a
sphere inside another spherical hole. The adhesion force for
this case can be calculated by modifying the standard
equation used to calculate adhesion force between two
spheres [11]. To compute the adhesion force at first we need
to derive the energy of the system in figure 6. By neglecting
the repulsive term we consider the energy of the two
molecules with distance r to be:
5
It is evident that the energy of the system is only
dependent to their distance. By assuming the interactions to
be additive and non-retarded, the area of equi-distance points
from the center of sphere can be found to be:
sin
6
2
If we substitute then we will have:
7
8
We can derive as well:
9
2.86
Roughness (nm)
b)
a)
Ra
Rq
Using these two integration areas the energy can be
computed as follows:
2.16
b)b)
a)
a)
2.42
1.44
1.36 1.06
1.28
1.22
0.889
Measuredsurface
surface
Measured
0.636 0.736 0.788
0.867
0.891
0.369
0.624
0.573
0.481 0.541
0.273
Real surface
Real surface
Figure 3. Surface roughness versus graphene thickness. We can see a
constant rise in surface roughness as the thickness of graphene layers
increase.
Figure 5. a) Tip- radius scale surface roughness and its effect. b) Simple
model of tip-sample for the adhesion force measurement. Increasing the
surface roughness in this regime will increase the radius of curvature of
hemispherical hole and effective contact area which increasess the adhesion
force between the tip and the sample.
is usually two to five orders of magnitude smaller than the
second term and can be neglected in most cases especially if
R2 is much bigger than R1. To compare the flat substrate and
spherical hole, practical assumptions can be made for the
variables in formula 16 and formula 4. For a case where R2=
30 nm, R1= 20 nm, C= 9 nm and D= 1 nm then using formula
16 the adhesion force will be
-6.6A, where A is the
hamaker constant, which is twice of the adhesion force in the
flat substrate case with the same gap size and radius using
equation 4 that is equal to -3.3A. This example shows that
tip-radius scale roughness can double the adhesion force
easily and surface roughness acts an important role in the
adhesion force measurement.
Figure 6. Tip-radius scale surface roughness model and integration volumes.
The tip can be modeled by a sphere and the topology as a hemispherical hole.
10
11
1
.
2
1
12
2
1
1
V.
CONCLUSION
The adhesion force of graphene samples were measured
for different thickness and correlated to the surface
roughness. Surface roughness can have different effects on
adhesion force between the sample and AFM tip in subnanometer range and in tip-radius scale range. At subnanometer range surface roughness reduces adhesion force
while at tip-radius scale it will increase the adhesion force. It
was shown that the tip-radius scale roughness can double the
adhesion force compared to flat surface, so surface roughness
significantly affects the adhesion force.
12
2
13
3
2
3
14
3
The force can be computed as:
15
2
2
3
2
3
16
Where A is the hamaker constant, C is the distance
between the center of spherical hole to the center of sphere
and R1 and R2 are the radius of the sphere and spherical hole
respectively. In the equation 16 the first term in the brackets
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