Encyclopedia of Systems and Control

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Dynamics and Control of Active Microcantilevers

  • Michael G. RuppertEmail author
  • S. O. Reza Moheimani
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DOI: https://doi.org/10.1007/978-1-4471-5102-9_184-2
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Abstract

The microcantilever is a key precision mechatronic component of many technologies for characterization and manipulation of matter at the nanoscale, particularly in the atomic force microscope. When a cantilever is operated in a regime that requires the direct excitation and measurement of its resonance frequencies, appropriate instrumentation and control is crucial for high-performance operation. In this entry, we discuss integrated cantilever actuation and present the cantilever transfer function model and its properties. As a result of using these active cantilevers, the ability to control the quality factor in order to manipulate the cantilever tracking bandwidth is demonstrated.

Keywords

Atomic force microscopy Flexible structures Piezoelectric actuators and sensors Vibration control Q control Negative imaginary systems 

Introduction

The microcantilever is a precision mechatronic device and an integral component of a myriad of systems which have collectively enabled the steady progress in nanotechnology over the past three decades. It is a key component and sits at the heart of a variety of instruments for manipulation and interrogation at the nanoscale, such as atomic force microscopes, scanning probe lithography systems, high-resolution mass sensors, and probe-based data storage systems. Despite its apparent simplicity, the fast dynamics, particularly when interacting with nonlinear forces arising at the micro- and nanoscale, have motivated significant research on the design, identification, instrumentation, and control of these devices.

One of the most important applications of the microcantilever is in atomic force microscopy (AFM) Binnig et al. (1986). In dynamic AFM, the cantilever is actively driven at its fundamental resonance frequency while a sample is scanned underneath a sharp tip located at its extremity. As the tip interacts with the sample, changes occur in the oscillation amplitude, phase, and/or frequency which relate to surface features and material properties. By controlling the vertical interaction between the tip and the sample with a feedback controller, 3D images with atomic resolution can be obtained Giessibl (1995).

Examples of cantilevers used for AFM are shown in Fig. 1. The most conventional approach is to use a passive rectangular cantilever with an extremely sharp tip at the end etched from single-crystal silicon as shown in Fig. 1a. In order to oscillate the cantilever, a piezo-acoustic bulk actuator is typically used which is clamped to the chip body Bhushan (2010). While this approach is simple and effective, it leads to highly distorted frequency responses with numerous structural modes of the mounting system appearing in the frequency response as can be seen in Fig. 1c. This renders the identification and subsequent analysis of the actual cantilever dynamics exceedingly difficult and model-based controller design nearly impossible.
Fig. 1

(a) Image of a commercially available passive AFM microcantilever (Asylum Research High Quality AFM probe, image courtesy of Oxford Instruments Asylum Research, Santa Barbara, CA) and sharp tip (User:Materialscientist, Wikimedia Commons, CC BY-SA 3.0). (b) Image of an AFM cantilever with integrated piezoelectric transducer (Bruker, DMASP) for integrated actuation/sensing. (c) Frequency response measured with the AFM optical beam deflection sensor of a standard base-excited passive cantilever. (d) Frequency response and estimated transfer function model (1) of a cantilever with integrated piezoelectric actuation

In contrast, active cantilevers with integrated actuation and sensing on the chip level provide several distinct advantages over conventional cantilever instrumentation Rangelow et al. (2017). Most importantly, these include clean frequency responses, the possibility of down-scaling, parallelization to cantilever arrays, as well as the absence of optical interferences. While a number of integrated actuation methods have been developed over the years to replace the piezo-acoustic excitation, only electro-thermal and piezoelectric actuation can be directly integrated on the cantilever chip. An example of such a cantilever is shown in Fig. 1b which features an integrated piezoelectric transducer which can be used for actuation, deflection sensing, or both Ruppert and Moheimani (2013). As a consequence, this type of cantilever yields a clean frequency response, such as shown in Fig. 1d, for which frequency domain system identification yields very good models suitable for controller design.

Microcantilever Dynamics

Active microcantilevers allow the utilization of second-order transfer function models to accurately model their dynamics. Moreover, these models have favorable properties for robust vibration control which will be discussed in the following.

Cantilever Model

When a voltage is applied to the electrodes of the piezoelectric layer of a cantilever such as in Fig. 1b, the transfer function from actuator voltage V (s) to cantilever deflection D(s) of the first n flexural modes can be described by a sum of second-order modes Moheimani and Fleming (2006)
$$\displaystyle \begin{aligned} G_{dv}(s)&=\frac{D(s)}{V(s)}\\&=\sum_{i=1}^{n}\frac{\alpha_i\omega_i^2}{s^2+\frac{\omega_i}{Q_i}s+\omega_i^2}+d, \quad \alpha_i\in\mathbb{R}{} \end{aligned} $$
(1)
where each term is associated with a specific vibrational mode shape (shown in the inset of Fig. 1d), quality (Q) factor Qi, natural frequency ωi, and gain αi. The feedthrough term d takes into account the modeling error associated with limiting the sum to modes within the measurement bandwidth.
For the first eigenmode of the cantilever (i = 1), the poles of (1) are given by
$$\displaystyle \begin{aligned} p_{1,2}&=-\frac{\omega_1}{2Q_1}\pm j\omega_1\sqrt{1-\frac{1}{4Q_1^2}}. \end{aligned} $$
(2)
The real part of the eigenvalues characterizes the amplitude transient response time or cantilever tracking bandwidth as
$$\displaystyle \begin{aligned} f^c&=\frac{f_1}{2Q_1}. {} \end{aligned} $$
(3)

