Proceedings of the American Control Conference, art. no. 4586984, pp. 3194-3200 (2008)

Control Techniques in High-speed Atomic Force Microscopy
Toshio Ando

Abstract—Biological macromolecules are highly
dynamic. Their functions results from dynamic structuralchanges and dynamic interactions with other molecules.
The visualization of the dynamic processes at high spatial
and temporal resolution appears to be straightforward tounderstanding the functional mechanisms of biologicalmacromolecules. Such visualization is only possible by theatomic force microscope (AFM) with a high-speed
imaging capability, since dynamic biomolecular processes
generally occur on a millisecond timescale. Therefore, wehave been improving both the scan speed as well as the tip
force exerted on the sample by developing various devises
and techniques. The latest version of our high-speed AFM
can capture biological processes at an imaging rate of
30-100 ms/frame without disturbing their functions.
Here, we report the devices and techniques that realized
this performance.

A

I. INTRODUCTION

FM was invented in 1986 by Binnig et al. [1] and opened
the door to the visualization of nanometer-scale worlds
in liquids [2]. This unique capability was received withexcitement by researchers of biological sciences as
biomolecules only show vital activities in aqueous solutions.
Before the AFM era, the high-resolution visualization ofindividual biopolymers (proteins, DNA) was only possible byelectron microscopy in a vacuum environment. However, theAFM’s unique capability does not seem to have contributed
significantly to answering many biological questions.
One of the essential features of biological systems is“dynamics”. The functions of biological systems areproduced through dynamic processes that occur inbiopolymers, biosupramolecules, organelles, and cells.
Therefore, what required of AFM for biological sciences isthe ability to rapidly acquire successive high-resolution
images of individual biomolecules at work. This is solely
because this type of imaging is impossible to perform using
other techniques. However, the imaging rate of conventional
AFM is too slow to observe dynamic behavior of active
biomolecules. Thus, endowing AFM with high-speed

Manuscript received September 12, 2007. This work was supported in partby JST , and a Grant-in-Aid for Basic Research S from the Ministry of
Education, Culture, Sports, Science, and Technology of Japan.
T. Ando is with the Department of Physics, Kanazawa University,
Kanazawa 920-1192 Japan (phone: +81 76-264-5663; fax: +81 76-264-5739;
e-mail: tando@kenroku.kanazawa-u.ac.jp).

imaging capability is expected to have a revolutionary impact
on biological sciences.
The low imaging rate of AFM mainly originates in that
AFM uses the mechanical detection of the sample surface andthe mechanical scanning of the sample stage. The sample
surface is detected by its interaction with a sharp tip attachedto the free end of a soft cantilever, and this interaction isreflected on the cantilever’s mechanical behavior. The
relative position between the tip and the sample is changed byscanning the sample stage in the three dimensions. To
increase the surface detection speed, we first need a scannerwith high resonant frequencies as well as small cantilevers
with high resonant frequencies and small spring constants.
After having these devices, we need control techniques torealize fast and accurate positioning of the scanner. Since
biological samples are fragile and dynamic biomolecularinteractions are more delicate, we further need control
techniques to maintain the tip-sample interaction force at asmall level without sacrificing the scan speed.
At present, digital mode controllers are not sufficiently
fast for realizing high-speed AFM capable of imaging at avideo rate of 30 ms/frame. Therefore, recently reported
sophisticated control algorisms [3]-[5] are not applicable tothe video-rate high-speed AFM. We have been exploring
simple control techniques [6]-[8] that can be implemented inanalog circuits. These controllers have enabled the maximum
imaging rate of 30-50 ms/frame, together with other devices
optimized for high-speed scanning. The tip force acting on
the sample has been reduced greatly. These capacities
allowed us to observe biological processes that involved
weak protein-protein interactions [9]-[11]. In this article, I
review our studies that realized the high-speed AFM with thiscapability, focusing on the devises and techniques.
II. THEORETICAL CONSIDERATIONSIn the development of high-speed AFM apparatuses, it isimportant to have practical guidelines that can quantitatively
indicate how each device performance affects the scan speed
and the imaging rate. An early theoretical consideration of thescan speed limit in contact mode AFM was given in [12].
Concerning taping mode AFM, the dependence of feedback
bandwidth on various factors has been quantitatively
described [13]. Numerical simulations have also been
performed for this purpose, including the effect of the
dynamics of the tip-sample interaction [14]. However, they
are not sufficient as practical guidelines. Here, we derive theProceedings of the American Control Conference, art. no. 4586984, pp. 3194-3200 (2008)

