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www.sciencemag.org/content/348/6235/679/suppl/DC1
Supplementary Materials for
Adhesion and friction in mesoscopic graphite contacts
Elad Koren, Emanuel Lörtscher, Colin Rawlings, Armin W. Knoll, Urs Duerig* *Corresponding author. E-mail: [email protected]
Published 8 May 2015, Science 348, 679 (2015) DOI: 10.1126/science.aaa4157
This PDF file includes: Materials and Methods
Supplementary Text
Figs. S1 to S11
Tables S1 to S3
References
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Materials and Methods
Sample preparation
High quality highly oriented pyrolytic graphite (HOPG, µmash, ZYA grade, 0.4o mosaic spread)
samples of cm size and with a thickness of approximately 2 mm were used as substrates.
The graphite structures were fabricated from a freshly cleaved HOPG substrate by means of
anisotropic oxygen plasma etching. In a first step Pd/Au metal masks were deposited onto the
HOPG surface. The metal masks were fabricated by means of electron beam lithography and
lift-off using a 100 nm thick poly-methyl-methacrylate (MICRO CHEM 950 PMMA) resist
layer. The metal masks consist of 15 nm of Pd as adhesion promotor to the HOPG surface and
20 nm of Au as top layer and the metal layers were deposited by means of thermal evaporation.
Prior to evaporation a short oxygen plasma cleaning was applied in order to remove organic
residues on the exposed HOPG surface. The mesa structures with a height between 50 nm and
60 nm emerge during the plasma etch which selectively thins down only the unprotected HOPG
area. Reactive ion etching was performed on a Oxford Plasma Lab 80 plus tool. Oxygen pres-
sure, 20 mbar, and bias potential, 200 V, were optimized to obtain a low etch rate of 11 nm
min−1 for good control of the mesa height and to minimize chemical anisotropic side wall etch-
ing. Electron microscopy inspection of the fabricated mesas showed negligible under etching of
the graphite structures at the metal masks and only a minor side wall taper towards the HOPG
substrate (Fig. 1D). We also exposed some of the fabricated structures to a hydrogen plasma
but we did not see a measurable difference in the adhesion and friction characteristics in these
samples in comparison to the untreated ones.
Lateral force measurements
We used a commercial atomic force microscope (AFM, Bruker Dimension V) with the ”Nanoman”
nano-manipulation software to mechanically shear the mesas under ambient conditions and Pt/Ir
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coated commercial AFM probes (SCM-PIT, Bruker Ltd., nominal spring constant 2.8 N/m). For
a lateral force measurement the mesa is sheared in a direction perpendicular to the cantilever
long axis and the lateral deflection signal is recorded. Fabricated mesas are selected for the slide
experiments by imaging of the structured HOPG sample in the tapping mode. The mechanical
slide is performed in the following order: (i) The tip is approached onto the mesa surface close
to a center position with a load force of 50nN. (ii) A current pulse of 1 mA for 1s is applied
between tip and sample in order to cold-weld the metallic tip apex to the Au top surface of
the metal mask on the HOPG mesa. This establishes a strong mechanical contact. (iii) The
applied normal force is released. (iv) Lateral sliding is performed at a tip displacement velocity
of 50nm/sec and the lateral force is sampled every 0.05 nm of sliding distance.
Force calibration
The lateral force calibration involves a first calibration step for the optical lever sensitivity
Sl = x/Vl which relates the lateral tip motion along the x-axis to the optical detector signal
Vl and a second calibration step for determining the lateral spring constant cl which relates the
lateral motion x to the applied lateral force Ftip at the tip apex. The lateral force is then given
by Ftip = cl × Sl × Vl. The optical lever sensitivity is directly measured in each experiment
by recording friction loops signals on high adhesion Au samples and equating the slope of the
measured curves at the turning points with Sl. The calibration procedure for cl, described in
detail in the Supporting Text section, is significantly more complex. In a nutshell, the lateral
stiffness c−1l is determined by the series action of the in-plane bending stiffness k−1l and the
torsional stiffness c−1Φ of the cantilever, viz. c−1l = k
−1l +c
−1Φ . The parameters cΦ/`
2tip where `tip
denotes the tip length and kl are related to the normal spring constant kn via materials parameters
and cantilever dimensions (Eqs S5, S6). The material parameters chosen in the calibration
correspond to a Si cantilever fabricated from a [100] wafer with its long axis pointing along
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a 〈110〉 direction. The cantilever dimensions are obtained from scanning electron and optical
micrographs. The vertical spring constant kn is determined by means of the thermal noise tuning
method (28), as implemented in the Dimension V AFM, and a resonance spectrum analysis as
proposed by Sader (29). In our measurements the extracted spring constants are always smaller
for the Sader method by 4-22%, which is typically observed also by other groups (30, 31). For
the purpose of this experiment we take the mean value of the two methods for the normal spring
constant and we take the difference to be representative for the error. Independently, cΦ/`2tip is
also determined from an analysis of the torsion resonance spectrum known as the Sader torsion
method (32). Also here we found that the values of cΦ/`2tip obtained by the Sader method are
smaller by a few % than the corresponding calculated ones. As for kn we take the mean value of
the two methods for the torsional spring constant and we take the difference to be representative
for the error. In summary, we find that the lateral force constant is in the range from 75 N/m to
99 N/m for the four different cantilevers used in the experiments, labeled Tip A to Tip D in the
manuscript, and the estimated relative calibration error is on the order of 7.5% (Table S3).
Linear fit in Figure 3B
A standard linear least square fit routine was applied on the logarithmic representation of the
data. The quoted variance of F0 is obtained by converting the one σ error from the logarithmic
to the linear scale.
