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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Electrostatic-driven microelectromechanical systems devices, in most cases, consist of couplings of such energy domains as electromechanics, optical electricity, thermoelectricity, and electromagnetism. Their nonlinear working state makes their analysis complex and complicated. This article introduces the physical model of pull-in voltage, dynamic characteristic analysis, air damping effect, reliability, numerical modeling method, and application of electrostatic-driven MEMS devices.

Micro-Electro-Mechanical Systems (MEMS) are an electromechanical integrated system where the feature size of components and the actuating range are within the micro-scale. Unlike traditional mechanical processing, manufacturing of MEMS device uses the semiconductor production process, which can be compatible with an integrated circuit, and includes surface micromachining and bulk micromachining. Due to the increasingly mature process technology, numerous sophisticated micro structural and functional modules are currently available. Therefore, greater optimized performance of the devices has been developed. Electrostatic-driven MEMS devices have advantages of rapid response, lower power consumption, and integrated circuit standard process compatibility. Among the present MEMS devices, many are electrostatic-driven MEMS devices, such as capacitive pressure sensors [

Due to its simplicity of design and process, as well as convenience of integration with the integrated circuit processes to form a single-chip system, the electrostatic principle is commonly employed in sensing of MEMS or drive modules. However, due to the interaction between electrostatic force and structural behavior, namely the electromechanical coupling effects due to the coupling of multiple physical fields, such as stress fields and electrical fields, and since the system is nonlinear, instability of the pull-in often results, which leads to failures including stick, wear, dielectric changing, and breakdowns. Many studies have focused on common applications of electrostatic principle in MEMS devices, including: the instability when pull-in phenomenon occurs [

Accurate modeling the electrostatic microstructures is very challenging in virtue of the mechanical-electrical coupling effect and the nonlinearity of the structure and electrostatic force. Effects such as the non-ideal boundary conditions, fringing fields, pre-deformation due to the initial stresses, and non-homogeneous structures further complicate the modeling, as shown in

The aforementioned three review articles provide complete idealized models based on the assumptions of ideal fixed boundaries, homogeneous structures, and without pre-deformations, while this work provides models considering the non-ideal boundary conditions, non-homogeneous structures, and pre-deformations due to initial stresses. Over the past few decades’ development, MEMS technologies are now capable of manufacturing many microsensing or actuating components employing standard Complementary-Metal-Oxide-Semiconductor (CMOS) processes, which are the so-called CMOS-MEMS. There is a gap between prototype and commercial product that must be filled out, namely the reliability testing of MEMS devices. This is exactly what the aforementioned three review articles lack. The main benefit of CMOS-MEMS is batch production using the well-developed standard CMOS facilities. However, apart from the electrical testing of circuits, the MEMS-side still requires the mechanical testing of microsensing or actuating components. The performance of microdevices depends on the constitutive properties of the thin-film structural materials of which they are made. It is known that thin-film properties can differ from bulk material ones. As a result, certain material properties are critical in device performance, which must be monitored in manufacturing to ensure the repeatability from device to device and wafer to wafer. However, the mechanical property extraction methods available in the literature for MEMS fabrication require additional measurement and actuating equipments or complicated test structure designs, which are not compatible with standard CMOS metrology technologies. To be compatible with CMOS metrology technologies, the best choice of test and pickup signals are both electrical. In the past decade, the mechanical property extraction for MEMS by electrostatic structures was developed. The following sections present the quasi-static pull-in physical model of MEMS devices, dynamic response analysis of microstructures, air damping effects, breakdown mechanism analysis of the components, numerical simulation, and the application on inline mechanical properties extraction of microstructures.

As shown in

The abscissa is the ratio of the beam deformation (g_{0}-g) to the initial gap (g_{0}) between beam and substrate while the ordinate is the forces. The spring force is proportional to the deformation of beam while the electrostatic force, namely the electrical force, is proportional to the square of the deformation. When the drive voltage is weak, the spring force can contend with the electrostatic force and keep the system in a stable equilibrium state. However, when the drive voltage achieves a critical value, the spring force can no longer contend with the electrostatic fore and throws the beam off balance. The critical value of drive voltage is referred to as pull-in voltage. The system is under an unstable equilibrium state at pull-in. The physical model of

From 1994 to 1997 Senturia

The beam is with length _{a}_{0}, as shown in

Based on the Euler’s beam model and minimum energy method, the pull-in voltage _{PI}_{0} denoted the initial stress, and

