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MECHANICAL DYNAMICS AND THERMALLY-INDUCED

INTERMODULATION IN AN OHMIC CONTACT-TYPE MEMS SWITCH FOR RF AND MICROWAVE

APPLICATIONS

A Thesis Presented

by

Zhijun Guo

to

The Department of Electrical and Computer Engineering

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in the field of

Electrical Engineering

Northeastern University Boston, Massachusetts

August, 2007

Table of Contents

Page ii

HTable of Contents

HTable of Contents .............................................................. ii

Abstract.............................................................................. v

List of Figures.................................................................. vii

List of Tables .................................................................. xiii

Acknowledgement .......................................................... xiv

Chapter 1. Introduction.................................................... 1

Chapter 2. Background of RF MEMS Switch................ 3

2.1 History and Development of MEMS Technology............................3

2.2 RF MEMS Switch ............................................................................5

2.2.1 Operation and Category of RF MEMS Switch ...............................................................................5

2.2.2 Performance and Characteristics of RF MEMS Switches ............................................................11

2.2.3 Applications..................................................................................................................................12

2.2.4 Failure Mechanisms and Reliability Issues...................................................................................14

References ............................................................................................18

Chapter 3. Mechanical Dynamics of a MEMS Switch 22

3.1 Dynamic Response of MEMS Switch ............................................22

Table of Contents

Page iii

3.2 Finite Element Analysis (FEA) ......................................................26

3.3 Lumped Parameter Modeling of a Cantilever Beam ......................27

3.4 Geometry of the Microswitch.........................................................30

3.5 Finite Element Modeling................................................................32

3.6 Electrostatic Actuation ...................................................................33

3.7 Squeeze-Film Damping ..................................................................34

3.8 Effect of Perforation .......................................................................39

3.9 Nonlinear Contact Model with Adhesion .......................................43

3.10 Dual-Pulse Scheme for Actuation ................................................45

3.11 Results and Discussion .................................................................50

3.11.1 Simulation Results ......................................................................................................................50

3.11.2 Comparisons Between Experiments and Simulations ................................................................58

Chapter 4. Intermodulation Distortion......................... 70

4.1 Intermodulation Effect....................................................................71

4.2 Theoretical Analysis of Intermodulation Distortion.......................74

4.3 Thermally-Induced PIM in MEMS Switch ....................................77

4.4 Design of a Model System .............................................................80

4.4.1 Design Considerations ..................................................................................................................80

4.4.2 Microfabrication ...........................................................................................................................81

4.4.3 Mathematical Analysis .................................................................................................................84

Table of Contents

Page iv

4.5 Results and Discussion ...................................................................96

4.5.1 Model Predictions.........................................................................................................................96

4.5.2 Static and Transient Electrical Resistance ....................................................................................98

4.5.3 Comparison Between Experiment and Simulation .....................................................................102

4.5.4 Prediction of Intermodulation in an RF MEMS Switch .............................................................106

References ..........................................................................................109

Chapter 5. Summary and Future Work ..................... 112

5.1 Dynamic Simulation.....................................................................112

5.2 Intermodulation Distortion ...........................................................114

Appendix A.................................................................... 117

Abstract

Page v

Abstract

RF MEMS switches have demonstrated superior electrical performance compared

with semiconductor switches. However, the failure mechanisms of the microswitch are

not yet fully understood.

We first developed a full dynamic model based on the built-in capabilities of

ANSYS® in combination with a finite difference method for squeeze-film damping. The

model includes the real cantilever structure, electrostatic actuation, the 2-D non-uniform

squeeze-film damping effect, and a nonlinear spring to model the contact tip impact on

the drain.

Meanwhile, we developed an analytical model for designing a dual-pulse

actuation scheme for the microswitch in an effort to optimize its dynamics during

operation, i.e. fast closing, minimum bouncing and oscillation, and gentle contact or

reduced impact force. Simulation results show that switch bounce has been dramatically

reduced or completely eliminated by using the open-loop dual-pulse actuation method.

Moreover, the impact forces have also been reduced as a result of the reduced velocity on

initial contact. The experiment is consistent with the simulation. However, it is found that

the reduction in bounce is very sensitive to the pulse voltages and the times of the dual-

pulse.

Second, the thermally-induced intermodulation distortion has been investigated

both theoretically and experimentally in a test structure. It is shown that the thermally-

induced intermodulation distortion can be predicted from the device geometry, the

thermal and electrical conductivities of the materials, and the difference frequency of a

Abstract

Page vi

two-tone input signal. The intermodulation is largest in the low difference frequency

limit. As the difference frequency is increased to a value which is comparable to the

reciprocal of the thermal time constant of the device, the intermodulation distortion starts

to decrease rapidly, approaching zero at high difference frequencies. In the high

frequency regime, the thermal conductivity of the substrate is the dominant material

property for intermodulation distortion.

The predictions agree well with the experimental measurements. The derived

intermodulation formulations have also been applied to an Ohmic contact RF MEMS

switch. The resulting technique can be conveniently used to predict the thermally-induced

intermodulation and provide guidelines for reducing it in MEMS, NEMS or other

devices.

List of Figures

Page vii

List of Figures

Figure 2-1 An example of a typical three terminal MEMS switch..................................... 6

Figure 2-2 A metal-to-metal contact-type RF MEMS switch ............................................ 9

Figure 2-3 (a) An example of capacitive MEMS RF switch and (b) the electrical CRL

circuit ................................................................................................................................ 10

Figure 2-4 Schematic representation of switches in a series and shunt configuration ..... 10

Figure 2-5 (a) and (b) broadside MEMS switches, (c) inline MEMS switch ................... 11

Figure 3-1 Dynamic behavior of a RF MEMS switch, the step curves are for the step

voltage for actuation. The traces are recorded using oscilloscope which show the transient

‘in contact’ and ‘out of contact’ after actuation [see Reference (3)] ................................ 25

Figure 3-2 Side view of a typical cantilever beam ........................................................... 27

Figure 3-3 The lumped mechanical model for a cantilever beam. ................................... 28

Figure 3-4 Gap of the cantilever vs. applied voltage ........................................................ 29

Figure 3-5 The electrostatic force and spring force vs. normalized gap for a voltage-

controlled electrostatic actuator. ....................................................................................... 30

Figure 3-6 SEM micrograph of the Northeastern University MEMS switch. .................. 31

Figure 3-7 The top view as well as the dimensions of the Northeastern University RF

MEMS switch where w1 = 80 µm, w2 = 10 µm, w3 = 16 µm, w4 = 30 µm, L1 = 30 µm and

L2 = 24 µm. ....................................................................................................................... 32

Figure 3-8 The side view of the microswitch where h1 = 6.3 µm, h2 = 0.6 µm and h3 =

0.38 µm. ............................................................................................................................ 32

Figure 3-9 Grid of finite elements of half of the switch for ANSYS® simulation. .......... 33

List of Figures

Page viii

Figure 3-10 Electrostatic force between two parallel plates ............................................. 34

Figure 3-11 Schematic representation of the finite difference method............................. 38

Figure 3-12 The displacement of the microswitch contact tip vs. the contact force. ....... 45

Figure 3-13 (a) Lumped spring-mass system, (b) a typical profile for a dual-pulse

actuation method, and (c) the desired gradual close for a dual-pulse actuation ............... 46

Figure 3-14 The relationship between the contact force, where ta is the actuation time, ton

is the turn-on time, and Fa is the applied force. Note that ta and ton are normalized to the

period of the first natural frequency, and Fa is normalized to a force Fth which

corresponds to threshold voltage. ..................................................................................... 49

Figure 3-15 The actuation time, ta, and the turn-on time, ton, for a dual voltage pulse

method as a function of actuation voltage Va. Note that ta and ton are normalized to the

period of the first natural frequency, and Va is normalized to the threshold voltage........ 49

Figure 3-16 Contact tip displacements of the switch at actuation voltages of (a) 70V, (b)

74V, and (c) 81V............................................................................................................... 51

Figure 3-17 The simulated contact tip velocity as a function of time for an actuation

voltage of 81V................................................................................................................... 52

Figure 3-18 The top view as well as the dimensions of the Northeastern University RF

MEMS switch where w1 = 80 µm, w2 = 10 µm, w3 = 16 µm, w4 = 30 µm, L1 = 30 µm and

L2 = 24 µm. ....................................................................................................................... 53

Figure 3-19 Comparison of displacements at different locations of the switch (see Figure

3-7) with an actuation voltage of 74 V. ............................................................................ 53

Figure 3-20 (a) Electrostatic force, Fe, (b) squeeze-film damping force, Fd, and (c) the

ratio, ⎜Fd/Fe⎜, of their relative values with an actuation voltage of 74 V. ........................ 54

List of Figures

Page ix

Figure 3-21 Evolution of the squeeze-film pressure distribution across the actuator at an

actuation voltage of 74 V. ................................................................................................. 55

Figure 3-22 Comparison of the simulated microswitch contact tip displacement for cases

with and without the slip-flow effect ................................................................................ 56

Figure 3-23 Impact forces, together with the static contact forces, of the switch with

actuation voltages of (a) 70V, (b) 74 V, (c) 78 V, and (d) 81 V, respectively. ................ 57

Figure 3-24 Displacement of the contact tip using a dual pulse actuation, Va = 88 V, ta =

0.8, Vh = 67 V, and ton = 1.05 µs. The inset shows the impact force for this dual pulse

actuation. The static force for a single-step actuation voltage of 67 V gives a static force

of 15 µN. ........................................................................................................................... 58

Figure 3-25 A schematic representation of the circuit and instruments used for

experimental measurement. .............................................................................................. 59

Figure 3-26 Switch voltages (solid lines) measured by oscilloscope and the corresponding

single step actuation voltages (dotted lines) of 70 V, 74 V, and 81 V.............................. 60

Figure 3-27 Close and open times versus actuation voltage, where Tc1, To1, Tc2, To2 are 1st

close time, 1st open time, 2nd close time, and 2nd open time, respectively. The scattered

dots are experimental results and the lines are from simulations. .................................... 61

Figure 3-28 Comparison between the simulated and measured opening and closing times

for an actuation voltage of 81 V. The horizontal axis is the number of closings or

openings of the switch. ..................................................................................................... 62

Figure 3-29 Comparison between simulation (a) and experiment (b) for a dual pulse

actuation, the insets show the corresponding pulses......................................................... 63

List of Figures

Page x

Figure 3-30 Oscilloscope traces of the switch voltage for a dual voltage pulse actuation

with V h = 74 V, and 81 V, respectively. The inset shows the corresponding actuation

dual voltage pulses. ........................................................................................................... 63

Figure 3-31 Oscilloscope traces of the switch voltage for dual voltage pulses: (a1)

[0.95Va, ta, 0.95Vh, ton], (a2) [Va, ta, Vh, ton], and (a3) [1.05Va, ta, 1.05Vh, ton], where Va =

1.35 Vth, Vh = 1.03 Vth, ta = 0.5 µs and ton = 0.8 µs............................................................ 64

Figure 3-32 Oscilloscope traces of the switch voltage for dual voltage pulses: (b1) [(Va,

0.89ta, Vh, 0.89ton], (b2) [Va, ta, Vh, ton], and (b3) [(Va, 1.11 ta, Vh, 1.11ton], where Va =

1.35 Vth, Vh = 1.03 Vth, ta = 0.5 µs and ton = 0.8 µs............................................................ 65

Figure 3-33 Simulated contact tip displacement of the switch at pressures of 1 atm and 10

atms for an actuation voltage of 74 V. .............................................................................. 66

Figure 4-1 Schematic representation of a nonlinear system ............................................. 74

Figure 4-2 Generation of harmonics in a nonlinear system.............................................. 75

Figure 4-3 Generation of IMD (2nd and 3rd order) in a nonlinear system ......................... 75

Figure 4-4 The 3rd order intermodulation power and output power versus input power. 77

Figure 4-5 The geometry and dimensions of the device, not to scale (dimensions in µm).

........................................................................................................................................... 81

Figure 4-6 The wafer-level layout of the device............................................................... 82

Figure 4-7 The die-level layout of the device................................................................... 82

Figure 4-8 The layout of the device.................................................................................. 82

Figure 4-9 The process flow of the fabrication of the device ........................................... 83

List of Figures

Page xi

Figure 4-10 (a) SEM micrograph of the fabricated device. (b) Cross-sectional view of a

device, not to scale, where W1 = W3 = 160 µm, W2 = 12 µm, H1 = 1062 Å, H2 = 500 µm

and H3 = 1 µm................................................................................................................... 84

Figure 4-11 The three-dimensional view of the device on a pryex glass substrate .......... 85

Figure 4-12 The cross-sectional device-on-substrate schematic showing the heat

generated by tungsten as uniformly distributed over a semicircle with a radius of half the

width of the device, i.e. r1 = W2/2, and is transferred to the ambient through conduction.