Linear fit in Figure 4A
A weighted least squares fit was applied to the data assigning to each data point i a weight
inversely proportional to the expected measurement uncertainty 1/∆F 2i . The measurement
uncertainty, standard deviations of the data points are indicated by crossbars, comprises a term
∆F 2r,i = S2∆r2 due to the measurement error of the radius of the HOPG mesas, ∆r = 10 nm
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(indicated by the horizontal bar), where S = 2×σ is the slope of the linear fit and a calibration
error of the lateral force measurement ∆F 2c,i = F2P,i × γ2tip (indicated by the vertical bar)
where γtip denotes the relative calibration error of the lateral force constant of the cantilever
sensor used in the experiment referenced as Tip A-C. The respective values are γtip = 10%,
13.1%, and 23.8% for Tip A, B, C, respectively (Table S3). The variance of the slope fit is
calculated assuming statistical independence of the radius errors and of the calibration errors
for the different cantilever sensors.
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Supporting Text
Rotation locking
Another type of a meta-stable structure is obtained from the 6-fold in-plane crystal symmetry
in graphite which leads to stable locking positions at integer intervals of 60◦ rotation angle
(5, 7). To demonstrate rotational lattice locking on the mesoscopic scale we use the cylindrical
structure connected to a rectangular beam section (Fig. S1A). The beam serves as a mechanical
lever arm to induce a rotational motion by placing the AFM tip on the top side of the lever and
applying a mechanical force perpendicular to the beam axis on a circular path. The cylindrical
part acts as a pivot for the rotation axis. Although the actuation force is applied transverse to the
beam, the structure remains anchored at the center position of the circle by virtue of adhesion
interaction which locks the rotation axis to the center of the circular section. The simulated
adhesion energy profile (Fig. S1B) exhibits local energy minima at multiples of 60◦ of rotation.
Therefore, the mobile upper part tends to lock in these preferred positions as shows in the AFM
images for 60◦ rotation and for 120◦ rotation. We note that a free rotational bearing with no high
symmetry positions could be realized by using different materials for the two mesa sections e.g.
graphene and boron nitride.
Simulation of Bilayer Graphene Sliding
We used the analytical model developed by Kolmogorov et al. (9) to calculate the potential
landscape and the forces acting on the atoms for circular graphene bilayer stacks with radii r
from 4 nm up to 15 nm. The model accurately predicts the experimentally measured mean value
of the interface energy. In the simulation, the bottom reference layer is centered at a hollow site
position and the coordinate frame is oriented such that the x-axis points along an arm-chair
orientation and correspondingly the y-axis points along a zig-zag orientation of the graphite
lattice. The second layer is initially positioned in an AB stacking configuration with respect to
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the bottom layer at a fixed vertical offset of 0.335 nm corresponding to the interlayer spacing in
bulk graphite (Fig. S2A). Mesa sliding is simulated by laterally displacing the circular top
layer along one of the coordinate axes. We calculate the binding energy for each atom in
the overlapping area as a function of the sliding distance and from the resulting energy map
we derive the lateral forces acting on the atoms by taking the derivative of the energy with
respect to sliding distance. For a commensurate system with 0o rotation between the sheets
the sliding force exhibits giant fluctuations which scale with the overlap area as a result of the
commensurate motion of the atoms. The fluctuation amplitude is on the order of 35 pN per
atom yielding a maximum force at the beginning of the slide of 0.12 µN for a radius of 6 nm
and 84 µN for a mesa radius of 100 nm. These values are orders of magnitude larger than the
respective mean line tension forces obtained from the simulation and measured experimentally.
In a next step the top sheet is rotated anti-clockwise by an angle Φ around the center position.
As a result a Moire superstructure which is isomorphic to the graphite lattice emerges consisting
of domains with approximate AA and AB stacking (Fig. S2A). The lattice constant of the super-
structure is L = a/√
2− 2 cos Φ where a = 0.142 nm is the in-plane graphite lattice constant
and the superstructure is rotated anti-clockwise by an angle Φ/2 with respect to the fixed bottom
layer. Figure S2B shows the binding energy per atom for the rotated interface. As intuitively
expected the binding energy is largest in the AB stacking domains, ' −45 meV per atom and
smallest in the AA stacking domains, ' −17 meV per atom.
Figure S2C shows the forces along the x-direction acting on the atoms in the system. The
sign convention is chosen such that a positive force acts opposite to the x-axis and a negative
force acts along the x-axis. Force maxima and minima cluster at interstitial sites between AB
and AA domains along the vertical Moire axis. In addition, one also finds localized force
extrema at the periphery of the double layer. The net force acting on the top layer is given by
the overall sum of the atomic forces. Therefore the alignment of the force Moire pattern with
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respect to the overlap area is the decisive factor for the net force that is observed when the upper
layer is laterally displaced. From a geometrical analysis one finds that the Moire pattern shifts
upwards by one Moire period L along an axis which is tilted by an angle −Φ with respect to
the y-axis for a lateral displacement by the graphite lattice period a. One thus expects that the
net force exhibits fluctuations with a repeat period on the order of a.
Figure S3A shows the total lateral force along the slide direction as a function of the sliding
distance calculated for a bilayer system with a radius of 6 nm and a rotation angle of 5o between
the layers . The force exhibits quasi periodic fluctuations with a peak amplitude on the order of
2 nN. There is not just one period in the signal as is the case for a commensurate system with
Φ = 0. The most prominent features have a period of 0.225 nm ' 1.58 × a and 0.425 nm
' 3×a and one also observes a beat modulation of the envelope as a result of the fractional ratio
of ' 1.9 of the two periods. The force Moire pattern corresponding to the points 1 - 7 in Fig.
