Bhushan B. Handbook of Micro/Nano Tribology, Second Edition

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experimentally. The lack of such a system had slowed progress in tribology considerably. Recently,

however, with the development of such techniques as the surface force apparatus, the quartz microbalance,

and most recently the SFM, we consider that such simple systems can be prepared, which in turn has

also triggered theoretical interest and progress. In recent years this has led to a new ﬁeld, termed

nanotribology, which is one of the subjects of the present book.

Within this new ﬁeld, the SFM and the scanning force and friction microscope (SFFM), which is

essentially an SFM with the additional ability to measure lateral forces, have probably drawn the most

attention, even though in some respects, namely, reproducibility and precision, the surface force apparatus

as well as the quartz microbalance might at the moment be superior. Presumably the interest which has

accompanied the SFFM is due to its great potential in tribology. The most dramatic manifestation of

this potential is its ability to resolve the atomic periodicity of the topography and of the friction force as

the tip moves over a ﬂat sample surface.

An important feature of modern tribological instruments is that wear can be excluded down to an

atomic scale. Under appropriate experimental conditions this is true for the SFFM as well as for the

surface force apparatus and the quartz microbalance. In general, wear can lead to friction, but it is known

that wear is usually not the main process that leads to energy dissipation. Otherwise, the lifetime of

mechanical devices — a car, for example — would be only a fraction of what it is in reality. In most

technical applications — excluding, of course, grinding and polishing — the lifetime of devices is fun-

damental; therefore, surfaces are needed where friction is not due to wear, even though in some cases

wear can actually reduce friction. Research in wearless friction of a simple contact is thus of technical as

well as of fundamental interest. From a fundamental point of view, wearless friction of a single contact

is possibly the conceptually simple and controlled system needed for the well-established interplay

between experiment and theory: development of models and theories which are then tested under well-

deﬁned experimental conditions.

Four features makes the SFFM a unique instrument as compared with other tribological instruments:

1. The SFFM is capable of measuring simultaneously the three most relevant quantities in tribological

processes, namely, topography, normal force, and lateral force.

2. The SFFM has a resolution which is orders of magnitude higher than that of classical tribological

instruments. Topography can be determined with nanometer resolution, and forces can be mea-

sured in the nanonewton or even piconewton regime.

3. Experiments with the SFFM can be performed with and without wear. However, due to its imaging

capability, wear on the sample is easily controlled. Therefore, operation in the wearless regime,

where tip and sample are only elastically but not plastically deformed, is possible.

4. In general, an SFFM setup can be considered a single asperity contact (see, however, Section 6.3.3).

While some instruments used in tribology share some of these features with the SFFM, we believe that

the combination of all these properties makes the SFFM a unique tool for tribology. Of these four features,

the last might be the most important one. Of course, it is always valuable to be able to measure as many

quantities with the highest possible resolution. The fact that an SFFM setup is a simple contact — which

can also be achieved with the surface force apparatus — is a qualitative improvement as compared with

other tribological systems, where it is well known that contact between the sliding surfaces occurs at

many, usually ill-deﬁned asperities.

Classic models of friction propose that the friction is proportional to the real contact area. We will

see that this seems to be also the case for single asperity contacts with nanometer dimension. It is evident

that roughness is a fundamental parameter in tribological processes (see Chapter 4 by Majumdar and

Bhushan). On the other hand, a simple gedanken experiment shows that the relation between roughness

and friction cannot be trivial: very rough surfaces should show high friction due to locking of the

asperities. As roughness decreases, friction should decrease as well. Absolutely smooth surfaces, however,

will again show a very high friction, since the two surfaces can approach each other so that the very

strong surface forces act between all the atoms of the surfaces. In fact, two ideally ﬂat surfaces of the

same material brought together in vacuum will join perfectly. To move these surfaces past each other,

the material would have to be torn apart. This has been observed on a nanoscale and will be discussed

in Section 6.3.1.3.

