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1040 M. Amrein

tially be overcome by choosing an appropriate imaging mode (inter-

mittent contact mode, see below). The properties of the stylus may also

be changed to avoid formation of a water bridge between sample and

tip. Knapp et al. (1995) have established a method for rendering the

probes hydrophobic. The probes became coated with a Teﬂ on-like

polymer through glow discharge in a hexaﬂ uoropropene (HFP, Hoechst

AG, Frankfurt, Germany) atmosphere (Guckenberger et al., 1994; Knapp

et al., 1995).

2.4 The Feedback Loop of the AFM

During scanning, the sample topology causes the stylus to move up or

down and the cantilever is deﬂ ected accordingly. The deﬂ ection is

measured by means of the optical pointer and compared to the set

point in the regulator. The set point represents the loading force chosen

by the user. Deviation from the set point is relayed to the driver of the

scanner and the height of the sample with respect to the stylus is read-

justed, etc. Hence, the scanning process translates the sample topology

into a time-dependent signal. The signal may be decomposed (by the

Fourier transform) into a spectrum of sine waves. Each sine wave of

the spectrum is distinct based on its amplitude and frequency. The

smaller the features are in the plane of the sample and the faster the

probe is scanned over the sample, the higher the frequencies. The taller

the features of the sample, the higher the amplitude of the respective

frequencies. Each frequency is also distinct by its phase. The phase

describes how much one sine wave is shifted with respect to all other

sine waves. The quality of the feedback loop refers to how accurate

phase and amplitude is relayed through feedback loop as a function

of the frequency.

The feedback loop is only able to respond adequately up to a certain

speed (frequency). Above that speed, the stylus does no longer trace

properly the sample topology or the feedback becomes unstable. When

the probe is scanned fast over the sample, for example, the speed of

the feed-back loop may be too low to adjust the tip to sample distance

in time. This results in variable and at times high interaction forces

(Weisenhorn et al., 1993).

The quality of the feedback loop with respect to speed and accuracy

depends on the quality of its elements. The elements of the feedback

are the regulator, the cantilever spring, the scanner element that moves

the sample up and down, and also the interaction between the sample

and the apex of the stylus and the sample topology itself (Figure

16–14).

2.4.1 Stability of the Feedback Loop

Each element of the feedback loop has its own transfer function or

frequency response. It ampliﬁ es the signal in a frequency-dependent

manner. Moreover, it relays the signal with a delay. A given delay

relates to a larger and larger phase shift with increasing frequency. The

phase shifts from all elements add up. At a certain frequency the total

phase shift will have become too large and causes the feedback loop

to react in the wrong direction, i.e., instead of moving up, the scanner

Chapter 16 Atomic Force Microscopy in the Life Sciences 1041

would move down, etc. Only the frequency range below this limiting

frequency must be used for imaging. The higher frequencies must be

suppressed by an appropriate setting of the regulator. Otherwise, the

feedback loop will oscillate. Because the frequency response of the

feedback loop may be different for each new sample and probe and

may even change during imaging, the regulator needs to be regularly

adjusted by the user. Most of the time, the regulator allows the ampli-

tude for all frequencies to be adjusted proportionally (P-regulator) and

in a frequency-dependent way (integral-regulator).

The bandwidth (the usable frequency range) of an AFM is usually

limited either by the scanner or the probe in contact with the sample.

The optical pointer and the regulator should not pose a limitation to

speed or accuracy of the feedback loop.

2.4.2 Elements of the Feedback Loop

2.4.2.1 The Scanner

Subatomic-precision scanners are the key to the success of STM, AFM,

and nanotechnology in general. Such scanners are based on piezo-

ceramic elements. They deﬂ ect in an electrical ﬁ eld. Within a certain

range, the deﬂ ection is approximately linear to the applied voltage. A

scanner is characterized by its scan range (x,y), the z range, and the

frequency response. The scan range of the actuator limits the size an

image can have. The z range puts a limit on the maximum specimen

corrugation the scanner can trace. The ﬁ rst scanners used were tripods,

with each leg representing one axis of space (Binnig and Rohrer, 1982).

