Mariateresa Napoli

Characterization of electrostatically coupled microcantilevers

The use of tightly packed arrays of probes can achieve the much desirable goal of increasing the throughput of scanning probe devices. However the proximity of the probes induces coupling in their dynamics, which increases the complexity of the overall device. In this paper we analyze and model the behavior of a pair of electrostatically and mechanically coupled microcantilevers. For the common case of periodic driving voltage, we show that the underlying linearized dynamics are governed by a pair of coupled Mathieu equations. We provide experimental evidence that confirms the validity of the mathematical model proposed, which is verified by finite element simulations as well. The coefficients of electrostatic and mechanical coupling are estimated respectively by frequency identification methods and noise analysis. Finally, we discuss parametric resonance for coupled oscillators and include a mapping of the first order coupled parametric resonance region.

Understanding mechanical domain parametric resonance in microcantilevers

In this paper we present a mathematical model for the dynamics of an electrostatically actuated micro-cantilever. For the common case of cantilevers excited by a periodic voltage, we show that the underlying linearized dynamics are those of a periodic system described by a Mathieu equation. We present experimental results that confirm the validity of the model, and in particular illustrate that parametric resonance phenomena occur in capacitively actuated micro-cantilevers. The combined parametric/harmonic mode of operation is investigated as well and experimental data are provided.

A Capacitive Microcantilever: Modelling, Validation, and Estimation Using Current Measurements

cantilever. For the common case of cantilevers excited by a periodic voltage, we show that the underlying linearized dynamics are those of a periodic system described by a Mathieu equation. We present experimental results that confirm the validity of the model, and in particular, illustrate that parametric resonance phenomena occur in capacitively actuated micro-cantilevers. We propose a system where the current measured is used as the sensing signal of the cantilever state and position through a dynamical observer. By investigating how the best achievable performance of an optimal observer depends on the excitation frequency, we show that the best such frequency is not necessarily the resonant frequency of the cantilever. @DOI: 10.1115/1.1767851# The recent advances in the field of miniaturization and micro fabrication have paved the way for a new range of applications, bringing along the promise of unprecedented levels of performance. In particular, scanning probe devices have proven to be extremely versatile instruments, with applications that range from surface imaging at the atomic scale @1#, to ultra high density data storage and retrieval @2#, and to biosensors @3,4#, to cite but a few. The working principle for most of these devices is based on a measurement of displacement. As an example, consider imaging in atomic force microscopy: the topography of a sample is reconstructed from the displacement of the cantilever-probe, caused by the interaction forces with the sample @5,6#. In biosensors applications the displacement of a cantilever can be related to the binding of molecules on the ~activated! surface of the cantilever beam, and is therefore used to compute the strength of these bonds, as well as the presence of specific reagents in the solution under consideration @7,8#. It is clear that the sensitivity of these devices strongly depends on the smallest detectable motion, which poses a constraint on the practically vs. theoretically achievable performance. In order to make the gap between the two smaller, while at the same time providing compactness of devices and faster dynamics, much of the research effort has been focused on the design of integrated detection schemes. The most common solutions for integrated detection make use of the piezoresistive, @9,10#, piezoelectric @11‐13#, thermal expansion @14# or capacitive effects @15‐17#. A major advantage of capacitive detection, is the fact that it offers both electrostatic actuation as well as integrated detection, without the need for an additional position sensing device. The common scheme used in capacitive detection is to apply a second AC voltage at a frequency much higher than the mechanical bandwidth of the cantilever. The current output at that frequency is then used to estimate the capacitance, and consequently the cantilever position. This sensing scheme is the simplest position detection scheme available, however, it is widely believed to be less accurate than optical levers or piezoresistive sensing. The device that we propose is an electrostatically actuated microcantilever. More precisely, in our design the microcantilever constitutes the movable plate of a capacitor and its displacement is controlled by the voltage applied across the plates. In order to measure the cantilever displacement, we propose a novel scheme that avoids the use of a high frequency probing signal by the use of a dynamical state observer, whose input is the current through the capacitive cantilever. For the purpose of implementation, this scheme offers significant advantages as it involves simpler circuitry. By using an optimal observer, or by tuning the observers gains, it is conceivable that a high fidelity position measurement can be obtained, thus improving resolution in atomic force microscopy applications. In this paper, we present a model for this electrostatically actuated microcantilever. Using simple parallel plate theory and for the common case of sinusoidal excitation, it turns out that its dynamics are governed by a special second order linear periodic differential equation, called the Mathieu equation. We produce experimental evidence that validates the mathematical model, including a mapping of the first instability region of the Mathieu equation. The optimal observer problem that was formulated also in @18# is solved here following a different and simpler approach. This optimal design is then used as an analysis tool to select the frequency of excitation that corresponds to the best achievable observer performance. In other words, the optimal observer design is used to actually design the system ~rather than the observer! ,b y selecting the excitation frequency that produces the least estimation error. Interestingly, it turns out that this frequency is not necessarily the resonant frequency of the cantilever, and it depends on the statistics of the measurement and process noise.