The arrows illustrate the isotropic nature of heat conduction, r2 = H2 + H3, not to scale. 86

Figure 4-13 The circuit configuration in which the microstructure is in series with a load

where RS and RL are for source resistance and load resistance, respectively. RSW represents

the resistance of the device that is variable with input power. ......................................... 93

Figure 4-14 (a) The electrical resistance variation showing a sinusoidal-type variation

with a frequency of 2ω, i.e. R = sin(4πft+∆). (b) The input sinusoidal signal with a

frequency of f = 3.2 kHz, i.e. I = I0sin(2πft). .................................................................... 97

Figure 4-15 Variation of the resistance of the device as a function of the frequency. The

input power for a 50 ohm load is 40 mW. ........................................................................ 98

Figure 4-16 The third-order intermodulation distortion of the device as a function of

difference frequency ∆f = f2 - f1, f2 = 10 MHz. The input power for a 50 ohm load is 40

mW.................................................................................................................................... 98

Figure 4-17 The electrical resistance of the device as a function of the measuring current

using a four point probe test setup .................................................................................... 99

Figure 4-18 Block diagram of the measurement system for the transient electrical

resistance of the microscale devices ............................................................................... 101

List of Figures

Page xii

Figure 4-19 The transient electrical resistance of the device with different applied

voltages ........................................................................................................................... 102

Figure 4-20 Block diagram of the experimental setup for the two-tone intermodulation

measurement, where f1 and f2 are two tone signals and SSPA is for solid-state power

amplifier. This figure is provided by Professor Elliot Brown from University of

California at Santa Barbara. ............................................................................................ 103

Figure 4-21 Output spectrum of the intermodulation distortion with respect to the total

input power of the device for cases: (a) Pin = 72 mW, (b) Pin = 36 mW, and (c) Pin = 18

mW, where f1 = 10 MHz, ∆f = f2 - f1 = 6.4 kHz. The measurements were conduced by

Professor Elliot Brown from University of California at Santa Barbara ........................ 105

Figure 4-22 Comparison of the modeled third-order intermodulation distortion with

experimental measurement at different power levels, the frequency of the first tone signal

is f1 = 10 MHz, the difference frequency is ∆f = f2 - f1 = 6.4 kHz. The measurements were

conduced by Professor Elliot Brown from University of California at Santa Barbara... 105

Figure 4-23 The solid model of a quarter of the Ohmic contact-type RF MEMS switch

......................................................................................................................................... 107

Figure 4-24 The simulated electrical resistance of the microswitch as a function of

current which flows through the switch.......................................................................... 107

Figure 4-25 Intermodulation sideband power relative to input power as a function of

power transmitted by switch ........................................................................................... 108

List of Tables

Page xiii

List of Tables

Table 2-1 Comparison of RF MEMS Actuation Mechanism ............................................. 9

Table 3-1 Flow Regimes and Their Knudsen Number ..................................................... 36

Table 4-1 Physical Properties of Device Materials Used in the Model ............................ 84

Acknowledgement

Page xiv

Acknowledgement

I would like to take this chance to express my deepest thanks and gratitude to my

supervisor, Professor Nick McGruer, for his continuous support and guidance throughout

my research in the past five years. His wide knowledge, dedication, and enthusiasm in

research deeply impressed me and taught me what a true scientific researcher should be. I

would also like to express my greatest thankfulness to my advisor, Professor George

Adams. His kind help and wholehearted support are indispensable for the completeness

of my thesis and have benefited me a lot. They support me in every possible way to

enhance my academic capabilities and skills to the highest level. I learned a lot of lessons

and values from their great personality. Professor Elliot Brown from University of

California at Santa Barbara is also greatly appreciated for his help with intermodulation

testing of our fabricated devices. Without his help, this thesis can not be completed.

I would also like to thank my committee member Dean Paul M. Zavracky for his

valuable comments and suggestions. His attendance to my thesis defense is greatly

appreciated, although he has an extremely busy schedule as Dean of School of

Technological Entrepreneurship.

Also, I would thank all faculty, staff and students in the microfabrication lab for

their helpful discussions and friendship.

August 6, 2007

Chapter 1.Introduction

Page 1

Chapter 1. Introduction

This thesis deals with microelectromechanical systems (MEMS) switch

technology for radio frequency (RF) and microwave frequency applications. Since RF

MEMS switches hold great potential for replacement of the existing semiconductor-based

switches as the next-generation switching components in both industrial and military

applications, RF MEMS switches technology has received considerable attention.

However, the RF MEMS switches still have problems such as long-term reliability which

are being intensively investigated. Therefore, the emphasis of this thesis is placed on the

understanding of the dynamics, which are relevant to the reliability of the switch, and the

thermally-induced intermodulation effect in micro-/nano-scale micromechanical devices

for RF and microwave application. The intermodulation distortion due to Ohmic heating

is not well understood and it may become significant when RF MEMS switches are used

for high-power applications which require high fidelity of the signals.

In the first part, the development of a comprehensive mechanical dynamic model

will be the focus of the MEMS switch dynamic study. This model will include all

important aspects such as the real geometry, squeeze-film damping, contact, etc. that are

relevant to the performance of the microswitch. The goal of the dynamic model of the

microswitch is to simulate its dynamic response during operation for a better

understanding of the switch dynamics. Furthermore, the model can be utilized as a design

tool to predict or to optimize the dynamic performance of the Ohmic contact-type switch.

Chapter 1 Introduction

Page 2

The second part of this thesis is on the intermodulation effect due to the Ohmic

heating in microscale mechanical devices. The work consists of development of

analytical models and experimental verification of the predicted results. The emphasis for

the intermodulation effect is on the fundamental understanding of this signal distortion as

a function of difference frequency, materials properties, etc. It is aimed at deriving some

closed-form expressions for convenient prediction of intermodulation distortion in micro-

/nano- scale structures. The organization of this thesis is shown as follows:

Chapter 1 is the outline of the thesis and the primary content and structure of this

thesis is presented. The background of RF MEMS switch technology will be given in

Chapter 2, with an emphasis on the current status of RF MEMS switches and the major

problems which hinder the widespread application of the RF MEMS switches.

Mechanical dynamics of the RF MEMS switches will be concentrated on in Chapter 3.

This includes previous work about modeling and simulation of RF MEMS switches and

development of the comprehensive dynamic model in this thesis. The comparison

between the simulated results and measurements will also be made. In Chapter 4, an

introduction to the intermodulation effect will be first given, then the development of the

analytical model is described, followed by the design, fabrication and testing of the

fabricated micromechanical structures. And last, Chapter 5 is a summary of the thesis and

the future work.

Chapter 2. Background of RF MEMS switch

Page 3

Chapter 2. Background of RF MEMS

Switch

This section provides an overview of the technology of MEMS with an emphasis

on RF MEMS switches. We summarize the current status of the development for RF

MEMS switch and identify the issues which may hinder the widespread applications of

RF MEMS switches.

2.1 History and Development of MEMS Technology

MEMS is the acronym of Micro-Electro-Mechanical Systems. As its name

implies, MEMS is a technology which deals with devices in multiple physical domains

on a micrometer scale. In other words, devices manufactured by using MEMS technology

could involve combined disciplines such as electronic, electrical, mechanical, optical,

material, chemical, and fluids engineering.

The development of this emerging MEMS technology involves integrating

mechanical elements with conventional microelectronics using silicon-based

micromachining technology. The compatibility of MEMS technology with silicon-based

integrated circuits (IC) enables electronics to sense or control environments on the

same chip. The mechanical advantages of MEMS components allow microelectronics to

operate with improved electrical performance. MEMS devices gather information from

its environment by measuring mechanical, acoustics, thermal, biological, optical,

Chapter 2. Background of MEMS

Page 4

magnetic and chemical phenomenon. The MEMS devices can also be utilized to react to

changes in that environment through the mechanical movements of the MEMS actuators

by responding, moving, pumping, positioning and directing. The low cost of MEMS

devices is enabled by batch fabrication which often adopts the infrastructure for IC

fabrication.

In the 1980s, the basic ideas about MEMS were developed although the progress

was slow. The first MEMS device with demonstrated functionality was a gold resonating

MOS gate structure1DPT. The MEMS devices have found applications in the field of sensors

and actuators for automobiles, inkjet printers, and photo projectors. Typical MEMS

devices which were developed in the early days were resonating MOS gate structures1,

surface micromachined switches 2 , crystalline silicon based torsional scanning

micromirrors 3 PT, microaccelerometers4DPT, silicon micromachined gyroscopes5

DPT, inkjet printer

headsD

6DPT, and piezoresistive silicon-based MEMS pressure sensors.7

With the development of advanced technology for micro/nano-fabrication and the

appearance of information technology (IT) in the 1990s, devices made by means of

MEMS technology have found a great variety of potential applications. One of the most

attractive applications for MEMS devices is that for RF and microwave/millimeter

integrated circuits. RF MEMS technology has been used to manufacture

micromechanical devices which exhibit superior electrical performance over

conventional counterparts, as discussed before. RF MEMS devices are used in systems in

which directing, switching, varying, and routing of signals or reconfiguration of the

system are required.


Page 5

The replacement of conventional devices or supplement conventional devices

with RF MEMS devices enables the operation of systems with enhanced performance. To

date, RF MEMS technology has already been utilized to implement high quality

devices/components such as switchesTPD

8DPTP

-DDDTD

14DTP, high Q varactors (variable capacitor)TPD

15DPT, high Q,

highly linear inductors,TPD

16DPT and RF resonatorsTPD

17DPTP

-DDTD

19DTP circuits such as filtersTPD

20DPTP

,TD

21DTP, voltage-

controlled oscillators (VCO) PD

22DPTP

,TD

23DTP, low-loss phase shifters TPD

24DPTP

-DDTD

26DTP, and subsystems/systems

e.g. high-efficiency power amplifiersTPD

27DPT, phased array antennas P

23

,P TPD

28DPT and reconfigurable

antennas.29

2.2 RF MEMS Switch

In this section, we will give an overview of microswitches which are intended to

be used for applications in the RF, microwave and millimeter wave regimes. This

includes operation principles, classifications, characteristics, and applications with an

emphasis on promised functionality and the reliability concerns. Also, we will summarize

the current status of RF MEMS switches and identify the issues which must be addressed

properly before they are widely accepted as a mainstream product in industry.

2.2.1 Operation and Category of RF MEMS Switch

RF MEMS switches are devices that use mechanical movement to achieve an

open (“break”) or short (“make”) circuit condition in an RF transmission line or an

antenna. As an example, a three terminal electrostatically actuated MEMS switch is

shown in Figure 2-1. In the 1990s, a MEMS switch, although it was far from mature and

had poor reliability, designed for microwave applications was demonstrated by Dr Larry


Page 6

Larson at Hughes Research LabsTPD

30DPT. A group at Northeastern University, sponsored by

Analog Devices Inc, developed an electrostatically actuated, normally open switch that

consists of a surface micromachined electroplated gold cantilever beam and three

electrical terminals: drain, source and gate. When an actuating voltage is applied to the

gate, the resulting electrostatic force deflects the beam, causing its free end to move

against the contacts. By adding a fourth terminal, the design becomes a relay in which

two terminals are used for actuation while the other two are switched.

Figure 2-1 An example of a typical three terminal MEMS switch

When the switch is used as a part of a circuit, the cantilever beam is pulled down,

and the switch closes, ‘making’ a closed circuit. When the beam is lifted up by the

restoring force, the circuit ‘breaks’, thus an open circuit forms. This simple “break” and

“make” mechanism of the microswitch makes it technologically feasible and viable as an

emerging new device.

RF MEMS switches are generally classified according to the actuation mechanism,

contact type, and configuration in a circuit. Actuation mechanisms for MEMS switches

are diverse and invoke several physical phenomena that produce a mechanical movement

from a different physical domain. The primary actuation methods are: electrostatic,

Anchor Cantilever

RF out

RF in Actuation electrode Contacts


Page 7

piezoelectric, thermal, electromagnetic, and bimetallic. The various actuating

mechanisms offer different voltage and current handling capabilities, require different

power levels to actuate, and operate at different speeds. Electrostatic designs are the

fastest and draw the least control power, while thermal actuation delivers high power

handling and larger actuating forces. The following gives a brief description about the

mechanisms and the pros and cons for any individual mechanism.

Electrostatic: this mechanism is the commonly used actuation scheme in RF

MEMS mainly due to its ease of technological implementation, no off-state power and

very little power consumption during switching, and compatibility with normal CMOS

processing. It involves the creation of Coulomb force elicited by the positive and/or

negative charges, set by applied voltages between certain mechanical structures. For an

actuation with considerable electrostatic force, most devices requires a large voltage,

usually 30V or higher. For handheld devices such as cellular phone in wireless

communication applications, one has to build a CMOS integrated up-converter to

increase the usually used 5 volt control voltageDPT. Attempts are also made to reduce the

actuation voltage by novel structure designs TPD

31DPTP

-DD

TD

33DTPor by using other actuation mechanisms.

Piezoelectric actuation: this actuation mechanism takes advantage of the inverse

piezoelectric effect: a voltage across certain surfaces of a ferroelectric material, e.g. PZT

(Lead Zirconate Titanate, piezoelectric ceramic material), causes elastic deformation of

the materials, which gives larger contact force for a smaller actuation voltage in contrast

with electrostatic actuation. The RF MEMS switch using piezoelectric actuation has

shown good performance for a low actuation voltage 34DPTP

,TD

35DTP.


Page 8

Electromagnetic actuation: Electromagnetic methods of actuation rely on

aligning the magnetic moment in a magnetic material, usually soft magnetic materials, by

an external magnetic field. The magnetostatic force exerted by the external magnetic field

on the switch can turn the switch ON or OFF, depending on the direction of the applied

current. This is a novel method and has some advantages compared to other methods but

requires special processing involving magnetic materials TPD

36DPTP

-46DT

P. Among the RF MEMS

switches, the design by MicroLab shows promising for applications since it overcomes

the large power consumption of conventional magnetically actuated switches.

Electrothermal: Electrothermal actuation involves using two materials with

different thermal expansion coefficients. When the materials are heated, the composite

beam bends away from the material with the higher thermal expansion coefficient TPD

47DPT, thus

providing mechanical movement. Another thermal method employs shape memory alloys

(SMA), which involves a solid phase change for some special materials. At low

temperatures, the SMA has a martensitic crystalline structure, which is more flexible and

allows relatively large elastic deformations. When the temperature is raised,

transformation to austenitic phase takes place and the material loses its flexibility and

thus the strain is recovered. Currently, these thermal methods have not been very popular

despite the latching properties due to the required power consumption and slow

switchingTPD

48DPTP

,TD

49DTP.

