FINITE ELEMENT (QUASI-STATIC) MODEL OF ELECTROSTATIC MICROELECTROMECHANICAL CANTILEVER

ADRIANA LAI MOOK KIM

A thesis submitted

in fulfillment of the requirement for the degree of

Master of Engineering

Faculty of Engineering

UNIVERSITI MALAYSIA SARA W AK

2009

ACKNOWLEDGEMENT

Firstly, I would like to thank my supervisors, Assoc. Prof. Dr. Awangku Abdul Rahman

and Dr. Wallace Wong for their encouragement and guidance. I would also like to thank

my friend, Dr. William Pao for introducing the world of finite element and for his

guidance in developing the moving mesh algorithm. My heartfelt gratitude goes to my

family for their patience and support. Finally, I would like to thank my friends who

have encouraged me with their kind words and prayers without which I would not have

succeeded.

r

ABSTRACT

A finite element model has been developed to enable an initial assessment of the

analysis of an elastic-electrostatic system of an electrostatic micromechanical cantilever

(which is often referred to simply as a cantilever) for microelectromechanical systems

(MEMS) design performance and evaluation. The model comprised of an electrostatic

and a mechanical solv~. The electrostatic solver derived the electrostatic force on the

cantilever from the potential difference applied between the cantilever and a fixed

reference plate. The force is then fed to the mechanical solver that computed the

deformation of the cantilever. Solutions to the solvers were attained iteratively. It was

assumed that the system was in quasi-static equilibrium for each iterative step, as the

time taken for redistribution of electric charges on the surface of the cantilever was

much shorter in comparison to the mechanical reaction time of the cantilever. For time

saving, a "virtual neutral axis" and a 'moving mesh' algorithm were incorporated in the

modeL The "virtual neutral axis" algorithm generated the neutral axis for the cantilever

from the nodal points on the boundary of the cantilever, from the meshing of the

electrostatic domain. The "moving mesh" algorithm eliminated the need for continuous

transfer of mesh information between the solver and the mesh generator, by updating

the "legacy" nodal information from the initial mesh with the displacement vector

obtained from the mechanical solver. The model was useful to analyse and to simulate

the reaction of the cantilever due to the change in electric field, to simulate pull-in effect

and to estimate pull-in voltage. The simulation time to estimate the equilibrium position

of the cantilever was approximately 4 seconds and approximately 15 seconds to

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estimate a pull-in for a 1000-element mesh. For the simulation of a 4000-element mesh

in batch mode with 1 V voltage step, the time taken to determine the pull-in was one

and half hours. Unlike conventional finite element model, this model has a simple setup

and the parameters like voltage, tolerance and Young's modulus can be easily

controlled by the user.

111

,...-­

ABSTRAK

Suatu model unsur terhingga telah dibina untuk memberi pentaksiran pertama analisis

penilaian dan prestasi rekabentuk sistem mikroelektromekanik (MEMS) yang

berasaskan sistem elastik-elektrostatik bagi julur mikromekanik elektrostatik. Model int

terdiri daripada penyelesai elektrostatik dan mekanik. Penyelesai elektrostatik

menerbitkan daya elektrostatik dartpada beza keupayaan elektrik di antara julur dan

plat rujukan. Daya elektrostatik ini disalurkan kepada penyelesai mekanikal untuk

menentukan ubah bentuk lentur struktur julur. Hasil penyelesaian untuk penyelesai­

penyelesai int dicapai melalul proses lelaran. Sistem int diandaikan berada dalam

keseimbangan kuasi-statik bagi setiap lelaran kerana masa yang diambil untuk

pengagihan semula cas-cas elektrik adalah lebih singkat berbanding dengan masa

tindak balas mekanikal julur . Algoritma "paksi neutral maya" dan algoritma "jejaring

bergerak" telah diperkenalkan di dalam model int untuk memendekkan masa analisis.

