Electrostatic Pull-in Instability Research Articles

An experimental study for characterization of size-dependence in microstructures via electrostatic pull-in instability technique

This paper experimentally investigates the size-dependent effective Young's modulus Eeff of aluminum (Al) clamped-clamped microbeams using the electrostatic pull-in instability technique. This study presents an experimental characterization of the so-called “length scale parameter” in couple stress theory and surface elasticity. Eeff is retrieved from the measured pull-in voltage, of the clamped-clamped beams with different dimensions, via an electromechanically coupled equation. Measurement results show a strong size dependence of Eeff for the Al beams in small sizes. The Young's modulus increases monotonously as the beams become thinner. The experimental observations are consistent with the published modeling results of the size effects, in which couple stress theory and surface elasticity theory are taken into consideration. The presented experimental method has substantial advantages such as precise adjustable magnitude of the non-contacting force and a lower cost over the other approaches used for characterization of micro-/nanoelectromechanical systems. This simple and reproducible method can be extended for characterization of various materials with different sizes and boundary conditions.

Study on the size dependent effective Young modulus by EPI method based on modified couple stress theory

Various experimental and theoretical researches have been shown the size-dependence behavior of the effective Young modulus (EYM) in the micron and sub-micron scales. One of the most accurate methods is the electrostatic pull-in instability (EPI) method that is based on the bending of the classical beam under the electrostatic force. In this paper, the modified couple stress theory (MCST) is employed to calculate the EYM of silicon nanocantilevers. The MCST compensates the inability of the classical continuum mechanic to predicting the size-dependent behavior of the nano-scale structures. Next, as a case study the EYM of silicon nanocantilevers have been calculated and results compared with classical EPI. The governing equation is solved by the Galerkin method and obtained results show significant size-dependent behavior in the EYM. From the other hand, a new value for the material length scale parameter is introduced based on the dimension of the crystal or grain size of the material.

EM radiation from electrostatic nonlinear pull‐in instability of MEMS

Electrodynamics theory tells that EM waves are generated from accelerated charged particles. Conventional radiators such as antennas are understood to emit waves because of the current's distribution and accumulation in the capacitor, where electrons are accelerated. In MEMS mechanisms, charges can be accelerated by the mechanical motion, which suggests that an MEMS structure can also be a radiator. The electrodynamics to theoretically analyse the EM wave radiation from MEMS is used. During the electrostatic nonlinear pull-in instability MEMS, where the electrostatic gap has become narrower than the 2/3 of the initial gap, the distributed charges are accelerated at a very high rate so as to generate significant power radiation in the form of EM waves. The MEMS's mechanical accelerations are analysed to calculate the transient radiations in both spatial and frequency domains.

A new approach using hybrid power series – cuckoo search optimization algorithm to solve electrostatic pull-in instability and deflection of nano cantilever switches subject to van der waals attractions

A hybrid Power Series (PS) and Cuckoo Search via L´evy Flights (CS) optimization algorithm (PS-CS) method is utilized to obtain a solution for the deflection and pull-in instability of a nano cantilever switch in the presence of the van der Waals attractions, electrostatic forces and fringing filed effects. In order to obtain a relation for deflection of the beam, a trial solution including adjustable coefficients, satisfying the boundary conditions of the governing, is proposed. The cuckoo search optimization algorithm is executed to find the ad-justable parameters of the trial solution satisfying the governing equation of the nanobeam. The results are compared with the available results in the literature as well as numerical solution. The results indicate the remarkable accuracy of the present approach. The minimum initial gap and the critical free standing detachment length of the nano actuator that does not stick to the substrate due to the van der Waals attractions, as an important parameter in pull-in instability of the nano switches, is calculated. Utilizing the results of the PS-CS, the stress distribution inside the nano actuator is determined at the onset of the pull-in instability.

Modeling the influence of the Casimir force on the pull-in instability of nanowire-fabricated nanotweezers

The Casimir force can strongly interfere with the pull-in performance of ultra-small structures. The strength of the Casimir force is significantly affected by the geometries of interacting bodies. Previous investigators have exclusively studied the effect of the Casimir force on the electromechanical instability of nanostructures with planar geometries. However no work has yet considered this effect on the pull-in instability of systems with cylindrical geometries such as nanotweezers fabricated from nanotube/nanowires. In our present work, the influence of the Casimir attraction on the electrostatic response and pull-in instability of nanotweezers fabricated from cylindrical conductive nanowires/nanotubes is theoretically investigated. An asymptotic solution, based on scattering theory, is applied to consider the effect of vacuum fluctuations in the theoretical model. The Euler–Bernoulli beam model is employed, in conjunction with the size-dependent modified couple stress continuum theory, to derive the governing equation of the nanotweezers. The governing nonlinear equations are solved by two different approaches, i.e., the modified Adomian–Padé method (MAD–Padé) and a numerical solution. Various aspects of the problem, i.e., the variation of pull-in parameters, effect of geometry, coupling between the Casimir force and size dependency effects and comparison with the van der Waals force regime are discussed.