In conclusion, it seems reasonable that for a better understanding of friction in macroscopic systems

one should ﬁrst investigate friction of a single asperity contact, a ﬁeld where the surface force apparatus

and, more recently, the SFFM have led to important progress. Macroscopic friction could then possibly

be explained by taking all possible contacts into account, that is, by adding the interaction of the individual

contacts which form due to the roughness of the surfaces.

As discussed above, three instruments can be considered to be “simple” tribological systems: the surface

force apparatus, the SFFM, and the quartz microbalance. All three represent single-contact instruments,

the last being in a sense an “inﬁnite single contact.” Since experiments with the surface force apparatus

are discussed in detail in Chapter 9 by Berman and Israelachvili, we will limit our discussion to the last

two and mainly to the SFFM. Accordingly, in the next section we will describe the main features of an

SFFM, then present experiments which we feel are especially relevant to friction on an atomic scale, and

ﬁnally try to explain these experiments in a more theoretical section.

6.2 Instrumentation

An SFM (Binnig et al., 1986) and an SFFM (Mate et al., 1987) consist essentially of four main components:

a tip which interacts with the sample, a force-sensing element which detects the force acting on the tip,

a piezoelectric element which can move the tip and the sample relative to each other in all three directions

of space, and control electronics including the data acquisition system as well as the feedback system

which nowadays is usually realized with the help of a computer. A detailed description of the instrument

can be found in this book in Chapter 2 by Marti. Therefore, we will limit the discussion of the instrument

only to the ﬁrst two components, the tip and the force-sensing element, which we consider especially

relevant to friction on an atomic scale. For many applications a thorough understanding of how the

SFFM works is essential to the understanding and correct interpretation of data. Moreover, in spite of

the impressive performance of this instrument, the SFFM is unfortunately still far from being ideal and

the experimentalist should be aware of its limitations and of possible artifacts.

6.2.1 The Force-Sensing System

The force-sensing system is the central part of an SFFM. Usually, it is made up of two distinct elements:

a small cantilever which converts the force acting on the tip into a displacement and a detection system

which measures this often very small displacement. The force is then given by

where

is the force constant of the cantilever and

∆

the displacement which is measured. The fact that

the force is not measured directly but through a displacement has important consequences. The ﬁrst one

is evident: for an exact determination of the force, the force constant has to be known precisely and this

is quite often a problem in SFM. Another implication is that an SFM setup is not stiff. If a force acts on

the tip, the cantilever bends and the tip moves to a new equilibrium position. Therefore, especially in a

strongly varying force ﬁeld, the tip position cannot be controlled directly. Moreover, a spring in a

mechanical system subject to friction forces can modify its behavior substantially (see the Chapter 9 by

Berman and Israelachvili). This is specially important in SFM: since the resolution is limited by the

minimum displacement that can be measured, a force measurement gives high resolution if the force

constant is low. With a low force constant, however, the tip–sample distance is less easily controlled.

Finally, for a low force constant the properties of the system are increasingly determined by the force

constant of the macroscopic cantilever and not by the intrinsic properties of the tip–sample contact,

which is the system to be studied. Therefore, a reasonable trade-off between resolution and control of

the tip–sample distance has to be found for each experiment. Although some schemes, such as feedback

Fc=⋅∆

control of the cantilever force constant (Mertz et al., 1993) and displacement controlled SFMs (Joyce and

Houston, 1991; Houston and Michalske, 1992; Jarvis et al. 1993; Kato et al. 1997), have been proposed

to avoid this problem, up to now these schemes have not been commonly used.

6.2.1.1 The Cantilever — The Force Transducer

The cantilever serves as a force transducer. In SFFM not only the force normal to the surface, but also

forces parallel to it have to be considered; therefore, the response of the cantilever to all three force

components has to be analyzed. In principle, the cantilever can be approximated by three springs

characterized by the corresponding force constants. Within this model, the tip is attached to the rest of

the rigid microscope through these three springs, one in each direction of space. The force acting on the

tip causes a deﬂection of these springs. To determine the force and the exact behavior of the microscope,

their spring constants have to be known.