As discussed above, each actuation mechanism has its own advantages and

disadvantages. One may choose the actuating mechanism for benefiting a specific

application while tolerating the drawbacks associated with it. A table by Rebeiz 50 is

reproduced in


Page 9

Table 2-1 Table 2-1 to summarize the main characteristics of the above mentioned

mechanism.

Table 2-1 Comparison of RF MEMS Actuation Mechanism

Voltage (V)

Current (mA)

Power (mW) Size Switching

time (µs)

Contact force (µN)

Electrostatic 20-80 0 0 small 1-200 50-1k Electrothermal 3-5 5-100 0-200 large 300-10k 500-1k Magnetostatic 3-5 20-150 0-100 medium 300-1k 50-200 Piezoelectric 3-20 0 0 medium 50-500 50-200

MEMS switches can also be categorized as metal-metal contact or Ohmic

contactTPD

51 and metal-insulator-metal, or capacitive coupling 52

DPT, based on the contact

characteristic during switching. The metal-metal contact switches use metal to metal

direct contact to achieve an Ohmic contact, as shown in Figure 2-2TPD

53DPT. This type of switch

Figure 2-2 A metal-to-metal contact-type RF MEMS switch

can be used in a broad frequency range from DC to W band (75 – 111GHz).

The capacitive switch utilizes a thin dielectric layer between two metal electrodes to

achieve a closed circuit, as shown in Figure 2-3. This switch is an example of practical

MEMS capacitive shunt MEMS switches and was developed by Goldsmith13 et al at

Raytheon (formerly Texas Instruments). This switch is based on a fixed-fixed metal (Al


Page 10

or Au) beam design. The anchors are connected to the coplanar-waveguide (CPW)

ground plane, and the membrane is, therefore, grounded. As its name implies, this type of

switch is only applicable to high frequency signals.

Figure 2-3 (a) An example of capacitive MEMS RF switch and (b) the electrical CRL circuit

Due to its intrinsic contact characteristics, a capacitive MEMS switch has to be

designed to have a large contact area for smaller insertion loss, but large contact area

results in poor isolation. Therefore, a trade-off has to be made for optimized performance

of capacitive switches.

In addition, MEMS RF switches may be grouped as series and shunt types from

the configuration topology in a circuit, as shown in Figure 2-4.

Figure 2-4 Schematic representation of switches in a series and shunt configuration


Page 11

The broadside and the inline switch for contact-type switches are shown in Figure

2-5 54DPT. The actuation of the broadside switch is in a plane that is perpendicular to that of

the transmission line, while the inline switch is actuated in the same plane as the transmi-

Figure 2-5 (a) and (b) broadside MEMS switches, (c) inline MEMS switch

ssion line.

2.2.2 Performance and Characteristics of RF MEMS

Switches

Much attention has been paid to RF MEMS switch technology since the first

micromechanical membrane-based switch was demonstrated by Petersen using

electrostatic actuation 55 . This is mainly due to the fact that conventional switching

devices such as GaAs-based metal-semiconductor field effect transistors (MESFETs) and

PIN diodes for high-speed switching can not meet the demanding requirements for RF

applications. For instance, silicon FETs can handle high power signal at low frequency,

but the performance drops off dramatically as frequency increases; others, such as GaAs

MESFETs work well at moderately high frequencies but only at low power levels. For


Page 12

frequency greater than 1 GHz, these semiconductor switches have a large insertion loss

(typically 1- 2 dB) in the closed circuit state and a lower electrical isolation (typically 20

– 25 dB) in the open-circuit state. Also, the inherent junction capacitance of the

semiconductor based switches exhibits a larger nonlinear current versus voltage behavior,

leading to larger intermodulation distortion. However, the MEMS switches have a 3 P

rdP

order input intercept point (IP3) better than 65 dBm 54. This low loss, high isolation, and

high linearity are advantages of conventional electromagnetically-actuated mechanical

relays. On the other hand, like semiconductor switches, the MEMS switches have

smaller size, less weight, and fast switching in contrast to the electromagnetically

actuated mechanical relays. Therefore, MEMS switches combine the merits of both

semiconductor switches and mechanical relays.

2.2.3 Applications

As mentioned above, RF MEMS switches have low insertion loss, high isolation,

and high linearity for RF applications, compared with semiconductor-based solid-state

switches. At the same time, RF MEMS switches occupy little space, are not sensitive to

acceleration, have extremely low power consumption, have an extremely high cutoff

frequency of 20 – 80 THz, in contrast to 0.5 – 2 THz for MESFETs and 1.0 – 4.0 THz for

PIN diodes50 and are compatible with low cost silicon based IC technology. So, RF

MEMS switches have potential applications in a wide variety of areas. RF MEMS

switches can be used as a discrete switching component to switch signals. RF switches

can also be used as the building blocks of circuits such as phase shifters, which are

suitable for modern communications, automotive, and defense applications, low-loss


Page 13

tunable circuits (matching networks, filter, etc) and high performance automatic

instrument testing systems, or subsystems or systems such as reconfigurable phased-array

antennas. Due to the cost of hermetic packaging of MEMS switches, the switches may

first be used in defense and high-value commercial applications. The following details

some example applications of RF MEMS switches:

Band switching and T/R Duplexers (TDD) in mobile phone or cellular phones56

Almost all the cellular or mobile phones on the market use a transmit/receive (T/R)

switch, or a band switch, and/or duplexers to interface the antenna and the chipset. The

use of any one or a combination of switching devices depends on the number of bands,

which is determined by the cellular phone system operator. Currently, compound

semiconductor such as GaAs and PIN diodes switches provide a reasonably good solution

to switching due to their power handling and flexibility. The overall performance of the

mobile phone or cellular phone could be greatly improved after RF MEMS switches

replace semiconductor-based counterparts in a multiband switching networks or T/R

switches in a T/R duplexer.

High frequency high Q digitized capacitor banks and phase-shifting networks

8:

The semiconductor switches, e.g. back-biased Schottky diodes, which are commonly

used in digital capacitor banks, have a low Q factor (Q ~ ωC/G in microwave and

millimeter wave applications). The RF MEMS switch may provide a high Q factor for

high frequency applications due to its inherent low loss characteristics.

Phase shifting is a popular control function at microwave and millimeter wave

frequencies. The reduction of occupation area and increase in accuracy in time-delay

phase shifting can be achieved using RF MEMS switches. One approach is to use a


Page 14

coplanar-waveguide transmission line periodically with RF MEMS switches equally

distributed along the lineTPD

57DPT.

Applications in the defense area include phased array antennas, phased-array

radar, and satellite communications58 . Antennas used in military airborne crafts are

required to be able to handle high-data rates and possess large steering angles at

frequencies as high as Ku band (12.2 – 12.7 GHz). State-of-the-art phased array antennas

(PAA) are generally used for this application. The constructive interference of radiation

at PAA is realized through a high efficiency time-delay phase-shifting network, which

can be made possible through RF MEMS switches due to their intrinsically low insertion

loss and low-power consumption.

Other applications of RF MEMS switches are in automotive smart antenna, anti-

collision airbags, automotive GPS systems, base-stations for cellular phones, automatic

instrumentation, wireless LAN’s, data communications, digital personal assistants,

Bluetooth devices, etc.

2.2.4 Failure Mechanisms and Reliability Issues

As can be seen from the preceding discussions, the main driving force for much

effort on research and development of RF MEMS switches is their superior electrical

performance compared with existing semiconductor-based switches. As an emerging

technology, besides some inherent drawbacks with RF MEMS switches such as slow

switching speed, there are still concerns associated with RF MEMS switch technology.

To better understand the current status and potential problems, the following provides a

brief description of the issues related to the long-term reliability of microswitches, and


Page 15

identifies some specific aspects which must be addressed before the RF MEMS switch is

widely accepted.

Compared with other actuation mechanisms, electrostatic actuation has the

advantages of being fast, easy to implement, and having virtually no power consumption.

However, electrostatic discharge (ESD) may cause failures to MEMS devicesTPD

59DPTP

- DDTD

61DTP. The

sudden build-up of a static charge on the MEMS device may result in potentials of over

one thousand volts, causing parts of the actuator or contact melt and weld together, which

may lead to the failure of the switch. It is generally recommended that proper precautions

should be taken before transport or handing of RF MEMS switches.

In general, electrostatically actuated MEMS switches use a relatively high

actuation voltage, usually on the order of 20 - 120V. From an application perspective,

high actuation voltages are not desired. To reduce the actuation voltage, one may use the

following methods: 1) increase the actuation area, 2) decrease the gap between the

electrodes, although this may decrease the electrical isolation during opening, 3) design

switches which have lower spring constant.

Alternatively, one may also provide an intermediary step that enables an RF

MEMS switch to operate at much lower voltages. A dc-dc voltage converter and

controller may be integrated with a high-voltage RF MEMS device to create a low-

voltage solution.

In addition to the above aspects which are relevant to RF MEMS switch

technology, another major concern about RF MEMS switches is its long-term reliability.

So far, the failure mechanisms are not completely understood, although it is observed that

the failure of a well-designed MEMS switch associated with mechanical malfunction


Page 16

such as mechanical fatigue or even fracture is not usually a problem. It is also found that

most failures of current RF MEMS switches are associated with their contacts. The

reasons for mechanical failure at contact are very diverse and complicated. This is due to

the fact that contributing factors from different physical domains may have different

effects on failures. For instance, a simple Ohmic contact type switch may fail as a result

of a permanent stiction, or fail to open. The stiction may be caused by the increased

adhesive force during cycling, or due to degradation of contact with a larger contact area,

The second mode of failure associated with contact is the increase of resistance at the

contact after cycling. The switch is considered to fail if the contact resistance is larger

than a few ohms during operation.

It is believed that the reliability of the switch could be enhanced if one can

address the following issues properly:

(1) Contact materials: minimum adherence force at the contact interfaces is

desired for a better contact, near zero adherence force would be ideal;

(2) Actuation scheme: an optimized actuation scheme gives an optimum dynamic

behavior in terms of low impact force, reduced bounces;

(3) Thermal issues: low temperature of the switch is anticipated even when

handling high power;

(4) Resistance increase: it is often related to the chemically contaminated or

physically damaged contact.

In this thesis, we will deal with items (2) and (3). To study the dynamics of the

switch, we have used a finite element package ANSYS® and a finite difference method to

develop a comprehensive dynamic model. This model includes the complete structure of


Page 17

the switch, squeeze-film damping, nonlinear contact, etch holes, and adherence force.

Afterwards, we use the model to optimize the dynamic performance of the switch. Also,

the simulated results are compared with the experiments. We also need to establish a

thermal model to investigate the thermally-induced intermodulation. Specifically, we first

build an analytical model to quantitatively examine the intermodulaton effect and design

the test device, and subsequently, make measurement on the fabricated device. Also, we

applied the developed method to predict the intermodulation distortion for a RF MEMS

switch. The intermodulation is caused primarily by Ohmic heating, since it is found that

the intermodulation caused by the change in contact resistance from the change in contact

force from the signal is much smaller than the thermally-induced intermodulation62.


Page 18

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Chapter 3. Dynamics of Microswitch

Page 22

Chapter 3. Mechanical Dynamics of a

MEMS Switch

In this chapter, we will develop a comprehensive dynamic model using ANSYS®

(a software package based on the finite element method) in combination with a finite

difference method. First, we give a brief introduction to work on dynamics of MEMS

devices with an emphasis on RF MEMS switches. Then, we describe the modeling based

on finite element analysis, and after that we will describe models which are parts of the

comprehensive model for simulating dynamics of the switch This model includes solid

modeling of the switch using ANSYS®, electrostatic actuation, non-uniform squeeze-film

damping based on the Reynolds equation including compressibility and slip-flow, effects

of perforation of the beam on damping, nonlinear elastic contact and adherence force

during unloading. Finally, we present the experimental measurements and make

comparisons between the simulated results and the experimental measurements.

3.1 Dynamic Response of MEMS Switch

As mentioned in Chapter 1, MEMS switches promise to replace conventional

solid-state switches in many high frequency applications due to their enhanced

performance. For these applications, MEMS switches must be designed to be able to

operate for 1 to a few hundred billion cycles. The reliability of MEMS switches is

believed to be strongly connected to the dynamics of the actuation. It has been


Page 23

experimentally observed that most failures occur at the contact, either because of stiction

due to large adherence force, or due to a substantial rise of the electrical resistance.

Impact force can flatten and increase the area of the contact leading to increased

adherence force. Contaminated contact and/or damaged contact resulting from fracture,

pitting, hardening, etc may cause switch resistance to increase. It is generally assumed

that if the contact resistance of the switch is 5 Ω or more, which corresponds to an

insertion loss of 0.5 dB in a 50 Ohm environment, the switch fails.

In general, the characterization of mechanical dynamics of the switch includes

actuation and release time, switching speed, impact force at contact, and bounce. All of

these properties are critical for the successful development of RF MEMS switches. But

among them, switching speed, impact force and bounce may be most critical, because

they are most relevant to the reliability of the switch.

During operation, the contact tip on the cantilever beam makes contact with the

drain, or signal transmission line. Before making steady contact, the contact tip usually

bounces several times due to the elastic energy stored in the deformed materials of the

actuator. The existence of bouncing behavior increases the effective closing time of the

switch. Meanwhile, the contact may be damaged by the impact force. This instantaneous

high impact force may induce local hardening or pitting of materials at the contact area.

The switch contact may also stick to the drain because of large adherence forces caused

by high impact force. Also, the bounces may facilitate material transfer, or contact wear-

out, which is not desired for a high-reliability switch. It has been experimentally observed

that the switches bounce a few times before making permanent contact1 -DPTDDDDDTD

5DTP. Elimination,

or at least reduction, of bounces is highly desirable for microswitches to operate with


Page 24

longer lifetime and better performance. To control the dynamic behavior of the switch, it

is necessary to develop full dynamic models to simulate the dynamic response of the

microswitch.

