Microelectromechanical Systems Structures Research Articles

Implementation of Highly Reliable Contacts for RF MEMS Switches.

A contact is the key structure of RF MEMS (Radio Frequency Microelectromechanical System) switches, which has a direct impact on the switch's electrical and mechanical properties. In this paper, the implementation of highly reliable contacts for direct-contact RF MEMS switches is provided. As a soft metal material, gold has the advantages of low contact resistance, high chemical stability, and mature process preparation, so it is chosen as the metal material for the beam structure as well as the contacts of the switch. However, a Pt film is used in the bottom contact area to enhance the reliability of the contact. Three kinds of contacts with various shapes are fabricated using different processes. Particularly, a circular-shaped contact is obtained by dry/wet combined processes. The detailed fabrication process of the contacts as well as the Pt film on the bottom contact area are given. The experimental test shows that the contact shape has little effect on the RF performance of the switches. However, the circular contact shows better reliability than other contacts and can work well even after 1.2 × 109 cycles.

(Invited) SAE-MEMS Technology for Electrostatic Vibrational Energy Harvesters

Energy harvesting [1] is expected to be a self-powering technology for small wireless sensor nodes in the next generation Internet-of-Things (IoT). In particular, small vibrational energy harvesters (VEHs) based on micro-electro-mechanical systems (MEMS) technology, which are battery-free and can generate electricity even at night or in the dark, are being investigated worldwide for next-generation power sources for small wireless IoT nodes [2,3]. Electrostatic MEMS VEHs, which use semi-permanent charge-holding dielectrics called "electrets," are useful for power generation using environmental vibrations because they have a higher output power density at lower frequencies than other methods [4]. Further miniaturization, higher performance, and higher productivity will be required to use electret-type MEMS VEHs as power sources for compact wireless IoT nodes and other devices. Although the integration of MEMS VEHs with power management circuits enables miniaturization, higher performance, and improved productivity, the monolithic integration of electret-type MEMS VEHs and electronic circuits is an unexplored area in conventional technology. Conventional electrostatic MEMS VEHs involve corona discharge, electron beam irradiation, X-ray irradiation, or high-temperature treatment as electret charge processing, which imposes restrictions on the design and manufacturing of MEMS structures and the integration of MEMS and electronic circuits. Recently, self-assembled electrets (SAEs), which are electrets that do not require any charging process, have been developed using materials for organic electroluminescent devices [5]. The surface charge density of SAEs is equivalent to those of previously reported electrets for VEHs, and thus SAE is expected to improve the performance of VEH and simplify the manufacturing process. However, there have been no reports of MEMS devices using SAEs.Our proposed method is the first technology to form SAEs inside MEMS structures using only a vacuum evaporation process at room temperature, and we have succeeded in actual vibration power generation [6]. This technology can be integrated into existing semiconductor processes because the electret formation requires only room-temperature deposition, making it possible to monolithically integrate electret-type MEMS VEHs and electronic circuits on the same substrate. In addition, since the SAE surface potential is proportional to the film thickness, the amount of electricity generated can be further increased by increasing the thickness of the SAE film. Therefore, this technology is expected to accelerate the miniaturization, performance, and productivity improvement of MEMS environmental vibration power generation devices. Akinaga, “Recent advances and future prospects in energy harvesting technologies,” Jpn. J. Appl. Phys. 59 (2020) 110201.D. Mitcheson, E. M. Yeatman, G. K. Rao, A. S. Holmes, and T. C. Green, “Energy Harvesting from Human and Machine Motion for Wireless Electronic Devices” Proc. IEEE 96 (2008) 1457–1486.Suzuki, “Recent progress in MEMS electret generator for energy harvesting,” IEEJ Trans. Elec. Electron. Eng. 6 (2011) 101-111.Toshiyoshi, S. Ju, H. Honma, C.-H. Ji, and H. Fujita, “MEMS vibrational energy harvesters,” Sci. Technol. Adv. Mater. 20 (2019) 124-143.Tanaka, N. Matsuura, and H. Ishii, “Self-Assembled Electret for Vibration-Based Power Generator,” Sci. Rep. 10 (2020) 6648.Yamane, H. Kayaguchi, K. Kawashima, H. Ishii, and Y. Tanaka, “MEMS post-processed self-assembled electret for vibratory energy harvesters,” Appl. Phys. Lett. 119 (2021) 254102.