A cantilever is a complex mechanical system; therefore, calculation of these force constants can be a

difﬁcult problem (Neumeister and Drucker, 1994; Sader, 1995), in some cases requiring numerical

computation. Most SFFM experiments are done with rectangular cantilevers of uniform cross section,

since they have a higher sensitivity for lateral forces than triangular ones, which are commonly used in

SFM. Moreover, for rectangular cantilevers the relevant force constants can be calculated analytically. We

will limit the following discussion to these cantilevers. The equation describing the deﬂection of a

cantilever is (see, for example, Feynmann, 1964)

(6.1)

where

is the Young’s modulus of the material,

∫

the moment of inertia of the cantilever and

(

) the bending moment acting on the surface which cuts the cantilever at the position

(

) in the

direction perpendicular to the long axis of the cantilever (see Figure 6.1). For a cantilever of rectangular

cross section of width

and thickness

the moment of inertia is

/12. Solving Equation 6.1 with

the correct boundary conditions one ﬁnds the bending line

(6.2)

FIGURE 6.1

Geometry and coordinate system for a typical cantilever. Its length is

its width

, its thickness

and

the tip length

tip

. The

-axis is oriented in the direction corresponding to the long axis of the cantilever. Forces act

at the tip apex and not directly at the free end of the cantilever. This induces bending and twisting moments as

discussed in the main text.

′′

()

⋅

()

zy My EI,

()













−













⋅

⋅⋅

where

is the length of the cantilever and

the force acting at its end. From this bending line, the force

constant is read off as

(6.3)

This is the “normal” force constant in a double sense: it is the force constant associated with a deﬂection

in a direction normal to the surface, and also the force constant generally used to characterize a cantilever.

However, other force constants are also relevant in an SFM and an SFFM setup. Exchanging

and

the above equation gives the force constant corresponding to the bending due to lateral force

(see

Figure 6.1):

(6.4)

where

is the normal force constant (Equation 6.3). Since the lateral force acts at the end of the tip and

not at the end of the cantilever directly, this force exerts a moment

tip

which twists the cantilever.

This twisting angle

causes an additional lateral displacement

∆

tip

of the tip. The corresponding

force constant is (Saada, 1974)

where

is the shear modulus and



1 for cantilevers that are much wider than thick (



), which

is the usual case in SFFM. It is useful to relate this force constant to the normal force constant

. With

the relation

/2(1 +

) and assuming a Poisson factor

⅓

, one obtains

(6.5)

Both lateral bending and torsion of the cantilever contribute to the total lateral force constant which

is calculated from the relation of two springs in series (see Section 4.3.1, Equation 6.22):

(6.6)

The last case is that of a force

acting in the direction of the long axis of the cantilever (

-direction).

This force induces a moment

tip

on the cantilever which causes it to bend in a way similar but

not equal to the bending induced by a normal force. Solving Equation 6.1 one ﬁnds the new bending line:

(6.7)

This bending has two effects. First, the tip is displaced an amount,

˜z

(

) = (3/2) · (

tip

) · (

) in

the

direction. Second, the tip is displaced an amount

tip

in the

direction, where

is the

()

⋅⋅

=⋅⋅













cEt

bend

=⋅⋅

























⋅

tors

tip

=⋅⋅

⋅

tors

tip tip

()













⋅













⋅

 .