Most dynamic models on MEMS switches account for only certain aspects of the

switch such as the squeeze-film damping, but contact characteristics and adhesions of the

microswitches during operation are not taken into account. For instance, Czaplewski et

al. 6 used a dynamic model to predict the dynamics of a Ohmic RF MEMS switch. But

the contact, squeeze-film damping, and adhesion effects have not been taken into account

in this model. The analytical analysis presented by Steeneken et al. 4 about the dynamics

of a capacitive RF MEMS switch mostly deals with the squeeze-film damping as well as

the slip-flow effects. Recently, Granaldi and Decuzzi 7 presented a one-dimensional

dynamic model which mainly focuses on the switching time and bouncing of a cantilever

based microswitch. In this model, the squeeze-film damping and the spring restoring

force have been lumped into two parameters, thus it does not take into account the

nonuniformity across the actuator and the nonlinearity of the damping force. Gee et al.8

presented a one-dimensional dynamic model and examined the effect of the dynamics of

the switch on its opening time. In that model, they used a fourth-order beam deflection

equation and included the adhesion force due to both van der Waals type forces and

metal-to-metal bonds. The one dimensional dynamic model developed by McCarthy et

al.3 based on a finite difference method for squeeze-film damping was used to simulate

the dynamics of the RF MEMS switch both before and after the contact. In that model,

the squeeze-film damping effect and a simple spring contact have been included, and the

spring shows the bouncing features after initial contact, as shown in Figure 3-1. It is seen


Page 25

that the number of bounces increase with increasing actuation voltage, resulting in longer

time to close. But the nonuniformity and nonlinearity of the squeeze-film damping as

well as the bowing of the microswitch has been neglected.

In this work, we develop a model which will cover almost all important aspects

pertaining to the dynamics of the switch. This includes the complex two-dimensional (2-

D) geometry, squeeze-film damping, compressibility, slip-flow, and the effect of

perforation of the mobile structures, nonlinear contact, and adhesive force during

unloading. This reveals the dynamic response of the switch both before and after closure.

Furthermore, we develop an open-loop actuation strategy for operation of the switch with

enhanced performance. We measure the dynamic response of the microswitch. And last, a

comparison between the modeling and the experimental measurement is made. The

following will present the development of the models in more detail.

Figure 3-1 Dynamic behavior of a RF MEMS switch, the step curves are for the step voltage for actuation. The traces are recorded using oscilloscope which show the transient ‘in contact’ and ‘out

of contact’ after actuation [see Reference (3)]

T im e after actuation (µs)

0 10 20

Sw

itch V

olta

ge

(V)

0 .0

0.1

0.2

0.3

0.4

0.5

Actu

atio

n V

olta

ge

(V)

0102030405060


0 10 20

Sw

itch V

olta

ge

(V)

0 .0

0 .1

0.2

0.3

0.4

0.5

Actu

atio

n V

olta

ge

(V)

0102030405060


0 10 20

Sw

itch V

olt

0 .0

0.1

0.2

0.3

0.4

0.5

Actu

atio

n

0102030405060


Page 26

3.2 Finite Element Analysis (FEA)

The finite element method is a numerical technique which has been used to solve

complex nonlinear problems in fields of research such as mechanical structures, fluid

mechanics, heat transfer, vibrations, electric and magnetic fields, acoustic engineering,

civil engineering, aeronautic engineering, and even in weather forecasting. The common

characteristic of FEA is the mesh descretization of a continuous domain into a set to

discrete sub-domains. In doing analysis of solid mechanics, a complex solid structure is

divided into a finite number of elements, and these elements are connected at points

called nodes. The stresses of each element are balanced by those of neighboring elements

and ultimately by the forces exerted on the exterior or at the boundaries. The

displacement of each node is determined by the overall displacement constrained by the

boundary conditions. Compared with analytical methods, FEA allows the simulation of a

generally complex geometry, and examination of the three-dimensional effects both

locally and globally.

In the modeling and simulation of dynamics of the RF MEMS switch, we used

ANSYSP

®P version 10.0, a FEA package from ANSYS Inc. The procedure of performing

simulation involves building solid model, material property designation, meshing, set-up

of boundary conditions, solving and post-processing. Before we go into the details of the

simulation, we need to introduce the aspects associated with the dynamics of the switch

such as lumped-parameter modeling, geometry and dimensions, electrostatic actuation,

squeeze-film damping, effect of etch holes, nonlinear contact, and adhesion.


Page 27

3.3 Lumped Parameter Modeling of a

Cantilever Beam

Cantilever beams are often used as actuators in MEMS devices. The reasons

include the better understanding of the mechanical behavior and ease of fabrication. For

instance, cantilever beams are used in some inline series RF MEMS switches and

broadside switches, as discussed in Chapter 2. For applications of moving switches,

adjusting elements, valves and grippers, a DC voltage is applied, whereas for resonant

devices, an AC component is added to the driving voltage to excite the harmonic motions

of the beam. A simple cantilever beam is shown in Figure 3-2.

Figure 3-2 Side view of a typical cantilever beam

Since one end of the cantilever beam is free standing, the residual stress within the

beam is released. However, the released unloaded beam can also be deformed by the

nonidealities, which gives rise to take-off angle, and the existence of the stress gradient

over the cross section of the cantilever, which creates curvature of the released part of the

beam. Thus, the total deflection curve of an unloaded beam mainly consists of two

components: the take-off angle and the curvature.

The first natural resonance frequency of a cantilever beam in transverse vibration

as shown in figure is governed by the general equation9

Cantilever beam

g


Page 28

eff

eff

MK

fπ21

0 = (3-1)

where KBeffB and MBeffB are the effective stiffness or spring constant and mass of the beam,

The effective spring constant of a cantilever-type structure depends on the force

distribution over the beam, Young’s modulus, and geometry 10. The effective mass for a

uniform cantilever beam is MBeffB = (33/140) M, where M is the mass of the cantilever

beam11.

The static and dynamic behavior of a cantilever beam, as shown in Figure 3-2,

with electrostatic actuation, can be modeled using a simplified lumped one dimensional

mass-spring system with a voltage-controlled parallel-plate capacitor, as shown in Figure

3-3 .

Figure 3-3 The lumped mechanical model for a cantilever beam.

As can be seen from Figure 3-3, the bottom electrode is fixed and the top

electrode having a mass of MBeffB is suspended by a spring with stiffness of KBeffB and a

damper with damping constant b. In the following static analysis, the damping effect has

been neglected for simplification. The normalized gap with respect to the initial gap

versus the applied voltage which is normalized with respect to the pull-in voltage is

shown in Figure 3-4.

Keff b

VMeff

g


Page 29

Figure 3-4 Gap of the cantilever vs. applied voltage

It can be seen that the system becomes unstable at g = (2/3)gB0 B due to the existence

of a forward feedback. At equilibrium when g > (2/3)gB0B, the electrostatic force pulling the

upper electrode down balances the spring restoring force which pulls the electrode upTPD

12DPT.

If the sign convention is assigned a positive sign for forces that increase the gap, the net

force on the upper electrode at voltage V and gap g is:

)(2 02

2

ggkgAVFnet −+

−=

ε (3-2)

where gB0 B is the gap at zero volts and zero spring extension. For this system to be stable at

the equilibrium point, the net force, FBnetB = 0, and the derivative of Eqn (3-2) has to be

less than or equal to zero. Then, at pull-in we have:

3

2

2 PI

PI

gAVk ε

= (3-3)

032 gg PI = (3-4)

A

kgVPI ε27

8 30= (3-5)

Stable

Unstable


Page 30

To better understand the pull-in phenomenon, we normalized the voltage to the

pull-in voltage as PIVV /=ν , and the displacement to 0/1 gg−=ς . At equilibrium, we

can get:

ςς

ν=

− 2

2

)1(274 (3-6)

The normalized force, the left hand side of Eqn (3-6) as a function of normalized

gap ζ with a variable voltage as a parameter, is shown in Figure 3-5. It can be seen that

there exist two equilibrium states for ν ≤ 1, and one of them is stable. The stable

equilibrium point is specified by the condition that the derivative of Eqn (3-2) is negative.

When ν = 1, the system is at pull-in state, and when ν > 1, the system becomes unstable,

as discussed above.

Figure 3-5 The electrostatic force and spring force vs. normalized gap for a voltage-controlled electrostatic actuator.

3.4 Geometry of the Microswitch

The microswitch under investigation was fabricated at Northeastern University using the

standard micromachining technology. The details of the fabrication process can be found


Page 31

in the doctoral dissertation by Majumder 13. The switch is based on a cantilever-beam

type mechanical structure, as shown in Figure 3-6. The source, the actuator and the drain

of the microswitch is made of electroplated gold, and the gate is sputtered gold.

Figure 3-6 SEM micrograph of the Northeastern University MEMS switch.

The source end of the microswitch is attached to the substrate. The contacts

indicated on the figure make contact with the lower drain metallization (barely visible) in

the on-state.

The cantilever beam is actuated through the electrostatic force between the top

electrode, i.e. actuator, and the bottom electrodes, i.e. gate. The initial separation

between the top and bottom electrode is 0.6 µm before actuation. The top view along

with the dimensions of the microswitch is shown in Figure 3-7. The side view along with

the dimensions of the microswitch is shown in Figure 3-8.

ActuatorSource Drain

Gate

Contact


Page 32

Figure 3-7 The top view as well as the dimensions of the Northeastern University RF MEMS switch

where w1 = 80 µm, w2 = 10 µm, w3 = 16 µm, w4 = 30 µm, L1 = 30 µm and L2 = 24 µm.

Figure 3-8 The side view of the microswitch where h1 = 6.3 µm, h2 = 0.6 µm and h3 = 0.38 µm.

3.5 Finite Element Modeling

ANSYS® is a well established simulation tool which utilizes finite element

techniques. The properties of MEMS switches can be examined both locally and globally

using ANSYS®. The top and side views of the switch are shown in Figure 3-7 and Figure

3-8. Only half of the switch is simulated by utilizing the symmetry of the switch. The

electrode and beam of the switch are discretized to rectangular structures, i.e. regular

mapped mesh grids, as shown in Figure 3-9, which are used for both electrostatic

actuation and implementation of the finite difference method to solve the Reynolds

equation for the squeeze film damping. The rest of the microswitch is meshed using free

meshing. Element solid45 is used for the whole mechanical three-dimensional structure,

whereas surface element surf22 is used for the surface which is subject to electrostatic

and squeeze-film damping forces. Element link8 is used to simulate the contact between

Source Gate

h1

h2 h3

Drain

w3

w4

w2

A

L2

L1

w1

BBeamFixed

Fixed


Page 33

the contact tip and the drain of the switch. The total number of elements is 634 consisting

of 598 solid45, 35 surf22 and 1 link8 element. There are three layers through the

thickness.

Figure 3-9 Grid of finite elements of half of the switch for ANSYS® simulation.

3.6 Electrostatic Actuation

As discussed above, electrostatic actuation is one of the most popular actuation

mechanisms for MEMS devices. The main reasons are its near zero-power consumption

and its ease of implementation. The electrostatic force between two parallel plates is

established through the Coulomb force on oppositely polarized charges. The charges at

the surface of two conductors are accumulated by an electric field, which is created by a

voltage applied to the plates with a distance of h, as shown in Figure 3-10 . Note that the

fringing effect has been neglected in the model.

ActuatorBeam


Page 34

Figure 3-10 Electrostatic force between two parallel plates

The pressure between two parallel plates separated by a distance g is given as:

2

2

2gVFELEε

= (3-7)

where ε B0B is the permittivity of free space, V is the voltage difference between the

electrodes, and g is the distance between the electrodes. In applying the electrostatic force

to the elements of the switch, we assume that the forces between two opposite elements

of opposite electrodes can be approximated by the electrostatic force between the two

parallel plates. This is because the gap is much smaller than the length of the switch, thus

the local two opposite elements is close to be parallel.

3.7 Squeeze-Film Damping

MEMS devices which are electrostatically actuated often have a large electrode

area and a smaller gap between electrodes, which gives a large electrostatic force and fast

speed. Such devices exhibit a damping force. The damping forces originate from

deformed structural materials, or damping from the viscosity of the surrounding fluid.

The damping mechanisms associated with these damping forces are called structural and

squeeze-film damping, respectively. In the latter case, the damping force is due to the fact

that a displacement of small magnitude has to squeeze air out of the narrow gap. The

V hE field


Page 35

viscosity of the air limits the flow rate, which gives rise to a pressure at the surface of the

moving electrode. The distribution of the gas film pressure varies across the electrode

surface. The total damping force, which affects the mechanical dynamics, and ultimately

the design and control of the device, is often known as squeeze-film damping.

As early as the 1960s, LangloisTPD

14DPT and Gross TPD

15DPT investigated the squeeze-film

damping phenomenon from a theoretical perspective. Griffin 16DPT and Blech 17 linearized

the Reynolds equation for it to be suitable for structures which undergo vibrations of

small amplitude. The linearized Reynolds equation is widely utilized in analyzing

squeeze-film damping effects. Since the Reynolds equation is derived from Navier-

Stokes equations, which describes viscous, pressure and inertial mechanisms in fluid

mechanics, it holds true only under certain circumstances. The assumptions are as follows:

1) inertial effect is negligible; 2) the surfaces move perpendicular to each other; 3) the gas

thin film is isothermal; 4) the gap, i.e. g, dimension is much smaller than the lateral

dimensions, W and L, thus pressure does not vary across gap.

In general, the force due to squeeze-film damping effect consists of two

components: 1) spring force due to the compressibility; 2) the dissipative force arising

from the viscous flow. The relative importance of the two components in squeeze-film

effect is measured by the squeeze number. For a two-dimensional system, the squeeze

number is related to the geometry and the properties of the gas film as follows 18:

20

212gPL

a

ωµσ = (3-8)

whereµ is viscosity of air gas, L is the lateral dimension of the moving structure, ω is the

frequency of oscillation of the structure, PBa B is the ambient air pressure, and gB0 B is the initial


Page 36

gap between the two electrodes. If the squeeze number is small, the dissipative damping

force is dominant over the spring force, otherwise the spring force dominates.