Two-dimensional synchronous motion modulation MEMS structure for suppressing 1/f noise in magnetoresistive sensors

Magnetoresistive (MR) sensors have great application prospects in the field of weak magnetic field detection due to their high sensitivity, small size, and low power consumption. However, 1/f noise greatly limits the low-frequency detectivity of MR sensors. In order to suppress 1/f noise, this paper proposes a two-dimensional synchronous motion modulation (TDSMM) structure based on microelectromechanical systems (MEMS). This structure can effectively reduce 1/f noise by modulating the frequency of the measured magnetic field in the high-frequency band. Theoretical analysis and finite element simulation were conducted on three different modulation models: TSDMM, magnetic flux concentrators motion modulation, and MR components longitudinal motion modulation. The results showed that the modulation efficiency of the TDSMM reached as high as 127%, which is currently the highest value in MR-MEMS sensors. The TDSMM MEMS structure has been successfully manufactured, and the resonant frequency of the transverse resonator is twice that of the longitudinal resonator, enabling extremely high modulation efficiency. The noise spectral density of giant-magnetoresistive components on a silicon-on-insulator substrate was tested, and the noise level in the high-frequency band was three orders of magnitude lower than that in the low-frequency band. These results position MR-MEMS sensors with TDSMM structures as highly competitive candidates in the field of ultra-weak magnetic field detection.

Enhancement of cooling performance in MEMS by modifying 3D packaging structure: A design, integration, analysis and test study

The heat dissipation of three-dimensional (3D) integrated Micro-Electro-Mechanical Systems (MEMS) has garnered considerable attention in scientific research and electronic applications in recent years. As power demands increase, device sizes decrease, and energy consumption is limited, the use of liquid and forced air cooling techniques is diminishing. In light of the growing emphasis on achieving carbon neutrality, there is renewed interest in passive cooling methods, which present a new imperative for environmentally-friendly thermal management of chips in the field of green energy. While the concept of designing thermal paths to mitigate heat in chips is not novel, it has become increasingly important due to the rapid accumulation of heat, reduction in chip volume, and the need to minimize energy consumption and carbon emissions. Consequently, the implementation of appropriate cooling structures within the available space of the chips has emerged as an appealing approach to enhance passive cooling. In order to tackle this challenge, four cooling structures, namely embedding network structure, overlaying extension platform, overlaying micro-pillar array, and overlaying surface microstructures, have been employed in the packaging process to augment the passive cooling capabilities of the chips. In this study, a 3D packaging model is developed to analyze the thermal characteristics of MEMS under the influence of power, TSV voltage, and heat source area using the finite element method. Additionally, the cooling structures are optimized based on the cooling performance analysis, taking into account the self-remaining space within the system and the heat source area factor. The application of four different cooling structures in 3D MEMS leads to enhanced cooling performance. In the event that liquid cooling and forced air cooling are not utilized, the system experiences a decrease in maximum temperature by 3.66 K, resulting in a corresponding decrease in temperature difference by 43.79 %. These reductions significantly contribute to the mitigation of thermal deformation and stress in MEMS, thereby enhancing the thermal reliability of the system. To verify the accuracy of the thermal simulation, a thermal test is conducted on the embedded network structure under simulated conditions. The thermal simulation, which has been validated through thermal testing, indicates that the implementation of four cooling structures within the self-contained area of MEMS has a notable effect on reducing temperature. This finding holds promise for its potential application in 3D MEMS.

Fabrication of hollow composite structure microfluidic double-end clamped beam

Double-end clamped beam is one of the most important miniaturized structures in micro-electromechanical systems (MEMS). In recent years, they have achieved extensive application in a variety of fields, including environmental monitoring, food detection, and disease diagnosis, since they have high sensitivity, high throughput, good specification and fast response. To meet the market demand for precise three-dimensional microstructures and tiny device sizes，the rationalization of the process design of micro-nano technology is essential. In previous research, the function of the microcantilever sensor was finally realized through eight lithography cycles. High performance piezoresistive microcantilever arrays were successfully developed with a yield of 90% in our work. In this paper, a hollow composite microfluidic double-end clamped beam structure is proposed, a reasonable process flow is developed. Simulations show that the hollow microfluidic structure’s sensitivity can reach the order of 10–12 g, and the quality factor is nearly stable above 10,000. It is crucial for the development of double-end clamped beam sensors because it offers a low-cost, portable, and rapid detection method for the detection of biomolecules.