quantitative relationship between the feedback bandwidth
and the various factors involved in AFM devices and thescanning conditions, based on an idea previously presented
for the derivation [7].
A. Acquisition Time and Feedback Bandwidth
Suppose that an image is acquired in time period T overscan range W × W with N scan lines, then the scan velocity Vs
in the x-direction is given by Vs = 2WN/T. Assuming that asample has a sinusoidal shape with periodicity λ, the scanvelocity Vs requires a feedback operation at frequency f =Vs/λ to maintain the tip-sample distance. The feedback
bandwidth fB should be greater than or equal to f and therefore
be expressed asf B ≥ 2WN / λT(1).

Eq.1 gives a relationship between the image acquisition time
T and the feedback bandwidth fB. For example, for T = 30 ms
with W = 240 nm and N = 100, the scan velocity is 1.6 mm/s.
When λ is 10 nm, fB ≥ 160 kHz is required for this scanvelocity.
B. Feedback Bandwidth as a Function of Various Factors
Although not described here, the relationship between the
open-loop phase delay Ф(ω) and the closed-loop phase delayφ(ω) is given by φ(ω) = 2Ф(ω), provided the feedback gain
is maintained ~1. The phase delay Ф is given by 2πfΔτ, whereΔτ is the total time delay in the open-loop and f is the feedback
frequency. In tapping mode AFM, the main delays are thereading time of the cantilever oscillation amplitude (τd), thecantilever response time (τc) , the z-scanner response time (τs),
the integral time (τI) of error signals in the feedback controller,
and the parachuting time (τp). Here, “parachuting” means thatthe cantilever tip completely detaches from the sample
surface at a steeply inclined region of the sample, andthereafter, time elapses until it lands on the surface again. It
takes at least a time of (1/2fc) to measure the amplitude of a
cantilever that is oscillating at its resonant frequency fc. The
response time of second-order resonant systems such as
cantilevers and piezoelectric actuators is expressed as Q/πf0,
where Q and f0 are the quality factor and resonant frequency,
respectively. The feedback bandwidth is usually defined by
the feedback frequency that results in a phase delay of π/4. On
the basis of this definition, the feedback bandwidth fB isapproximately expressed as

(

)

⎞
f ⎛
2Qc 2Qs f c
f B = α c / ⎜⎜1 +++ 2 f c τ p + τ I ⎟⎟
8 ⎝
π
πf s
⎠

(2),

where fs is the z-scanner’s resonant frequency; Qc and Qs are
the quality factors of the cantilever and z-scanner,respectively. α represents a factor related to the phasecompensation effect given by the D component in the
proportional- integral-differential (PID) feedback controller.
The parachuting time is a function of various parameterssuch as the sample height h0, the free oscillation amplitude A0of the cantilever, the amplitude set point r, the phase delay φ,and the cantilever’s resonant frequency. Its analyticalexpression was previously obtained under roughapproximation [7] asτ p ≈ [(tan β ) / β − 1] / f c(3),where ß is cos −1 [2 A0 (1 − r ) / {h0 sin(φ / 2 )}] . However,experimental estimations of the feedback bandwidth imposesa modification on ß asβ ≈ cos −1 [2 A0 (1 − r ) / {5h0 sin (φ / 2 )}] (4).

It is difficult to theoretically estimate the integral time (τI)
with which the optimum feedback control is attained. Its
experimental estimation approximately resulted inτ I ≈ 4h0 sin (φ / 2 ) / ( A0 f c ). The feedback bandwidths
obtained theoretically and experimentally are shown in Fig.1
as a function of r and 2A0/h0.

Fig.1. Feedback bandwidth as a function of the set point (r) and the ratio(2A0/h0) of the free oscillation peak-to-peak amplitude to the sample height.
The number attached to each curve indicates the ratio 2A0/h0. The feedback
bandwidths were obtained under following conditions: the cantilever’s
resonant frequency, 1.2 MHz; Q factor of the cantilever oscillation, 3; theresonant frequency of the z-scanner, 150 kHz; Q factor of the z-scanner, 0.5.
Black lines, experimentally obtained feedback bandwidths using a mock
AFM; gray lines, theoretically derived feedback bandwidths.

