International Conference on Control, Automation and Systems 2008 Oct. 14-17, 2008 in COEX, Seoul, Korea

Design and Control of Tapping Mode Atomic Force Microscope in Liquid Utilizing Optical Pickup System Wan-Lin Hu1 , Shao-Kang Hung2 and Li-Chen Fu1 1

Electrical Engineering Department, National Taiwan University, 10617 Taipei, Taiwan, R. O. C. (Tel : +886-2-2363-5251#236; E-mail: [email protected]) 2 Institute of Physics, Academia Sinica, 11529 Taipei, Taiwan, R. O. C. (Tel : +886-2-2788-0058#4022; E-mail: [email protected])

Abstract: In this work, we propose a fluid tapping mode atomic force microscopy (AFM) implemented by a DVD pickup head. The use of DVD pickup head minimizes the volume of the hardware system, and thus reduces the measurement error caused by heat expansion. In order to realize the system mentioned above, we design a Q-controller to modulate the interaction force between the tip and the sample. Increasing the quality factor will overcome the problem with high damping ratio in the fluid which makes the probe hard to oscillate. Because of the reduction of the tip-sample force, the sample surface will not be hurt by the tip. Therefore, we can use the AFM to scan soft sample, and obtain more realistic topography. Traditionally, people use proportion-integration controller to control the system. Users need to tune this kind of controller manually, and hence the quality of the scan images is highly related to users’ experiences. To overcome this problem, we design an adaptive sliding-mode controller to improve the scanning capability and robustness.For testing the system capability, we will have a series of numerical simulations. Keywords: atomic force microscope (AFM), tapping mode, Q-control, adaptive sliding-mode control

1. INTRODUCTION

loop to change the damping coefficient of the whole system. Adding a feedback loop is a more theoretical and flexible method, which is referred to Q-control. In this work, a novel tapping mode AFM utilizing CD/DVD pickup head [6][7][8][9][10] operation in liquid is developed. Using an optical pickup system could reduce the complex light path system, lessen the volume of the whole system, and hence minimize the sensing error induced from temperature change. This work makes the AFM mentioned above more widely applied, such as in life science field. In order to obtain more real topography and reduce the hysteresis effect, we apply Q-control to the system. An adaptive controller is also designed for the z-axis motion, thus control parameters can be automatically adjusted to handle unknown samples. There are four sections in this paper. Section describes the system design from the view of hardware and software. Then in section , important control issues containing Q-control and adaptive sliding-mode control are discussed. Numerical simulation is also proposed in this chapter. In the final section, we will sum up achievements in this thesis.

Since the scanning tunneling microscope (STM) was in-vented in 1981 [1], the fields of application of this new method have expanded rapidly. However, STM is only suitable for samples of conductive surface, and it is proposed as a method to measure forces as small as 10−18 N. For this reason, atomic force microscope (AFM) was proposed [2]. As the name described, AFM will be used on an atomic scale and can investigate both conductors and insulators. AFM systems can be employed in a broad spectrum of applications. It has been widely used in, for example, surface physics, semiconductor industry, biomedical science, etc. Several operation modes are used in AFM, among them, contact mode and tapping mode [3] are most commonly used. In the contact mode AFM, the cantilever with tip is dragged across the surface along each scan line. The slope at the cantilever free-end is measured as a feedback and then sent to the control system. On the other hand, in the constant frequency tapping mode AFM, the cantilever is oscillated vertically near its resonance frequency, so that the tip makes contact with the sample surface only briefly in each cycle of oscillation. One of the benefits of tapping mode AFM is that the sample accepts small lateral force from the tip. This advantage makes this kind of AFM suitable for biological application. If we want to obtain the image of a living cell (or other bio-sample), culture the sample in its native surrounding is necessary. A tapping mode atomic force microscopy operation in liquid was proposed by P. K. Hansma et al. and C. A. J. Putman et al. [4][5]. However, the oscillating cantilever will serve high damping force in the liquid. To solve this problem, we can apply higher energy to the cantilever beam or add a feedback

2. SYSTEM DESIGN 2.1 Hardware design The experimental setup of an AFM driven in the amplitude modulation mode (tapping mode) is shown in Figure 1. The cantilever is driven at a fixed frequency by a constant sinusoidal signal originating from a function generator, and the resulting oscillating amplitude is detected by a lock-in amplifier. The PC-based controller we designed will use the signal to adjust the z-scanner. The scanning range is 35μm × 35μm × 9μm in x-direction, y-direction, and z-direction, respectively.