Optimal Control of Arrays of Microcantilevers

In this paper we will present a model for an array of microcantilevers that are used in Atomic Force Microscopy and nano-scale manufacturing. The microcantilevers are connected to each other through a common base, and are individually actuated. The sensors are also integrated on each microcantilever. We consider the problem of controlling a tightly packed array of identical microcantilevers that are dynamically coupled. This system is an example of a spatially-invariant system with a distributed array of sensors and actuators. We exploit the spatial invariance of the problem to design optimal H 2 controllers for this array. An analytic expression for the optimal controller is derived in the transformed domain, and estimates of the coupling range of the controller is obtained.

Mathematical modeling, experimental validation and observer design for a capacitively actuated microcantilever

We present a mathematical model for the dynamics on an electrostatically actuated micro-cantilever. For the common case of cantilever excited by a periodic voltage, we show that the underlying linearized dynamics are those of a periodic system described by a Mathieu equation. We present experimental results that confirm the validity of the model, and in particular, illustrate that parametric resonance phenomena occur in capacitively actuated microcantilevers. We propose a system where the current measured is used as the sensing signal of the cantilever state and position through a dynamical observer. By investigating how the best achievable performance of an optimal observer depends on the excitation frequency, we show that the best such frequency is not necessarily the resonant frequency of the cantilever.

A novel sensing scheme for the displacement of electrostatically actuated microcantilevers

We present the design and implementation of a novel sensing scheme to reconstruct the displacement of electrostatically actuated microcantilevers that are used in atomic force microscopy. Our approach proposes to estimate rather than measure directly the displacement, by means of an observer that uses the current through the capacitive cantilever as an input. In particular, we formulate the observer problem as an H/sub /spl infin// optimal filtering problem for periodic systems. We show here our first experimental results regarding the implementation of this sensing scheme, which includes a custom made off-chip circuit to measure very small currents (few p/spl Lambda/) at high frequencies (/spl ap/ 100 kHz).

Modeling and observer design for an array of electrostatically actuated microcantilevers

We present a mathematical model for the dynamics of an array of capacitively actuated microcantilevers. We propose a system where the current measured at each cantilever is used as the sensing signal of the cantilever state through an observer. We show that such an array is a spatially invariant system with distributed control and sensing. For the common case of periodically excited cantilevers, we show that the underlying dynamics are those of a periodic system described by a Mathieu equation. We exploit the spatial invariance of the problem to design an optimal distributed observer, where the temporal periodicity is handled using the lifting technique.

Dynamics of mechanically and electrostatically coupled microcantilevers

In this paper we study the behavior of a pair of electrostatically and mechanically coupled microcantilevers. For the case of cantilevers excited by a periodic voltage, we show that the underlying linearized dynamics are governed by a pair of coupled Mathieu equations. We provide experimental evidence that confirms the validity of the mathematical model proposed, including in particular a mapping of the first order coupled parametric resonance region. The mechanical coupling coefficients are identified from experimental data, and their values are shown to match well those obtained by finite element methods.

A novel observer based sensing scheme for the displacement of electrostatically actuated microcantilevers

We present the design and implementation of a novel sensing scheme to reconstruct the displacement of electrostatically actuated microcantilevers that are used in atomic force microscopy. Our approach proposes to estimate rather than measure directly the displacement, by means of an observer that uses the current through the capacitive cantilever as an input. In particular, we formulate the observer problem as an H/sub /spl infin// optimal filtering problem for periodic systems. We show here our first experimental results regarding the implementation of this sensing scheme, which includes a custom made off-chip circuit to measure very small currents (few pA) at high frequencies (/spl ap/ 100 kHz).

Optimal control of arrays of microcantilevers

We present a model for an array of microcantilevers that are used in atomic force microscopy and nano-scale manufacturing. The microcantilevers are connected to each other through a common base, and are individually actuated. The sensors are also integrated on each microcantilever. This system is an example of a spatially-invariant system with a distributed array of sensors and actuators. We exploit the spatial invariance of the problem to design optimal H/sub 2/ controllers for this array. An analytic expression for the optimal controller is derived in the transformed domain, and estimates of the coupling range of the controller is obtained.