Review on the Modeling of Electrostatic MEMS

1. Introduction

Figure 1. Nonlinear electromechanical coupling systems.

2. Development of Related Studies on Electro-Mechanics for MEMS Devices

2.1. Physical Model of Quasi-Static Pull-in Voltage

As shown in Figure 1, when there is an electrical potential difference between the beam and substrate, namely the drive voltage, the electrostatic attractive force will attract the beam downward while the elastic restoring force of the beam, namely the spring force, will restore the beam upward. In brief, the electrostatic attractive force and the elastic restoring force are kept in an equilibrium state. However, the deformation of the beam would cause a distribution change in its surface charges; thus, the electric field would redistribute and rework on the beam until a new equilibrium state is achieved. Therefore, Figure 1 is a nonlinear electromechanical coupling system. Figure 2 shows the variations of electrostatic force and spring force versus to the deformation of the beam.

Figure 2. The electrical and spring force for voltage-controlled parallel-plate electrostatic actuator [93].

The abscissa is the ratio of the beam deformation (g₀-g) to the initial gap (g₀) 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 Figure 1 can be modeled by the analytical model shown in Figure 3 which consists of a parallel-plate capacitor suspended by a spring. The pull-in occurs when the deformation exceeds the one third of the initial gap of the parallel-plate. Thus, the key of analyzing electro-mechanics for MEMS devices is the study of the quasi-static pull-in properties.

Figure 3. A discrete model of an equivalent spring and a parallel-plate capacitor [80].

From 1994 to 1997 Senturia et al. published a series of research findings on electro-mechanics for MEMS devices. In 1994, Senturia [94] simulated a microbridge-shaped beam driven by electrostatic force as a lumped model of an equivalent spring and a parallel plate capacitor (Figure 3) in order to achieve the fundamental function forms of pull-in voltage, geometric dimension, and material parameters. Then in 1997, they proposed M-Test [8] technology, using semiconductor process technology to create three different micro test structures, namely, the microcantilever, the microbridge-shaped beam, and the microsector plate. They also produced micro- beams of different lengths to measure their pull-in voltage, respectively, in order to generalize the correcting factors based on those measured data and simulated numbers, and eventually modified the functional form based on the discrete model. In 2002, Pamidighantam et al. [19] simulated the system as a discrete system of an equivalent spring and a parallel plate capacitor. The stiffness of the equivalent spring and equivalent area of the parallel plate capacitor were obtained by CoventorWare, a commercial simulation software, in order to gain the pull-in voltage relation of the micro-bridge-shaped beams; however, the deviation was as high as 18%. In 2003, O’Mahony et al. [22] also analyzed the microbridge-shaped beams subjected to electrostatic loads using CoventorWare, and inferred the numerical solution of the microbridge-shaped beams by taking fringing capacitance effect, the efficiency of the plate-like phenomenon, and different boundaries into consideration. In 2005, Lishchynska et al. [95] derived the numerical solution of the pull-in voltage of cantilever beams, using the CoventorWare simulation software, and achieved an error within 4%. From 2004 to 2008, Krylov et al. published a series of research findings [7,12–14] on pull-in behavior of microstructures. In 2004, Krylov et al. [14] studied the transient nonlinear dynamics of microbeams subjected to electrostatic force, and developed a model based on the Galerkin procedure with normal modes which considered the effects about the distributed nonlinear electrostatic forces, nonlinear squeezed film damping, and rotational inertia of a mass carried by the beam. In 2006, Krylov et al. [13] developed simple expressions for electrostatic pressure with higher order corrections, mainly related to the curvature and slope of the electrode by using the perturbation theory. The results showed that tuning the ratio of the mechanical pressure, the string was with different pull-in behavior. In small initial pre-stress case, bistability of the string occurred. In 2008, Krylov et al. [7,12] reported on theoretical and experimental investigation of a multistability phenomenon in initially curved clamped-clamped microbeams subjected to a distributed electrostatic force. The results showed that the pull-in voltage of clamped-clamped with initially curved flexible was lower than a straight beam. From 2006 to 2009, Hu and co-workers proposed an approximate analytical model, taking into consideration such elements as initial stress, fringing capacitance effect, and elasticity boundaries [29,34,35,96,97]. For example, reference [34] presented an approximate analytical model to the pull-in voltage of a microbridge with elastic boundaries. The elasticity boundaries of the microbridge can be treated as a beam with torsional spring at both ends, and the conceptual diagram of a microbridge is shown in Figure 4.