Actuators used today in AFM are often made of single piezoceramic

tubes (Binnig and Smith, 1986). The bending of the tube allows the

probe or the sample to be moved in two dimensions (on approximately

a spherical surface). The length adjusts the sample-to-probe distance.

Piezoceramic tube scanners are superior in many respects to piezo-

ceramic tripods that were used in the beginning of SPMs. The tube

scanners are symmetrical with respect to the z axis and therefore are

Figure 16–14. The feedback loop of the AFM. The boxes symbolize the fre-

quency response of the elements of the loop.

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less susceptible to thermal drift. They can be made much smaller to

obtain a scan range similar to the tripods. Most recent microscopes

also have scanners with an open frame for access from above and

below the sample. In these cases, the x,y scanner is separate from the

elements moving the scanner up and down.

The frequency response determines how fast the actuator can

respond to an applied voltage. Today’s scan units are a compromise

with respect to these properties. On the one hand, a high range and

sensitivity are accomplished by increasing the size of the actuators; on

the other hand, larger scanners are slow to react. The bandwidth (i.e.,

the usable frequency range) of the scanner is limited by its ﬁ rst mechani-

cal resonant frequency (also natural frequency) in the z direction and

this depends on the size and shape of the element as well as material

properties (Table 16–1). An almost linear relation between an applied

voltage and the deﬂ ection is achieved only when the actuator is used

well below its resonant frequency. Close to the resonance, the sensitiv-

ity rises sharply and, even more important, the deﬂ ection becomes

increasingly delayed with respect to the driving voltage (i.e., there is a

negative phase shift between the driving voltage and the deﬂ ection).

As discussed above, too much of a delay is not acceptable, because the

probe could then no longer trace the relief of a specimen in real time.

To date the scanner is usually the slowest element in the feedback

loop of the AFM. It thus determines the bandwidth of the whole instru-

ment. From this condition a slow scan speed results and, hence, a long

image acquisition time that not only can make the SPM a tedious task,

but also puts a corresponding limit on the time domain for the observa-

tion of dynamic events. This is surprising, since there are ways to

strongly increase the bandwidth of AFM by a faster scanner without

limiting the scan range in any direction (Figure 16–15).

2.4.2.2 The Probe and the Sample

The dynamic response of the probe is more difﬁ cult to understand than

the frequency response of the scanner. Prior to establishing contact with

the sample, the probe (cantilever) may be treated as a beam with one

end ﬁ xed and one end free. Establishing the frequency response is

straightforward, as described in Table 16–1. The smaller and the stiffer

the cantilever, the higher its ﬁ rst resonant frequencylies. Note that the

frequency response of a free cantilever is always established prior to

imaging in the intermittent contact mode (tapping mode). However,

after establishing contact, the beam is in an intermediate state between

a beam with both ends ﬁ xed and one end ﬁ xed. Moreover, contact to the

Chapter 16 Atomic Force Microscopy in the Life Sciences 1043

Table 16–1. Natural frequencies for the elements of a scanner.

Natural frequency (Hz)

Longitudinal Lateral Torsional

Beam with quadratic cross section: one end ﬁ xed

(E = Young’s modulus, r = density,

4 ρρ

= 0.1615

ρρ

L = length, and a = edge length

Tube with att ached mass: one end ﬁ xed

A = cross-sectional area, m = mass,

M = attached mass,

(

)

ππ

MmL

0.23

ππ

(

)

ππ

l = area-moment of inertia,

d = outer diameter, w = wall thickness,

= moment of inertia,

= moment of inertia of attached mass, and

42 2

ππ

−−−−





























md d

42 2





























−−−−

ddw=

32 2.6

ππ

−−−−

(

)









From Harris (1987).

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sample is complicated. The cantilever and the sample are coupled via

the force interaction between the tip and the sample and the compliant

sample itself. Mechanically, the situation resembles a series of three

dampened springs, the cantilever, the tip sample interaction, and the

viscoelastic sample. There is no analytical solution for this situation

(Budó, 1976) and the frequency response is best determined experimen-

tally (Figure 16–16). Smaller and stiffer cantilevers will still lead to a

broader bandwidth of this element of the feedback loop.