S3A is shown in Fig. S3B. At the force maximum 1 the positive force patches (red) outnumber
the negative ones (blue) by approximately one. As the top layer is moved to position 2, the
Moire pattern moves upwards as explained above. As a result, two new negative force patches
start to enter the overlap area from the bottom thus canceling the overall force in-balance. At
position 3, the positive force patch at the top of the overlap area has escaped from the overlap
area and the negative patches at the bottom have almost completely entered from below leading
to an overall negative net force. Note that the Moire pattern has an approximate anti-reflection
symmetry with respect to the vertical symmetry axis of the overlap area in this short period
regime. In the long period regime, points 4 - 6, the Moire pattern has now reflection symmetry
with respect to the vertical axis. As a result, the force modulation is caused by 2 patches entering
from the bottom or leaving at the top thus creating an overall larger force modulation. At
point 7 the Moire pattern has reverted to the anti-reflection symmetric state. However, because
of the shape of the overlap area the overall force modulation is less prominent. The average
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mean force indicated by the dashed curve is calculated form the line tension Eq. 5 using the
experimentally determined value of σ = 0.226 Jm−2 for the mean interface energy. The curve
fits the simulated data extremely well which provides an experimental proof that the model
potential captures the interface interaction with good accuracy.
The above mechanism suggests that the amplitude of the force modulation should depend
on the rotation angle since the size of the Moire patches scales with the Moire period. From
the simulations we find that the mean value of the force fluctuations scales as Φ−1.5 (Fig. S4A)
which means that the force fluctuation become smaller with increasing angle but the decrease
is somewhat steeper than expected from the Moire period which roughly scales as Φ−1. Also
note that the scaling levels off as the rotation angle approaches 30o which corresponds to the
maximum rotational misalignment that can be realized at a graphite interface. We also studied
the dependence of the force fluctuations on the structure size. The mean value of the force
fluctuations is defined here as the average value of the deviation of the simulated force from the
mean line tension force evaluated over a sliding distance from x = 1 nm to x = r, viz. ∆FS =∫ r1nm
(FS(x)−FL(x))/(r−1 nm) dxwhere FL(x) is given by Eq. 5 with σ = 0.227 Nm−2. Fig.
S4B shows the mean amplitude of the force fluctuation for sliding along the x-axis determined
from simulations of bilayer structures with radii of 4 nm, 6 nm, 10 nm, and 15 nm and rotation
angles of 2o, 5o, 10o, and 30o. The simulated data points confirm the fractional scaling predicted
for incommensurate sliding. In fact, the data points for a 10o rotational mismatch follow exactly
the scaling law observed in the experiment for the mean friction force as indicated by the dashed
line.
The simulated sliding assumes a perfectly rigid control of the layer displacement. In that
sense, there is no real energy dissipation involved since the forward and backward scans are
deterministic and reversible. Energy dissipation arises from a Tomlinson mechanism due to the
compliance of the sliding actuator and also due to the in-plane lattice compliance. The latter is
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neglected here because of the huge value,' 1 TPa, of the in plane elastic modulus (33). Energy
dissipation and thus friction arises from the fact that the actuated top surface spontaneously
jumps from a marginally stable position to the next stable equilibrium position whenever the
stiffness of the actuator is less than the negative value of the displacement force gradient along
the slide direction. Therefore, the energy dissipation expressed in terms of a friction force
basically scales with the magnitude of the force fluctuation. Thus one may conclude from Fig.
S4B that the average rotational misalignment between the sliding mesas in the experiment was
in the range from 5o to 10o.
Experimentally we observe a much wider distribution of the friction force values as well as
randomness which cannot be reconciled in terms of the simulations which assume a fixed rota-
tion angle as well as a fixed position orthogonal to the slide direction. Based on the observed
statistical pattern of the measured friction force (Fig. 3A) and the Φ−1.5 scaling of the force
fluctuations one infers that the rotation angle is not a fixed quantity but it is subject to random
fluctuations on the order of 2o to 5o. This mechanism introduces an unpredictable stochastic
element rendering trace and retrace paths intrinsically statistically independent. This interpreta-
tion is corroborated by a model simulation in which we allow the top layer to move orthogonal
to the slide direction and to change the rotation angle in order to minimize the interface energy
during sliding. The simulation is implemented in the following way: Assuming sliding along
the x-axis and starting from an initial position X0 = (x0, y0,Φ0) the interface energy E0 is cal-
culated. In a next step, the top layer is moved along the x-axis by an amount ∆x = 0.002 nm.
Interface energies E1ij are calculated for 9 virtual displacement options in the y-Φ space at the
new position X1ij = (x1 = x0 + ∆x, y0 + i∆y,Φ0 + j∆Φ where the indices i,j are taken from
the set {-1,0,1} and ∆y = 0.01 nm denotes the step size along the y-axis and ∆Φ = 0.3o
denotes the step size for a rotation with respect to the symmetry axis of the overlap area. For
each path option (i, j) the energy difference ∆Eij = E1ij − E0 is calculated and a path prob-
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ability Pij = exp(−∆Eij/(kBT )) is assigned to allow for thermal fluctuations. The actual
path (i1, j1) is chosen at random according to the path probability. The model thus entails an
intrinsic thermally activated randomness. However, we also observe randomness in the sliding
process even at T = 0 K due to a sporadic degeneracy in the energy matrix ∆Eij .