One of the important characteristics associated with squeeze-film damping is the

slip-flow effect, which may dramatically change the damping force. This becomes more

important when the gap thickness (i.e. characteristic length) is comparable to the mean

free path of the gas molecules and the tangential component of the gas velocity at the

boundary is no longer zero.

Fluid or gas flows are generally categorized based on the Knudsen number. The

Knudsen number is defined as the ratio of the mean free path, LBmB, of a fluid to the

characteristic length, LBcB, of the flow region:

c

mn L

LK = (3-9)

Also, the mean free path of a typical gas is inversely proportional to the pressure 19D The

flow regimes which follow different principles are listed in Table 3-120.

Table 3-1 Flow Regimes and Their Knudsen Number

Flow Regimes KBn B number Continuum flow < 0.001 Slip flow 0.001 ~ 0.1 Transition flow 0.1 ~ 10 Molecular flow > 10

The switches under investigation have large, closely-spaced electrodes for

actuation. The gas between the electrodes, which are moving perpendicular to each other

during operation, is assumed to be compressible and isothermal. The mean free path of

air molecules at one atmosphere is about 62 nm, the initial gap between two electrodes is


Page 37

600 nm. The Knudsen number is 0.1, so the slip-flow has to be taken into account in

calculating the damping force.

Without including the slip-flow effect, the Reynolds equation under isothermal

conditions for two parallel plates can be written as TPD

21DPTP

,TD

22

th

yph

yxph

x ∂∂

=∂∂

∂∂

+∂∂

∂∂ )(12)()( 33 ρµρρ (3-10)

where p is pressure, ρ is density, h is the distance between the electrodes, µ is the gas

viscosity is equal to 1.82 × 10-5 Pa⋅s for air at room temperature, and t is time. If the gas

is assumed to be ideal, the ideal gas law states that the gas density is proportional to the

pressure. The compressibility of the gas film is included by assuming the density is

proportional to the pressure for the idealized isothermal air gas. After including the slip-

flow effect, Eqn (3-10) can be modified to

tph

yph

yP

xph

xP

ypph

yxpph

x amam ∂∂

=∂∂

∂∂

+∂∂

∂∂

+∂∂

∂∂

+∂∂

∂∂ )(12)(6)(6)()( 2233 µλλ (3-11)

Eqn (3-11) shows that the pressure due to squeeze-film damping effect is a function of

position, height and time, that is, p = p(x, y, t). It is assumed that the pressure, also the

velocity of the gas molecule, is only dependent on x and y coordinates and not a function

of z.

As discussed above that the gas between the electrodes is modeled as a

compressible, continuous fluid, and undergoes an isothermal process during operation.

But, during operation of the MEMS switch, the damping force is not uniform since the

cantilever undergoes a movement which is location dependent. The equation has been

rewritten using the forward finite difference method, which was schematically illustrated


Page 38

in Figure 3-11, such that the pressure at a specified location and in time t + 1 is

determined by the pressures of the four nearest elements at time t.

Figure 3-11 Schematic representation of the finite difference method

The detailed expansion of Eqn (3-11) using finite difference method is described below.

⎭⎬⎫

⎩⎨⎧

∂∂

−∂∂

∆=

∂∂

∂∂

−+ 2/13

2/133 )()(1)( ii x

pphxpph

xxpph

x (3-12)

where x

ppxppppppp ii

iiiiiii ∆−

=∂∂

+=+= ++−−++

12/112/112/1 )()(

21),(

21

So, the expression for )( 3

xpph

x ∂∂

∂∂ can be further expanded as

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−++−

−++

∆=

∂∂

∂∂

−−−

+++

)())((

)())(()(16

1)(1

311

13

112

3

iiiiii

iiiiii

pphhpp

pphhppxx

pphx

(3-13)

After simple mathematical manipulation, Eqn (3-13) can be reduced as

)()())(()(16

1)( 21

231

31

2212

3−−++ −+−+−

∆=

∂∂

∂∂

iiiiiiii pphhhhppxx

pphx

(3-14)

After the similar expansion of other terms in Eqn (3-11), the finite difference format of

Reynolds equation including compressibility and slip-flow can be written

1,

,,1

,

1, )(

12 +++ ++++

∆= t

ji

tjit

jitysf

txsf

ty

txt

ji

tji h

hppppp

htp

µ (3-15)

∆y

∆x

i, j - 1

i + 1, j

i, j + 1

i - 1, j

i, j


Page 39

where [ ] [ ]

2

2,

2,1

3,,1

2,

2,1

3,,1

)(16)()()()()()(

xpphhpphh

pt

jit

jit

jit

jit

jit

jit

jit

jitx ∆

−++−+= −−++ (3-16)

[ ] [ ]

2

2,

21,

3,1,

2,

21,

3,1,

)(16)()()()()()(

ypphhpphh

pt

jit

jit

jit

jit

jit

jit

jit

jity ∆

−++−+= −−++ (3-17)

[ ] [ ]

2,,1

2,,1,,1

2,,1

)(4)()(

xpphhpphh

Ppt

jit

jit

jit

jit

jit

jit

jit

jima

txsf ∆

−++−+= −−++λ (3-18)

[ ] [ ]

2,1,

2,1,,1,

2,1,

)(4)()(

ypphhpphh

Ppt

jit

jit

jit

jit

jit

jit

jit

jima

tysf ∆

−++−+= −−++λ (3-19)

where ∆t, ∆x and ∆y are the time increment, and the elemental distances in the x and y

directions, respectively. The tjih , term represents the distance of the element (i, j) from

the top to the bottom electrode at time step ‘t’. This explicit solution given by Eqn (3-15)

gives accurate results as long as the time step, ∆t, is sufficiently small for given values of

the spatial finite difference grid, ∆x and ∆y. In the simulation, the required time step is on

the order of nanosecond for a converged solution. Based on the preceding formulation, a

sequential simulation program has been developed for the transient simulation of the

dynamic response of the microswitch.

3.8 Effect of Perforation

The existence of holes on the cantilever may increase the switching speed of the

MEMS switch by reducing the squeeze-film damping, and facilitate the release of the

structures during fabrication. Usually, electrostaticly driven actuators have a relatively

large surface area, which may create problems in releasing them by etching processes.


Page 40

On MEMS devices, particularly in cases where actuators of large area are used,

there may be some distributed etch holes on the actuator. The etch holes are generally

used to reduce the squeeze film damping as well as to facilitate the fabrication process

during the release process. Although there are no etch holes on the actuator of the current

version of the Northeastern University MEMS switch, we developed formulations which

account for the etch hole effect on squeeze-film damping in the following analysis.

The effect of etch holes on the damping has been included in analyzing the

dynamics of the planar microplateTPD

23DPTP

- 26DTP microscannerTPD

27DPT, microaccerometerP

28

,TD

28DTP, and

micromirrorTPD

29DPTP

,TD

30DTP. It was found that the number of the etch holes is more important than

the size of the holes in reducing the damping force. However, all models used an

equivalent damping coefficient in the Reynolds equation to calculate effect of the etch

holes on damping. In our work, we model the gas flow through the etch holes as a steady

fluid flow with slip flow boundary conditions. The advantage of this analysis is that the

effect of etch holes can be calculated using a finite difference method, thus can be

incorporated in the analysis in the previous section.

To include the effect of the etch holes in the compressible Reynolds equation, the

following formulations were performed. According to Gross31, the pressure p(x,y) inside

the gap can be written in terms of the velocity components u, v and w of gas molecules as

2

2

2

2

)(

)(

zv

zv

zyp

zu

zu

zxp

∂∂

=∂∂

∂∂

=∂∂

∂∂

=∂∂

∂∂

=∂∂

µµ

µµ (3-20)

where u and v are the velocity components in the x and y directions, respectively. The

absolute pressure p varies only with x and y. Assume that the velocity of the gas at the


Page 41

bottom electrode and the bottom edge of holes with a gap height h is zero, that is, the

boundary conditions are

00

00

0

0

==

==

==

==

hzz

hzz

vv

uu (3-21)

After integrating Eqn (3-20) twice with boundary conditions of Eqns (3-21) we have

)(

21

)(21

2

2

zhzypv

zhzxpu

−∂∂

=

−∂∂

=

µ

µ (3-22)

According to the continuity equation32T, we have

0)()()( =∂∂

+∂∂

+∂∂

+∂∂ w

zv

yu

xtρρρρ (3-23)

Integrating Eqn (3-23), we have:

∫∫ ∂∂

+∂

∂+

∂∂

−=∂

∂ hh

dzty

vxudz

zw

00

])()([)( ρρρρ (3-24)

The left hand side (LHS) of Eqn (3-24) can be written as:

)()(0

0

Vthwdz

zw h

h

αρρρ+

∂∂

==∂

∂∫ (3-25)

where α is the area fraction of etch holes, V is the mean velocity in etch holes, and ρ is

the mass density. For our model, the values for α are 0.0428 and 0.0828, respectively, for

the front and end parts of the cantilever beam.

According to Munson et al., the mean velocity of gas, V, in the steady flow of a

pipe is given as 33


Page 42

lR

ppV a

µαβ

αβ

8

)(

2

=

−= (3-26)

where R is the radius of the etch holes. In our case we assume we have 3 µm × 3 µm

square holes, R is the equivalent radius of a circular hole with the same area as the square

holes. P is the absolute pressure and PBa B is the pressure of the ambient environment, µ is

the viscosity of the gas, l is the height of the etch holes.

Substituting Eqn (3-26) into Eqn (3-25), we have:

)()(

0a

h

ppthdz

zw

−+∂∂

=∂

∂∫ βρρρ (3-27)

Substituting Eqn (3-22) into the right hand side (RHS) of Eqn (3-24), and after

integration over z from 0 to h, we have

thdz

t

yph

ydz

yv

xph

xdz

xu

h

h

h

∂∂

=∂∂

∂∂

∂∂

−=∂

∂

∂∂

∂∂

−=∂

∂

∫

∫

∫

ρρ

ρµ

ρ

ρµ

ρ

0

3

0

3

0

)(12

1)(

)(12

1)(

(3-28)

Substituting Eqn (3-28) into (3-24), we have

)()(12)()( '33appp

tph

ypph

yxpph

x−+

∂∂

=∂∂

∂∂

+∂∂

∂∂ βµ (3-29)

h

dπαβ2

3 2' = (3-30)

where d is the length of the edge of the square holes.


Page 43

From Eqns (3-29) & (3-30), it can be seen that the effect of etch holes on the

pressure has been quantitatively associated with its geometry and pressure. Accordingly,

the pressure in the finite difference form including the slip-flow terms can be written as:

)(12

')(12 ,,11

,

,,1

,

1, a

tji

tjitt

ji

tjit

jitysf

txsf

ty

txt

ji

tji ppp

ht

hh

ppppphtp −

∆−++++

∆= +++

+

µβ

µ (3-31)

where txp , t

yp , txsfp and t

ysfp take the same forms as those in Eqns (3-16) - (3-19).

3.9 Nonlinear Contact Model with Adhesion

When the switch is actuated, the contact tip on the cantilever makes contact with

the drain. It is observed that typically only a few asperities with radius of curvature of

about 0.1 - 0.2 µm make contact with the bottom drain 13. The contact between the switch

tip on the upper beam and the drain can be modeled as the interaction between an

equivalent rigid spherical bump and a compliant flat surface. The widely used contact

models with adhesion are the Johnson-Kendall-Roberts (JKR) 34 and Derjaguin-Müller-

Toporov (DMT)35 models. The former is most appropriate for a larger radius bump, large

adhesion energy, and more compliant contact materials. The latter is best applied to cases

where interaction occurs between small and more rigid bumps with low adhesion energy.

A more quantitative dimensionless parameter, µ, defined by Tabor36 is used to determine

the regions of validity of the two models. This parameter is given as

3/1

30

2*

2

⎟⎟⎠

⎞⎜⎜⎝

⎛=

zERwµ (3-32)

where R is the radius of curvature of the bump and is 1.37µm, which corresponds to a

contact radius of 0.34 µm at a contact force of 1 mN for elastic deformation, w is


Page 44

adhesion energy, E* is the effective Young’s modulus which is defined as 1/E* = (1-

ν12)/E1 + (1-ν2

2)/E2, and z0 is the equilibrium spacing of the surfaces in the Lennard-Jones

potential (typically z0 ≈ 0.28 nm for metals37).

For the case µ > 3, corresponding to larger radius bumps, lower Young’s moduli

and higher adhesion energy, the JKR theory is more applicable. On the other hand, if µ <

0.2, the DMT is more appropriate. In the microswitch, the electroplated Au is used as the

contact material, the Young’s modulus E = 42.4 GPa38 , Poisson’s ratio ν = 0.44, surface

energy γ = 1.37 J/m2 39 and adhesion energy w = 2γ. Substituting these values into Eqn.

(7), we found µ = 5.7, which indicates that the Au - Au contact is in the JKR regime.

Notice that we take the Young’s modulus value for electroplated gold from reference

[21]. It is reported that the Young’s modulus of the electroplated gold ranges from 41.9

GPa to 52.3 GPa40. In Section 3.1, it will be seen that the use of E = 42.4 GPa for

electroplated gold is a reasonable approximation for the real value.

