Some Design Considerations on the

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model uses Euler-Bernoulli beam theory for fixed-fixed end type beams. ... Normally, the electrostatic force is approximately proportional to the inverse of the ...
Institute of Physics Publishing doi:10.1088/1742-6596/34/1/029

Journal of Physics: Conference Series 34 (2006) 174–179 International MEMS Conference 2006

Some Design Considerations on the Electrostatically Actuated Fixed-Fixed End Type MEMS Switches

Hamed Sadeghian Mech. Eng. Dept., Urmia university,Urmia, Iran, [email protected] Ghader Rezazadeh Mech. Eng. Dept., Urmia university,Urmia, Iran [email protected] Ebrahim Abbaspour Sani Elec.Eng. Dept., Urmia university, Urmia, Iran [email protected] Abstract. The nonlinear electrostatic pull-in behaviour of MEMS Switches in micro-electromechanical systems (MEMS) is investigated in this article. We used the distributed model when the electrostatic pressure didn’t apply at the whole of the beam and applied only in the mid-part of the beam. In this part the electrostatic area is different from two other parts. The model uses Euler-Bernoulli beam theory for fixed-fixed end type beams. The finite difference method was used to solve the nonlinear equation. The proposed model includes the fringing effects of the electrical field, residual stress and varying electrostatic area effects. The numerical results reveal that the profile deflection of the MEMS Switch may not only influence the distribution of the electrostatic force but also considerably change the nonlinear pull-in voltage. Keywords: MEMS Switches; pull-in voltage, fringing field, residual stress

1. Introduction Today because of the advantages of electrostatic actuators, such as, favourable scaling property, lower driving power, large deflection capacity, relative ease of fabrication, and others, have led their being more widely applied the electrostatic-actuator applications in micro-electromechanical systems (MEMS). The MEMS switch is one of the most important devices in such systems. Fixed-fixed beams under voltage driving are widely used in many MEMS sensors and actuators, including MEMS switches [1]. These MEMS devices are relatively simple to design and fabricate as well as to integrate on a chip with CMOS circuits. However, voltage driving may exhibit an inherent instability situation, © 2006 IOP Publishing Ltd

174

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known as the pull-in phenomenon. This effect may be either derogatory or useful, depending on the application. On the other hand, with the growth of micromachining process technologies for Microelectromechanical systems (MEMS), there has developed a need for simple, accurate, and standardized process monitoring and material property extraction capability (e.g., Young’s modulus, residual stress, and fracture strength) at the wafer level, fixed-fixed beams under the voltage driving are commonly used for test structures. It is therefore necessary to study the pull-in phenomenon of the voltage driving fixed-fixed beams and develop advanced models of pull-in voltage. Previously, static analyses have been performed to determine the beam equilibrium position for a given voltage using both lumped parameter and finite difference methods [2-4], but these models cannot capture exact model, such as fringing field, residual stress and varying section effects. An analytical expression of the pull-in parameters is given about the MEMS switches in [5]. A lumped two degrees of freedom pull-in model is presented in [6] for a direct calculation of the electrostatic actuators. The effect of fringing field has great influence on the behaviour of MEMS switches, and even causes the failure of devices [7]. Also the residual stress is very important and inevitable to the device. The pull-in phenomenon widely exists in many micromachined devices that require bi-stability for their operation, such as MEMS switches [8]. The present paper aims to study the behaviour of MEMS switches considering the fringing field effect, residual stress and variative electrostatic area.

2. Modelling of MEMS Switches Figure 1 shows a MEMS switch with fixed–fixed beam suspended above a ground plane. During the application of a voltage to the deformable beam and a fixed electrode, a position-dependent electrostatic force distribution is created and pulls the deformable beam toward the curved electrode. An isolation layer or other structure is required to prevent a short circuit following pull-in contact. In our nonlinear distributed model electrostatic pressure is applied only to a certain part of a beam and in this part the section is different from other sections. In proposed model, we compare different electrostatic areas (20%, 30% and 40%) and their effect in pull-in voltage. In addition we considered that the area of the beam above substrate as shown in figure 1 is different from the other two sections.

Figure 1- The schematic diagram of the fixed-fixed end type MEMS switches with varying section When a driving voltage is applied between the electrodes, the electrostatic force deflects the fixedfixed beam. Normally, the electrostatic force is approximately proportional to the inverse of the square of the distance between the electrodes. When the voltage exceeds the critical voltage, the fixed-fixed beam is suddenly pulled into the electrode. The micro switch configuration is modeled using EulerBernoulli beam theory with a constant cross-sectional area along the length of the each part of the beam. The nonlinear electromechanical coupled governing differential equations for a fixed–fixed beam, as shown in figure 1, enhanced with a first-order fringing correction and residual stress is presented in [9].

176

~ d 4u d 2u EIˆ2 4  Tr 2 dx dx

q( x)

(1)

where u(x) is the deflection of the beam, E is the effective Young’s modulus, for which w t 5t

the

~ ~ effective modulus E can be approximated by the plate modulus E \ (1  Q 2 ) ; otherwise E is Young’s modulus E and I 1 , I 2 the effective moment of inertia of the cross-section which are wide relative to their thickness and width, and equals to

w t3 w1t 3 and 2 respectively, t is the film thickness, w1 and 12 12

w2 is the width of each part of the beam, a, is the length of left and right part of the beam, L is the length of the beam and q(x) is electrostatic pressure applied per unit length of the beam. H 0V 2 w2

q ( x)

2[G  u ( x)] 2

[ H ( x  a ))  H ( x  ( L  a ))](1  f f )

(2)

where H(x) is Heaviside function and V is the applied voltage between the movable/ground plates on the fixed substrate, G is the initial gap between the movable/fixed plates and H 0 is the permittivity of air and f f is the fringing field correction. A uniform magnetic field cannot drop abruptly to zero. In actual situation, there is always a ‘‘fringing field’’ existing, and a more realistic Situation including ‘‘fringing field’’ modification. If fringing field effect is taken into account, the first order fringing-field correction [10, 11] is denoted as: ff

0.65

G u w

(3)

Residual force can be expressed as : Tr

Vˆwt

where Vˆ

(4) V 0 (1  Q ) for fixed-fixed beam, and V 0 is the biaxial residual stress [10].