III. MECHANICAL DEVICES
A. Small Cantilevers
The small cantilevers recently developed by Olympus are
made of silicon nitride and are coated with gold of thicknessof ~20 nm [15]. They have a length of ~6 μm, a width ~2 μm,
and a thickness ~90 nm, which results in the resonant
frequencies of ~3.5 MHz in air and ~1.2 MHz in water, aspring constant kc ~0.2 N/m, and Qc ~2.5 in water. Therefore,
their response time τc (= Qc/πfc) is 0.66 μs in water. The tip is
not sufficiently small (the apex radius, ~17 nm) for thehigh-resolution imaging of biological samples. We usuallyattach a sharp tip on the original tip by electron beam
deposition in naphthalene gas, which can be sharpen to ~4 nm
by plasma etching in argon or oxygen gas. To detect thecantilever’s oscillation amplitude at every half oscillation
period, we developed a peak-hold method; the peak andbottom voltages are captured and then their difference is
output as the amplitude [16]-[17]. The amplitude detectionProceedings of the American Control Conference, art. no. 4586984, pp. 3194-3200 (2008)

time τd (=1/2fc) is 0.42 μs for our small cantilevers with aresonant frequency of ~1.2 MHz in water.
B. Sample-stage Scanner
We use a flexure-based stage for the x- and y-scanners[10]. The flexure stage is produced by monolithic processing
to minimize the number of resonant elements. The slowest
y-scanner displaces the x-scanner, and the x-scannerdisplaces the z-scanner, on top of which a sample stage isplaced. The maximum displacements at 100 V of the x- andy-scanners are 1 μm and 3 μm, respectively. The x-scannershows resonant peaks at 45 kHz, 65 kHz, and higher
frequencies, as shown in Fig. 2. The peak at ~20 kHz does not
originate from vibrations of the x-scanner but from themeasurement method used.

damping, the x-scanner moved approximately in a triangle
waveform [Fig. 3(b)]. This damping was also verified byimaging mica surfaces at 33 frames/s over a scan range of 240
× 240 nm2 with 100 scan lines (this imaging speed
corresponds to an x-scan frequency of 3.3 kHz). When no
active damping was applied, striation patterns parallel to they-axis appeared [Fig. 3(c)]. The periodicity of the patterns
indicates that the vibrations had a frequency of ~60 kHz,
which corresponded with the frequency of the main resonantpeak shown in Fig. 2. When active damping was applied, thestriation patterns disappeared [Fig. 3(d)]. Here, note that theresonant vibrations in the x-scanner also produce vibrationsin the z-direction.

Fig. 2. Frequency spectra of mechanical response of the x-scanner.

Two z-piezoelectric actuators (maximum displacement, 2
μm at 100 V; self-resonant frequency, 360 kHz) are used toalleviate the z-scanner vibrations; impulsive forces arecountered by the simultaneous displacements of thez-piezoelectric actuators of the same length in the counter
direction. In this arrangement, the counterbalance works
effectively below the resonant frequencies. The vibration
phase changes sharply around the resonant frequencies, andtherefore, a slight difference in the mechanical properties of
the two actuators disturbs the counterbalance. To solve thisproblem, we are now attempting to use a different design for
the z-scanner, as mentioned later.

Fig. 3: Effect of feedforward active-damping control on thex-scanner vibrations. (a) X-scanner displacement driven bytriangle waves without damping. (b) X-scanner displacement
driven by triangle waves compensated by the feedforward
control. (c) Mica surface images acquired at 33 frames/s (c)without and (d) with damping.