Coupling effect of strain gradient strengthening and thermal softening on the microscale plastic behavior of metallic materials

Size effects and thermal effects together determine the microscale plastic response of metallic structures in high-integrated microelectromechanical systems (MEMS) at different temperatures. To investigate the microscale plastic behavior of metallic materials at different temperatures, a thermo-mechanical coupled microscale plasticity model is developed. In the mechanical part of this model, the framework of conventional plasticity theory is maintained, and the plastic strain gradient is introduced as an internal variable to increase the tangential hardening modulus without the introduction of higher-order stress and higher-order boundary conditions. The influence of temperature on the mechanical parameters (e.g. the yield strength) is calibrated by experiment results. In the thermal part of this model, the heat generation in the plastic deformation stage is calculated by the difference between the plastic work and the hardening stored energy influenced by the plastic strain gradient. Due to the strong nonlinearity of the coupled equations, a finite element solution algorithm that combines the radial return method and the staggered solution scheme is proposed. The effectiveness of this model and its solution algorithm is verified by comparisons with the experiment results and a numerical benchmark example. Finally, taking the tensile behavior of a plate with a hole in its center as an example, the coupling effect of strain gradient strengthening and thermal softening on the microscale plastic behavior of metallic materials is investigated. The results show that the microscale plastic behavior of metallic materials at high temperatures depends on the competition between thermal softening and strain gradient strengthening. Our study provides a theoretical basis and a reliable simulation method for the design of MEMS at different temperatures.

Mechanically resilient, alumina-reinforced carbon nanotube arrays for in-plane shock absorption in micromechanical devices

Microelectromechanical systems (MEMS) are of considerable interest due to their compact size and low power consumption when used in modern electronics. MEMS devices intrinsically incorporate three-dimensional (3D) microstructures for their intended operations; however, these microstructures are easily broken by mechanical shocks accompanying high-magnitude transient acceleration, inducing device malfunction. Although various structural designs and materials have been proposed to overcome this limit, developing a shock absorber for easy integration into existing MEMS structures that effectively dissipates impact energy remains challenging. Here, a vertically aligned 3D nanocomposite based on ceramic-reinforced carbon nanotube (CNT) arrays is presented for in-plane shock-absorbing and energy dissipation around MEMS devices. This geometrically aligned composite consists of regionally-selective integrated CNT arrays and a subsequent atomically thick alumina layer coating, which serve as structural and reinforcing materials, respectively. The nanocomposite is integrated with the microstructure through a batch-fabrication process and remarkably improves the in-plane shock reliability of a designed movable structure over a wide acceleration range (0–12,000g). In addition, the enhanced shock reliability through the nanocomposite was experimentally verified through comparison with various control devices.

Global static and dynamic tristability in electrostatically actuated initially curved and weakly coupled double microbeams

In the present study, an expanded dynamic model of an electrostatically actuated double beam metastructure, composed of geometrically different beams, is formulated and solved. The two beams are connected via an elastic bridge in the form of a linear spring, forming a weak coupling. A reduced order (RO) model is derived using Galerkin’s decomposition with buckling eigenmodes of a straight beam taken as the base functions. Through a case study of three different structures, it is found that weakly coupled and geometrically different sub-structures can manifest global tristability when introduced to a nonlinear, displacement-dependent, electrostatic actuation whilst exhibiting bistability under mechanical, displacement-independent, load. The stability of the structures was established via three different methods. One being the combined eigenvalues based calculation, known as the energy method. The second is a direct Lyapunov stability sweep. And the third method involves direct quasi-static finite elements (FE) based simulations, corroborating the combined stability properties of the structures. In addition, it was found that electrostatic tristability can be manifested in different configurations, with statically inaccessible stable branches. To corroborate the behaviour of the electrostatically actuated structures, quasi-static loading sequences were performed on the RO model, complemented by the dynamic trapping method, to account for all three stable equilibria. Furthermore, it was found that in contrast to rigidly coupled double beam structures, the current model can account for globally stable latching points as well. The study has therefore shown that the energy method can be used to characterise the stability of the proposed metastructure, while also revealing new abilities that can be manifested in a seemingly simple structure. Such findings can promote the usage of such structures in multi-valued micro-electromechanical systems (MEMS) devices, promoting further miniaturisation.