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(a)

(c)

Fig. 1 (color online) Schematic drawing of the experimental setup of the AFM system using the constant excitation mode. A PC-based controller is used to generate the xy-trajectory and to reduce the tracking error in z-axis. The red part is inserted for Q control. This experimental setup exhibits an additional feedback loop containing an amplifier and a phase shifter, which are shown in red.

(a)

(b)

Fig. 3 Scanning electron microscopy images of (a) the probe tip (b) tip apex used in this work. The nominal thickness, mean width, and length of the cantilever are 4 μm, 30 μm, and 125 μm, respectively. (c) is the picture of the probe.

(b)

Fig. 2 (a) The front view, and (b) The right side view of the fine tuning mechanism.

Fig. 4 Photographic picture of the main parts of the AFM system.

The kernel elements of the designed AFM system is the pickup head, which is used to measure the cantilever deflection. The relative positions of different components may change due to temperature or other factors, but the laser beam should focus on the tip exactly. Besides, the probe may wear away by scanning. Hence, a fine tuning mechanism (see Figure 2), which is made by wire cut electrical discharge machining (WEDM), is designed to solve this problem. It contains a flexure structure, a high precision adjustment screws (AJS8-100-02H, Newport), a piezoelectric element (bimorph), a magnet, and a magnetic mount. The piezoelectric element is to oscillate the cantilever in tapping mode operation; the magnetic mount is to fix the probe; the remaining parts are to ensure the ability to reposition the probe whenever the probe is exchanged. The cantilever reveals high Q-value property in the frequency response, so that a slight shift in the driving frequency may induce significant decrease in oscillation amplitude. To prevent such problem, a function generator based on direct digital synthesized (DDS) technology is used in the system. During the calibration process, the FE signal is monitored to confirm the exact vertical position. The probe is placed between the DVD pickup head and the sample. The choice of probe depends on the sam-

ple property and the operation mode of AFM. Here, the probe we choose is NANOSENSORS PointProbe Plus RT–NCHR, which is designed for the non-contact mode and tapping mode AFM (see Figure 3). The resonant frequency of the probe is ranging from 204 kHz to 497 kHz. Due to the manufacturing deviation, every probe has its own spectrum. According to the sweep-sine identification experiment, the resonant frequency of the probe used in this research is found to be 313.4 kHz. The photographic picture of the main parts of the implemented AFM system. 2.2 Software design The Real-Time Windows Target Version 2.5.1 of Matlab Simulink (Mathworks Inc.) is a tool to develop the image scanning program and the controller algorithm. It is a PC solution for real-time prototyping and testing. After creating a model and compiling it, the Simulink model will be transferred into executable C/C++ code. All the control tasks are implemented on this platform in real time.

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3. CONTROLLER DESIGN

Now, we include the tip-sample interaction into our model. The tip experiences long-range attractive forces described by a van der Waals term and a short-range repulsive forces arising from Pauli principle. The longrange attraction is shown as follows:

3.1 Q-controller design When AFM scan operates in liquids, the high drag force applied to the tip needs to be considered. As previous researches have shown [11][12], the model is simplified by assuming tip as a sphere with radius R and sample as a flat surface. Considering the drag force on a smooth sphere immersed in a uniform flow, the drag force depends on the sphere diameter (D) and velocity (V ), the fluid density (ρ), and fluid viscosity (μ). To identify the drag coefficient in such condition, the dimensionless parameter Reynolds number defined in (1) needs to be discussed. ρV D (1) Re = μ

