# Dynamics and Control of Active Microcantilevers

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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).

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

*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)

*Q*

_{i}, natural frequency

*ω*

_{i}, and gain

*α*

_{i}. The feedthrough term

*d*takes into account the modeling error associated with limiting the sum to modes within the measurement bandwidth.

*i*= 1), the poles of (1) are given by

### Example 1

For a standard tapping-mode microcantilever (e.g., Budget Sensors, TAP190-G) with a fundamental resonance frequency of *f*_{1} = 190 kHz and a natural Q factor of 400, the tracking bandwidth is only approximately *f*^{c} = 238 Hz.

### System Property

*G*of a system is said to be NI if it is stable and satisfies the condition Petersen and Lanzon (2010)

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 *G*_{dv}(*s*) is NI since *α*_{1}, *ω*_{1}, *Q*_{1} > 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

*γ*

_{c,i},

*ζ*

_{c,i}, and

*ω*

_{c,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

*γ*

_{c,i},

*ζ*

_{c,i},

*ω*

_{c,i}> 0, that is, robust stability is achieved if such a controller is used to dampen a negative imaginary mode of the cantilever.

*n*= 1) (1) in positive feedback with the PPF controller (

*m*= 1) (5) can be written as

*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

*γ*

_{c}and \(\omega _c^2\) using the Schur complement as

### Controller Design

*γ*

_{c},

*ζ*

_{c}, 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

## Application to Atomic Force Microscopy

*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.

## 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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