11 11

cc c

xx x

tot bend tors

tip

=+=⋅





































.zy

()

⋅

tip

bending angle

˜z

′

(

), which follows from Equation 6.7. The corresponding force constant for bending

due to the force

is then

(6.8)

We note that the displacement of the tip in the

-direction due to a force

implies that the model

describing the movement of the tip by three independent springs is not completely correct. The correct

description of an SFM setup is in terms of a symmetric tensor

which relates the two vectors force

∆

and displacement

The terms

corresponds to the displacement

tip

of the tip in the

-direction due to bending

induced by a normal force F

(Equation 6.3). If the off-diagonal terms are neglected, the relation between

forces and displacements is determined by the diagonal terms, the three force constants, which can then

be related to three independent springs.

We ﬁnally note that usually the cantilever is tilted with respect to the sample. This directly affects the

relation between the different components of the forces, and has to be taken into account if the tilting

angle is signiﬁcant (Grafström et al., 1993, 1994; Aimé et al., 1995).

6.2.1.2 Measuring Forces

Force is a vector and therefore in our three-dimensional world it has three components. A classical SFM

measures the component normal to the surface, while an SFFM measures at least one of the components

parallel to the surface. Since normal force and lateral force are usually intimately related, the simultaneous

measurement of both is fundamental in tribological studies. In fact, nowadays practically all commercial

SFMs offer this possibility. The optimum solution is, of course, the determination of the complete force

vector, that is, of all three force components, and in fact such a system has been proposed (Fujisawa et al.,

1994) but is not widely used. As described in Chapter 2 by Marti, the simultaneous detection of normal

force and the x-component of the lateral force is easy with the optical beam deﬂection technique (Meyer

and Amer, 1990b; Marti et al., 1990), see Figure 6.2. Since this detection technique is most commonly

used in SFFM, we will brieﬂy recall some of its properties. A very particular feature of the optical beam

deﬂection technique is that it is inherently two dimensional: the motion of the reﬂected beam in response

to a variation in orientation of the reﬂecting surface is described by a two-dimensional vector. In the

case of SFFM, if the cantilever and the optical components are aligned correctly, and if the sample is

scanned perpendicular to the long axis of the cantilever (x-axis), then normal and lateral forces cause

motions of the reﬂected beam which are perpendicular to each other (see Figure 6.3). This motion can

then easily be measured with a four-segment photodiode or a two-dimensional position sensitive device

(PSD).

Another important feature of the optical beam deﬂection method is that unlike other detection

techniques, angles and not displacements are measured. Moreover, due to the reﬂection properties, the

angles that are detected on the photodiode are twice the bending or twisting angles of the cantilever.

This has to be taken into account when signals are converted into forces.

One consequence of measuring angles instead of displacements is that, in the case of a lateral force

acting on the tip, only the displacement corresponding to the torsion of the cantilever is detected.

However, the tip is also displaced due to lateral bending which does not result in a variation of the

==⋅













⋅

tip

∆∆=

−













=⋅













222

200

0332

0321

ccc

lltw

ll l l

xx xy xz

yx yy yz

zx zy zz

tip

measured angle. Therefore, this motion is not detected. Depending on the calibration procedure used,

this might lead to errors in the estimation of the lateral force when the cantilever is displaced more due

to bending than due to torsion. From Equations 6.4 and 6.5 we see that this is the case for cantilevers

with t/w  l

tip

/l.

The technique for measuring friction forces with the optical beam deﬂection method just described

assumes scanning in a direction perpendicular to the long axis of the cantilever (x-axis). However, friction

forces can also be measured in the other direction parallel to the surface (Radmacher et al., 1992; Ruan

and Bhushan, 1994a). In this different mode for measuring friction, the sample is scanned back and forth

in a direction parallel to the long axis of the cantilever (y-axis). As discussed previously, the friction force

acting at the end of the cantilever then bends it in a similar way as when induced by a normal force.

From Equations 6.2 and 6.7 the bending line corresponding to the back-and-forth scan can be calculated.

One obtains

Note that the sign of F

depends on the scan direction. A technique is needed to discriminate between

bending due to a normal force and bending due to a lateral force. The friction force changes sign when

the scanning direction is reversed, while the normal force remains unchanged; therefore the difference

signal corresponds to the effect caused by friction and the mean signal is due to the normal force. It

should be noted, however, that usually the microscope is operated in the so-called constant-force mode.