Figure 4. (a) Physical model, (b) Analytical model [34].

The beam is with length L, anchor height L_a, width b, thickness h, and initial gap g, which subjected to a driving voltage V, resulted in a position-dependent deflection w(x). The equivalent torsional spring constant k can be expressed by the following equation [34]:

$$k=\frac{M}{\theta }=\frac{4E({2I}_a^{2}L+I\cdot I_a{L}_a)}{{2I}_a{L}_{a}L+I_a{L}_a^2}$$

(1)

where I denoted the area moment of inertia, the subscript a denoted the anchor, and E denoted Young’s modulus. The reactive bending moment M, and the rotation angle θ were derived from a frame subjected to a uniform distributed load P₀, as shown in Figure 5:

$$M=\frac{P_o{L}^2}{12\left(1+0.5\frac{I}{I_a}\frac{L_a}{L}\right)}$$

(2)

$$\theta =\frac{P_o{L}^2}{24E\left(2\frac{I_a}{L_a}+\frac{I}{L}\right)}$$

(3)

Figure 5. (a) Frame subjected to distributed load P₀, (b) Free body diagram [34].

Based on the Euler’s beam model and minimum energy method, the pull-in voltage V_PI of a micro-bridge was [34]:

$$V_{\mathit{PI}}^2=\frac{{\sigma }_0{\int }_0^{L}2\mathit{bh}{({\varphi }^{\prime })}^2\mathit{dx}+E{\int }_0^{L}2I{({\varphi }^{″})}^2\mathit{dx}+({8\mathit{EI}}_a/L_a){({\varphi }^{\prime })}_{x=L}^2}{\epsilon (c_1+{2c}_2{\eta }_{\mathit{PI}}+{3c}_3{\eta }_{\mathit{PI}}^2)}$$

(4)

where σ₀ denoted the initial stress, and ϕ was first natural mode of a beam having torsional spring at both ends given by [132]:

$$\varphi (x)=\left[\text{sin}\left(\frac{\beta x}{L}\right)-\text{sinh}\left(\frac{\beta x}{L}\right)\right]+\gamma \left[\text{cos}\left(\frac{\beta x}{L}\right)-\text{cosh}\left(\frac{\beta x}{L}\right)-\frac{{\beta L}_a}{2L}\text{sinh}\left(\frac{\beta x}{L}\right)\right]$$

(5)

$$\gamma =\frac{\text{sinh} \beta -\text{sin} \beta }{\text{cos} \beta -\text{cosh} \beta -\frac{{\beta L}_a}{2L}\text{sinh} \beta }.$$

(6)

Besides, β should satisfie the following equation:

$${\left(\frac{\mathit{kL}}{\mathit{EI}}\right)}^2+\frac{\mathit{kL}}{\mathit{EI}}\frac{2\beta (\text{sin} \beta \text{cosh} \beta -\text{cos} \beta \text{sinh} \beta )}{1-\text{cos} \beta \text{cosh} \beta }+\frac{{2\beta }^2\text{sin} \beta \text{sinh} \beta }{1-\text{cos} \beta \text{cosh} \beta }=0.$$

(7)

Equation (4) shows an obvious physical meaning. The first term shows that the pull-in voltage was dependent on initial residual stress, the second one was dependent on beam flexibility, and the third one was dependent on elastic boundary condition. The findings had greater physical significance than the previous studies we motioned above, which employed numerical methods in studies of electro-mechanics for MEMS Devices.

2.2. Microstructure Dynamic Response Analysis

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 [75,98]. Younis et al. [75] presented a reduced-order model to analyze the behavior of microbeams actuated by electrostatic force. The model was obtained by discretizing the distributed-parameter system using Galerkin procedure into a finite-degree-of-freedom system, which considered the effects about moderately large deflections, dynamic loads, linear and nonlinear elastic restoring forces, the nonlinear electrostatic force generated by the capacitors, and the coupling between the mechanical and electrostatic force. However, under the reduced order method, it would be very difficult to analyze the dynamic actions of the devices when air damping is taken into consideration; while other related parameters are still obtained through the numerical method, thus, analysis efficiency has not been improved. To solve this problem, Clark [99] determined an effective method for analyzing the dynamic actions of MEMS devices, by which the complicated system is divided into several basic structural units, and then the equivalent circuit model, consisting of those basic structures, is built using the simulation protocols between similar systems. Wen [100] employed a model analysis method based on linear disposal near the bias point in order to analyze the AC small signal in frequency domain of beams. Zhang [101] set an electrostatic-driven cantilever beam equivalent to a single-degree-of-freedom model to conduct its analytical form, and used a feedback mechanism to realize the coupling. Time and frequency domain analyses were conducted. In the time domain analysis, stronger driving voltage leads to more obvious beams overshoot. In the frequency domain analysis, the natural frequency of the cantilever beams would gradually reduce with an increase of voltage. Hu [80] established an analysis model for dynamic characteristics and stability of electrostatic-driven devices, and found that the stiffness of a microstructure will be softened periodically with the frequency of applied voltage. The variation of stiffness increases with the magnitude of applied voltage (Figure 6).