B. Feedback Type of Active Damping for Z-scanner

IV. ACTIVE DAMPING
It is impossible to manufacture a scanner that exhibits noresonance around the harmonic frequencies contained in its
driving signals. Therefore, the elimination of unwanted
vibrations from the scanner is a key to realizing high-speed
AFM. Furthermore, the bandwidth of any mechanicalsystems is limited by its dimensions and materials, and hence,some manipulation techniques are also required to overcome
this limitation and expand the scanner bandwidth.
A. Active Damping of X-scanner
The active damping of the x-scanner is easy because theirscan speed is not high, and their scan waves are known
beforehand and periodic, and therefore, feedforward control
for active damping can be implemented in a digital mode[18]-[20]. When the line scan was performed at 3.3 kHz
without damping, its displacement exhibited vibrations [Fig.
3(a)]. When it was driven with inverse compensation

Fig.4. Active Q-controller for the z-scanner. The mock z-scanners are
RLC circuits characterized with the transfer functions similar to those ofz- and z’-scanners.

For the active damping of the z-scanner, we cannot use
feedforward control in digital modes as its scan waves are
unpredictable and the scan speed with high-speed AFM is too
fast for digital driving signals to be calculated in real time.
The active Q-control is well known as an active damping
technique and has been often used to control the quality factor
of cantilevers [21]-[24]. When this control is applied to thez-scanner, its displacement or speed has to be detected.
However, it is difficult to do this. This problem is solved byusing a mock scanner characterized by the same transfer
function as the z-scanner (Fig. 4) [6]. Using this technique,Proceedings of the American Control Conference, art. no. 4586984, pp. 3194-3200 (2008)

we achieved a bandwidth of 150 kHz (Fig. 5) and a quality
factor of 0.5, which resulted in a response time of 1.1 μs.
However, the higher resonant vibrations are not damped well.
This is because the two main elemental resonators areconnected in parallel judging from the phase spectra [Fig.
5(b)].

~ = 1 / ⎛⎜ 1 − a − b ⎞⎟ω⎜Q ω Q ω Q ω ⎟
1 1
2 2⎠
⎝ 0 0

(8),

~ / ⎛⎜ a + b ⎞⎟ + 1
g = cω
⎜Q ω Q ω ⎟
1 1⎠
⎝ 2 2

(9),

Where c is given byω2
1
1
c = 2 + 2 − 20 2
ω1
ω2
ω1 ω2
⎡ ω0
1
ωω
1 ⎛ aω1 bω2 ⎞⎤⎜⎟⎥+− 1 2 −+⎢
ω0ω1ω2 ⎣⎢ Q1Q2
ω0
Q0 ⎜⎝ Q2
Q1 ⎟⎠⎦⎥(10),

The effect of this controller H(s) is depicted in Fig. 6.

Fig. 5. Frequency spectra of mechanical response of the z-scanner. (a)Gain spectra, (b) phase spectra. The gray-line curves represent the
response without feedback Q-control, and the black-line curves represent
the response with feedback Q-control.

When the elemental resonators are connected in series,
we can use multiple Q-controllers connected in series, each ofwhich contain a mock scanner characterized by the transfer
function representing one of the elemental resonators.
However, when they are connected in parallel, active
damping becomes more difficult. Although we have not yet
examined experimentally, there is a solution to this problem.
Here, we consider only the case where two elemental
resonators are connected in parallel. The feedback
Q-controller H(s) is expressed as
H (s ) ==1
1−
G1 (s ) + G2 (s ) R (s )
1
2

2

b ω2
a ω1
+
ω
ω
s 2 + 1 s + ω1 2 s 2 + 2 s + ω2 2
Q2
Q1

⎛ s2
⎞
s
−⎜ 2 +
+ 1⎟
⎜ω
⎟
Q
ω
0 0
⎝ 0
⎠

(5),

where a and b are the weight factors of two resonators (hence,
a+b = 1). To eliminate the highest order term from H(s), ω02
should equal aω12+bω22, which approximately results in~ (6).
H(s) ~ [(band-pass filter) × g – 1] × s/ ω

~ , and the gain parameter g are
The band-pass filter BP(s), ω
respectively expressed as
⎛ a
b ⎞
⎜⎜
⎟s
+
Q2 ω2 Q1ω1 ⎟⎠⎝(7),
BP (s ) =
⎛ a
ω0 2
b ⎞
2
⎜
⎟
+
+
s
s
+
1
⎜Q ω
⎟
ω12 ω2 2⎝ 2 2 Q1ω1 ⎠Fig.6. Effect of feedback control for active damping on the resonant
system that consists of two elemental resonators connected in parallel.
This effect was theoretically calculated. (a) Bode plots of the resonant
system before damping. (b) Bode plots of the resonant system after
damping.