Figure 16–15. (Left) The frequency response of a typical commercial scanner (the J scanner from

NanoScope III, digital instruments). The scanner may be used up to approximately 4 kHz. Above, the

phase shift is too large for stable imaging. (Right) A scanner developed in the group of the author.

The scan range and range in the z direction are comparable to the J-scanner. However, the usable fre-

quency range is one to two orders of magnitude wider (Knebel et al., 1997).

Figure 16–16. (Left) Frequency response of a silicon nitride cantilever (100 µm long; c = 0.1 N/m;

Olympus Ltd., Tokyo, Japan) in contact with a mica sample. The loading force was 1 nN. The frequency

limit posed by this element of the feedback loop lies at about 10 kHz, above which the phase shift has

become too great for a stable feedback loop. (Right) Frequency response of the closed feedback loop

of a NanoScope III (digital imaging, Santa Barbara, CA) equipped with a J-scanner. The same cantilever

has been used. The measurement was carried out in water. Instead of scanning across the mica sample,

the sample was oscillated, mimicking a corrugated sample. The oscillation was continuously changed

from low frequency (mimicking broad objects) to a higher frequency (representing smaller objects).

The amplitude of the oscillation was constant over the total measuring range. The regulator of the

microscope was adjusted as for imaging to optimally track the oscillation of the sample. The amplitude

shown corresponds to the output of the feedback for image acquisition.

Chapter 16 Atomic Force Microscopy in the Life Sciences 1045

2.4.3 Frequency Response of the Closed Feedback Loop

For imaging, all elements work together in the closed feedback loop

(Figure 16–16A). The frequency response of the closed feedback loop

is therefore a direct measure of the quality and trustworthiness of

AFM imaging. In the example shown in Figure 16–16B, lower frequen-

cies (larger object details) up to about 1.5 kHz are all represented with

about the same amplitude. Higher frequencies up to about 2.5 kHz

(smaller object details) are ampliﬁ ed increasingly more strongly by the

system. Hence, in an image these details would be more prominent

than they should be. Very small details would no longer be traced. For

proper interpretation of the sample structure at high resolution, images

should be taken at various scan speeds.

2.5 Noise

The instrument’s noise degrades the accuracy with which the topology

is traced. For a well-designed instrument, from all the elements of the

feedback, the thermal motion of the cantilever is the major source of

noise. The cantilever “produces” one k

T per resonance interval (k

Bolzmann’s constant and T is the absolute temperature). This will

cause the cantilever typically to oscillate by a few Angstroms. When

the stylus rests in ﬁ rm contact on the sample, the thermal motion of

the cantilever is strongly reduced and will not affect resolution directly.

However, the thermal energy now creates a ﬂ uctuation of the loading

force. In the case of a soft, susceptible sample, this in turn will create

a marked ﬂ uctuation of the sample deformation. The thermal force

noise within the context of the simple harmonic oscillator model is

given by (Sarid, 1991)

∆F

kBk

= 23

where ∆F is the rms force noise, B the bandwidth at which the micro-

scope operates, T the absolute temperature, k the spring constant of the

cantilever, ω

the radial resonant frequency, and Q the mechanical

quality factor. Q parameterizes the sharpness of the resonance, being

the ratio of the resonant frequency to the full-width at half-height of

the peak. In air or vacuum, very high Q (on the order of thousands)

can be achieved. In water, Q is much lower. As an example, for a typical

cantilever used to scan a molecular sample in an aqueous solution

(k = 0.1 N/m, ω

= 20 kHz, Q = 1, B = 10 kHz) the thermal force noise is

5–20 pN. From the equation above it follows that an ideal cantilever for

low-noise operation should have a low spring constant, but still a high

resonant frequency. For a given size of a cantilever, however, the stiffer

the lever, the higher the resonant frequency. This dilemma can be

overcome by employing smaller (shorter) cantilevers (Hansma, 1996).