The above algorithm was used for calculating the sliding force for a circular bilayer structure
with a radius of 5 nm. The initial rotation angle was set to 10o and the slide was performed along
the x-axis. The resulting sliding force (Fig. S5A) is strikingly different from the one obtained
for a perfectly rigid slide (Fig. S3A). The sliding force appears to be much more random and in
particular, one obtains short lived force spikes just as observed in the experimental friction data
(Fig. 3A). The randomness in the sliding force results from a quasi random walk process in the
Φ-y phase space when the system tries to minimize the overall energy during the slide. Figs S5B
and S5C show the evolution of the lateral displacement y and the rotation angle Φ, respectively.
Initially, the y-motion is confined to small excursions mainly due to the large overlap area which
provides a strong restoring line tension force. Towards the end of the slide, this confinement
force becomes less effective and correspondingly large y-fluctuations are obtained. Due to the
line tension force effect, we also expect that the off-axis excursions are smaller in larger radius
structures. The rotation angle on the other hand is less constrained by the surface interaction.
Therefore, one sees already at an early stage large excursions by as much as 6o from the initial
starting point. Owing to the quasi-random walk nature of the fluctuations we see substantial
correlation in the angular fluctuations with a correlation period of roughly 5 nm. We performed
several simulations with different initial rotation angles between 5o and 30o and we obtained
qualitatively the same characteristics as shown in Fig. S5.
Additional rotational and translational degrees of freedom have been considered before in
describing superlubricity friction. Experimentally, it was found that small graphene flakes re-
vert to a commensurate orientation after a short sliding distance, thereby destroying superlu-
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bricity (23). The stability of superlubricity sliding was theoretically investigated by de Wijn et
al. (24). In studying the dynamics of small graphene flakes with a radius from 0.5 nm to 1.9 nm
they found stable periodic orbits in the incommensurate state, which become more unstable,
viz. driving the system towards a commensurate interface, with increasing temperature. On
the other hand, the authors also observed that superlubricity becomes robust even at elevated
temperature with increasing system size. We also performed simulations of small systems of
comparable size and the results confirm the previously published characteristics. The transition
to the random dynamics discussed above typically occurs when the sliding interface comprises
approximately 3 Moiree periods. We found in our simulations that as the structure size in-
creases, the potential landscape becomes flat in an average sense and the complexity of the
local structure increases such that the system no longer feels the weak attraction towards peri-
odic orbits or the commensurate state. As such, superlubricity becomes a stable property even
at very low sliding speeds and at high temperature.
The power spectral density of the force fluctuations reflects the correlations due to the quasi-
random walk nature of the sliding path in the y-Φ space. Indeed we experimentally observe a
power law scaling with an exponent -1.5 (Fig. 3E) and the scaling is compatible with the
simulated data (Fig. S6A, note that the simulated low spatial frequency data is not reliable due
to finite size effects). Intriguingly, one still finds strong spectral components at a spatial period
of approximately 0.2 nm as expected from the short period fluctuations for a rigid scan (Fig.
S6B). The same period is also observed in the experimental power spectrum which exhibits
additional structure up to a period of 0.4 nm. The fact that we experimentally observe spectral
features which can be traced back to the lattice interaction provides additional evidence that
measured friction signal is genuinely due to a superlubricity mechanism arising from rotated
lattice sliding.
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Calibration of the Cantilever’s Lateral StiffnessIntroduction
In this section we describe our measurement of the force constant cl which relates the lateral
force Ftip applied to the apex of the cantilever’s tip to its lateral displacement ltip (Fig. S7):
Ftip = clltip (S1)
The force constant cl is used in combination with the lateral sensitivity of the AFM’s optical
lever system to determine Ftip from the AFM’s “friction” signal.
Unfortunately the direct measurement of cl is challenging. We therefore begin by expressing
cl in terms of stiffness constants which may be more readily determined. For typical lever
geometries the effect of the tip’s deformation on cl will be negligible in comparison to that of
the cantilever. The referred load exerted on the cantilever (Fig. S7) by Ftip is composed of a
moment and a force. For the range of forces occurring in this experiment the total deformation
of the cantilever may be obtained as the linear sum of these two loads. Consequently we write
for the torsional stiffness of the cantilever which relates the rotation of the cantilever about its
long axis φ to the applied moment Tφ:
Tφ = kφφ (S2)
and the lateral bending stiffness of the beam kl. If for convenience we define (Fig. S7)
cφ =kφl2tip
(S3)
the lateral stiffness at the tip apex may obtained as:
1
cl=
1
kl+
1
cφ(S4)
The remainder of this document will be devoted to the measurement of kl and kφ.
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Governing Equations
A cantilever, whose length L significantly exceeds the maximium dimension of its cross section
can be accurately described by Euler-Bernoulli beam theory (34). This theory provides simple
analytical results for the bending stiffness and resonant frequencies of the beam in terms of the
material properties and beam geometry. Additionally under normal measurement conditions the
cantilever will be in thermal equilibrium with its surroundings. Thus the well known equipartion
result will apply which provides a relationship between the magnitude of the random motion of
the cantilever and its stiffness (35). The fluctuation-dissipation theorem relates the amplitude of
the thermal motion to the observed macroscopic damping of the driven cantilever motion. This
result has been applied to the case of cantilevers of rectangular cross-section for which b � h
by Sader et al. (36) with a later theoretical correction by Paul and Cross (37).
Direct evaluation of the stiffness using the beam equations requires the accurate measure-
ment of the the cantilever dimensions and material properties. Conversely the thermal motion
of the cantilever may be readily and accurately measured in an AFM. Thus the stiffness may be
more accurately calculated if these measurements are used to substitute out unknown material
and geometrical parameters in the Euler-Bernoulli equations.