Algoritma "paksi neutral maya" menjanakan paksi neutral untukjulur mikro daripada

nod-nod sempadan pada julur mikro; hasil daripada proses jejaring domain

elektrostatik. Algoritma "jejaring bergerak" mengelakkan keperluan proses

pemindahan maklumat jejaring yang berterusan di antara penyelesai dan pejana

jejaring unsur terhingga, dengan menggemaskinikan maklumat nod-nod "legasi"

daripada maklumat jejaring yang terdahulu dengan vektor arifakan nodyang dijanakan

oleh penyelesai mekanik. Model tni adalah berguna untuk analisis dan menyelakukan

tindak balas julur dengan perubahan medan elektrik, menyelakukan kesan "pull-in"

dan menganggarkan beza upaya "pull-in ".Masa analisis untuk menganggarkan posisi

lV

keseimbangan julur adalah kira-kira 4 saat dan masa untuk menganggarkan kesan

"pull-in" adalah kira-kira 15 saat untuk analisis jejaring 1000 elemen.Masa analisis

untuk jejaring 4000 elemen dalam "batch mode" dengan perbezaan beza upaya 1 V,

untuk menganggarkan kesan "pull-in" adalah kira-kira satu jam tigapuluh mini!.

Model unsur terhingga ini amat mudah dikendalikan and pengguna dapat menukar

nilai-nilai seperti voltan, toleransi dan modulus Young dengan senang bila

dibandingkan dengan model unsur terhingga yang sedia ada.

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TABLE OF CONTENTS

ACKNOWLEDGEMENT

ABSTRACT

ABSTRAK

LIST OF FIGURES

LIST OF TABLES

GLOSSARY

INTRODUCTION

1.1 Micro-Electro-Mechanical Systems (MEMS)

1.2 Research Motivation

1.3 Research Objectives

1.4 Research Methodology

1.5 Thesis Structure

MATHEMATICAL MODEL

2.1 Governing Equations for Electrostatics

2.1.1 Maxwell's Stress Tensor

2.2 Governing Equations of Solid Mechanics

2.3 Scaling, Mechanics and Electrostatics

FINITE ELEMENT FORMULATION

3.1 Overview of Finite Element Modeling

3.2 Discretization

3.3 Electrostatic Solver Fonnulation

i

ii

iv

viii

ix

x

1

1

7

12

12

16

17

17

18

19

20

25

25

28

33

VI

3.4 Mechanical Solver Fonnulation 36

3.5 Coupling of Electrostatic and Mechanical 41

3.6 Mesh Morphing 43

3.7 Convergence Criteria 47

3.8 Error in Approximation 49

3.9 Postprocessing 50

MODEL VERIFICATION 52

• 4.1 Introduction 52

4.2 Displacement of the Cantilever Along Its Length 52

4.3 Displacement of the Cantilever with Various Thicknesses 53

4.4 Pull-in Voltage of the Cantilever with Various Thicknesses 56

4.5 Number ofIterations to Convergence 59

CONCLUSION AND FURTHER WORK 62

REFERENCES 66

LIST OF PUBLICATIONS 72

APPENDIX A 73

APPENDIXB 74

APPENDIXC 81

APPENDIXD 94

APPENDIXE 95

APPENDIXF 97

Vll

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 2.1

• Figure 3.1 i

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure B.l

Figure B.2

LIST OF FIGURES

Page

DTUsat, a picosat created by Technical University of Denmark 3

Idealized electrostatic-elastic system 8

A schematic of an elastic-electrostatic cantilever system l3

Algorithm of the solver 15

Parallel plates that are displaced perpendicular to each other 22

Spatial geometry of cantilever 29

Triangulated electrostatic domain 30

A zoomed in view of the mesh surrounding the cantilever 30

The extrapolated virtual neutral axis 32

Beam element in local and global coordinates 40

Displacement of cantilever beam along its length 53

Displacement of the cantilever beam with different thickness along its

length with an applied potential 52Y. 54

Result ofdeflection of the cantilever beam by [37] 55

Deflection of the cantilever beam versus applied potential 57

Comparison ofpull-in voltage prediction between finite element model

and the MLPG by [13] 58

Number of iterations to convergence versus pull-in voltage 59

Integral for the region of interest 76

A triangular element 79

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LIST OF TABLES

Page

Table 1.1 MEMS products example [4] 4

Table 1.2 Comparison among various actuation/sensing methods [7] 5

Table 4.1 Normalized pull-in position versus thickness 61

Table B.1 Relation between the nodal coordinates and natural coordinates 78

ix

ASCII

CAD

CAGR

CSM

, DLP

DTUSAT

FEA

FEM

GLV

IC

MEMS

MOEMS

MST

GLOSSARY

American Standard Code for Information Interchange

Computer Aided Design

Compound Annual Growth Rate

Computational Solid Mechanics

Digital Light Processing

Denmark Technical University (Danmarks Tekniske

Universitet) Satellite

Finite Element Analysis

Finite Element Modelling

Grating Light Valve

Integrated Circuit

Micro-Electro-Mechanical Systems

Micro-Optics-Electro-Mechanical Systems

Maxwell's Stress Tensor

x

CHAPTER 1

Introduction

1.1 Micro-Electro-Mechanical Systems (MEMS)

Micro-electro-mechanical systems (MEMS) refer to an integration of

mechanical and electrical systems at the micrometer scale. It is fabricated usmg

technology adapted from the integrated circuit (IC) batch processing technology.