Wet bulk micromachining characteristics of Si{110} in NaOH-based solution

Silicon wet bulk micromachining is an extensively used technique in microelectromechanical systems (MEMS) to fabricate variety of microstructures. It utilizes low-cost etchants and suitable for batch process that made it popular for industrial production. The etch rate and the undercutting at convex corner significantly affect the productivity. In wet anisotropic etching-based micromachining, Si{110} wafer is employed to fabricate unique shape geometries such as the microstructures with vertical sidewalls. In this research, we have investigated the etching characteristics of Si{110} in 10 M sodium hydroxide without and with addition of hydroxylamine (NH2OH). The main objective of the present work is to improve the etch rate and the undercutting at convex corners. Average surface roughness (R a), etch depth, and undercutting length are measured using a 3D scanning laser microscope. Surface morphology of the etched Si{110} surface is examined using a scanning electron microscope. The incorporation of NH2OH significantly improves the etch rate and the corner undercutting, which are useful to enhance the productivity. Additionally, the effect of etchant age on the etch rate and other etching characteristics are investigated. The etch rate of silicon and the undercutting at convex corners decrease with etchant aging. The results presented in this paper are very useful to scientists and engineers who use silicon wet anisotropic etching to fabricate MEMS structures using bulk micromachining. Moreover, it has great potential to promote the application of wet etching in MEMS.

Sensing Devices for Detecting and Processing Acoustic Signals in Healthcare.

Acoustic signals are important markers to monitor physiological and pathological conditions, e.g., heart and respiratory sounds. The employment of traditional devices, such as stethoscopes, has been progressively superseded by new miniaturized devices, usually identified as microelectromechanical systems (MEMS). These tools are able to better detect the vibrational content of acoustic signals in order to provide a more reliable description of their features (e.g., amplitude, frequency bandwidth). Starting from the description of the structure and working principles of MEMS, we provide a review of their emerging applications in the healthcare field, discussing the advantages and limitations of each framework. Finally, we deliver a discussion on the lessons learned from the literature, and the open questions and challenges in the field that the scientific community must address in the near future.

Compensation of the Stress Gradient in Physical Vapor Deposited Al1-xScxN Films for Microelectromechanical Systems with Low Out-of-Plane Bending.

Thin film through-thickness stress gradients produce out-of-plane bending in released microelectromechanical systems (MEMS) structures. We study the stress and stress gradient of Al0.68Sc0.32N thin films deposited directly on Si. We show that Al0.68Sc0.32N cantilever structures realized in films with low average film stress have significant out-of-plane bending when the Al1−xScxN material is deposited under constant sputtering conditions. We demonstrate a method where the total process gas flow is varied during the deposition to compensate for the native through-thickness stress gradient in sputtered Al1−xScxN thin films. This method is utilized to reduce the out-of-plane bending of 200 µm long, 500 nm thick Al0.68Sc0.32N MEMS cantilevers from greater than 128 µm to less than 3 µm.

Vertical and Lateral Etch Survey of Ferroelectric AlN/Al1−xScxN in Aqueous KOH Solutions

Due to their favorable electromechanical properties, such as high sound velocity, low dielectric permittivity and high electromechanical coupling, Aluminum Nitride (AlN) and Aluminum Scandium Nitride (Al1−xScxN) thin films have achieved widespread application in radio frequency (RF) acoustic devices. The resistance to etching at high scandium alloying, however, has inhibited the realization of devices able to exploit the highest electromechanical coupling coefficients. In this work, we investigated the vertical and lateral etch rates of sputtered AlN and Al1−xScxN with Sc concentration x ranging from 0 to 0.42 in aqueous potassium hydroxide (KOH). Etch rates and the sidewall angles were reported at different temperatures and KOH concentrations. We found that the trends of the etch rate were unanimous: while the vertical etch rate decreases with increasing Sc alloying, the lateral etch rate exhibits a V-shaped transition with a minimum etch rate at x = 0.125. By performing an etch on an 800 nm thick Al0.875Sc0.125N film with 10 wt% KOH at 65 °C for 20 min, a vertical sidewall was formed by exploiting the ratio of the planes and planes etch rates. This method does not require preliminary processing and is potentially beneficial for the fabrication of lamb wave resonators (LWRs) or other microelectromechanical systems (MEMS) structures, laser mirrors and Ultraviolet Light-Emitting Diodes (UV-LEDs). It was demonstrated that the sidewall angle tracks the trajectory that follows the of the hexagonal crystal structure when different c/a ratios were considered for elevated Sc alloying levels, which may be used as a convenient tool for structure/composition analysis.