C. Feedforward Type of Active Damping for Z-scannerIn the feedforward type of active damping, the z-scanner
is driven through a circuit with a transfer function 1/G(s),where G(s) is the transfer function of the z-scanner. However,for a complicated G(s), it is very difficult to design an electric
circuit characterized by 1/G(s). We invented a circuit that can
automatically produce the inverse transfer function (APTIF)
for a given transfer function. [8, 25]. The transfer function
K(s) of the circuit shown in Fig.7(a) is given by
K (s ) = 1 / [1 + g (M (s ) − 1)] ; here, M(s) represents a mockscanner. In the case of g = 1, a complete inverse transfer
function, 1/M(s), can be realized. However, delays occur inthe electronic components such as the operational amplifiers
used in the circuit. Therefore, the gain factor g must be lessthan 1, and consequently, the complete inverse transfer
function cannot be realized. In fact, K(s)M(s) has a large
resonant peak at a frequency higher than the peak frequenciesof M(s). However, this peak can be eliminated by a Notch
filter. When the disagreement between K(s) and 1/M(s) is
significant, we can reduce the difference by placing a phasecompensator (1 + differential operation) immediately before
or after the mock scanner and by using a multiple-loop circuit,as indicated in Fig.7(b). The difference is more reduced byincreasing the number of the nested loops. We examined theeffectiveness of the APTIF method by applying it to aProceedings of the American Control Conference, art. no. 4586984, pp. 3194-3200 (2008)

Fig. 7. Circuit diagram for APITF. G(s) represents a given transfer
function for a real resonant system. M(s) represents a transfer function ofa mock resonator for G(s). (a) Single-loop circuit type. K(s)approximately gives 1/M(s). (b) Nested circuits for improving theapproximation.

z-scanner under development. In this z-scanner, a single
piezoelectric actuator is used, and is only held at the rims andcorners of a plane perpendicular to the displacement direction.
The piezoelectric actuator used is similar to that used for thescanner mentioned above. The z-scanner exhibited large
resonant peaks at 370 kHz and 540 kHz, as indicated with ablack-line curve in Fig. 8(a). The resonant frequency of 370
kHz is similar to that of the free oscillation of thepiezoelectric actuator (410 kHz). Judging from the phase
spectrum [the black-line curve in Fig. 8(b)], the two
resonators are connected in parallel. Thus, we applied
feedforward active damping using a double-loop APITF
circuit containing a mock scanner composed of two LRC
circuits connected in parallel. The resulting gain and phase
spectra are shown with gray-line curves in Fig. 8(a) and 8(b),respectively. The peak at 370 kHz was completely removed
and the frequency that gave a 90° phase delay was increased
from 370 kHz to 500 kHz.

signal (difference between the cantilever’s peak-to-peakoscillation amplitude Ap-p and the set point As) saturates at(2A0 – As) = 2A0(1 – r). The saturated error signal becomes
smaller as r increases, and hence, the parachuting time (τp) is
prolonged. The feedback gain cannot be increased to shorten
τp, since a large gain induces overshoot at uphill regions of
the sample, resulting in the instability of the feedback
operation. To solve this problem, we devised a new PID
controller named “dynamic PID controller” in which the gain
parameters are automatically altered depending on theoscillation amplitude [7]. A threshold level Aupper is set
between As and 2Ao (or is set to As). We usually set a value ofAupper that is much closer to As than 2A0. When Ap-p exceeds
Aupper, the gain parameters are increased. This method
significantly enhances the feedback bandwidth, particularly
when the set point r is close to 1 (see dotted-line curves in Fig.
9). The feedback bandwidth is independent of r, provided r is
less than ~0.9, indicating that parachuting does not occur.
The superiority of the dynamic PID control is also clear from
captured images [Fig. 10(b) and 10(c)] of a mock sample withsteep slopes [Fig. 10(a)].

Fig. 9. Feedback bandwidth as a function of the set point r. Solid-line
curves: feedback bandwidths measured with a conventional PID
controller; dotted-line curves: feedback bandwidths measured with the
dynamic PID controller. These curves are aligned from top to bottom
according to the ratio 2A0/h0 = 5, 2, 1, and 0.5.

