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

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Marti, O. Ò"AFM Instrumentation and Tips"Ó

Handbook of Micro/Nanotribology.

Ed. Bharat Bhushan

Boca Raton: CRC Press LLC, 1999

AFM Instrumentation

and Tips

Othmar Marti

2.1 Force Detection

2.2 The Mechanics of Cantilevers

Compliance and Resonances of Lumped Mass Systems •

Cantilevers • Tips and Cantilevers • Materials and

Geometry • Outline of Fabrication

2.3 Optical Detection Systems

Interferometer • Sensitivity

2.4 Optical Lever

Implementations • Sensitivity

2.5 Piezoresistive Detection

Implementations • Sensitivity

2.6 Capacitive Detection

Sensitivity • Implementations

2.7 Combinations for Three-Dimensional Force

Measurements

2.8 Scanning and Control Systems

Piezotubes • Piezoeffect • Scan Range • Nonlinearities,

Creep • Linearization Strategies • Alternative Scanning

Systems • Control Systems

2.9 AFMs

Special Design Considerations • Classical Setup • Stand-

Alone Setup • Data Acquisition • Typical Setups • Data

Representation • The Two-Dimensional Histogram

Method • Some Common Image-Processing Methods

Acknowledgments

References

Introduction

The performance of AFMs and the quality of AFM images greatly depend on the instruments available

and the sensors (tips) in use. To utilize a microscope to its fullest, it is necessary to know how it works

and where its strong points and its weaknesses are. This chapter describes the instrumentation of force

detection, of cantilevers, and of the instruments themselves.

2.1 Force Detection

Atomic force microscopy (AFM)(Binnig et al., 1986) was an early offspring of scanning tunneling micros-

copy (STM). The force between a tip and the sample was used to image the surface topography. The

force between the tip and the sample, also called the tracking force, was lowered by several orders of

magnitude compared with the proﬁlometer (Jones, 1970). The contact area between the tip and the

sample was reduced considerably. The force resolution was similar to that achieved in the surface force

apparatus (Israelachvili, 1985). Soon thereafter atomic resolution in air was demonstrated (Binnig et al.,

1987), followed by atomic resolution of liquid covered surfaces (Marti et al., 1987) and low-temperature

(4.2 K) operation (Kirk et al., 1988). The AFM measures either the contours of constant force, force

gradients or the variation of forces or force gradients, with position, when the height of the sample is

not adjusted by a feedback loop. These measurement modes are similar to the ones of the STM, where

contours of constant tunneling current or the variation of the tunneling current with position at ﬁxed

sample height are recorded.

The invention of the AFM demonstrated that forces could play an important role in other scanned

probe techniques. It was discovered that forces might play an important role in STM. (Anders and Heiden,

1988; Blackman et al., 1990). The type of force interaction between the tip and the sample surface can

be used to characterize AFMs. The highest resolution is achieved when the tip is at about zero external

force, i.e., in light contact or near contact resonant operation. The forces in these modes basically stem

from the Pauli exclusion principle that prevents the spatial overlap of electrons. As in the STM, the force

applied to the sample can be constant, the so-called constant-force mode. If the sample

-position is not

adjusted to the varying force, we speak of the constant

-mode. However, for weak cantilevers (0.01 N/m

spring constant) and a static applied load of 10

–8

N we get a static deﬂection of 10

–6

m, which means that

even structures of several nanometers height will be subject to an almost constant force, whether it is

controlled or not. Hence, for the contact mode with soft cantilevers the distinction between constant-

force mode and constant

-mode is rather arbitrary. Additional information on the sample surface can

be gained by measuring lateral forces (friction mode) or modulating the force to get

dF/dz,

which is

nothing else than the stiffness of the surfaces. When using attractive forces, one normally measures also

dF/dz

with a modulation technique. In the attractive mode the lateral resolution is at least one order of

magnitude worse than for the contact mode. The attractive mode is also referred to as the noncontact

mode.