Design and Applications of Integrated Transducers in Commercial CMOS Technology

Monolithic integration of Microelectromechanical Systems (MEMS) directly within CMOS technology offers enhanced functionality for integrated circuits (IC) and the potential improvement of system-level performance for MEMS devices in close proximity to biasing and sense circuits. While the bulk of CMOS-MEMS solutions involve post-processing of CMOS chips to define freely-suspended MEMS structures, there are key applications and conditions under which a solid, unreleased acoustic structure composed of the CMOS stack is preferred. Unreleased CMOS-MEMS devices benefit from lower barrier-to-entry with no post-processing of the CMOS chip, simplified packaging, robustness under acceleration and shock, stress gradient insensitivity, and opportunities for frequency scaling. This paper provides a review of advances in unreleased CMOS-MEMS devices over the past decade, with focus on dispersion engineering of guided waves in CMOS, acoustic confinement, CMOS-MEMS transducers, and large signal modeling. We discuss performance limits with standard capacitive transduction, with emphasis on performance boost with emerging CMOS materials including ferroelectrics under development for memory.

Investigation of nonlinear squeeze-film damping involving rarefied gas effect in micro-electro-mechanical systems

In this paper, the nonlinear squeeze-film damping (SFD) involving rarefied gas effect in the micro-electro-mechanical systems (MEMS) is investigated. Considering the motion of structures (beam, cantilever, and membrane) in MEMS, the dynamic response of the structure is affected greatly by the SFD. In the traditional model, a viscous damping assumption that the damping force is linear with the moving velocity is used. As the nonlinear damping phenomenon is observed for a micro-structure oscillating at a high velocity, this assumption does not hold and will cause error results for predicting the response of the micro-structure. Meanwhile, due to the small size of the device and the low pressure of the encapsulation, the gas in MEMS is usually rarefied gas. Therefore, to correctly predict the damping force, the rarefied gas effect must be considered. To study the nonlinear SFD problem involving the rarefied gas effect, a kinetic method, i.e., discrete unified gas kinetic scheme (DUGKS), is introduced in this paper. Also, based on the DUGKS, two solving methods, i.e., a traditional decoupled method (Eulerian scheme) and a coupled framework (arbitrary Lagrangian-Eulerian scheme), are adopted. With these two methods, two basic motion forms, i.e., linear (perpendicular) and tilting motions of a rigid micro-beam, are studied under forced and free oscillations. For a forced oscillation, the nonlinear SFD phenomenon is investigated. For a free oscillation, in the resonance regime, some numerical results at different maximum oscillating velocities are presented and discussed. Besides, the influence of oscillation frequency on the damping force or torque is also studied, and the cause of the nonlinear damping phenomenon is investigated.

A monolithic three-dimensional thermal convective acoustic vector sensor with acoustic-transparent heat sink.

An acoustic vector sensor can directly detect acoustic particle velocity based on the measured temperature difference between closely spaced heated wires. For the detection of velocity in three dimensions, an integrated three-dimensional (3 D) sensor is desired, but it remains challenging in MEMS (Micro-Electro-Mechanical System) manufacturing. Here, a novel monolithic 3 D acoustic vector sensor is proposed, which is constructed using in-plane distributed wires assembled with acoustically transparent heat sink. The planar MEMS structure of the proposed sensor makes it easy to be fabricated and packaged. This work offers a new method for the design of acoustic vector sensors and other thermal convection-based MEMS sensors.

Optimum Design and Test of a Novel Bionic Electronic Stethoscope based on the Cruciform Microcantilever with Leaf Microelectromechanical Systems Structure

AbstractThis paper proposes a heart sound sensing structure and system by combining the traditional auscultation structure and human‐like ear eardrum gas–solid coupling sound conduction process. This system is based on the lever pickup mechanism of eardrum vibration and the traditional stethoscope pickup amplification, microelectromechanical system (MEMS) technology, and the detection principle of piezoresistive and Wheatstone bridge. The optimal size and process parameters of the biomimetic sensor structure are given through theoretical calculation and numerical simulation on Comsol and SRIM. Computer numerical control, 3D printing, MEMS technology, and 3D heterogeneous integration technology are used to complete the precise processing and encapsulation of the proposed structure. After the performance tests, results show that the optimal structure can collect directional signals, with a bandwidth from 10 Hz to 1 kHz, which can effectively cover the range of the heart frequency. The signal‐to‐noise ratio is improved by 2.3 dB compared with the 3M stethoscope. An electrocardiogram and phonocardiogram synchronization detection system is developed with this structure. The structure and system can be effectively applied to the high‐quality collection and intelligent recognition of heart sound data sets. The recognition accuracy can reach 98.5%. It is highly effective for early screening and intelligent diagnosis of cardiovascular diseases.