Example 1

For a standard tapping-mode microcantilever (e.g., Budget Sensors, TAP190-G) with a fundamental resonance frequency of f1 = 190 kHz and a natural Q factor of 400, the tracking bandwidth is only approximately fc = 238 Hz.

System Property

An important property of the cantilever transfer function for Q factor control (Q control) is its negative imaginary (NI) nature. In general, the transfer function G of a system is said to be NI if it is stable and satisfies the condition Petersen and Lanzon (2010)
$$\displaystyle \begin{aligned} j\bigl[G(j\omega)-G^*(j\omega)\bigr]\geq 0 \quad \quad \forall \omega>0. {} \end{aligned} $$
(4)

Moreover, G is strictly negative imaginary (SNI) if (4) holds with a strict inequality sign. It is straightforward to show that the first mode of Gdv(s) is NI since α1, ω1, Q1 > 0. If additionally, collocated actuators and sensors are available, this property can be extended to multiple modes of the cantilever Ruppert and Yong (2017). The benefit of this structural property results from the negative imaginary lemma which states that the positive feedback interconnection of an SNI plant and an NI controller is robustly stable if and only if the DC-gain condition G(0)K(0) < 1 is satisfied Lanzon and Petersen (2008). In other words, even in the case where the resonance frequency, modal gain, and/or damping of a cantilever mode change, as long as the NI property and DC-gain condition hold, the closed-loop system remains stable.

Microcantilever Control

In dynamic-mode AFM, the output of the vertical feedback (z-axis) controller resembles the surface topography only when the cantilever is oscillating in steady state. Therefore, a fast decaying transient response is highly desirable. Reduction of the cantilever Q factor using an additional feedback loop enables it to react faster to sample surface features. This, in turn, allows the gain of the regulating z-axis controller to be increased which improves the cantilever’s ability to track the surface topography.

Q Control Using Positive Position Feedback

The positive position feedback (PPF) controller of the form
$$\displaystyle \begin{aligned} K_{\mathrm{PPF}}(s)&=\sum_{i=1}^{m}\frac{\gamma_{c,i}}{s^2+2\zeta_{c,i}\omega_{c,i}s+\omega_{c,i}^2}, {}\end{aligned} $$
(5)
with γc,i, ζc,i, and ωc,i being the tunable controller parameters is an example of a model-based controller which can be used to control the Q factor of m modes of the cantilever. From its structure, it can be verified that it fulfills the negative imaginary property if γc,i, ζc,i, ωc,i > 0, that is, robust stability is achieved if such a controller is used to dampen a negative imaginary mode of the cantilever.
The closed-loop system of the cantilever first mode (n = 1) (1) in positive feedback with the PPF controller (m = 1) (5) can be written as
$$\displaystyle \begin{aligned} \ddot{z}&+\underbrace{\begin{bmatrix}\frac{\omega_1}{Q_1} & 0 \\ 0 & 2\zeta_{c}\omega_{c} \end{bmatrix}}_{E=E^T}\dot{z}\\ &+ \underbrace{\begin{bmatrix}\omega_1^2 & -\gamma_c\alpha_1\omega_1^2 \\ -\gamma_c\alpha_1\omega_1^2 & \omega_{c}^2-\gamma_c d \gamma_c \end{bmatrix}}_{K=K^T}z=0. {}\end{aligned} $$
(6)
In order to show stability of the closed loop (6), E and K need to be positive definite. While E>0 is given due to the NI property of plant and controller, for K to be positive definite, (6) can be rewritten into an LMI in the variables γc and \(\omega _c^2\) using the Schur complement as
$$\displaystyle \begin{aligned} \begin{bmatrix}\omega_1^2 & -\gamma_c\alpha\omega_1^2 & 0 \\ -\gamma_c\alpha\omega_1^2 & \omega_{c}^2 & \gamma_c \\ 0 & \gamma_c & d^{-1}\end{bmatrix}&>0. {} \end{aligned} $$
(7)
Using the constraint (7), a feasible set of controller parameters can be calculated to initialize the controller design Boyd et al. (1994). This is demonstrated in the next subsection.