We will ﬁrst try to estimate the forces between atoms to get a feeling for the tolerable range of

interaction forces and, derived from them, the compliance of the cantilever.

For a real AFM tips the assumption of a single interacting atom is not justiﬁed. Attractive forces like

van der Waals forces reach out for several nanometers. The attractive forces are compensated by the

repulsion of the electrons when one atom tries to penetrate another. The decay length of the interaction

and its magnitude depend critically on the type of atoms and the crystal lattice they are bound in. The

shorter the decay length, the smaller is the number of atoms which contribute a sizable amount to the

total force. The decay length of the potential, on the other hand, is directly related to the type of force.

Repulsive forces between atoms at small distances are governed by an exponential law (like the tunneling

current in the STM), by an inverse power law with large exponents, or by even more complicated forms.

Hence, the highest resolution images are obtained using the repulsive forces between atoms in contact

or near contact. The high inverse power exponent or even exponential decay of this distance dependence

guarantees that the other atoms beside the apex atom do not signiﬁcantly interact with the sample surface.

Attractive van der Waals interactions on the other hand, are reaching far out into space. Hence, a larger

number of tip atoms take part in this interaction so that the resolution cannot be as good. The same is

true for magnetic potentials and for the electrostatic interaction between charged bodies.

A crude estimation of the forces between atoms can be obtained in the following way: assume that

two atoms with mass

are bound in molecule. The potential at the equilibrium distance can be

approximated by a harmonic potential or, equivalently, by a spring constant. The frequency of the

vibration

of the atom around its equilibrium point is then a measure for the spring constant

(2.1)

where we have to use the reduced atomic mass. The vibration frequency can be obtained from optical

vibration spectra or from the vibration quanta h

(2.2)

As a model system, we take the hydrogen molecule H

. The mass of the hydrogen atom is

= 1.673

-27

kg and its vibration quantum is h

= 8.75

–20

J. Hence, the equivalent spring constant is

560 N/m. Typical forces for small deﬂections (1% of the bond length) from the equilibrium position are

∝

–10

N. The force calculated this way is an order of magnitude estimation of the forces between

two atoms. An atom in a crystal lattice on the surface is more rigidly attached since it is bound to more

than one other atom. Hence, the effective spring constant for small deﬂections is larger. The limiting

force is reached when the bond length changes by 10% or more, which indicates that the forces used to

image surfaces must be of the order of 10

–8

N or less. The sustainable force before damage is dependent

on the type of surfaces. Layered materials like mica or graphite are more resistant to damage than soft

materials like biological samples. Experiments have shown that on selected inorganic surfaces such as

mica one can apply up to 10

–7

N. On the other hand, forces of the order of 10 to 9 N destroy some

biological samples.

2.2 The Mechanics of Cantilevers

2.2.1. Compliance and Resonances of Lumped Mass Systems

Any one of the building blocks of an AFM, be it the body of the microscope itself or the force measuring

cantilevers, is a mechanical resonator. These resonances can be excited either by the surroundings or by

the rapid movement of the tip or the sample. To avoid problems due to building or air-induced oscilla-

tions, it is of paramount importance to optimize the design of the scanning probe microscopes for high

resonance frequencies; which usually means decreasing the size of the microscope (Pohl, 1986). By using

cubelike or spherelike structures for the microscope, one can considerably increase the lowest eigenfre-

quency. The eigenfrequency of any spring is given by

(2.3)

where

is the spring constant and

eff

is the effective mass. The spring constant

of a cantilevered beam

with uniform cross section is given by (Thomson, 1988)

(2.4)

where

is the Young’s modulus of the material, l

the length of the beam, and

the moment of inertia.