The adhesion force for a JKR contact is 1.5πwR. According to JKR theory, the

contact radius, a, of a rigid sphere on a compliant flat surface with adhesion as a function

of load is given as 41

[ ]23 )3(63 wRwRPwRPKRa πππ +++= (3-33)

where K is the contact modulus and is equal to 4E*/3, and R is the effective radius of

curvature of the contact tip between Au and Au contact [16]. The adhesion force based on

JKR model is about 17.8 µN. Subsequently, the penetration, or displacement, of the

sphere relative to the drain can be written as:

Kaw

Ra

382 πδ −= (3-34)


Page 45

It is clear that the penetration depends nonlinearly on the external force. This nonlinear

behavior is more reasonable than a linear spring given the fact that the contact area tends

to increase as the contact force increases, leading to a nonlinear stiffness which increases

with increasing deformation. As discussed above, contact is a very complex phenomenon,

i.e. elastic, elasto-plastic or even fully plastic deformation may all be involved. But for

simplicity, we assume that all plastic deformations occur during the first contact and,

subsequent loading and unloading are assumed to be purely elastic. To implement this

nonlinear elastic contact behavior, we used the link element Link8 in ANSYS® to

simulate the contact using Eqns (3-33) - (3-34).

Figure 3-12 The displacement of the microswitch contact tip vs. the contact force.

3.10 Dual-Pulse Scheme for Actuation

As discussed above, the long-term reliability of the MEMS switch is a major

concern. The mechanical dynamics of the switch are related to the reliability and

performance of the switch, because the impact force and bounces of the switch during

contact may deteriorate the contact physically and/or chemically. Meanwhile, switch


Page 46

failure may be caused by increased stiction which results from repeated scrubbing and

flattening during operation. One way to improve the reliability of the switch is to tailor

the actuation waveforms such that a minimum impact force can be reached and thus

reduce the chances of creating local pitting and contact hardeningPD

42DPT. In the actuation of

PIN diode, a dual-pulse actuation method is often used. The first large current pulse is

injected into the wide depletion region and the device is turned on very quickly.

Afterwards, a lesser quiescent current maintains the device in the on-state TPD

43DPT. For the

MEMS switch, a similar idea may be applied to gently close the switch. The idea behind

this method is that a large actuation pulse is first applied to the switch, after a short period

of time, denoted as tB1 B, the pulse is turned off such that the speed of the switch is ideally

zero when it barely touches the bottom electrode at time tB2 B. The dynamic behavior of the

microswitch has been modeled using a simple lumped spring-mass damper system under

a contact force. A schematic representation of such system together with the pulses is

Figure 3-13.

Figure 3-13 (a) Lumped spring-mass system, (b) a typical profile for a dual-pulse actuation method,

and (c) the desired gradual close for a dual-pulse actuation

Notice that in this model, we used a constant force for modeling convenience

instead of a voltage to actuate the switch for simplicity. If we neglect the dependence of

the electrostatic force on the gap, the force is proportional to the square of the actuation

d

k

0 t1 t2

m

0

F

tt2 t1

F0

Fh

(a) (b)

0

d

tt2 t1

(c)


Page 47

voltage. A voltage profile desired to eliminate bounce for gate actuation is shown in

Figure 3-13. A constant force of FB0 B, is first applied until time tB1B, and it will be removed

between time tB1 B and tB2B, which corresponds to the closing time of the microswitch. At time

t B2B, a second constant contact force FBh B is used to hold the switch. The second force of

smaller amplitude can reduce the impact force while maintaining a reasonable large

contact force. Notice that the velocity of the switch at time t B2B is expected to be close to

zero.

They can be expressed as follows:

⎩⎨⎧

><

=

−−=

0100

)(

)]()([)( 10

xx

xH

ttHtHFtF (3-35)

The displacement x for a system without damping )(tFkxxm =+&& ,

mkn =ω under load F(t) can be written as 44

110 ],cos)(cos1[)( ttttt

kFtx nn >−−−= ωω (3-36)

0)(,)(22== == tttt dt

tdxdtx (3-37)

Take the boundary conditions into Eqn (3-37), we have the following relationships

⎥⎦

⎤⎢⎣

⎡−

−= −

1

112 cos1

sintan2 t

ttn

nn

ωω

πτ (3-38)

⎥⎦

⎤⎢⎣

⎡= −

0

12 2

cos2 F

dkt n

πτ (3-39)


Page 48

where nω and nτ are the angular frequency and period of the first mode of vibration. We

define FBthB = kd/3 as the threshold force, at which the snap down occurs. Then, Eqn (3-39)

becomes

⎥⎦

⎤⎢⎣

⎡= −

0

12 5.1cos

2 FFt thn

πτ (3-40)

In general, a voltage is used to actuate the gate of the microswitch. For a mass-

spring system, the voltage and force is related as follows

2

20

2dAVF ε

= (3-41)

From Eqn (3-41), it is seen that F is a function of voltage squared. So, Eqn (3-40) can be

approximated as

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛= −

2

0

12 5.1cos

2 VVt thn

πτ (3-42)

Notice that Eqn (3-42) was derived with an assumption that F is constant. To keep a

constant force, we need to vary voltage with distance d, as seen in Eqn (3-41). We have

neglected this nonlinear effect to derive Eqn (3-42).


Page 49

Figure 3-14 The relationship between the contact force, where ta is the actuation time, ton is the turn-on time, and Fa is the applied force. Note that ta and ton are normalized to the period of the first natural frequency, and Fa is normalized to a force Fth which corresponds to threshold voltage.

Figure 3-14 shows the relationship between contact force and time based on Eqns (3-38)

and (3-40). As a first order of approximation, the voltage is assumed to be proportional to

the square root of the electrostatic force. Figure 3-15 is the relationship of actuation

voltage normalized with respect to the threshold voltage as a function of time, based on

Eqns (3-38) and (3-42)(3-40).

Figure 3-15 The actuation time, ta, and the turn-on time, ton, for a dual voltage pulse method as a function of actuation voltage Va. Note that ta and ton are normalized to the period of the first natural

frequency, and Va is normalized to the threshold voltage.


Page 50

3.11 Results and Discussion

As can be seen from above, a dynamic model which includes aspects that are

believed to be most relevant to the dynamics of the MEMS switch has been developed. In

the following, we will present the simulation results of the MEMS switch. To compare

with the experimental measurements, we first introduce the measurement setup, then

present the experimental measurement of the microswitch, and lastly, a comparison will

be made between the simulated and measured results.

3.11.1 Simulation Results

The switch used in the simulation is fabricated using electroplated gold. The

dimensions of the switch and the gap between the tip and drain as well as the initial gap

between the two electrodes were obtained using Zygo NewView 6000, as shown in

Figure 3-7 and Figure 3-8.

The switch is simulated at atmospheric pressure and at room temperature. We first

simulated the modal behavior of the switch. It was found that the first resonant frequency

of vibration is about 349 kHz in the open position and 1.77 MHz in the closed state. The

measured resonant frequency of vibration is 346 kHz in the open position. Notice that the

Young’s modulus for electroplated gold which is used in simulation is from the literature,

as discussed in Section 3.8. The excellent agreement between the measured and simulated

resonant frequencies of vibration suggests that the Young’s modulus value we used for

electroplated gold in the simulation is a good approximation to the real value. The

corresponding time periods are 2.86 µs and 0.57 µs, respectively. The simulated


Page 51

threshold voltage (Vth) is about 65 V, which is in agreement with the measured values of

about 63 - 66 V.

Figure 3-16 shows the simulated displacement of the contact tip of the switch with

actuation voltages of 70 V, 74 V and 81 V. The corresponding initial contact times are

1.62 µs, 1.34 µs, 1.24 µs respectively. It is seen that the switch closes faster with larger

actuation voltage. However, the switch bounces with this single-step actuation and the

Figure 3-16 Contact tip displacements of the switch at actuation voltages of (a) 70V, (b) 74V, and (c) 81V.

number and magnitude of bounces increase with increasing actuation voltage. For each

case the magnitude of the bounces decreases with time due to the squeeze-film damping

effect.

The speed of the microswitch is important since it is related to the momentum of

the microswitch during operation. The velocity of the microswitch contact tip is obtained

as the first derivative of the contact tip displacement relative to time when the actuation

voltage is 81 V and is shown in Figure 3-17. Note that a positive velocity corresponds to

motion away from the surface. From Figure 3-17, the average acceleration of the

microswitch contact tip before the initial contact is about 44000 g. Notice that the


Page 52

horizontal lines indicate the switch remains closed while the vertical lines for the sudden

change of switch from close to open state. The velocity when the switch moves toward

the lower drain is negative, otherwise it is positive.

Figure 3-17 The simulated contact tip velocity as a function of time for an actuation voltage of 81V.

Figure 3-19 shows the displacements of the contact tip and of locations A and B,

as labeled in Figure 3-7 with an actuation voltage of 74 V. It can be clearly seen from the

motion of A in Figure 3-7 that the switch bends about 30 nm in the width direction while

the tips are in contact with the drain. The small difference of the motion between location

A and the tip is due to cross-bending of the actuator area due to the electrostatic force

exerted on it. In contrast, the displacement of location B is about half of that of location

A, due to the large stiffness of the switch in its closed position. For clarity, Figure 3-7 is

repeated here.


Page 53

Figure 3-18 The top view as well as the dimensions of the Northeastern University RF MEMS switch

where w1 = 80 µm, w2 = 10 µm, w3 = 16 µm, w4 = 30 µm, L1 = 30 µm and L2 = 24 µm.

Figure 3-19 Comparison of displacements at different locations of the switch (see Figure 3-7) with an actuation voltage of 74 V.

The total squeeze-film damping force, the total electrostatic force on the actuator,

and the ratio of the absolute values of these forces are shown in Figure 3-20. It is seen

that both the electrostatic force and the squeeze-film damping force oscillate. This

behavior is caused by the bouncing dynamics of the switch after initial contact. We first

simulated the modal behavior of the switch. It was found that the first resonant frequency

of vibration is about 349 kHz in the open position and 1.77 MHz in the closed state. The

measured resonant frequency of vibration is 346 kHz in the open position. From Figure

3-20 (b), it can be seen that the damping force becomes negative when the switch start to

w3

w4

w2

A

L2

L1

w1

BBeam


Page 54

bounce. Figure 3-20 (c) shows the ratio of the absolute values of the damping and

actuation forces. It is interesting to see that the damping force may significantly affect the

dynamics of the switch, since the damping force is dissipative and is as much as 13.5 %

of the electrostatic force for an actuation voltage of 1.1 times the threshold voltage. Also,

it can be seen that the damping force reaches its maximum just before the contact tip of

the switch makes initial contact with the drain. At this moment the gap attains its

minimum value and the speed is its maximum.

Figure 3-20 (a) Electrostatic force, Fe, (b) squeeze-film damping force, Fd, and (c) the ratio, ⎜Fd/Fe⎜, of their relative values with an actuation voltage of 74 V.

In order to see the evolution of the squeeze-film damping force, the distribution of

the squeeze-film pressure with respect to the atmosphere pressure, i.e. the gauge pressure,

across the gate area for an actuation voltage of 74 V is shown Figure 3-21. Before the

initial contact of the switch at times P1, P2, and P3 of Figure 3-21 (a), the maximum

pressure is located at the center of the gate, and it moves toward the contact tip edge. This

is because the local speed of the movable electrode near that edge is larger than that at

other locations and the corresponding separation is also smaller. After the contact tip

starts to bounce off the drain, the pressure at the edges first becomes negative and then


Page 55

the negative pressure spreads to the middle of the gate when the contact tip reaches the

maximum bounce [see P4 and P5 in Figure 3-21 (a), (d) and (e)]. This means that the

squeeze-film damping force initially resists closing and then resists it from bouncing off.

Also, it is worth noting that the pressure at the center of the gate is positive even after the

switch bounces off the drain [see the gauge pressure distribution of point P4 in Figure

3-21 (d)]. This is due to the fact that air is compressible and the pressure due to

compression is greater than due to viscosity. From Figure 3-21 (c) it can be seen that the

0 1 2 3 4-0.4

-0.3

-0.2

-0.1

0.0

0 8 16 2480

70

60

50

40

3020

10

0

(e) @ P5(d) P4(c) @ P3

(b) @ P2(a) @ P1

02356891112

Gate Width (µm)

Gat

e Le

ngth

(µ

m)

0 8 16 2480

70

60

50

40

3020

10

0

03691215182124

Gate Width (µm)

0 8 16 240

10

20

3040

50

60

70

80

Gat

e Le

ngth

(µ

m)

-12-10-9-7-6-4-3-102

Gate Width (µm)0 8 16 24

80

70

60

5040

30

20

10

0

-10-8-7-6-5-4-2-101

Gate Width (µm)0 8 16 24

80

70

60

5040

30

20

10

0

0918263544536170

Gate Width (µm)

P5

P4

P3

P2

P1

Tip

Dis

plac

emen

t (µ

m)

T ime (µs)

Va=74 V

Figure 3-21 Evolution of the squeeze-film pressure distribution across the actuator at an actuation voltage of 74 V.

maximum gauge pressure occurs at the center of the gate and has a magnitude of about 60

kPa, i.e. approximately 60 % greater than atmospheric pressure.The slip-flow effect

becomes important when the minimum gap of the device is on the order of the

micrometer or less. To study the effect of slip-flow on the dynamic behavior of the

microswitch, the displacement of the microswitch tip for cases where the slip-flow terms

in Eqn.(3-15) are and are not included, are shown in Figure 3-22 for an actuation voltage

of 70 V. It is clearly seen from Figure 3-22 that both the switching speed and the duration


Page 56

of bounces are reduced while the magnitude of bounce increases when the slip-flow

effect is included.