The effect of the thermal noise on the image quality will also be

reduced by operating the microscope at a lower bandwidth. But this

comes at the cost of reduced scan speed.

It is notable that noise is not evenly spread over the entire frequency

spectrum. For large Q, there is a large reduction of the thermal noise

1046 M. Amrein

away from the resonant peak. In water, the thermal noise is quite

evenly distributed over the whole frequency range. At low frequencies,

other sources of mechanical motion become progressively more impor-

tant. This is referred to as 1/f noise and is notable foremost in contact

mode imaging (see below).

2.6 Imaging Modes

2.6.1 Contact Mode

In contact mode, the tip is permanently touching the sample. On a rela-

tively solid sample, the position of the cantilever in the z direction is

well deﬁ ned and is affected only by force noise. It has therefore pro-

vided the best resolution to date on molecular samples. Contact mode

imaging works best with densely packed samples, to provide stability

against the sideways push by the tip. On soft samples, small cantilevers

with a low spring constant are desirable. For operation at minimal

force, the force during scanning is reduced until the cantilever takes

off from the sample and then increases it just enough to touch back

down and trace the topology properly.

2.6.2 Noncontact Mode

In noncontact mode, the probe tip is excited to oscillate near the reso-

nant frequency. To chose the excitation frequency, it is necessary to

perform a sweep from low to high frequencies and observe the ampli-

tude and phase of the cantilever oscillation. The fundamental bending

frequency of the cantilever will stand out as a prominent peak and the

Chapter 16 Atomic Force Microscopy in the Life Sciences 1047

excitation frequency chosen to be close by (on the lower-frequency

ﬂ ank of the peak). In close vicinity to the sample but prior to contact,

the interaction between the tip and the sample will change the reso-

nance conditions for the lever (as the cantilever now enters an inter-

mediate state between a lever free on one side and a lever ﬁ xed on both

sides). This in turn will cause the amplitude to decrease, because the

driving frequency is now further away from the resonant frequency.

The shift in resonant frequency is a function of the force gradient

(dF/dz) between the tip and the sample. The microscope is operated to

keep the amplitude reduction at a deﬁ ned level and, hence, the tip at

a deﬁ ned distance to the sample. Although noncontact imaging prom-

ises to be the most subtle mode for highly deformable, weakly immo-

bilized objects, it has not become popular so far in life science

applications because it does not usually allow for stable high-resolution

tracing of an interface of a biological sample. This might be because

the tip is kept at a distance to the sample where it still can oscillate

considerably by thermal noise and instrument instabilities. Very stable

microscopes, very small cantilevers, and very sharp tips with a high

aspect ratio may, in the future, bring about a strong improvement for

this potentially attractive imaging mode.

2.6.3 Intermittent Contact Mode (Tapping Mode, Dynamic

Force Microscopy)

Intermittent contact mode AFM in a certain sense combines the advan-

tages of noncontact and contact AFM. The probe oscillates as in the

noncontact operation. But the set point for the amplitude is now chosen

so that the tip can make brief contact with the sample in each cycle of

its oscillation. As in the noncontact mode, there are no lateral forces

excerpted on the sample. As in contact mode AFM, the local topological

height is determined accurately by the well-deﬁ ned contact point. This

mode has therefore had success in imaging well-separated molecules

bound to an underlying support rather weakly that did not withstand

the lateral forces of contact mode imaging. It is notable, however, that

the loading force during contact is not smaller than with contact mode

AFM. Intermittent contact mode has resulted in resolution of molecular

samples on the order of 1 nm (Figure 16–17).

As in noncontact mode, intermittent contact mode AFM in air is

sensitive to a shift in resonance frequency that then brings about a

change in amplitude of the oscillation. In buffer, resonance effects are

small an and the microscope is largely sensitive to the restriction of

the oscillation brought about by the physical barrier posed by the

sample. Its sensitivity is thus limited by thermal ﬂ uctuations and is

not signiﬁ cantly enhanced by operating at resonance. In water, it is

desirable to choose as small an amplitude as possible to minimize

disturbance of the sample. In practice, the overall oscillation amplitude

is set to the smallest value that provides stable imaging. Then the

set point amplitude change is increased (increasing the resolution) to

the largest value that does not damage the sample. However, stable

operation requires amplitude sufﬁ cient to pull the tip out of attractive

interactions.