FvdW (z) = −

where d is the tip-sample distance and a0 is the intermolecular distance. The effective module E ∗ can be calculated from the following equation: (1 − νt2 ) (1 − νs2 ) 1 = + ∗ E Et Es

(2)

The dynamics of the probe can be modeled as a second order system. In such 2nd order system, the Q factor represents effects of simplified viscous damping or drag. Define the damping coefficient b by the following equation, ˙ FD = bz(t)

(8)

where νt and νs are the Poisson ratios of the tip and sample; Et and Es are the Young’s modulus constants of the tip and sample, respectively; E ∗ is the effective value of the Young’s modulus constant between the tip and the sample. The mathematical form of tip-sample forces is a highly nonlinear term. In order to analyze the tapping mode AFM experiment, we need to focus on the steady-state solution derived from eq. (5). For further analysis, we expand the tip-sample force into the Fourier series. The first term of this Fourier series reflects the averaged tip-sample force over one full oscillation cycle. To calculate the responsible amplitude A and phase φ of a tapping mode AFM including tip-sample force and Q-control, the first harmonic term of the Fourier series is inserted into the equation (5). Using the following definitions,  2fd 1/fd Fts [z(t), z(t)] ˙ cos(2πfd t + π)dt Iodd (zc , z) = kc z 0  2π 1 = Fts [z(τ ), z(τ ˙ )] cos(τ )dτ (9) πkc z 0

(3)

where z(t) is the tip position at time t. The formula for the Q factor is √ 2πf0 m mkc = (4) Q= b b  where m, kc and f0 = kc /m/2π are the effective mass, the spring constant, and the eigenfrequency of the cantilever, respectively. The concept of Q-control is to add a feedback loop to the system such that the Q factor and the damping force acting on the cantilever by the surrounding medium can be changed. The signal generated from the Q-control circuit is the retarded amplification of the displacement, i.e. the tip position z is measured at the retarded time t − t0 and amplified by a gain factor g [13][14]. Substituting the drag force term with the Q factor, the dynamic equation can be expressed as eq. (5). 2πf0 m z(t) ˙ + kc z(t) + gkc z(t − t0 ) m¨ z (t) + Q ˙ = ad kc cos(2πfd t) + Fts [z, z]

(6)

where H is the Hamaker constant, and zc is tip-sample distance when the tip is at equilibrium position. The negative sign stands for attractive force. As the tip approaches to the sample, the ionic repulsion will occur. The attractive adhesion and the repulsive force can be combined using the Derjaguin–Muller–Toporov (DMT) theory.  for d ≥ a0 FvdW√(z) (7) Fts (z) = 4 ∗ 3/2 E R(a − d) + F (a ) for d < a0 0 vdW 0 3

Using the parameters of water and the probe introduced above, the resulting Reynolds number is 6.28510−3 which is far less than 1, which means inertia forces are small compared to viscous forces. Therefore, the flows is characteristically laminar flow. The drag force FD can be defined as follows: FD = 3πμV D

HR 6(zc + z)2

Ieven (zc , z) (5)

where ad is the constant excitation amplitude at a fixed frequency fd . Fts stands for the (nonlinear) tip-sample interaction force. The relationship between time shift t0 and the phase difference θ0 shown in Figure 5 is θ0 = 2πfd t0 . The control goal is to use the active feedback term gkc z(t − t0 ) to compensate the damping force (2πf0 m/Q).



=

2fd kc z

=

1 πkc z

1/fd

0

 0

2π

Fts [z(t), z(t)] ˙ sin(2πfd t + π)dt Fts [z(τ ), z(τ ˙ )] sin(τ )dτ

and set fd2 + g cos(2πfd t0 ) − Iodd (zc , z) f02

(11)

1 fd − g sin(2πfd t0 ) + Ieven (zc , z) Q0 f0

(12)

α1 = 1 − α2 =

785

(10)

The results are ad A=  2 α1 + α22 tan φ =

α2 α1

ˆ0 , and ˆb are estimated plant parameters; w ˆc is the esa ˆ1 , a timate of the constant uncertainty, and sat(•) is the saturation function defined as follows: ⎧ if s > ε ⎨ 1 s if −ε ≤ s ≤ ε (24) sat(s) = ⎩ ε −1 if s < −ε

(13) (14)

which describe the shape of frequency response. The controlled gain factor and time shift will influence in both amplitude and phase.