Figure 3-22 Comparison of the simulated microswitch contact tip displacement for cases with and without the slip-flow effect

After the contact tip of the switch makes contact with the drain, the instantaneous

contact force can be much larger than the static contact force. This is because the speed of

the switch is not zero when the contact between the contact tip and the drain occurs. In

the ANSYS® simulation, the instantaneous contact force along with the static contact

force at different actuation voltages are calculated and are shown in Figure 3-23. Here we

refer to the instantaneous contact force as the impact force. It is found that the maximum

impact forces are 5.6, 4.9, 4.5, and 4.2 times the static contact forces for actuation

voltages of 70V, 74V, 78V, and 81V, respectively. The ratio between the impact and

static forces decreases with increasing actuation voltage. This suggests that the higher

speed of the actuator causes a nonlinearly larger squeeze-film damping force for higher

actuation voltage, resulting in smaller ratio between the impact and static forces.


Page 57

Figure 3-23 Impact forces, together with the static contact forces, of the switch with actuation voltages of (a) 70V, (b) 74 V, (c) 78 V, and (d) 81 V, respectively.

The control of the dynamics of the switch is important for its proper operation.

Some closed loop feedback control mechanisms are used to control the dynamics of a

moving mass45, but this procedure often needs additional circuitry to implement. An

alternative approach to control the dynamics of the switch is the open-loop tailored

actuation waveform method. A simple version of this mechanism is the dual voltage

pulse control method45. However, a thorough investigation is needed to understand and to

make this tailored waveform actuation more efficient for controlling the dynamics of the

switch. By using the simplified results we have obtained in Section 3-10 as shown in

Figure 3-15, the simulation result using a dual-voltage pulse (Va = 88 V, ta = 0.8, Vh = 67

V, and ton = 1.05 µs) is shown in Figure 3-24. Compared with the single-step actuation,

this dual pulse can eliminate the bounce with a moderate impact force while maintaining

a fast switching speed. It is worth noting that the observed impact force oscillates with a

frequency of 1.2 MHz, which is smaller than the natural frequency of 1.77 MHz in the

closed state, after the contact tip is maintained in permanent contact with the drain. The

oscillating feature can be ascribed to the mechanical dynamics of the microswitch. The


Page 58

maximum impact force for this dual-pulse actuation is about twice the static force for the

same holding voltage, and is about one third of the impact force (~ 96 µN) for the same

single-step actuation voltage of 67 V. This result indicates that the bounce for a dual-

pulse actuation can be completely eliminated whereas the impact force is still larger than

the static force for the same holding voltage.

Figure 3-24 Displacement of the contact tip using a dual pulse actuation, Va = 88 V, ta = 0.8, Vh = 67 V, and ton = 1.05 µs. The inset shows the impact force for this dual pulse actuation. The static force for a

single-step actuation voltage of 67 V gives a static force of 15 µN.

3.11.2 Comparisons Between Experiments and

Simulations

The experimental work has been performed on the switches which were

developed and fabricated at Northeastern University. The measurement circuit is shown

in Figure 3-25. An arbitrary waveform generator (Agilent 33220A Function /Arbitrary

waveform generator, 20 MHz) for programmed waveforms and a power amplifier (Apex

PA85A model) for large actuation voltages are used. The voltage across the switch is

recorded with an Agilent Infiniium 54830B oscilloscope: 2 Channels, 600 MHz,


Page 59

sampling rate of up to 4 GSa/s. The total rise time, including the rise time of the function

generator and that of the amplifier for a single step voltage, is about 200 ns.

Figure 3-25 A schematic representation of the circuit and instruments used for experimental measurement.

The tests were conducted in room air where the resistance of the switch is

approximately an order of magnitude higher than in dry nitrogen. This finite resistance of

the switch causes the measured switch voltage to be nonzero after being closed. The

waveforms recorded in the oscilloscope basically show the closed and open status of the

switch. When the switch is in contact with the drain, the switch voltage is reduced;

otherwise it is a constant value of 500 mV. The measured results are shown in Figure

3-26. The traces show the instantaneous open and close state of the microswitch. When

the switch is open, the switch voltage is 0.5 V, otherwise, the voltage is less than 0.5 V.

The important features are that the switch bounces with a single step voltage actuation,

and the numbers of bounces increase with increasing magnitude of the actuation voltage.

Power Supply 500 mV

Oscilloscope

Function Generator

Resistor 50 ohm

Apex PA 85A Switch


Page 60

Figure 3-26 Switch voltages (solid lines) measured by oscilloscope and the corresponding single step actuation voltages (dotted lines) of 70 V, 74 V, and 81 V.

To compare the simulation results with the measurements of the switch dynamics

behavior, we plotted the initial contact time, Tc1, initial open time, To1, second contact

time Tc2 and second open time To2, as shown in Figure 3-27. The scattered points are

obtained from the experimental measurements. Since the response speed of the

oscilloscope is sufficiently fast, we neglect the circuit effect on the results. It seems that

the simulation results show an excellent agreement with the experiments for high

actuation voltages. The small discrepancy of the second open and close at low voltages

suggests that a more sophisticated contact model is needed to better predict the bouncing

dynamics of the switch.


Page 61

Figure 3-27 Close and open times versus actuation voltage, where Tc1, To1, Tc2, To2 are 1st close time, 1st open time, 2nd close time, and 2nd open time, respectively. The scattered dots are experimental

results and the lines are from simulations.

In the experiment for dual voltage pulse actuation, as shown in the inset of Figure

3-29(a), the values for Va, Vh, ta and ton are 1.5 Vth, 1.05 Vth, 0.5 µs, and 0.8 µs,

respectively, and are used in the function generator. Notice that since the first eigenperiod

of the switch is so short and the circuit has a finite rising and falling time, the expected

square shapes of the dual voltage pulses have been changed to triangle-like, as shown in

the inset of Figure 3-29(b). Notice that the observed peaks and valleys for time between 0

and 1.5 µs are caused by charging and discharging of the capacitor formed between the

actuator and the gate (Cag), which is coupled to the capacitor formed between the actuator

and the drain (Cad). Since these two capacitors have a common terminal, i.e. the actuator

of the switch, the charging or discharging of capacitor Cag will automatically charge or

discharge the capacitor Cad. In addition, the instantaneous current through the capacitor is

proportional to the rate of voltage change, i.e.dtdvCi = where C is the capacitance. To

include this effect, the waveforms are shaped to ensure that the work which is done on the


Page 62

switch for both actuation waveforms is maintained about the same in the experiments. It

is clearly seen from Figure 3-29 that the simulated result is in an excellent agreement

with the experiment.

Figure 3-28 compares the timing of closing and opening events for both

simulations [Figure 3-16 (c)] and experiments (see Figure 3-26) at V = 81 V.

Figure 3-28 Comparison between the simulated and measured opening and closing times for an actuation voltage of 81 V. The horizontal axis is the number of closings or openings of the switch.

In the practical application of the switch, higher contact force is desired for

smaller contact resistance. In the dual voltage approach, one can intentionally increase

the voltage and thus the contact force once the switch gets closed permanently, as shown

in Figure 3-30. In this graph, the oscilloscope traces of the switch voltage are shown for

dual voltage pulses with varying holding voltage of 74 V and 81 V. The inset shows the

corresponding actuation pulses. It can be seen that for a higher holding voltage, e.g. 81 V,

the recorded switch voltage is lower than the one for lower holding voltage, e.g. 74 V,

due to a reduction of contact resistance induced by a higher contact force or holding

voltage.


Page 63

Figure 3-29 Comparison between simulation (a) and experiment (b) for a dual pulse actuation, the insets show the corresponding pulses.

Figure 3-30 Oscilloscope traces of the switch voltage for a dual voltage pulse actuation with V h = 74 V, and 81 V, respectively. The inset shows the corresponding actuation dual voltage pulses.

It is common that fabrication process variations may exist even for a well-

established process for a device. The fabrication process variation may include the

performance degradation of the processing equipment and wafer-level non-uniformity or


Page 64

batch-to-batch inconsistency, which may cause the properties of the devices to vary

slightly for any individual device. To evaluate the sensitivity of the dual voltage pulse to

fabrication errors, we give a small variation of 5% of voltages, Va and Vh, and times of

11% ta, and ton before they are applied to the switch. Figure 3-31 shows the measured

results where the variations of the magnitude of the voltages Va and Vh are about 5 %, as

shown in Figure 3-31 (d).

Figure 3-31 Oscilloscope traces of the switch voltage for dual voltage pulses: (a1) [0.95Va, ta, 0.95Vh, ton], (a2) [Va, ta, Vh, ton], and (a3) [1.05Va, ta, 1.05Vh, ton], where Va = 1.35 Vth, Vh = 1.03 Vth, ta = 0.5 µs

and ton = 0.8 µs.

It is clear that the pulses (Va, Vh, ta and ton) with 5 % lower voltages are not

adequate to make the switch close [see Figure 3-31 (a1)]while the pulses with 5 % higher

voltages causes some bounces [see Figure 3-31 (a3)] although the number of bounces is

fewer than that in a single step actuation case. Similarly, the variations of times ta, and ton

of about 11 % also cause some bounces compared with the ideal dual pulse, as shown in

Figure 3-32. Notice that the 5 % voltage variation and 11 % time variation were chosen

for a small variation and distinguishability in the experiment. It is worth noting that the

dual pulse method is more sensitive to the threshold voltage than the time period. The


Page 65

observed results suggest that the dual pulse method may not be as efficient as expected in

terms of bounce elimination and impact force reduction if the switch parameters vary

significantly.

Figure 3-32 Oscilloscope traces of the switch voltage for dual voltage pulses: (b1) [(Va, 0.89ta, Vh, 0.89ton], (b2) [Va, ta, Vh, ton], and (b3) [(Va, 1.11 ta, Vh, 1.11ton], where Va = 1.35 Vth, Vh = 1.03 Vth, ta = 0.5

µs and ton = 0.8 µs.

As discussed above, squeeze-film damping plays an important role in determining

the dynamic response of the switch. One way to take advantage of the damping effect is

to design switches which have appropriate dimensions and geometries for critical

damping or moderate levels of damping during operation. Another alternative may to

intentionally increase the ambient pressure to some level for higher damping and better

operation, although this method may not be practical to implement for a switch. Figure

3-33 shows the simulation results of the dynamics of the switch which is actuated with a

single step voltage of 74 V but at pressures of 1 atmosphere and 10 atmospheres.


Page 66

Figure 3-33 Simulated contact tip displacement of the switch at pressures of 1 atm and 10 atms for an actuation voltage of 74 V.

It is obvious that the magnitude and length of time of bounces are greatly reduced

when the switch is operated at a pressure of about 10 times atmospheric pressure

compared with operating at atmospheric pressure while the closing speed has not been

dramatically affected. This behavior is because at high ambient pressure, the squeeze-film

acts more as an incompressible squeeze-film than as a compressible one. It is noted that

the method of using high pressure to increase damping force on microswitch may not be a

practical solution in real applications. But the observed responses of the microswitch to

high ambient pressure suggest that one can intentionally increase the damping force to

improve the microswitch performance. For instance, one can increase the damping area

and/or decrease the gap between the actuator and the ground to achieve a larger squeeze-

film damping force in a MEMS switch design.


Page 67

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TP

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17J. J. Blech, “On isothermal squeeze-film”, J. Lubrication Technology, V.105, 1983, pp615 – 620. 18T. Veijola, A. Pursula, and P. Raback, “Surface extension model for MEMS squeezed-film dampers”,

Proceedings of DTIP 2005, Montreux, June 1-3, 2005, pp. 405-410. 19S. Dushman and J. M .Lafferty, “Scientific foundations of vacuum technique,” Wiley, New York, 1962. 20Y. Yang, squeeze-film damping for MEMS structures, Master’s thesis, MIT, USA, 1998 21J. Starr, “Squeeze-film damping in solid-state accelerometer,” IEEE workshop on solid-state sensor

actuator (Hilton Head Island, SC, USA, June 4-7), pp.44-47,1990. 22M. H. Sadd and A. K. Stiffler, “Squeeze-film dampers: amplitude effects at low squeeze number,” J. Eng.

Ind. B V.97 pp.1366-1370, 1975 23E. S. Kim, Y. H. Cho, and M. U. Kim, “Effect of holes and edges on the squeeze-film damping of

perforated micromechanical structures,” MEMS’99 (Orlando, FL, Jan.1999) pp 296–301. 24 D. Homentcovschi and R. N. Miles, “Modeling of viscous damping of perforated planarmicrostructures:

applications in acoustics,” J. Acoust. Soc. Am. 116 2939 – 2947, 2004. 25M. Bao, H. Yang, Y. Sun and P. J. French, “Modified Reynolds equation and analytical analysis of

squeeze-film air damping of perforated structures,” J. Micromech Microeng. 13, pp.795–800, 2003. 26T. Veijola and P. Raback, “A method for solving arbitrary MEMS perforation problems with rare gas

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interactions and adhesion of gold contacts using a combined atomic force microscope and transmission

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Chapter 4. Intermodulation Distortion

Page 70

H


In this chapter, we will investigate the intermodulation distortion (IMD) effect

associated with MEMS/NEMS devices. The intermodulation effect to be examined is

different from the conventional intermodulation effect in a nonlinear device. In a

traditional nonlinear device, the nonlinear transfer function between the input and output

signals gives rise to components which have frequencies that are sums and differences of

multiple integers of the two fundamental signals for a two-tone input signal. In the

contact-type microswitch, the nonlinearity of the Ohmic contact between two metals is

not important. But the thermal response of the device to the input signals creates variable

electrical resistance in the switch due to the temperature dependence of the electrical

resistivity of the switch materials. The modulation of the signals by the variable

resistance at a difference frequency for a two-tone signal results in the generation of

intermodulation distortion. This mechanism is termed thermally-induced intermodulation

distortion and has been observed previously in waveguides. In this chapter, we will

discuss the origin and the underlying mechanism of the thermally-induced

intermodulation in more detail. In a model system, consisting of a very small co-planar

waveguide, we will first develop a thermal model to predict the temperature. Second, we

present the expression for calculating the intermodulation in a circuit. The effect of the

materials properties on intermodulation distortion is examined in great detail. Third, the

simulation results are compared with the experimental measurements for the co-planar


Page 71

waveguide system. Finally, as an example, the derived expressions are applied to predict

the intermodulation expected for an Ohmic contact-type RF MEMS switch.