Richard P. Feynman first introduced the idea ofMEMS in 1959 as the "problem

of manipulating and controlling things on a small scale" [1]. However, it took some

thirty years for the fabrication technology to advance and the first MEMS device to be

successfully fabricated.

The term MEMS was only adopted by the pioneer research group headed by

Prof. R. Howe in 1989 [2]. This emerging technology is also known as Micromechanics

in Japan, Microsystems Technology in Europe and at times, it is referred to as

nanotechnology [3]. MEMS encompasses system integration of microelectronics and

micro-mechanics, micro-optics (MOEMS) and micro-magnetic.

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•

Miniaturization is the most apparent advantage ofMEMS and probably the most

important driver of its development. Miniaturization reduces cost, as it decreases

material consumption and allows mass fabrication. Moreover, it also increases

applicability, as the reduction in size allows MEMS in places where a traditional system

would not be able to fit. For example, MEMS accelerometers which was developed to

replace traditional airbag sensor, has also been incorporated into many appliances such

as digital cameras for image stabilization and contact-less game controller integrated in

the latest handphones . ,

The integration of systems in MEMS also increased reliability in the product and

reduction in assembly cost [4].

MEMS technology has also enabled scientists and engineers to build things that

have been impossible or prohibitively expensive with other technology [3]. The use of

electric field to pump reactant around a chip (electro-osmotic effect); based on the

existence of drag force in the fluid, can only work in micro-scale [4]. In space

exploration and satellite technology, Picosats,(see Figure 1.1) the lkg and (10 x 10 x

10) cm3 MEMS-based satellites are used as inspector to monitor and to verify the

motions of complicated host satellites and providing feedback to satellite operators. Its

satellite-earth communication and propulsion capabilities, coupled with its negligible

mass, size and power consumption allows it to be launched using small launch vehicles

2

or piggybacked onto conventional satellites. This provides substantial cost savings on

space launching and space mission [5].

Figure 1.1 Drusat, a picosat created by Technical University of Denmark

In telecommunications networking, optical MEMS are employed as high speed

switches with switching speeds up to terabits, and improved system efficiency, that is an

increase in bandwidth and signal integrity; by allowing optical signals to be kept as

light.

MEMS can be found, for example, in hearing aids, systems for nerve

stimulation, dosage systems for medication, implanted dosage systems and minimally

invasive surgery. MEMS components currently on the market can be classified into six

categories, namely: pressure sensor, inertia sensor, micro-fluidicslbio-MEMS, optical

3

i

MEMSIMOEMS, RF MEMS and others. Table 1.1 lists some examples of MEMS

products [4].

Table 1.1 MEMS products example [4]

Product category Examples

Pressure sensor Manifold pressure (MAP), tyre pressure, blood pressure

Inertia sensor Accelerometer, gyroscope, crash sensor

Micro-fluidicslbio- Inkjet printer nozzle, micro-bio-analysis systems, DNA chips

Optical MEMS/ Micro-mirror array for projection (DLP), micro-grating array

MOEMS for projection (GL V), optical fibre switch, adaptive optics

RFMEMS High Q-inductor, switches antenna, filter

Others Relays, microphone, data storage, toys

Today, MEMS has developed into a thriving research field, with a growing

industry and an expanding market share. In 2003, MEMS reaped revenue of USD 4.2

billion; and is forecasted to reach the USD 30.0 billion mark by year 2010 [6]. The

tabulation of the forecast according to the application types can be found in Appendix

A. MEMS overall market value is still meagre in comparison to the USD 180 billion

integrated circuit (IC) industry. However there are two aspects of MEMS that make it

very attractive [4],

a) a projected annual growth of 18% for the foreseeable future, and

b) the value of a MEMS-based systems is on average 8 times more than the MEMS

chip price.

4

) I

While market forecast for MEMS are optimistic and advance fabrication

methods has enabled fabrication of micron size mechanical parts, much is to be learned

about microns scale physics [7].