Indirect stress and air-cavity displacement measurement of MEMS tunable VCSELs via micro-Raman and micro-photoluminescence spectroscopy

We employ micro-Raman spectroscopy to optically infer the stress experienced by the legs of a bridge-type microelectromechanical systems (MEMS) used in high contrast gratings tunable vertical cavity surface emitting lasers (VCSELs). We then employ micro-photoluminescence (PL) spectroscopy to indirectly measure the air cavity displacement of the same MEMS structure. Results from micro-Raman showed that electrostatically actuating the MEMS with a DC bias configuration yields increasing residual stress on the endpoints of the MEMS with values reaching up to 0.8 GPa. We simulated a finite element model via Comsol Multiphysics which agrees with the trend we observed based on our micro-Raman data. Our micro-PL spectroscopy showed that change in the air cavity of the VCSEL structure resulted in a change in the full width of the PL peak emitted by the layer consisting of four pairs of distributed Bragg reflectors. The change in the full width of the PL peak was due to the change in the optical cavity induced by displacing the MEMS via externally applied bias and agrees with our transfer matrix convolution simulation. These optical characterization tools can be used for failure analysis, MEMS design improvements, and monitoring of MEMS tunable VCSEL devices for mass production and manufacturing.

Mode coupling and locking of a Π-shaped cantilever resonator using laser-induced asymmetric modulation

Mechanical resonators, such as microcantilevers, demonstrate significant potential for use in information technology. Cantilevered beams of various geometries clamped at one end form the most ubiquitous structures in microelectromechanical systems that support multimode vibration for the detection, conversion, and processing of small signals. In this study, we demonstrate that the potential of these devices can be further extended by utilizing a strategy based on mode coupling and locking induced by asymmetric photothermal modulation. A cantilever was designed to have a Π-shape with a specific geometry such that the resonant frequencies of the two orthogonal modes are close to one another. Additionally, we show that mode coupling between the two modes, which are originally orthogonal to one another, can be achieved through laser-induced photothermal modulation. In particular, the two modes can be parametrically tuned to become degenerate through mode coupling with a significant increase in the quality factor from 112 to 839. This approach is universal and can be extended to improve the detection limits of microresonators in high-dissipation environments with enhanced signal-to-noise ratios.

Temperature-Dependent n-Type and p-Type Sensing Behaviors of CuO Nanosheets/MEMS to NO2 Gas

This study uses a micro-electromechanical system (MEMS) structure to fabricate a cupric oxide (CuO) nanosheet (NS) gas sensor. A CuO-NS with a size ranging from 100 to 500 nm is produced by chemica...

Analytical Deflection Profiles and Pull-In Voltage Calculations of Prestressed Electrostatic Actuated MEMS Structures

This paper presents a zeroth and first order perturbative analysis of prestressed electrostatic actuated Microelectromechanical systems (MEMS). Perturbation theory is used to calculate the deflection profile of various MEMS structures, from which the pull-in voltage is estimated using the weighed residual method, where both a Galerkin expression and a Dirac delta function have been used as weight functions. A prestressed circular Capacitative Micromachined Ultrasonic Transducer (CMUT) is used as the main example in this paper. This device is modeled as a circular clamped plate subjected to an electrostatic pressure. The calculated pull-in voltage has been compared with experimental data of highly prestressed CMUTs, where a relative error between -36% and -8% is observed for a model that does not include stress. The model that includes the residual stress lowers the range of the relative error to values between -5% and 21%. To improve the accuracy of the pull-in voltage estimate Richardson extrapolation has been calculated from the zeroth and first order estimates. The pull-in voltage models are compared with a Finite Element Model (FEM), where an overestimation, in the high stress regime, of 10% is observed for the zeroth order model, less than 5% for the Galerkin method and 3% for the Richardson extrapolation. [2020-0358]