Substituting eq. (22) into eq. (18), we have e¨ = h¨d + a1 h˙ + a0 h − buas + wc + wv = a˜1 h˙ + a˜0 h + w˜c + wv − ˜buas − λe˙ − κs − ηsat(s)

3.2 Adaptive sliding-mode controller design Using the dynamics of the plant: ¨ + a1 h˙ + a0 h = bu, h

(15)

(25)

where the estimation errors are defined as = a1 − aˆ1 = a0 − aˆ0 = b − ˆb = wc − wˆC

we assume that hd is a desired constant height between tip and sample, and h is the shrinking displacement contributed by applied voltage. The control goal is to change h to maintain the tip-sample distance in a desired distance hd . The tracking error can be defined as

a˜1 a˜0 ˜b w˜c

e ≡ hd − h

Applying appropriate gains λ, κ, and η can accelerate the convergence and force the state to zero in shorter time. Stability Analysis In the following stability analysis, an adaptation law has to be proposed in order to compensate the effect of estimation errors. Based on the adaptive control theory, a positive definite Lyapunov function candidate V is defined as: 1 1 1 1 1 V = s2 + a˜1 2 τ1−1 + a˜0 2 τ0−1 + ˜b2 τ2−1 + w˜c 2 τ3−1 (27) 2 2 2 2 2 where τ0 , τ1 , τ2 , and τ3 are positive constants. By differentiating the Lyapunov function candidate V , we obtain

(16)

Substituting eq. (15) into eq. (16), we have ¨ e¨ ≡ h¨d − h ¨ = hd + a1 h˙ + a0 h − bu

(17)

Because of the simplification during the modeling process, two additional disturbance terms need to be added into eq. (17) so that the system can be expanded as e¨ = h¨d + a1 h˙ + a0 h + bu + wc + wv

(18)

where wc and wv represent a constant and varying uncertainty, respectively. The varying uncertainty term is assumed to be bounded and satisfies the inequality wv  ≤ wmax , where wmax is a constant. A first order sliding surface s is chosen as s = e˙ + λe

˙ V˙ = ss˙ + a˜1 a˜˙1 τ1−1 + a˜0 a˜˙0 τ0−1 + ˜b˜bτ2−1 + w˜c w˜˙c τ3−1 (28) Substituting eqs. (19) and (25) into eq. (28), a more detailed form of V˙ can be derived as: ˙ V˙ = s(¨ e + λe) ˙ + a˜1 a˜˙1 τ1−1 + a˜0 a˜˙0 τ0−1 + ˜b˜bτ2−1 + w˜c w˜˙c τ3−1 = s(a˜1 h˙ + a˜0 h + w˜c + wv − ˜buas − λe˙ − κs − ηsat(s) + λe) ˙

(19)

where λ is a the positive parameter to be designed. The control goal is to make the state of the system reach the sliding surface above, or in the case of bounded tracking some regions around the surface. Under such condition, states are not affected by uncertainties; hence the error converges to zero. Take the time derivative of eq. (19) and substitute eq. (18) into that, we have s˙ = (h¨d + a1 h˙ + a0 h + bu + wc + wv ) + λe˙

˙ −1 + w˜ ˙ τ −1 + a˜1 a˜˙1 τ1−1 + a˜0 a˜˙0 τ0−1 + ˜b˜bτ ˜c 3 cw 2 −1 2 ˙ ˙ = −s[ηsat(s) − wv ] − κs + a˜1 (a˜1 τ + sh) 1

+

a˜0 (a˜˙0 τ0−1

˙ + sh) + ˜b(˜bτ2−1 + suas ) + w˜c (w˜˙c τ3−1 + s)