Besides,

In the movement process, beams are affected by the interaction between electrostatic force, elasticity-restoring force, and damping force. As a result, the equation of motion in coupling is often a simultaneous partial differential equation of an electrostatic force equation, an Euler beam equation, and an air-damping equation, which explain the dynamic actions of the devices in all three spatial dimensions. It would be very difficult to solve this equation using only a numerical method. Therefore, mathematical operations, such as state-variable analysis and basic function expansion methods, are usually employed to translate a partial differential equation of infinite dimensions into a system of ordinary differential equations of finite dimensions. This is known as the reduced order method [

The dynamical pull-in voltage may be lower than the static one (

Furthermore, the instable regions of the dynamical pull-in expand with the increasing of applied voltage, as shown in the dot-area of _{1}, _{3}, and

Microstructures, which move relatively along a vertical direction, are widely used in MEMS devices, such as microaccelerometers [

Veijola [_{eff}_{0} is the coefficient of viscosity under atmospheric pressure:

In 1999, Li

The development process of a MEMS system is complicated, involving product design, manufacturing, packing and systemic integration. Like an IC circuit and its common mechanical structures, MEMS devices can use computer aided design (CAD) to facilitate their performance, reliability, reduce the development cycle and costs. The main difference is that MEMS CAD is still a work in progress. For electronic products design, the technology of electronic design automation (EDA) serves as a platform to enable circuit designers to design and analyze, with the help of a computer and model libraries provided by foundries and related design kits, in order to complete the design, development, and testing of devices in the most economic and efficient method. SPICE, SABER, and Simulink are software commonly used. For mechanical products design, MDA (Mechanical Design Automation) has numerous and large-scale common software to aid design, manufacturing, and analysis, such as IDEAS, UGII, and ProPEngineer. A linking device between EDA and MDA is required for MEMS CAD to determine multiple physical coupling effects, which increases the development difficulties of MEMS CAD. Numerical simulation of MEMS devices mainly uses numerical methods, such as the finite element method, the boundary element method, and can simulate the actions of various structural components with high accuracy. Its drawbacks are high computation complexity and low analysis efficiency. Therefore, related studies have explored how to reduce the large amount of computations. Hung [

The suspended MEMS devices can always be treated as linear lumped model under the assumption of small displacement. The whole structure can be considered as linear massless springs (flexures parts) connected to the rigid mass (the proof mass).Then, one can use the FEA tools to obtain the spring constants and the equivalent mass [

The breakdown of MEMS devices means that the devices are unable to achieve their expected functions. Zhang [

In 1992, Bart [

Electrostatic-driven MEMS devices have been widely used in various sensing and actuating, and can be used in biosensors or to extract mechanical properties of thin film materials. In the past decade, all the important discoveries on the technology of calculated mechanical properties of thin film materials came from the research team of Senturia [_{PI}

_{PI}_{PI}_{0} and _{0} represent the beam width, Young’s modulus, thickness, area inertia moment of beam cross section, beam length, and the initial stress, and

The testing technology is able to conduct inline measurements and monitoring of wafer fabrication, and uses existing semiconductor measurement equipment, as they are adequate for semiconductor and MEMS processes.

Analysis of the electro-mechanics of electrostatic-driven MEMS devices is complex due to the coupling of several energy domains. Besides, the electromechanical coupling effects will cause the pull-in instability, nonlinear response, reliability issues during the system operation. This article has reviewed related literature on electrostatic-driven MEMS devices, including a physical model of quasi-static pull-in voltage about how the physical quantities, like residual stress, elastic boundary, structural flexibility, fringing field capacitance to affect pull-in voltage, dynamic characteristic analysis about the dynamic behavior when system operates, air damping effects about the relation between air damping coefficient and geometry of structure, reliability about the failure mode and failure mechanisms of various devices, numerical modeling method about how to generate the most effective reduced model to fit the real system, and application. By the further understanding of the interaction mechanisms of these significant topics, it is helpful for developing the optimization techniques and applications in MEMS field.

The authors are thankful for the financial support of our research from the National Science Council of Taiwan through the Grant No. 97-2221-E-002-151-MY3 and NSC-98-3111-Y-076-011.

Nonlinear electromechanical coupling systems.

The electrical and spring force for voltage-controlled parallel-plate electrostatic actuator [

A discrete model of an equivalent spring and a parallel-plate capacitor [

(a) Physical model, (b) Analytical model [

(a) Frame subjected to distributed load P_{0}, (b) Free body diagram [

The first modal stiffness variation [

Comparison between dynamic instability and static pull-in [

Dynamical instable region of a microcantilever subjected to an AC voltage [

SEM of a gyroscope (a) and its lumped model (b) [

SEM of a microbeam resonator (a) and its 1st modal shape simulated by CoventorWare (b) [

Tribology issues during micromotor operation [