Detection and Mitigation of Electrostatic Pull-in Instability in MEMS Parallel Plate Actuators

Electrostatic MEMS actuators are used in a wide variety of applications including micro-machined gyroscopes, high speed mechanical switches, variable capacitors, and vibration isolation devices. MEMS parallel plate actuators (PPAs) are simple to realize, yet complex nonlinear variable capacitors. If a DC voltage is applied in attempt to move the proof mass greater than 1/3 of the electrode rest gap distance, the device becomes unstable and the electrodes snap into contact. Most research into this pull-in phenomena is devoted to extending the operational range of motion past the 1/3 instability point. This usually involves the addition of complex external electronics. Many applications, however, only require that the actuator remain out of the pull-in region, and do not require an extended stable range of motion. If detection of the pull-in event is all that is required, then simpler solutions can be realized, minimizing the requirements on the driving signal. Once pull-in is reached, the velocity of the movable plate increases rapidly until the plates make contact. The decreasing distance causes a proportional increase in capacitance. To maintain a constant voltage across the plates, an inrush of current must flow into the actuator in response to the increased capacitance. This paper presents a method for detecting the inrush current using a transimpedance amplifier circuit to convert the current to a measurable voltage. Once pull-in is detected, the PPA is electrically shutdown to prevent damage to the actuator or the voltage source, thus mitigating pull-in. A simulation of the expected results was performed using a Simulink model for the actuator structure predicting the expected range of inrush current. This result was then verified using a silicon micro-machined PPA connected to the detection/mitigation circuit on a PCB. The experimental results follow closely with the simulation allowing precise control in mitigating the pull-in event.

A new approach to the electrostatic pull-in instability of nanocantilever actuators using the ADM–Padé technique

In this paper, the Adomian decomposition method and Padé approximants are integrated to study the deflection and pull-in instability of nanocantilever electromechanical switches. In a distributed parameter model, intermolecular forces, including Casimir forces, are taken into account considering their range of application. A closed form power series solution based on Adomian polynomials is obtained. The obtained analytic results are compared with numerical solution. The Adomian method is accurate for small deflections, but the results of a pull-in instability study demonstrate that the accuracy of the Adomian solution is not as good for small deflections. Thus, to increase the accuracy of the Adomian solution for the pull-in instability, the Adomian power series is converted to Padé approximants. The results of the present method are compared with the numerical results as well as those of the Adomian decomposition method and other methods reported in the literature. The results obtained using the ADM–Padé are remarkably accurate compared with the numerical results. The proposed technique can be easily extended to solve a wide range of instability problems. Finally, the minimum initial gap and the detachment length of the actuator that does not stick to the substrate due to the intermolecular attractions, which is an important parameter for the pull-in instability of a nanocantilever actuator, are calculated using Adomian–Padé approximants.

A new power series solution on the electrostatic pull-in instability of nano cantilever actuators

In this paper, a power series solution is used to study the deflection and pull-in instability of nano cantilever electromechanical switches using a distributed parameter model. In the modeling of intermolecular force the Casimir force and in the modeling of electrostatic force, the fringing field effect is taken into account. A closed form power series solution basis on symbolic computation is introduced to obtain a semi analytical solution for the distributed parameter model. The minimum initial gap and the detachment length of the actuator that does not stick to the substrate due to the Casimir attractions, as an important parameter in pull-in instability of a nano cantilever actuator, is calculated by obtained power series. Furthermore, the present method is capable of determining stress distribution of the nano beam actuator at the onset of instability. It is seen that the Casimir effect significantly reduce the maximum value of stress resultants at the onset of instability. The obtained results are compared with numerical results and the Adomian decomposition method. The obtained solution in compare with the Adomian decomposition method represents a remarkable accuracy.

Investigating the effect of Casimir and van der Waals attractions on the electrostatic pull-in instability of nano-actuators

This paper investigates the effect of dispersion (van der Waals and Casimir) forces on the pull-in instability of cantilever nano-actuators by considering their range of application. Adomian decomposition is introduced to obtain an analytical solution of the distributed parameter model. Dispersion forces decrease the pull-in deflection and voltage of a nano-actuator. However, the fringing field increases the pull-in deflection while decreasing the pull-in voltage of the actuator. The minimum initial gap and the detachment length of the actuator that does not stick to the substrate due to van der Waals and Casimir attractions were determined. Furthermore, the proposed approach is capable of determining the stress distribution of the actuator at the onset of instability. It is seen that Casimir and van der Waals attractions effectively reduce the maximum value of stress resultants at the onset of instability. The results indicate that Adomian decomposition is a reliable method for simulating nano-structures at submicrometer ranges.