To ensure ultimately bounded stability, the latent purpose is to guarantee V˙ ≤ 0. As a result, the adaptation law is designed as ⎧ ⎪ aˆ˙1 = −a˜˙1 = τ1 sh˙ − τ1 σ1 aˆ1 ⎪ ⎪ ⎨ aˆ˙ = −a˜˙ = τ sh − τ σ aˆ 0 0 0 0 0 0 (30) ˙ ˙ ˆ ˜ ⎪ b = −b = τ2 suas − τ2 σ2ˆb ⎪ ⎪ ⎩ ˙ wˆc = −w˜˙c = τ3 s − τ3 σ3 wˆc

(20)

The control law uas is designed here, such that the sliding surface will meet the sliding mode condition. Following this principle, eq. 20 becomes: ˙ aˆ0 h− ˆbuas + wˆc )+λe(21) ˙ −κs−ηsat(s) = (h¨d + aˆ1 h+ ˙ aˆ0 h+ h¨d + wˆc +λe+κs+ηsat(s))(22) uas = ˆb−1 (aˆ1 h+ ˙

such that ⎧ ⎪ a˜˙1 τ1−1 + sh˙ ⎪ ⎪ ⎨ a˜˙ τ −1 + sh 0 0 ˙ ˜ ⎪ bτ2−1 + suas ⎪ ⎪ ⎩ ˙ −1 w˜c τ3 + s

where κ and η are positive constants to be designed, η is a high gain constant used to bound the varying uncertainty wv , such that η ≥ wmax ,

(26)

(23)

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= σ1 aˆ1 = σ0 aˆ0 = σ2ˆb

= σ3 wˆc

(31)

(29)

Substituting eq. (30) into eq. (29), V˙ can be rewritten as follows: V˙ = −s[ηsat(s) − wv ] − κs2 + σ1 a˜1 aˆ1 + σ0 a˜0 aˆ0 + σ2˜bˆb + σ3 w˜c wˆc

(32)

From eq. (32), the range of each term is derived from σ1 a˜1 aˆ1

=

σ1 a˜1 (a1 − a˜1 )

=

−σ1 |a˜1 | + σ1 a˜1 a1

≤

 1 σ1 a21 + σ0 a20 + σ2 b2 + σ3 wc2 2   2 wmax ε w2 ε η |s| − = − + max − αV ε 2 η 4η  1 σ1 a21 + σ0 a20 + σ2 b2 + σ3 wc2 + 2  1 σ1 a21 + σ0 a20 + σ2 b2 + σ3 wc2 ≤ −αV + 2 2 wmax ε (38) + 4η

+

2 2

−σ1 |a˜1 | + σ1 |a˜1 ||a1 |

1 1 1 2 2 2 = −σ1 (|a˜1 | − |a1 |) + |a˜1 | − |a1 | 2 2 2 σ1  2 2 |a˜1 | − |a1 | (33) ≤ − 2 The following inequalities can be calculated in a similar way as eq. (33), : σ0  2 2 |a˜0 | − |a0 | σ0 a˜0 aˆ0 ≤ − 2 σ2  ˜ 2 2 |b| − |b| σ2 a˜2 aˆ2 ≤ − 2 σ3  2 2 |w˜c | − |wc | (34) σ3 a˜3 aˆ3 ≤ − 2 Substituting inequalities eqs. (33) and (34) into eq. (32), the relation can be rewritten as V˙ ≤ −s[ηsat(s) − wv ] − κs2 σ  σ1  0 2 2 2 2 |a˜1 | − |a1 | − |a˜0 | − |a0 | − 2 σ 2 σ2  ˜ 2 3 2 2 2 |b| − |b| − |w˜c | − |wc | (35) − 2 2 If 0 < α < min{2κ, σ0 , σ1 , σ2 , σ3 } is chosen, then V˙ ≤ −s[ηsat(s) − wv ] − αV  1 σ1 a21 + σ0 a20 + σ2 b2 + σ3 wc2 (36) + 2 Because the saturation function is used in the designed controller, to derive the upper bound of V˙ , the relationship between sliding surface s and boundary width ε needs to be discussed. Case 1. |s| > ε V˙ ≤ −s[ηsat(s) − wv ] − αV  1 σ1 a21 + σ0 a20 + σ2 b2 + σ3 wc2 + 2 = −η|s| + swv − αV  1 σ1 a21 + σ0 a20 + σ2 b2 + σ3 wc2 + 2 = −|s|(η − wmax ) − αV  1 σ1 a21 + σ0 a20 + σ2 b2 + σ3 wc2 + 2  1 σ1 a21 + σ0 a20 + σ2 b2 + σ3 wc2 (37) ≤ −αV + 2 Case 2. |s| ≤ ε ˙ V ≤ −s[ηsat(s) − wv ] − αV  1 σ1 a21 + σ0 a20 + σ2 b2 + σ3 wc2 + 2 η ≤ − |s|2 + |s|wmax − αV ε