In the present case, this mode is better called the constant-deﬂection mode, since the deﬂection (more

precisely, the bending angle) and not the (normal) force is kept constant. To maintain a constant

FIGURE 6.2 Schematic setup of the optical beam deﬂection method. With a four-segment photodiode, the two-

dimensional motion of the reﬂected beam is measured. Therefore, normal and lateral forces can be detected simul-

taneously. Bending of the cantilever due to a normal force causes a vertical motion of the reﬂected beam. Torsion of

the cantilever due to a lateral force causes a horizontal motion.

FIGURE 6.3 If the cantilever and the optical setup are aligned correctly, the motions n

and n

induced by normal

and lateral forces cause perpendicular movements r

and r

of the reﬂected laser spot on the photodiode. This is not

the case for arbitrary alignment of the optical axes.

zytot

tip

()













−













⋅+ ⋅













deﬂection, the feedback adjusts the height of the sample to correct for the difference in bending due to

friction while scanning back and forth; that is, the feedback adjusts the height so that

where z

′

tot+

(l)is the angle of the free end of the cantilever during the forward scan, z

′

tot–

(l)the angle of

the cantilever during the backward scan and ˜z'(l, F

) the angle at the free end of the cantilever induced

by a force F

according to Equation 6.7. The friction is related to the difference in height of the topographic

images corresponding to the back-and-forth scan. Solving the above equation for the friction force F

fric

as a function of the height difference ∆

between back-and-forth scan, one ﬁnally ﬁnds

with ∆

= z

tot+

(l) – z

tot–

(l).

6.2.2 The Tip

One of the great merits of the SFFM, the nanometric size of the contact, is on the other hand a serious

experimental problem, since it is almost impossible to characterize the tip and thus the contact down to

an atomic scale. Different schemes have been proposed to solve this problem. One possibility is to use

electron microscopy not only to image, but also to grow a well-deﬁned tip (Schwarz et al., 1997). If the

electron beam is focused on the tip, molecules from the residual gas are ionized and accelerated towards

its end, where they spread out due to their charge. The result is a well-deﬁned spherical tip end. A similar

procedure is to heat a very sharp metallic tip in high vacuum (Binh and Vzan, 1987; Binh and García,

1992). Surface diffusion will induce migration of atoms from regions of high curvature to regions of

lower curvature. Again, to control the process, an electron microscope is needed. As in the previous case,

this process will form a well-deﬁned and smooth tip (see Figure 6.4). These preparation methods are

very effective but also have disadvantages, the ﬁrst one being the immense effort needed to fabricate just

one single tip. Moreover, modiﬁcation of the tip during transfer, and, even more critical, during the

SFFM experiment due to wear cannot be excluded. To control possible wear, the tips should be imaged

before and after the measurement.

FIGURE 6.4 Well-deﬁned spherical tip ends of tungsten cantilevers produced by heating the cantilever as described

in the text. The formation of these tips is controlled by the balance between surface diffusion and surface energy. By

carefully tuning the experimental conditions, tip ends of different shapes can be obtained. (Courtesy of Augustina

Asenjo Barahona, Universidad Autónoma de Madrid.)

′

()

−

′

()

′

()

zlzl zlF

ytot+ tot-

fric

tip

=⋅ ⋅

∆ ,

Not only the geometry of the tip, but also its chemical composition is important for tribological studies,

since the tip represents half of the sliding interface. Commercial microfabricated cantilevers are usually

made of silicon or silicon nitride (Si

), which are oxidized on the surface under ambient conditions.

Therefore, most SFFM experiments are performed with this material. To vary the chemical composition

of the tip, a metal ﬁlm can be evaporated onto the cantilever (Carpick et al., 1996a) or alternatively thin

ﬁlms such as Teﬂon (Howald et al., 1995) or self-assembling monolayers (Ito et al., 1997) can be adhered

to the tip. Finally, the tip may even be biologically functionalized by attaching antibodies.