For a rectangular cross section with a width

(perpendicular to the deﬂection) and a height

one

obtains for

(2.5)

=ω













eff

Combining Equations 2.3 through 2.5, and we get the ﬁnal result for f :

(2.6)

The effective mass can be calculated using Rayleigh’s method. The general formula using Rayleigh’s

method for the kinetic energy

T of a bar is

(2.7)

For the case of a uniform beam with a constant cross section and length

one obtains for the deﬂection

(

) =

max

(1 – (3

/(2 l

) + (

)/(2l

). Inserting

max

into Equation 2.7 and solving the integral gives

(2.8)

and

for the effective mass.

Combining Equations 2.4 and 2.8 and noting that

ρl bh , where

ρ is the density of mass, one

obtains for the eigenfrequency

(2.9)

Further reading on how to derive this equation can be found in the literature (Thomson, 1988). It

is evident from Equation 2.9, that one way to increase the eigenfrequency is to choose a material

with as high a ratio

E /

. Another way to increase the lowest eigenfrequency is also evident in

Equation 2.9. By optimizing the ratio

h /l

one can increase the resonance frequency. However, it

does not help to make the length of the structure smaller than the width or height. Their roles will

just be interchanged. Hence, the optimum structure is a cube. This leads to the design rule, that

long, thin structures like sheet metal should be avoided. For a given resonance frequency the quality

factor should be as low as possible. This means that an inelastic medium such as rubber should be

in contact with the structure to convert kinetic energy into heat.

2.2.2 Cantilevers

Cantilevers are mechanical devices specially shaped to measure tiny forces. The analysis given in the

previous chapter is applicable. However, to understand better the intricacies of force detection systems

we will discuss the example of a straight cantilevered beam (Figure 2.1).

Ebh

eff eff

dx=

∂

()

∂













∫

mzt

∂

()

∂

−





































()

∫

max

maxeff

eff













3 ρ

The bending of beams with a cross section

(

) is governed by the Euler equation (Thomson, 1988):

(2.10)

where

is Young’s modulus,

(

) the ﬂexure moment of inertia deﬁned by

(2.11)

Equations 2.10 and 2.11 can be derived by evaluating torsion moments about an element of inﬁnites-

imal length at position

Figure 2.2 shows the forces and moments acting on an element of the beam.

is the shear moment,

the bending moment, and

(

) the position-dependent load per unit length. Summing forces in the

-direction, one obtains

(2.12)

Summing moments on the right face of the element gives

(2.13)

Finally, one obtains for the shear and bending moments

(2.14)

FIGURE 2.1 A

typical force microscope cantilever with a

length l

, a width

, and a height

. The height of the tip is

The material is characterized by Young’s modulus

, the shear

modulus

/(2(1 +

)), where

is the Poisson number,

and a density

FIGURE 2.2

Moments and forces acting on an

element of the beam.

EI x

zpx

()













()

I x z dydz

()

∫

dV p x dx−

()

= 0

dM Vdx p x dx−−

()( )

()

Combining both parts of Equation 2.14, one obtains the following result

(2.15)

Using the ﬂexure equation to express the bending moment, one obtains

(2.16)

Combining Equations 2.15 and 2.16, and one obtains the Euler Equation 2.10. Beams with a nonuni-

form cross section are difﬁcult to calculate. Let us, therefore, concentrate on straight beams. These

cantilever beams are widely used for friction mode as well as for noncontact experiments.

A force acting on the cantilever at a position x

can be handled by the Dirac function δ(x – x

), for

which one has

(2.17)

Hence, one sets

(2.18)

where l is the length of the cantilever. Integrating M twice from the beginning to the end of the cantilever,

one obtains

(2.19)

since the moment must vanish at the end point of the cantilever. Integrating twice more and observing

that EI is a constant for beams with an uniform cross section, one gets

(2.20)

The slope of the beam is

(2.21)

Evaluating this and Equation 2.20 at the end of the cantilever, i.e., for x = l, one gets

(2.22)

()

MEI

fx x x dx fx

()

−

()

−∞

∞

∫

px F

()

δ l

Mx xF

()

=−

()

⇒

()













−













′

()

== −













=−













dx EI

()

=−

′

()

=− =

()