Generally, MEMS devices can be classified according to its actuating and

sensing method. The main actuating and sensing methods together with their advantages

and disadvantages are shown in Table 1.2.

Table 1.2 Comparison among various actuation/sensing methods [7J

Types of Parameter Local DC Complex Linearity Issues Sensor circuits response

Piezoresistive Strain NO YES + +++ High temperature dependence,easy

to integrate Piezoelectric Force NO NO ++ ++ High sensitivity,

complex fabrication

Electrostatic Displacement YES YES ++ Poor Very simple, low temperature coefficients

Thermal Strain NO YES + Poor Cooling problems,

interference with electronics

Magnetic Displacement NO YES +++ + Very complex post fabrication

~ Facement NO YES +++ +++ Difficult implementation

Indicators: + less complex/linear, ++ more complex/linear, +++ most complex/linear

5

) I

Piezoresistivity sensing measures the change in the electrical current due the

change in the conductivity of the piezoresistive material under stress. It is easy to

integrate but its high dependence on temperature makes its application less desirable.

Piezoelectric actuation and sensing depend on the deformation of the piezoelectric

material under the influence of a electrical potential difference and the polarization of

the material under deformation. Its lack of DC response, high temperature sensitivity,

nonlinear working zones and hysteresis and therefore limits its applicability. Thermal

deformations of materials can also be used as thermal sensing and actuation. Difficulty

in isolating temperature changes to a fixed area and the possible interferences with

control electronics and other thermally dependent elements; prevents the use of this

method. The difficulty in scaling of magnetic forces and its construction leaves

magnetic actuation with limited applications. Optical actuation and sensing, known for

its non-interfering technology faces challenges in mass fabrication, as it requires the

integration of light source, building of reflecting surfaces and aligning the whole set-up,

is time consuming and no batch-fabrication implementation exist [7].

Electrostatic actuators are made up of two parallel surfaces that are applied with

an electrical potential difference. The actuation may be brought about by the mechanical

displacement of the parallel surface due to application of mechanical force. The change

in the proximity of the surfaces causes the electrostatic field between them to change

and hence the capacitance which can be sensed electrically. Alternatively, mechanical

displacement may be generated with a change in the electrical potential between the

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surfaces. In either case, the electrostatic device is basically a capacitor, which can be

easily built using the existing fabrication methods. Moreover, the mechanical structure

of the electrostatic actuator can be built on the same chip as the electrical circuitry; and

it has low temperature coefficient. This has made electrostatic as one of the most

popular actuating and sensing method, with the trade-off in linearity, small signal

changes which requires further amplification circuitry and vulnerability to parasitic

capacitance. The performance of electrostatic actuator and sensors are depending upon

, good understanding of the phenomena that takes place between the parallel surfaces [7]. !

The wide applications of elastic-electrostatic systems in MEMS also give rise to the

need to continuously evaluate and optimize the performance, and to prevent destructive

phenomena. This leads to the demand for efficient methods for analysis of these devices

[8].

1.2 Research Motivation

One of the biggest challenges in MEMS apart from the ability to manufacture

the mechanical parts on a micron scale is to understand and control the physical systems

behaviour on these scales. This requires an understanding of electrostatic, fluid,

electromagnetic, thermal, and mechanical forces on the micron scale in order to

understand the operation and function ofMEMS devices [7]. Two of the most important

parameters of interest in the design of electrostatically driven micro electromechanical

devices are the pull-in voltage and the travel range.

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An electrostatic-elastic system consists of an electric conducting, deformable

elastic structure, and a fixed reference structure, on which potential difference between

the elastic structure and fixed reference structure is applied. The dielectric medium,

usually air, fills the gap between them [9]. An idealized elastic- electrostatic system is

shown in Figure 1.2.

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Figure 1.2 Idealized electrostatic-elastic system

The electrical potential difference between the two parallel plates would give

nse to an attractive electrostatic surface force, IE on the surface of the elastic

deformable structure, which is the micromechanical cantilever, pulling it towards the

fixed reference structure as shown in Figure 1.2. This force applied stress on the elastic

deformable structure giving rise to a restoring elastic force,IM • At the equilibrium

8

\

1

position of the elastic deformable structure for an applied electrical potential, the

restoring elastic force balances the attractive electrostatic surface force.

