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: drg@zurich.ibm.com

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

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

1

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

2

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

3

(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.

4

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

5

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

6

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

7

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

8

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-

9

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-

10

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.

11

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.

12

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

13

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

14

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

15

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)

16

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