1048 M. Amrein

Two methods for exciting the cantilever oscillator are common,

acoustic drive (Hansma et al., 1994; Putman et al., 1994) and magnetic

drive. In acoustic drive an oscillator supplies a voltage drive to a piezo-

electric actuator that generates sound waves in the cantilever holder.

The frequencies are typically from tens of kilohertz to megahertz.

When the driving frequency is near a bending-mode resonance of the

cantilever, the cantilever is driven into an oscillation. A frequency scan

with acoustic drive results in many peaks, only some of which yield a

usable signal (Putman et al., 1994; Schäffer et al., 1996). This is because

the transmission of the acoustic wave is not uniform over the frequency

range. In magnetic drive, the cantilever is coated with a magnetic ﬁ lm,

or a magnetic particle is glued onto the end of the cantilever (Lantz

et al., 1994; Han et al., 1996). The oscillating magnetic ﬁ eld of a solenoid

in the vicinity of the cantilever causes the cantilever to oscillate.

In contrast to contact mode, which senses a force, intermittent contact

mode AFM is a technique that senses an interfacial stiffness. As fre-

quency is increased, intermittent AFM also becomes sensitive to

increased interfacial damping.

2.7 Optimizing AFM Design for Life Science Applications

The range of applications for AFM is very large, from atomic resolution

imaging on some crystal surfaces to imaging or manipulation of whole

cells. An AFM designed for working under high vacuum to look at

defects in atomic lattices has different basic requirements from one that

is designed predominantly for samples in liquid, with full optical

microscope integration as a priority. On biological samples, atomic

Figure 16–17. Latex-beads imaged in tapping mode. In the top portion of the

image, the set change in amplitude was chosen large and the beads were

deformed. In the lower portion, the set amplitude change was reduced, result-

ing in proper topographical imaging (1 µm × 1 µm).

Chapter 16 Atomic Force Microscopy in the Life Sciences 1049

resolution is generally not possible; for samples of macromolecules and

larger structures, the resolution is mainly deﬁ ned by issues such as the

tip–sample interaction and sample deformation. Then technical issues

such as integrated optics, ﬂ exibility of handling, and sample environ-

ment carry more weight.

2.7.1 Combining AFM and Optical Microscopy

Optical microscopy still dominates life science research, and tech-

niques for enhancing contrast, labeling, or staining to recognize and

study biological samples have been developed. It makes no sense to

throw away this vast reserve of information about samples that are to

be studied using the AFM, and indeed it is not necessary to compro-

mise the quality of optical images for light microscopy to be combined

with AFM. For a well-designed AFM it is possible to simultaneously

obtain high-contrast and high-resolution optical images, including

techniques such as laser scanning microscopy (LSM), total internal

reﬂ ection ﬂ uorescent (TIRF) microscopy, and ﬂ uorescence resonance

energy transfer (FRET) microscopy (Figure 16–18).

In one sense it is relatively straightforward to build a combined

optical and atomic force microscope, since the AFM can be built around

a standard inverted optical microscope. This allows the use of the

optics through the objective lens under the sample (if the sample is

transparent, which is often the case), while at the same time the AFM

can approach from above. In practice, care must be taken that the entire

system has the required mechanical stability, that sources of drift or

vibration are minimized, and that the “open” nature required of such

an AFM does not compromise its stability and reduce the resolution

achievable. There are also two potential problems that must be consid-

ered in the instrument design that relate directly to the optical imaging:

the optical quality of the support for the sample, through which the

optical image must be formed, and the light path to illuminate the

sample from above.

Figure 16–18. Images of a ﬁ xed 3T3 cell in buffer, on a coverslip. Left to right: Optical (differential

interference contrast, DIC) and simultaneous AFM (JPK Instruments, contact mode) topography and

feedback signal. (See color plate.)