The Euler-Bernoulli equations can be used to obtain the ratio between the stiffness normal
to the surface kn and the lateral stiffness kl for a rectangular cantilever as (38) (Chapter 6):
klkn
=b2
h2(S5)
Likewise for b� h (38) (Chapter 6)1:
kφkn
=4
3
G
EL2 (S6)
We used the Thermal Motion (28, 30, 35) method to obtain kn for use in Eqs S5 and S6.
1note that we do not specialise to the case of an isotropic material here
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As outlined in the review (30) an estimate of the uncertainty in the stiffness may be obtained
by comparing the stiffnesses obtained from the Sader and Thermal Motion methods. As such
we separately determined kn using the Sader method (29). The Sader method uses the following
equation to obtain the stiffness from the cantilever’s resonance frequency and quality factor:
kn = 0.1906ρfb2LQnω
2R,nΓ
ni (ωR,n) (S7)
where ρf is the density of air which was taken to be 1.18 kgm−3 and Γni is the hydrodynamic
damping function.
For the same reason we applied the Sader Torsion method (32) to directly measure kφ. kφ is
obtained from the measured Q factor and resonant frequency for the beam (Ql and ωR,l) as:
kφ =1
2πρfb
4LQlω2R,lΓ
li(ωR,t) (S8)
where Γli is the hydrodynamic function for the torsional vibrational mode.
Cantilever Geometry
Eqs S5-S8 assume that the beam is cuboidal. As can be seen in figure S8 this is not exactly
correct for our cantilevers (Bruker PIT SCM). The cantilever has a triangular end and the edges
of the cantilever are chamfered as a result of the etching process. Conveniently the centroid
of the triangular end section coincides with the tip apex. Thus we take the distance from the
root of the cantilever to the tip apex as the effective length of the cantilever L. As such it is
not necessary to discriminate between the normal stiffness of the end of the cantilever and the
normal stiffness at the tip apex. The effective width b of the cantilever is taken as the distance
between mid-points of the chamfer (Fig. S8).
The measured dimensions of the cantilevers are given in Table S1. We believe that our
system for assigning the effective dimensions is accurate ±1µm and ±5µm for the width and
length respectively. As such these uncertainties dominate the measurement uncertainties in the
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error analysis. The thickness of the cantilever and the tip length was measured using a Scanning
Electron Microscope to an accuracy of ±100nm.
Material Properties
Eq. S6 contains the ratio of the Young’s modulus E to the Shear modulus G of the can-
tilever. Care is required in computing this ratio since Silicon is an anisotropic material. We
have followed the guidance provided in ref. (39). Specifically we assumed that the cantilever
was fabricated from a [100] wafer with its long axis aligned with a 〈110〉 direction (parallel or
perpendicular to the wafer’s flat). This yielded:
E ≡ E110 = 169GPa, G ≡ G110 = 50.9GPa (S9)
Both sides of the cantilever are coated with PtIr the exact composition of which is not de-
tailed by the manufacturer. The presence of this coating will effect both the beam’s flexural
rigidity as well as its mass per unit length. However, it will not lead to an inaccurate mea-
surement of kn via the Thermal Motion or the Sader method. Likewise it will not lead to an
inaccurate measurement of kφ via the Sader method. Unfortunately, the coating is not consid-
ered in the derivation of Eqs S5 and S6 and its presence on the cantilever will introduce some
error at this stage. Fortunately, the Young’s Modulus for PtIr is within a factor of two of that of
Silicon (40) and in addition the coating is likely thin (O(10 nm)).
Effect of Tip
The calculations used to obtain Eqs S5-S8 neglect the effect of the tip on the dynamics of the
cantilever. To investigate the valididty of this assumption we compare the effective mass (29)
of the vibrating cantilever with the mass of the tip. We approximate the tip by a cone with a
base diameter Dt of 7µm and a height of ltip ' 12µm. The ratio of effective vibrating masses
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is therefore:
Rn =Vt
0.2427Vc= 3% (S10)
where Vt = πD2t ltip/12 is the volume of the tip and Vc = bhL is the volume of the cantilever
It is worth noting that the presence of this end mass will not effect the Thermal Motion
calibration of kn since the equipartition theorem still applies2. The effect of the tip mass on the
Sader method was investigated numerically by Allen et al. (41). For the mass ratio calculated
here they found that the error in determining kn from the Sader method would be less than 0.1%.
For the torsion case the ratio of the effective moments of inertia is given by (32):
Rt =Vt20
(2l2tip + 3D2t /4)
13π2Vcb2
= 3.7% (S11)
Thus we assume that as in the case of the normal mode calibration the effect of the tip on our
calculations is small.
Method
The measurement of the thermal motion was performed using the Bruker Dimension V Atomic
Force Microscope (AFM) in ambient conditions. For the thermal motion method the opti-
cal lever sensitivity in the normal direction was measured using a DC approach curve (30).
The slope to amplitude sensitivity conversion factor was taken from Beam theory as 1.08 (30).
The Thermal Motion and Sader method measurements of kn were performed using the AFM’s
“Nanoscope” control software 3. The cantilever dimensions were measured using a Scanning
Electron Microscope to an accuracy of ±100 nm and are shown in Table S1.
For the Sader measurement of kφ nine 0.7s long time series measurements of the photodiode
signals were recorded at a sampling rate of 6.25MHz. These sampling parameters ensured that
2Here we neglect the small change to the factor of 1.08 used to convert a static optical lever sensitivity to adynamic optical lever sensitivity as a higher order term.