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From the above discussions in Case 2, we can

12 and Case  1 σ1 a1 + σ0 a20 + σ2 b2 + σ3 wc2 + concluded that for V ≥V0 = 2α wmax2 ε ˙ , V ≤ 0. Therefore V and V˙ ∈ L∞ . According 4η to the control theory, the tracking error will converge to a residual set in order of σi , i = 0–3 and ε [15]. Therefore, the plant with uncertainties and bounded disturbance can be controlled by the adaptive sliding-mode controller designed as above stably. 3.3 Numerical simulation In this section, numerical simulations are performed and analyzed with Matlab Simulink. To simulate the adaptive sliding-mode controller, parameters of the plant are listed as following: a0 = 1.02 × 108 , a1 = 5.25 × 102 , b = 2.38; Parameters of the adaptive sliding-mode controller are: λ = 3640, κ = 1600, η = 1, τ0 = 10, τ1 = 5, τ2 = 500, τ3 = 50. The sample used in further experiments is a standard grating. The height of the step is 200 nm, the width of the step is 5 μm, and the distance between two steps is also 5 μm. The scanning speed is 50 m/s. Parameters of sliding-mode control should be designed to minimize the chattering phenomenon. The sliding-mode controller can reduce the tracking error quickly. On the other hand, the adaptive law enables the controller to be equipped with self-correcting capability that may improve performance significantly. As seen in Figure 5, the controlled z position approaches the desired value, and after 1.4 seconds the error is suppressed within 2 nm. The frequency response can give an overview of the oscillation properties of the system. The resonance curve of the cantilever of the AFM probe will be changed in shape while changing the Q factor. As shown in Figure 6, with the phase shift θ0 set to 90◦ and the effective Q factor, Qef f , increased to 300, the resonance peak of the amplitude curve is significantly enhanced.

4. CONCLUSIONS In this work, a tapping mode atomic force microscopy utilizing a CD/DVD pickup head operating in liquid has been proposed. To overcome the great drag force affecting on the cantilever beam applied by the fluid, we analyze the characteristics of the fluid flow, and deduce the effect of Q factor from the drag force. A model-based adaptive sliding-mode controller has been designed for z-scanner to reduce the tracking error. In numerical simulation, the maximum transient tracking error is less than

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(a)

(b)

Fig. 5 (color online) Numerical simulation of z-scanner tracking a standard grating. The reference (blue line) step height is 200 nm. (a) is the simulation result and (b) is the enlargement of (a).

Fig. 6 (color online) Amplitude vs. frequency curve with Q control. The eigenfrequency of the cantilever was assumed to be f0 = 330 kHz. Using the additional gain g = 0.5/Q, and the phase shift θ0 = 90◦ , the effective Q factor of the system increased from 30 (blue line) to 300 (green line) 3 nm, and the steady state error is less than 2 nm. This work opens a new way to study the biological material in their native environment. This novel mechanical and controller design have the ability to study changes in local surface properties with nano-scale spatial resolution.

ACKNOWLEDGMENT This work was supported by Taiwans National Science Council, NSC 96-2218-E-002-033. These supports are gratefully acknowledged.

REFERENCES [1]

G. Binnig, H. Rohrer, C. Gerber, and E. Weibel. Surface studies by scanning tunneling microscopy.

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