Pull-in is a phenomena caused by the difference in the rate of increase of the

elastic and electrostatic forces [10]. At potential differences below the pull-in threshold,

the change in electrostatic force on the elastic structure is balanced by the increased in

the elastic force. As potential difference increases, the displacement of the elastic

structure would increase; hence reducing the gap between the structure and the

reference potential plate, (g w) (see Figure 1.2). While the elastic force Increases

linearly with displacement (Equation (1-1», the electrostatic force Increases

proportionally to the inverse square of distance (Equation (1 -2»).

1M =Kw (1-1 )

where 1M is the elastic force of the restoring elastic force of the deformable

structure, in newton (N),

K is the elastic constant, in newton per meter (N/m), and

w is displacement ofthe deformable structure, in meter (m).

9

,..

(1-2)IE =kE (g-w)2

where IE is the electrostatic surface force on the deformable structure, in

newton (N),

k E is the electrostatic constant, in newton meter square per coulomb

V is the potential difference between the deformable structure and the

reference plate, in volt (V), and

(g - w) is the gap between the deformable structure and the reference

plate, in meter (m).

When the potential difference increased to a point where the growth of the

electrostatic force exceeds that of the elastic force, the system will not be able to reach a

point of equilibrium of forces; and the elastic structure is pulled towards the fixed

reference structure [10].

The estimation of pull-in voltage and the maximum possible displacement (also

known as the travel range) [11] of the micromechanical cantilever before pull-in is

crucial in the design of electrostatically actuated MEMSs. In some systems, such as

micromirrors and micro-resonators, the designer avoids pull-in in order to achieve stable

10

motions, while in switching applications the designer exploits this effect to optimize the

performance of the device [9].

Numerous models have been proposed for the study of the behaviour of the

electrostatic-elastic system; particularly focusing on the response of the elastic system to

the change in electric field and the pull-in phenomenon (see [10 - 14]).

The conventional finite element model implementation is a two-step process

consisting of dividing the electrostatic and mechanical domains into finite, non­

overlapping subdomains (also know as elements) which are joined together by boundary

points known as nodes. This process is known as meshing and it is carried out by

dedicated software called mesh generators. Mesh generators would generate a set of

information containing the meshed domains, for instance the number of nodes, nodal

coordinates, and number of elements, nodal connectivity, element type, and element

grouping. This information is then fed to a solver in which the mathematical solution to

the electrostatic and mechanical model is computed iteratively. The information is then

used to update the geometrical information of the mesh and is then transferred back to

the mesh generator where the domain is updated and re-meshed. The conventional full

finite element model therefore is time consuming and setting up the data files for the

analysis is tedious. Moreover, the response of the micromechanical cantilever is

dependent on its physical dimensions and its mechanical properties, such as Young's

modulus and Poisson's ratio.

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1.3 Research Objectives

The objectives of this research are to develop a simplified finite element model

for an elastic-electrostatic analysis and to develop a moving mesh algorithm, to enable a

fast and accurate modelling of multi-physics system (mechanical-electrostatic). The

model would also have the ability for batch voltage processing as well as single input

voltage processing, and allow the user to change the Young's modulus, the maximum

iterative steps as well as the width of the micromechanical cantilever. The research will

focus on:

• Development of an electrostatic solver,

• Development of a mechanical solver,

• Integration of the electrostatic and mechanical domain,

• Development of a virtual neutral axis algorithm to replace the meshing of the

mechanical domain, and

• Development of a moving mesh algorithm to replace conventional re-meshing

after every iterative step.

1.4 Research Methodology

The mechanical structure chosen for the model is a cantilever, with one fixed

end and one free moving end; suspended parallel to a fixed plate. The potential of the

fixed plate is held at zero electrical potential and the electrical potential on the surface

.."....-.

I \

l ,

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of the cantilever is fixed. This model was chosen because it is the common setup in the

electrostatic MEMS devices. The schematic setup for the elastic-electrostatic cantilever

system is as shown in Figure 1.3.

----'--v

T where QB: the mechanical domain (cantilever) QE: the electrostatic domain r B: the surface of the cantilever r G: the surface of the fixed plate

Figure 1.3 A schematic of an elastic-electrostatic cantilever system

The model consists of three main components:

• Solving the electrostatic field from the applied electric potential,

• Calculating the electrostatic force from the electric field vector, and

• Applying the electrostatic force on to the Navier's equation and solving for the

displacement vector of the cantilever.

Finite element method was adopted for the analysis of both the electrostatic and

mechanical domain. One-dimension (I-D) model was used to solve for the displacement

of the mechanical structure and a two-dimensional (2-D) model was used to solve for

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