Effects of size and defects on the elasticity of silicon nanocantilevers

The size-dependent elastic behavior of silicon nanocantilevers and nanowires, specifically the effective Young's modulus, has been determined by experimental measurements and theoretical investigations. The size dependence becomes more significant as the devices scale down from micro- to nano-dimensions, which has mainly been attributed to surface effects. However, discrepancies between experimental measurements and computational investigations show that there could be other influences besides surface effects. In this paper, we try to determine to what extent the surface effects, such as surface stress, surface elasticity, surface contamination and native oxide layers, influence the effective Young's modulus of silicon nanocantilevers. For this purpose, silicon cantilevers were fabricated in the top device layer of silicon on insulator (SOI) wafers, which were thinned down to 14 nm. The effective Young's modulus was extracted with the electrostatic pull-in instability method, recently developed by the authors (H Sadeghian et al 2009 Appl. Phys. Lett. 94 221903). In this work, the drop in the effective Young's modulus was measured to be significant at around 150 nm thick cantilevers. The comparison between theoretical models and experimental measurements demonstrates that, although the surface effects influence the effective Young's modulus of silicon to some extent, they alone are insufficient to explain why the effective Young's modulus decreases prematurely. It was observed that the fabrication-induced defects abruptly increased when the device layer was thinned to below 100 nm. These defects became visible as pinholes during HF-etching. It is speculated that they could be the origin of the reduced effective Young's modulus experimentally observed in ultra-thin silicon cantilevers.

Temperature sensitivity of silicon cantilevers’ elasticity with the electrostatic pull-in instability

In this paper the temperature effects on the effective Young’s modulus of [1 1 0] silicon cantilevers is analyzed and measured in the range of 25–100 ° C. The quasi-static electrostatic pull-in instability method developed recently for ultra-thin cantilevers [H. Sadeghian, C.K. Yang, J.F.L. Goosen, E. van der Drift, A. Bossche, P.J. French, F. van Keulen, Characterizing size-dependent effective elastic modulus of silicon nanocantilevers using electrostatic pull-in instability, Appl. Phys. Lett. 94 (22) (2009) 221903] is employed to measure the temperature sensitivity of the ultra-thin cantilevers (Dutch Patent App., No. P6027025NL). A temperature sensitivity of 81.3 ° C/V is obtained. The temperature sensitivity is mostly due to the temperature dependence of the effective Young’s modulus of silicon. It is shown that changes in geometrical dimensions due to the change in temperature can be neglected. The changes in the effective Young’s modulus due to the changes in the temperature are extracted using a thermo–electro-mechanical coupled system. The pull-in method showed substantial advantages over other methods used for the study of the thermal effects on micron and sub-micron structures. The method is reproducible for various temperatures and cantilevers’ dimensions and the results demonstrate a new concept for a temperature sensor with ultra-high sensitivity.

Temperature Sensitivity of Silicon Cantilevers with the Pull-in Instability Method

In this paper the temperature effects on [110] Silicon cantilevers is analyzed and measured in the range of 25 -100 °C. The quasi-static electrostatic pull-in instability method developed recently for ultra-thin cantilevers [“Characterizing Size-dependent Effective Elastic Modulus of Silicon Nanocantilevers Using Electrostatic Pull-in Instability”, Applied Physics Letters, Vol. 94(22), p. 221903, 2009] is employed to measure the temperature sensitivity of ultra-thin cantilevers. A temperature sensitivity of 81.3 °C/V is obtained. The temperature sensitivity is mostly due to the temperature dependence of the effective Young’s Modulus of silicon. It is shown that changes in geometrical dimensions due to the change in temperature can be neglected. The changes in the effective Young’s Modulus due to the changes in temperature are extracted using an electromechanical-coupled system. The pull-in method showed substantial advantages over other methods used for the study of the thermal effects on micron and sub-micron structures. The results demonstrate a new concept for a temperature sensor with ultra high sensitivity.

Characterizing size-dependent effective elastic modulus of silicon nanocantilevers using electrostatic pull-in instability

This letter presents the application of electrostatic pull-in instability to study the size-dependent effective Young’s Modulus Ẽ (∼170–70 GPa) of [110] silicon nanocantilevers (thickness ∼1019–40 nm). The presented approach shows substantial advantages over the previous methods used for characterization of nanoelectromechanical systems behaviors. The Ẽ is retrieved from the pull-in voltage of the structure via the electromechanical coupled equation, with a typical error of ≤12%, much less than previous work in the field. Measurement results show a strong size-dependence of Ẽ. The approach is simple and reproducible for various dimensions and can be extended to the characterization of nanobeams and nanowires.

Actuated elastomers with rigid vertical electrodes

This paper presents a novel design of actuated elastomers using rigid, vertical standing electrodes. The electrodes of high rigidity are designed to have small width, small gaps and large depth in order to produce large electrostatic forces at a moderate voltage, as well as to reduce the constraining effect on the soft elastomers. This novel design embodies lateral stacking to accumulate small strain into an adequate displacement for micro-actuation. Analytical and numerical analyses are performed to evaluate the force–displacement characteristics of the elastomer actuators. It is shown that actuation of the novel design is not limited by the electrostatic pull-in instability, and it is capable of travelling a large range at a high electric field. In addition, feasibility of fabrication processes is studied. It is shown that lateral stacking can readily be realized by filling liquid elastomers into deep, narrow and vertical trenches. A theoretical benchmark comparison with a conventional air-gap electrostatic comb drive indicates that the new elastomer actuator promises great flexibility, large attraction force and robustness against shock and dust blockage.