3NanoScope V Controller Manual NanoScope Software v 8 (2008)
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the torsional and normal resonant peaks were well resolved and occurred at frequencies signif-
icantly below the Nyquist frequency. The Fourier Transforms’ of these signals were calculated
and averaged. Finally a least squares fit of these transformed signals to the model:
|G(ω)|2 = A0|1− (ω/ωR)2 + iω/(ωRQ)|2
+N (S12)
was performed to determine the resonant frequencies ωR and the Q factors. The constant N
accounted for the presence of additive white noise in the signal. The values of Ql and ωR,l for
the first torsional and normal modes were then input into the Sader equations (Eqs S7 and S8)
to obtain the stiffnesses.
Uncertainty Estimation
In this section we outline the scheme we used to estimate the uncertainty in cl. For kn we
obtained the uncertainty directly from the discrepancy in the Sader and Thermal Motion mea-
surements. For each cantilever we define this error εn as:
εn =
12
(k
(thermal)n − k(sader)n
)12
(k
(thermal)n + k
(sader)n
) (S13)We then calculate the uncertainty ∆kn as the root mean square (RMS) value of this set of ob-
servations of the parameter εn. We used precisely the same approach to identify the uncertainty
in kφ.
The parameters kl and cl are calculated from kφ and kn using equations (S5) and (S4). We
assume that the errors are small, uncorrelated and without systematic offset. Specifically for
k = f(xi) where xi are the parameters we define:
xi = x̄i + δi, δi ∼ Pi (S14)
E[δi] = 0 (S15)
Cov[δi, δj] =
{σ2i , i = j
0, i 6= j(S16)
17
-
As such the well known error propogation result for a function f(x1, x2, . . .) applies:
σ2f =∑i
(∂f
∂xi
)2σ2i (S17)
where σf and σi are the standard deviations of f and the xi respectively. The uncertainties,
σb, σh and σl,tip, of the geometrical parameters are given in Table S1. Applying Eq. S17 to Eqs
S5-S4 yields the error propagation relationships for kl, cφ and cl as respectively:(σklkl
)2=
(σknkn
)2+(
2σbb
)2+(
2σhh
)2(S18)(
σcφcφ
)2=
(σkφckφ
)2+
(2σl,tipltip
)2(S19)(
σclcl
)2=
1(1 +
cφkl
)2 (σcφcφ)2
+1(
1 + klcφ
)2 (σklkl)2
(S20)
Results
The fit of Eq. S12 to the torsional resonant peak of cantilever A is shown in Fig. S9. From this
fit the value of the Q factor and resonant frequency characterising the peak were determined
as 805 and 663.6kHz respectively. The results for the measurement of the bending stiffness
(kn) and the torsional stiffness (kφ) are shown in Table S2. We observed an RMS value for the
uncertainty in the bending stiffness εn of:
σkn = 7.7% (S21)
This value is in reasonable agreement with that obtained by Cook et al. (30) when calibrating
a larger (N=10) sample of SCM PIT cantilevers. The authors observed a RMS value of εn for
these levers4 of 5.3%. Our estimated value for the uncertainty σφ calculated from the RMS
value of εφ is:
σkφ = 7.4% (S22)
4The definition of δ in (30) is related to our εn as δ = 2εn.
18
-
The values of kl, cφ and cl for the tips used in these experiments as well as the associated
uncertainties is shown in Table S3. It is worth noting that the relative uncertainty in cl is less
than the relative uncertainty in both kl and cφ. This is the familiar result for a pair of resistors
placed in parallel (cf. Eq. S4).
Shear force curves supporting Fig.3
In a first step a basal glide plane is created as described in the main text. The mobile top mesa
section is repeatedly sheared by a sliding distance of >100 nm starting from a 10 nm to 15 nm
off-center position. The trace and retrace directions correspond to sliding from the starting point
to the return point and vice versa. The trace direction points opposite to the line tension force
and the retrace direction points along the line tension force. The measured data for a 100 nm
radius mesa structure is shown in Fig. S10. The friction force curves in panels A’ - E’ have been
concatenated into one single curve in Fig. 3A whereby an offset has been applied to curves B’ -
E’ such that the mean friction force is 1.6 nN as in panel A’. The shear force and corresponding
friction force data for the mesas with radii from 150 nm to 250 nm is shown in Fig. S11.
19
-
Tables S1-S3 and Figures S1-S9
Cantilever Dimensions
Tip Experimental Structure L (µm) b (µm) h (µm) ltip (µm)
A 50 nm radius mesa 217±5 30±1 2.9 ±0.1 12.2 ±0.1B 100 nm, 200nm & 300nm radius mesas 218±5 30±1 2.9 ±0.1 11.5 ±0.1C 150 nm & 200 nm radius mesas 218±5 30 ±1 2.8 ±0.1 11.7 ±0.1D Circle & Beam Structure 218±5 30±1 2.7 ±0.1 11.9 ±0.1
Table S1: Dimensions of the SCM PIT Bruker cantilevers. The tip length ltip and the cantileverthickness h was measured using a Scanning Electron Microscope. The cantilever width b andlength L were measured using an optical microscope. The measurement uncertainty for b and Lis dominated by the uncertainty in the appropriate value for the effective width of an equivalentcuboidal cantilever.
kn kφ
Thermal Sader Mean εn5 Calc eq6 (S6) Sader Mean εφTip (Nm−1) (Nm−1) (Nm−1) (%) (10−8 Nm) (10−8 Nm) (10−8 Nm) (%)
A 1.22 1.17 1.19 2 2.29 2.17 2.23 3B 1.46 1.34 1.40 4 2.68 2.38 2.53 6C 1.88 1.54 1.71 10 3.24 2.59 2.91 11D 1.44 1.16 1.30 11 2.49 -7 - -
Table S2: Results part I: Results for the measurement of bending (kn) and torsional (kφ) can-tilever stiffnesses. The cantilever geometry is given in table S1.