In most SFFM experiments, the tip is not prepared. Instead, it is used as delivered on the commercial

microfabricated cantilever and some method is used to characterize the geometry of the tip end. One

possibility is to image a very sharp object in the normal SFM mode. If the radius of curvature of this

object is smaller than that of the tip, then the tip is imaged by this even sharper object (Sheiko et al., 1993).

Another possibility to characterize the tip is to assume a spherical tip end and estimate its radius

through the interaction with the sample. In most cases, this interaction is proportional to the tip radius.

Typically, to determine the tip radius the variation of the interaction is measured as the tip sample

distance is varied. If all other parameters are known, the tip radius can be extracted from a ﬁt to the

experimental data points. For adhesion in air due to a liquid meniscus one has for example

where R is the tip radius, γ the surface energy (for water 4πγ ≈ 0.88 N/m) and ϑ the contact angle, and

for van der Waals interaction

where A is the Hamaker constant and z the tip–sample distance. Other interactions which can be used

include electrostatic forces and friction force. The last option, however, is not very useful if the friction

force itself is to be investigated.

It should be noted that the determination of the tip using some interaction law can only be considered

an estimation, since on the one hand a spherical tip end is assumed a priori and, on the other hand,

macroscopic values are used for physical properties such as the surface energy, the contact angle, or the

Hamaker constant. On an atomic scale, the values of these properties might change or not even by

deﬁned. Finally, in air many interactions are affected by the ﬁlms adsorbed between tip and sample.

6.3 Experiments

This section will, of course, present the perhaps most dramatic progress in nanotribology, namely,

imaging the atomic periodicity of the lateral force as a sharp tip scans over a sample surface. However,

the scope of this section will be extended also to other experiments that shed light on the imaging

mechanisms and in general on tribological processes on an atomic scale, which are as yet very poorly

understood. Even the fact that the resolution of the atomic periodicity of some surfaces is quite easy —

with modern commercial instruments this should be standard provided the vibration isolation of the

instrument is good enough — is quite intriguing. In fact, from simple continuum theories for elastic

bodies one ﬁnds that the contact area between tip and sample is much larger than the atomic periodicity

that is measured. The Hertz theory (see Section 6.4.2) is commonly used to estimate the contact radius

between tip and sample. According to this model, the contact radius r

is given by

meniscus

=π

()

4 γϕcos ,

vdW

⋅⋅6

EEE

⋅

⋅+













⋅⋅

,

tip sample

where F

is the loading force, R the tip radius, and E* an effective modulus of elasticity (see Equation 6.16).

The approximation is valid assuming a Poisson ratio of ⅓. For an Si

tip on mica (E*  150 GPa),

with a sharp tip (R  25 nm) and a loading force of 20 nN, one ﬁnds r  2.0 nm, a contact radius

corresponding to an area of about 75 unit cells. Even for the hardest possible contact, namely, a dia-

mond–diamond contact, this radius is of the order of 1 nm. Therefore, the contact radius is usually much

bigger than the smallest features that are resolved, and the possible mechanisms leading to this apparent

high resolution should be explained.

To investigate the contrast mechanism, as well as the fundamental tribological properties of small

contacts, experiments have been made to measure the friction as the loading force is varied. Unlike for

macroscopic friction, the friction of nanoscale contacts does not increase linearly with load, but seems

to increase linearly with the contact area.

Another important feature of nanoscale contacts is that, although the forces are usually very small,

the interaction and thus the pressures are very high due to the small contact area. For the Si

–mica

contact above, one ﬁnds pressures of about 1.5 GPa. Generally, the pressures in nanoscale contacts can

be much higher than the bulk yield pressure and of the order of magnitude of the theoretical yield

pressure of defect-free materials (Agraït et al., 1996). Therefore, although the SFFM can be operated in

the wearless regime, care has to be taken not to exceed this regime if the aim is to investigate wearless

friction. Moreover, as the contact radius is decreased to increase resolution, this problem becomes more

critical.