Controller Design

An effective approach to controller design is to perform a pole-placement optimization Ruppert and Moheimani (2016). This method places the real part of the closed-loop poles \(\Re (p_{i,\mathrm {cl}})\) at a desired location \(p_{i,\mathrm {d}}=\frac {\omega _1}{-2Q_1^*}\) by selecting a desired effective Q factor \(Q_1^*\) and finding a solution to the optimization problem
$$\displaystyle \begin{aligned} \begin{aligned} &\underset{\gamma_c, \zeta_c, \omega_c}{\text{\underline{min}}} & & J\bigl(\gamma_c, \zeta_c, \omega_c \bigr ) \\ &~~\text{\underline{s.t.}} & &\begin{bmatrix}\omega^2 & -\alpha\omega_1^2\gamma_c & 0 \\ -\gamma_c\alpha\omega_1^2 & \omega_{c}^2 & \gamma_c \\ 0 & \gamma_c & d^{-1}\end{bmatrix}>0. \end{aligned}\end{aligned} $$
(8)
Here, the cost function
$$\displaystyle \begin{aligned} J\bigl(\gamma_c, \zeta_c, \omega_c \bigr )&=\sum_{i=1}^{4}\bigl(p_{i,\mathrm{d}}-\Re(p_{i,\mathrm{cl}}) \bigr)^2\\ &\quad +\sum_{i=3}^{4}\left\lvert p_{i,\mathrm{cl}}-p_{i-2,\mathrm{cl}}\right\rvert^2\end{aligned} $$
(9)
penalizes the difference between the desired and actual pole locations and between poles that correspond to the same open-loop poles. Notice that the objective function is nonlinear in the optimization variables γc, ζc, and \(\omega _c^2\), leading to a non-convex optimization problem. However in practice, initialization of the optimization problem with a feasible set calculated from (7) results in good convergence.

Example 2

Figure 2 demonstrates the effectiveness of this approach for reducing and increasing the Q factor of the first mode. In Fig. 2a, the Q factor was reduced from 268 to 10 by placing the closed-loop poles deeper into the left-half plane, and in Fig. 2b the procedure shifts the closed-loop poles closer to the imaginary axis to yield a Q factor of 720. Increasing the Q factor may be desirable in applications where the cantilever is naturally heavily damped, for example, when the AFM cantilever is used for imaging in liquid.
Fig. 2

(a) Q factor reduction and (b) Q factor enhancement: Bode plot of the open-loop (−) and closed-loop system (−) and pole-zero map of desired pole location (+ ), open-loop (×,∘), and closed-loop (×,∘) system

Application to Atomic Force Microscopy

In order to demonstrate the improvement in cantilever tracking bandwidth, we report a high-speed tapping-mode AFM experiment in constant-height mode. In this mode, sample features entirely appear in the cantilever amplitude image and any imaging artifacts are due to insufficient cantilever bandwidth. Using the model-based Q controller presented in the previous section, the Q factor of the fundamental resonance of the piezoelectric cantilever is reduced to as low as \(Q_1^*=8\), resulting in a measured tracking bandwidth of 3.3 kHz, as shown in Fig. 3a, b. With the native Q factor of Q = 222, this is more than an order of magnitude faster than the native tracking bandwidth of approximately 100 Hz. When imaging an AFM calibration grating (NT-MDT TGZ3), the increase in bandwidth manifests itself as better tracking of the rectangular surface features, as is visible in Fig. 3c. The controller was implemented on a Anadigm AN221E04 Field-Programmable Analog Array interfaced with a NT-MDT NTEGRA AFM which was operated at a scanning speed of 1 mm∕s.
Fig. 3

(a) Frequency response of the DMASP cantilever in open loop (blue) and for various closed loops to reduce the Q factor (stated in the legend). (b) Tracking bandwidths of the DMASP cantilever determined via drive amplitude modulation for the frequency responses shown in (a) and color coded accordingly. (c) Resulting constant-height AFM images in open loop and for various closed loops, color coded accordingly

Summary and Future Directions

When a cantilever with integrated actuation is used for tapping-mode atomic force microscopy (AFM), the negative imaginary nature of its transfer function model allows robust resonant controllers to be used to regulate the quality (Q) factor of the fundamental resonance mode. This is of advantage, as it allows an increase in cantilever tracking bandwidth and therefore imaging speed. With the continuing development of AFM techniques, which make use of the excitation and detection of not only one but multiple eigenmodes of the cantilever, the ability to control these modes and their responses to excitation is believed to be the key to unraveling the true potential of these multifrequency AFM methods.

Cross-References

Recommended Reading

For more details on the modeling and control of vibrations using piezoelectric transducers, we point the interested reader to the book Moheimani and Fleming (2006).

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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2020

Authors and Affiliations

  1. 1.The University of NewcastleCallaghanAustralia
  2. 2.The University of Texas at DallasRichardsonUSA

Section editors and affiliations

  • S. O. Reza Moheimani
    • 1
  1. 1.University of Texas at DallasDallasUSA