5Calculated using equation (S13)6Calculated using the mean value for kn.7Cantilever D was damaged during handling and it was not possible to calculate kφ using the Sader Torsion
method.
23
-
kl cφ = kφ/l2tip cl
Calc. eq (S5) σkl Calc eq. (S3) σkφ Calc eq (S4) σclTip (Nm−1) (%) (Nm−1) (%) (Nm−1) (%)
A 183 12 218 7.6 99 7.6B 150 12 202 7.6 86 7.9C 137 12 168 7.6 75 7.7D 160 12 176 7.6 84 7.4
Table S3: Results part II: Measured lateral stiffness of the cantilever kl and of the combinedcantilever and tip system cl.
-17
-16
-18
-10 -5 5 10 0 // //
////
Ad
he
sio
ne
ne
rgy
(a
J)
Rotation angle (deg)
500 nm
-10 -5 5 10 60 -10 -5 5 10 120
500 nm
1.2 mo60
o120
A
B
Figure S1: A Demonstration of rotation locking using a compound mesa structure consistingof a circular section with a radius of 300 nm for stabilizing the rotation axis and a rectangularlever arm with a length of 950 nm for applying a torque force: AFM images of the structurefor a 0◦, 60◦ and 120◦ rotation angle are shown. The structures are 60 nm tall and the glideplane is 15 nm above the substrate surface. The torque was applied by pushing with the AFMtip perpendicular to the lever arm. B Adhesion energy versus rotation angle from a modelsimulation of a cylindrical mesa with a radius of 6 nm (see ”Simulation of graphene sliding”section). Due to the 6-fold symmetry, stable locking positions are obtained at integer intervalsof 60◦ rotation angle.
24
-
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5 -1 -0.5 0 0.5 1 1.5
Fo
rce p
er
ato
m (
pN
)
o = 2 o = 5 o = 10
En
erg
y p
er
ato
m (
meV
)
L = 4.
01 nm
L = 1.
63 nm
L = 0.
81 n
m
0
-45
70
-70
0
B
Co = 2 o = 5 o = 10
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
x
y y
x
o = 10AAAA
AA
AB
AB AB
AB
L
A
Figure S2: A Left panel: Schematic of the double layer graphene stack in a commensurateAB stacking position. The bottom layer (blue circles) is centered at a hollow site. The x-axis points along an arm-chair orientation and correspondingly the y-axis points along a zig-zag orientation of the graphite lattice. The top layer is indicated by red circles. Right panel:Moire pattern obtained by rotating the top layer anti-clockwise by 10o around the center po-sition. The Moire pattern is isomorhic to the graphene lattice and it consists of patches withapproximate AA stacking and patches with approximate AB stacking with a lattice constant ofL = a/
√2− 2 cos Φ where a = 0.142 nm is the graphene lattice constant. B Energy per
atom for different rotation angles as indicated in the figure. C Lateral force along the x-axis peratom for different rotation angles corresponding. Positive and negative forces point oppositeand along the x-axis, respectively.
25
-
0 2 4 6 8 10 12 14
Sliding distance x (nm)
0
-1
-2
1
2
3
4
5
6
Sli
din
g f
orc
e F
(n
N)
S
1
3
4
6
752
1 2 3
4 5 6
7
A
B
r = 6 nmo = 5
x = 1.36 nm x = 1.415 nm x = 1.475 nm
x = 3.05 nm x = 3.125 nm x = 3.195 nm
x = 4.56 nm
Period0.225 nm
Period0.425 nm
Moire slide direction
To
p-l
ayer
slid
e d
irecti
on
To
p-l
ayer
slid
e d
irecti
on
To
p-l
ayer
slid
e d
irecti
on
To
p-l
ayer
slid
e d
irecti
on
Figure S3: A Sliding force versus sliding distance simulated for a bilayer structure with aradius of 6 nm and a rotation angle of 5o. The dashed line denotes the mean line tensionforce calculated according to Eq. 5 for a binding energy σ ,= 0.223 Jm−2 as measured in theexperiment. B Evolution of the force Moire pattern at the positions marked by the circles 1 - 7in A. Note that the Moire pattern shifts upwards while sliding from left to right. Red and bluecircles mark the patches with maximum and minimum force which leave the overlap area at thetop and enter the overlap area at the bottom, respectively.
26
-
0.01
0.1
1
10
1 2 5 10 20 30
Rotation angle(deg)
Mean
am
plitu
de o
f fo
rce
flu
ctu
ati
on
F
(n
N)
S
Number of interface atoms N
0 2 4 610 10 10 10
Mean
am
plitu
de o
f fo
rce
flu
ctu
ati
on
F
(n
N)
S
0.01
0.1
1
10measured
friction force
simulation
= o 2
o 5
o10
o30
F = 14 pN0
= 0.35
-1 0 1 210 10 10 10
Radius r (nm)
A
B
r = 6 nm
-1.5
Figure S4: A Log-log plot of the mean amplitude of the force fluctuations versus rotation anglefor a bilayer structure with a radius of 6 nm. B Log-log plot of the mean amplitude of the forcefluctuations versus radius of the bilayer structure for different values of the rotation angle asindicated in the figure. Note that the simulation for Φ = 10o yields exactly the same scaling aswas observed experimentally.