V. DYNAMIC PID CONTROL

Fig. 10. Pseudo-AFM images of a sample with rectangles with two
different heights. (a) A mock AFM sample. The images were obtainedusing a conventional PID controller (b) or using the dynamic PID
controller (c). These simulations with the mock AFM system were madeunder the following conditions: the cantilever resonant frequency, 1.2
MHz; quality factor of the cantilever oscillation, 3; the resonant
frequency of the z scanner, 150 kHz; and quality factor of the z scanner,
0.5; the line scan speed, 1 mm/ s; the line scan frequency, 1 kHz; theframe rate, 100 ms/ frame, the ratio of 2A0 to the total sample height = 1;
and r = 1.

To observe biological processes containing delicate
protein-protein interactions, the tip-sample interaction force
has to be minimized. The tapping force exerted on the sample
by the oscillating cantilever tip can be reduced by using anamplitude set point r close to 1. However, this set pointcondition induces “parachuting”, during which the error

Our most recent high-speed AFM can capture an image in
30-50 ms/frame, without significantly disturbing weakprotein-protein interactions. Some biological processes weresuccessfully captured on video. For example, the dynamic

Fig. 8. Frequency spectra of mechanical response of the z-scanner. (a) Gain
spectra, (b) phase spectra. Black-line curves represent the responses without
feedforward active damping, and gray-line curves represents the responses
with feedforward active damping.

VI. PRESENT CAPACITY AND LIMITATIONSProceedings of the American Control Conference, art. no. 4586984, pp. 3194-3200 (2008)

interaction process of chaperonin GroEL-GroES, which isregulated by the ATPase reaction, was clearly visualized. The
alternate association and dissociation of GroES at the tworings of GroEL were evident. Surprisingly, before thealternate switching took place, a GroES-GroEL-GroES
complex often appeared whose existence had been acontroversial issue for a long time. As demonstrated by thisobservation, high-speed AFM imaging can quickly answer tobiological questions.
To expand the scope of biological samples that can bestudied by high-speed AFM, the imaging rate must be
enhanced further and the tip-sample interaction force must be
reduced significantly. Presently, the cantilever is the factor
that prevents the achievement of both of these improvements.
Generally, to improve the resonant frequency, one must
compromise the stiffness and vice versa. The most advanced
small cantilevers developed by Olympus appear to havealmost achieved the ultimate goal of balancing these two
mechanical quantities. Considering their practical use,
doubling the resonant frequency to ~2.4 MHz in water seems
to be the upper resonant frequency limit, provided thecantilever compliance is not sacrificed.
VII. FURTHER REDUCTION OF TAPPING FORCE
Reduction of the tapping force must be achieved byincreasing the detection sensitivity of the tip-sample
interaction. The nonlinear impulsive tip-sample interaction
induces higher-harmonic vibrations of the cantilever. In theamplitude detection, these vibrations are neglected. Since theimpulsive force is exerted transiently in a short time, its peakforce is relatively large. This means that the peak force must
be a highly sensitive quantity. The force F(t) cannot be
detected directly because the cantilever oscillation gain islower at higher harmonic frequencies. F(t) can be calculated
from the cantilever’s oscillation wave z(t) by substituting z(t)into the equation of cantilever motion and then subtracting
the excitation signal (i.e., inverse determination problem)[26]-[27]. Figure 11(b) shows force signals F(t) that wereobtained by off-line calculation using an oscillation signal ofa small cantilever weakly interacting with a mica surface inwater. Since the time width of the impulsive force is narrow,
it appears to be difficult to capture the peak force using asample/hold circuit. Instead of capturing the peak force, we
can calculate it from the time when the cantilever oscillation
reaches the end of the bottom swing.

Fig. 11. Force signal (b) calculated from the oscillation signal (a) of a
cantilever interacting with a mica surface in water.

Our ultimate goal is the realization of noncontact
high-speed AFM in which we can use a much stiffer
cantilever with a significantly high resonant frequency.
Therefore, an ultra-fast scan speed must be achieved without
damaging the sample. The discussion on this issue ispresented in [28]-[29].
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