Figure 6. The first modal stiffness variation [80].

The dynamical pull-in voltage may be lower than the static one (Figure 7).

Figure 7. Comparison between dynamic instability and static pull-in [80].

Furthermore, the instable regions of the dynamical pull-in expand with the increasing of applied voltage, as shown in the dot-area of Figure 8. If the structure acts in small deformation condition, the beam can be considered as linear. On the contrary, for large deformations, non-linear structural effects have to be considered. For solving the non-linearity problem, one can use the finite element analysis (FEA) to obtain a non-linear cubic term. Moreover, it would be helpful to predict non-linear mechanical stiffness behavior [71,72,102,103]. The governing equation of a vibrating system with cubic stiffness non-linearity can be written as [50]:

$$m\ddot{x}+k_{1}x+k_3{x}^3+c\dot{x}=f(\upsilon )$$

(8)

where c, f(v), m, k₁, k₃, and x represent the equivalent damping coefficient, the equivalent external periodical force depending on drive voltage v, the equivalent mass, the equivalent linear stiffness coefficient, the equivalent non-linear stiffness coefficient, and the deformation respectively.

Figure 8. Dynamical instable region of a microcantilever subjected to an AC voltage [80].

2.4. Numerical/CAD Methods

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 [113] examined the methods to define grids, which are able to generate the most effective reduced model during the process of devices’ working. Stewart [114] developed a set of simulation methods, which can be used for microstructures in small vibraction situation. Swart [115] invented a computer-aided software, named AutoMM, which can automatically produce dynamic models for microstructures. Commercial FEA/BEA tools usually used for the MEMS design, like ANSYS, ABAQUS, Maxwell, CoventorWare, CFDRC, IntelliCAD, CAEMEMS, SESES ,and SOLIDIS. For example, Figure 9 shows the SEM of a gyroscope and its lumped model.

Figure 9. SEM of a gyroscope (a) and its lumped model (b) [50].

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 [118]. Besides, one can use these tools to solve the governing equations with given boundary conditions to know the mode shape of microstructures (Figure 10).

Figure 10. SEM of a microbeam resonator (a) and its 1st modal shape simulated by CoventorWare (b) [50].

2.5. Breakdown Mechanism Analysis of Microstructures

The breakdown of MEMS devices means that the devices are unable to achieve their expected functions. Zhang [117] discussed major breakdown modes and breakdown mechanisms of various MEMS devices, such as microswitches, micromotors, and comb drivers [118–124]. Mariani [87–92] studied the breakdown mechanism of multi-scale structures subjected to drop impacts using a finite element method. Komvopoulos [125] proposed that the major breakdown modes of MEMS devices are friction, wear, and stick (Figure 11), which should be avoided to improve the reliability of the devices.

Figure 11. Tribology issues during micromotor operation [38].

In 1992, Bart [126] proposed a dynamic model that involved analysis of kinetic friction, and found that the roughness of the surface has great impact on the friction force of the devices. Tai [127] established a model for friction movement to calculate the static friction coefficient of the devices, and suggested that due to the narrow gap between the rotor and the hub, contact wear is easily produced. Gabriel [122] found that the hub would be dramatically corroded or deformed at high speeds as the wear shortens the life span of the devices and limits their performance; however, a bracing structure could be adopted to avoid wear. Stick between the contact faces could prevent the devices from repeated operation or even working. Spengen [118] reported that the surface roughness of devices is a critical element that affects stick. Ren et al. [128] employed self-assembled monolayers (SAMs) to reduce the stick of devices surfaces.

3. Extracting the Mechanical Properties Utilizing Electromechanical Behavior of the Microstructures

4. Conclusions

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