27
-
0 2 4 6 8 10 -4
-2
0
2
4
Sliding distance x (nm)S
lid
ing
fo
rce
F (
nN
)S
0 2 4 6 8 10
Sliding distance x (nm)
-4
-2
0
2
4
6
8O
rth
og
on
al
dis
pla
ce
me
nt y
(n
m)
0 2 4 6 8 10
Sliding distance x (nm)
10
14
18
6Ro
tati
on
an
gle
(
de
g)
A
B
C
Figure S5: A Sliding force versus sliding distance simulated for a circular structure with aradius of 5 nm and an initial misfit angle of 10o for a thermally activated minimum energypath allowing for relaxations of the sliding top surface orthogonal to the slide direction (y-axis)and for relaxations of the misfit angle around a rotation axis at the center of the overlap area.B Orthogonal displacement of the sliding path and C rotation angle obtained in the thermallyactivated minimum energy simulation.
28
-
-2 -1 0 1 10 10 10 10
Spatial frequency f (1/nm)
-210
-110
010
-310
-410
2P
ow
er
sp
ectr
al d
en
sit
y (
nN
nm
)
-510
-1.5~ f
simulation
friction data
-2
0
2
4
6
8
10
x 10-3
0 0.1 0.2 0.3 0.4 0.5
simulation
friction data 3 x
Spatial period (nm)
2P
ow
er
sp
ectr
al d
en
sit
y (
nN
nm
)
A B
Figure S6: A Log-log plot of the power spectral density of the experimentally measured frictionforce fluctuation and of the simulated sliding force fluctuations. Note that the experimentalpower spectral density follows a power law scaling with an exponent of -1.5 reflecting thecorrelations in the signal. The same type of scaling also appears in the simulated signal. BClose-up view of the power spectral density with a spatial period less than 0,5 nm. Note thestrong peak at roughly 0.2 nm which corresponds to the short period observed in rigid slidingshown in Fig. S3A
A B
Figure S7: A Diagram showing the cantilever dimensions and the loading applied to the tip. BEquivalent load applied to the cantilever.
29
-
Figure S8: Optical microscope image of cantilever A. The departure of the cantilever geometryfrom that of a cuboid in the form of a chamfered edge and a triangular cantilever end are marked.The position of the tip apex is also shown. The effective dimensions we select for the cantileverlength L and width b are marked.
662 663 664 6650
10
20
30
40
f (kHz)
Vl
(µV
)
datafitted
Figure S9: Frequency content of the lateral signal obtained from the AFM’s photodiode in thevicinity the torsional resonant peak for cantilever A. The least squares fit of equation (S12) tothe data is shown by the red curve.
30
-
50
46
54
58
62
0 10 20 30 40 50 60 70 80 90
46
42
50
54
58
42
38
46
50
54
46
42
50
54
58
40
44
48
52
Sliding distance (nm)
Measu
red
sh
ear
forc
e (
nN
)
A
B
C
D
E
0 10 20 30 40 50 60 70 80 90
0
4
8
-4
-8
0
4
8
-4
-8
0
4
8
-4
-8
0
4
8
-4
-8
0
4
8
-4
-8
Measu
red
fri
cti
on
fo
rce (
nN
)
A'
B'
C'
D'
E'
Sliding distance (nm)
Figure S10: Shear force and friction force data supporting Figs. 3A,B: A - E Measuredshear force versus sliding distance for a shear displacement opposite to (trace, blue) and inthe direction (retrace, green) of the line tension force. A 100 nm radius mesa was repeatedlysheared along the same basal glide plane. The mean line tension force values (dashed line) areA 52.6 nN, B 51.5 nN, C 47.3 nN, D 51.7 nN, and E 56.8 nN. A’ - E’ Friction force (shear forcetrace - shear force retrace) derived from the shear force measurements. The five traces A’ - E’have been concatenated into one single trace in Fig. 3A. The mean friction force values (dashedline) are A’ 1.60 nN, B’ 1.68 nN, C’ 1.88 nN, D’ 1.78 nN, and E’ 2.05 nN.
31
-
128
120
112
2
4
6
0
-2
0
4
8
-4
-8
0
4
8
-4
-8
4
8
12
0
-4
70
68
72
74
76
66
66
64
68
70
72
62
2
4
6
0
-2
82
78
86
90
94
90
86
94
98
102
Me
as
ure
d f
ric
tio
n f
orc
e (
nN
)
Me
as
ure
d s
he
ar
forc
e (
nN
)
120
116
124
128
132
0 10 20 30 40 50 60 70 80 90
Sliding distance (nm)
0 10 20 30 40 50 60 70 80 90
Sliding distance (nm)
120
124
4
8
12
0
-4
4
8
12
0
-4
4
8
12
0
-4
A A'
B'
C'
D'
E'
F'
B
C
D
E
F
Figure S11: Shear force and friction force data supporting Fig. 3B: A - F Measured shearforce versus sliding distance for a shear displacement opposite to (trace, blue) and in the direc-tion (retrace, green) of the line tension force. Force traces are shown for two repeated shearexperiments for different mesas with radii of 150 nm (A,B, 200 nmm C,D, and 250 nm E,F.The mean line tension force values (dashed line) are A 71.1 nN, B 67.0 nN, C 87.2 nN, D 93.1nN, E 123.0 nN, and F 119.5 nN. A’ - F’. Friction force (shear force trace - shear force retrace)derived from the shear force measurements. The mean friction force values (dashed line) are A’2.37 nN, B’ 2.07 nN, C’ 2.78 nN, D’ 2.94 nN, E’ 3.33 nN, and F’ 4.56 nN.
32
-
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Adhesion and friction in mesoscopic graphite contacts