The strong interaction of tip and sample is usually considered a disadvantage of scanning probe

microscopy. In the case of the SFFM, however, strong interaction is evidently inherent to friction.

Moreover, in technologically relevant situations the interaction of the surfaces in contact is also very

strong. Sometimes, however, it is interesting to study friction when the interaction is weak. This is

achieved with quartz microbalance experiments where a thin ﬁlm is adsorbed on a moving substrate. As

will be discussed in more detail, the essential physics of the SFFM and the quartz microbalance is similar,

although the pressures and timescales are vastly different. In both cases, energy is dissipated as the

atomically corrugated surfaces are moved relative to each other. In the case of the quartz microbalance

interaction is very weak, while in the case of the SFFM this interaction is usually much stronger due to

long-range forces between tip and sample. In a certain way, a quartz microbalance experiment can be

interpreted as an SFFM experiment but with only a few last-tip atoms.

6.3.1 Atomic-Scale Imaging of the Friction Force

6.3.1.1 First Experiments

SFM was introduced in 1986 to measure the topography of nonconducting surfaces (Binnig et al., 1986).

Only 1 year later, the potential of the SFM to measure forces was applied successfully to image the atomic-

scale variation of the friction force as a sharp tip scans over a surface (Mate et al., 1987). Essentially, Mate

et al. had the simple but clever idea to turn their SFM around by 90° in order to measure the lateral force

instead of the normal force and so SFFM was born. They used an interferometric detection scheme to

measure the displacement of a tungsten wire. As shown in Figure 6.5, the free end of this wire was bent

and electrochemically etched to serve as a probing tip. With a typical length of 12 mm and diameters of

0.25 and 0.5 mm, they obtained force constants of 150 and 2500 N/m, respectively, which is considerably

high for SFM and SFFM standards. As a consequence, the loading force of the tip on the sample was in

the millinewton regime, which is also very high. In spite of this high load, atomic resolution of the friction

force on a HOPG sample was observed. Figure 6.6 illustrates how the lateral force varied as the sample

is scanned back and forth in a direction perpendicular to the long axis of the cantilever. Three of these

so-called friction loops are shown, each measured at a different loading force. Since this kind of curve

is quite general in SFFM, we will discuss them in some detail. At the beginning of each scan, which can

be considered to start either left or right, the tip ﬁrst sticks to the sample. Its position with respect to

the sample is therefore ﬁxed. Since the sample is moved, the cantilever is bent. As long as the lateral force

is lower than the force needed to shear the tip–sample junction, the signal corresponding to the lateral

force increases linearly with the scanned distance. However, at a certain critical force the junction is

sheared, the tip then “slips” into a new equilibrium position, and the lateral force decreases. In its new

position, the tip ﬁrst sticks until the critical force is reached again; then another “slip” will occur. This

process repeats itself as long as the scanning direction is not reversed. The number of discontinuous slips

therefore depends on the total scan range. If the scanning direction is reversed, the lever ﬁrst unbends;

FIGURE 6.5 SFM setup used to measure atomic-scale friction. An interferometer detects the small lateral deﬂection

of the cantilever due to the friction force between the tip and the sample. (From Mate, C. M. et al. (1987), Phys. Rev.

Lett. 59, 1942–1945. With permission.)

FIGURE 6.6 Variation of the lateral force between a tungsten tip and a graphite surface as the tip is scanned laterally

over the surface. Three of these so-called lateral force curves are shown for different loading forces. The lower curve

shows the typical stick-slip behavior most clearly. (From Mate, C. M. et al. (1987), Phys. Rev. Lett. 59, 1942–1945.

With permission.)