3. Pull-in analysis Due to the nonlinearity in the electrostatic pressure, an analytical solution is impractical to obtain and a numerical solution is sought. The solution to (1) is found using standard central finite difference approximations for 4th order derivative of u(x). The development of the finite difference model can be divided into four components. The Euler-Bernoulli beam equations are combined with the electrostatic pressure equation and with specially developed fringing field, residual stress and variative electrostatic area effects. From the point of view of analysis, this configuration presents some challenges. First the beam now consists of two different cross-sectional areas. In order to adapt the model to analyze a twosection geometry using finite differences, four artificial nodes, two on each side of the interface between the two sections, are required. There are four additional continuity conditions which state that the displacement, slope, bending moment, and shear force are continuous across this interface. Increment the applied voltage until pull-in occurs and relies on a discretized nodal array. The four spatial boundary conditions needed to find a solution are determined from the requirement that two end of the beam are fixed (zero displacement and zero slop).These conditions allow (1) to be expressed

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solely in terms of the nodal deflections ( u i , i = 1, 2,…N), where node 1 is located at the one end of the beam and node N is located at the other end of beam. The grid points represented by the displacements u 0 , u N 1 and u N  2 are image points needed for the finite difference approximations of the boundary conditions and field equations. The schematic diagram of the varying section fixed-fixed end type MEMS switch is shown in figure 1 By using of the energy method governing equation for the spring constant of the varying section fixed-fixed end type MEMS switch can be derived. For the fixed-fixed end type beam, by assuming no residual stress and concentrated force the spring constant KVF of the varying section fixed-fixed end type MEMS switch can be expressed as:

KVF

M

6 [ (12M a  a  6M a )  ] (12M 2b  b3  6M b 2 ) 2

a 2[  b 2] , 4(a [  b ] )

3

K

(5)

2

1 , EI1

]

1 , EI 2

[

K ] ,

(6)

Parameters used in this article were: Young’s modulus E was 169 GPa, the Poisson’s ratio u was .3, the length of beam was 800 µm, the width of the actuator b was 50 µm, the thickness t was 14.4 µm and the initial gap g was 1 µm, the Permittivity of air is 8.8541878u10-12 (F/m) and the width of electrostatic area is 100 µm. 4. Numerical Results and Discussion By considering the different length of the electrostatic area and residual stress and fringing field effects is observed that the exact pull-in voltage is different from the results of lumped models. Table 1 shows the results of the finite difference method and the lumped model. As it is indicated Table 1 there is a difference between results of the two models. There is an important phenomenon that with increasing the electrostatic area, pull-in voltage doesn’t decrease, because with increasing the electrostatic area the spring constant increases too. The displacement versus voltage characteristic for this geometry is shown in figure 2. These results are shown in table 1. (b is length of the electrostatic area).

Figure 2- Gap versus Vaoltage

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Table 1: pull-in voltage and center gap Model Type

Pull-in voltage Without effects

Pull-In Voltage With effects

Center Gap

Lumped Model

36.3221

36.3221

.6667

b=20%l

40.0508

40.4451

.6019

b=30%l

40.0818

40.5218

.6022

b=40%l

40.1253

40.9123

.6016

By solving governing equation, the deflection of the beam at pull-in voltage and normal voltage are plotted. Figure 3 shows the gap versus discretize beam. O is the non-dimensional applied voltage and equals

V V pull in

and are 0.8, 0.6 and 0.4. If the voltage increases up to pull-in, the beam will be

unstable and a large difference occurs between the deflection of O

0.8 and O

1.

Figure 3-deflection of fixed-fixed beam. Now by increasing the electrostatic area without increasing the stiffening of the beam, the pull-in voltage is decreased. Table 2 shows these results. It shows that in order to decrease the pull-in voltage by increasing the electrostatic area, the stiffening of the beam should be fixed. Table 2: pull-in voltage and center gap in different model by fixing stiffening of the beam Type Model

Pull-in voltage Without effects

Pull-In Voltage With effects

Center Gap

Lumped Model

36.3221

36.3221

.6667

a=20%l

30.4778

32.6567

.6023

a=30%l

29.0709

30.0567

.6022

a=40%l

27.5623

27.5707

.6024

5. Conclusions A comprehensive methodology to simulation model has been presented. The nonlinear coupled electromechanical differential equation with varying electrostatic area and tensile residual stress and

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fringing field effects has been investigated. Results have showed that, when the tensile residual stress is considered, the pull-in voltage is increased.By considering the first order fringing field effect has been shown the pull-in voltage is increased. The results solved for the proposed model reveal that the designed varying section electrodes and may substantially change the distribution of the electrostatic field, as well as the static response and the nonlinear pull-in behavior. The accurate based on our new approach will facilitate designers to select various parameters and influential factors within a wide range in their designs.

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