4.1 Intermodulation Effect

Intermodulation distortion refers to a physical phenomenon in which signals of

different frequencies transmit through an element or a system and produce new

frequencies that are sums and differences of the input frequencies.

If two tones of different frequency, f B1B and fB2B, are simultaneously applied as input

stimuli to a system, the overall output stimulus is a linear combination of the two

individual input stimuli. The amplitude of the two tones may be different. For a purely

linear system, the output response is simply the linear sum of the individual responses to

their associated pure input tones- no harmonics or other frequencies of any kind are

present. For a nonlinear system, the output waveform will have components which have

frequencies that are sums and differences of the input frequencies due to the nonlinear

mixing of the input signals.

Intermodulation distortion exists in devices, and circuits that contain nonlinear

devices. The devices include RF MOSFETsTPD

1DPT, GaAs MESFET’sTPD

2DPT microwave power

amplifiersTPD

3DPT, and PIN diodes due to the non-linear I-V characteristic at high frequencies

TPD

4DPTP

,TD

5DTP. For PIN diodes in high power applications, the effect of conductivity modulation by

the RF signal can be minimized by the stored charges, Qs = IBF Bτ, where IF is the forward

bias current, and τ is the minority carrier lifetime, in the wide I-regionTPD

6DPT. In comparison,

GaAs MESFETs are inherently lower power switching devices.


Page 72

As stated above, the intermodulation distortion in active semiconductor-based

components (amplifiers, etc) arises from nonlinear behavior due to the existence of p-n

junctions. The intermodulation distortion has also been observed in passive devices such

as microwave coaxial cavity filtersTPD

7DPT, duplexers, RF-cables, waveguides and antennas, and

is termed passive intermodulation distortion (PIM)TPD

8DPT It has been found that most of the

PIM’s occur due to the nonlinear contact and materials properties. The former refers to

any metallic contacts which have nonlinear current-voltage behavior, loose, oxidized and

contaminated metallic joints, electron tunneling, microdischarge, etc are typical examples.

PIM due to materials properties usually occurs in ferrites and ferromagnetic materials

such as nickel, cobalt etc.

Since the surfaces of any materials in contact have some degree of roughness at

the microscopic scale. Only a small portion of the nominal, or apparent, area of the

contacting surface is in real contact, and is termed as load bearing areaTPD

9DPT. Meanwhile, the

load bearing area can be further divided into metallic contact spots, quasi-metallic spots

separated by a thin film, and thick film coated area. For metallic contacts, the constriction

of current across the interface may dominate the flowing mechanism of the current,

whereas the electron tunneling effect predominate the behavior of the current flow across

the insulating thin film. The voltage breakdown phenomenon may occur for an interface

with thick insulating film. The PIM distortion has been observed in the metal-metal

contacts TPD

10DPT. The surface of most metals is covered by a thin layer of insulating materials.

The electrons may pass through the energy barrier created by the insulating film if they

can gain enough energy through thermal heating or electric fields. The electrons with

lower energy than the barrier can still have the probability of being able to overcome the


Page 73

energy hill. This phenomenon is called tunneling. The PIM phenomenon arising from the

tunneling effect has been experimentally investigatedTPD

11DPTP

,TD

12. The PIM effect due to the

nonlinearity of permeability in ferromagneticTPD

13DD

14DPT and ferrimagnetic materialsTPD

15DPTP

-DDTD

17DTPand

permittivity in dielectric materialsTPD

18 has also been reported.

MEMS devices designed for applications at RF, micro/millimeter wave

frequencies also exhibit intermodulation distortion effect. RF MEMS devices such as

SAW or FBAR filtersTPD

19DPT due to the nonlinear capacitance with gap displacement, silicon

beam resonatorTPD

20DPT due to nonlinear materials properties such as engineering Young’s

modulus, capacitive shunt switchesTPD

21DPT, variable capacitors TPD

22DPT,TPD

23DPT, micromechanical

resonatorsTPD

24DPT, cantilever-type MEMS switchesTPD

25DPT, etc. The intermodulation distortion in

most of the MEMS devices which contains a movable component is from an oscillatory

force with frequency less than the mechanical resonance frequency of the beam. In a

capacitive RF MEMS switch, the input signal causes the variation of the capacitance in

the up-state, which modulates the capacitance of the switch and results in the

intermodulation distortionTPD

26. For a GSM900 (global system mobile) outdoor base station,

the transmit power of a channel is typically 43 dBm and the receiver noise floor lower

than -110 dBmTPD

27DPT. It is not trivial for RF MEMS devices to fulfill this requirement. It has

recently been found that the self-heating appear to be important in generating

intermodulation distortion in microwave devices for high power multichannel

applicationsTPD

28DPTP

,TD

29DT

, TD

30DTP. The thermal effect in the active devices are calculated based on the

assumption that the drain current is reduced by thermal heating, and the decrease is

frequency dependent. Therefore, the basic idea about the thermal effect on the

intermodulation is the effect of the self-heating on the nonlinearitiy of the devices since


Page 74

the characteristic resistance and capacitance of the junctions are changed by the

dissipated power.

4.2 Theoretical Analysis of Intermodulation

Distortion

Traditionally, the analytical expressions for IMD have been derived from series

expansions of the nonlinear I-V characteristics. Taylor series are sufficient for a

memoryless model (i.e. without capacitances), while Volterra series have been used for

models with memoryTPD

31DPT, particularly, which is more suitable for analyzing distortion at

high frequencies used in communication systems. Figure 4-1 shows the input and output

characteristics of a typical nonlinear system.

Figure 4-1 Schematic representation of a nonlinear system

For an input signal )(tVVin = the output would be

...)()()( 332

21 +++= xVaxVatVaVout where the coefficient aB1 B describes the linear term

while coefficients aB2 B and aB3B characterize nonlinearity. For a single sinusoidal input signal

)cos( tAVin ω= the nonlinear system gives rise to harmonics which have frequencies of

the multiple times of the fundamental frequency, as shown in Figure 4-2.

Vin Vout


Page 75

Figure 4-2 Generation of harmonics in a nonlinear system

If two sinusoidal signals of equal magnitude, e.g. )cos()cos( 21 tAtAVin ωω += ,

are applied to nonlinear systems, one would get intermodulation distortions, as shown in

Figure 4-3 (see Reference 32) .

Figure 4-3 Generation of IMD (2nd and 3rd order) in a nonlinear system

The IMDs have frequencies, ωBMNB, which are the linear combinations of the

fundamental frequencies of ωB1B and ωB2B as follows:


Page 76

21 ωωω NMMN ±= (4-43)

where M, N = 1, 2, 3, …. Among the IMDs, the 3P

rdP 5th order IMD are technologically

more important than others for the performance of the system. For systems with broad

bandwidth, the intermodulation distortions of any order may affect the performance of the

systems. But, narrow bandwidth systems are only susceptible to the IMDs which fall

inside the passband. Bandpass filtering may be used to eliminate most of the unwanted

spurious signals. However, the 3P

rdP order intermoudlaiton products are usually too close to

the fundamental signals to be filtered out. As a practical example, in a cellular

communication system, if more than one signal from the transmitter is present in the

input to the receiver, IMD will be generated. Furthermore, if two signals are close enough

such as the 3 P

rdP order intermodulation product fall within the passband, it becomes

impossible to filter this distortion. In modern wireless communications systems, each cell

site often has multiple transmitters and receivers, The intermodulation distortion may

affect other receivers and/or transmitters operating near the transmitter frequency. In

particular, the third order intermodulation product may fall close enough to the carrier

signals and lies within the passband of the device in operation. In the cable between the

duplexer and the antenna in GSM base stations and in certain space applications where

the high power transmission and low power reception signals are carried simultaneously

in the same transmission line.

The third order intermodulation distortion denoted as IDB3 B is: 12

33 4/3 aAaID = .

The figure of merit for a nonlinear system is the third order input intercept point (IPB3B),

which is the input power in dBm for a point where the extrapolated output power is equal


Page 77

to that of the 3P

rdP order intermodulation distortion. The determination of IPB3B is shown in

Figure 4-4.

Figure 4-4 The 3rd order intermodulation power and output power versus input power

For RF MEMS switches the value of IPB3 B is 66 - 80 dBm, in contrast with 27 - 45

dBm for both FETs and PIN diodes 54.

4.3 Thermally-Induced PIM in MEMS Switch

As discussed in Chapter 2, the RF MEMS switches are believed to be very linear,

due to the absence of the nonlinear behavior of current vs. voltage curve. In a broadside

switch (see Figure 2-5), the signal path is decoupled from the actuation electrode.

Therefore, the intermodulation distortion due to the nonlinear variation of capacitance

arising from the motion of the electrode is also negligible. It is also known that the PIM

exists in a wide variety of passive devices although the exact underlying mechanisms are

not well understood. It is found that self-heating may be important under some

circumstances. As mentioned above, the intermodulation in active devices usually refers

to the nonlinear current vs. voltage curve. It is found that the thermal transients within the

active channel of the device in microwave active devices leads to deteriorated 3P

rdP order

Input power (dBm) IIP3

IMD3

Iout Pout(dBm)


Page 78

intermodulation distortion, particularly for smaller spacing between two tones TPD

33DPT.

WilcoxTPD

34DPT studied the PIM effect resulting from the thermal heating of a coaxial

waveguides walls. The instantaneous temperature variation is caused by the electrical

energy dissipation through thermal conduction with finite time constant. The interaction

of the conductance variation due to temperature variation with the electrical fields gives

rise to the third order intermodulation distortion. This effect is a pure PIM due to

thermally-induced variation of the electrical conductance in the coaxial waveguides.

Recently, the intermodulation distortion due to thermal effects was reported in an

RF MEMS switch 35DPT. The electrical resistance of the switch varies with the difference

frequency of a two-tone signal, resulting in a signal component with the difference

frequency. If the difference frequency between two input tones remixes either of the

fundamental frequencies, or two input tones are modulated by the resistance with a

frequency which is equal to the difference frequency, the 3P

rdP order intermodulation

frequencies may be produced. These 3P

rdP order intermodulation frequencies may also be

generated when the frequency fB1 B is mixed with the 2P

ndP harmonic of frequency fB2 B. But this

effect is smaller compared with the first mode since the second harmonic is much greater

than the inverse of the thermal time constant of the switch than the difference frequency.

It has been shown that the 3P

rdP order intermodualtion power for the switch can be written

as 35:

2

2'0

'03

)2(2 ⎥⎦

⎤⎢⎣

⎡+

=RR

RRPP

in

rd β (4-44)

For a particular switch, R is the load resistance of 50 Ω, R0’ is the resistance of

the switch, and β is the ratio of the switch resistance that has been increased by Ohmic


Page 79

heating to that at room temperature. By using Eqn (4-44), one can estimate the maximum

intermodulation distortion for RF MEMS switches or other MEMS devices. In addition, it

is reported that Ohmic heating exists in many MEMS devices. For instance, Chow et al.

36 reported the effect of self-heating on buckling, plasticity and power handling capability

in air-suspended RF MEMS transmission-line structures. They found the peak

temperature rise of 112 oC (average temperature of 73 oC) for a 0.5-W RF power input for

suspended CPW transmission lines. Mercado et al. 37 found that the temperature of the

RF MEMS switch contact could reach a few hundred degrees C for an input power of

0.36 W. As discussed above, due to Ohmic heating, an alternating signal may cause an

oscillating device temperature. The temperature-dependent electrical resistivity of the

device causes transient variation of electrical resistance with time-varying input signals,

yielding nonlinearity in the device. The modulation of the input signals by the variable

electrical resistance at the difference frequency for a two-tone signal may give rise to the

intermodulation distortion.

However, in the literature the quantitative analysis of this thermally-induced

intermodulation distortion in terms of materials properties and device geometry has not

been performed. To our best knowledge, the experimental study which is dedicated to

address the thermal-induced intermodulation distortion has not been conducted. In the

following, we will present both the theoretical and experimental investigation of the

thermally-induced intermodulation in a simple device. Then we extend the obtained

results to the RF MEMS switches.


Page 80

4.4 Design of a Model System

The need of high-power and broad-band transceivers in transmission line or

communication systems demands that the devices or systems of micro/nano meter scale

withstand being operated at an elevated temperature. Due to the intrinsic temperature

dependence of the thermal and electrical properties of materials, the electrical resistance

of the device varies with frequencies. The signals passing through the transmission line

may be modulated as a result of the variable electrical resistance of the device. For

example, for a two-tone signal the passive intermodulation may occur when the device

operates at high temperatures. In order to investigate the intermodulation effect arising

from the thermally modulated electrical resistance, we designed a simple system

consisting of a miniature coplanar waveguide which allows us to conveniently investigate

the thermally-induced intermodulation distortion.

4.4.1 Design Considerations

The refractory metal tungsten is used as the material of the model device because

it has a large temperature coefficient of resistivity and a high melting temperature, which

allows us to modulate the resistance of the devices, and thus the signal, in a wide range.

Contact pads are made of gold for ease of probing and for good contact between the gold

pads and the probe tips during the experiment. The device is made in a back-grounded

coplanar waveguide (CPW) configuration with a nominal characteristic impedance of 50

Ω. The CPW gaps for the devices are determined for a 50 Ω characteristic impedance for

a CPW transmission line using Txline, a shareware simulation program for transmissions

lines. The CPW grounds are tungsten, and the back-side ground is made of aluminum


Page 81

with a thickness of 1 µm. The tapered CPW transition angle of 30 degrees from a narrow

device to a wide pad is used for a smaller return loss 38. The geometry and dimensions of

the device are illustrated in Figure 4-5. The device was fabricated in Pyrex glass substrate

Figure 4-5 The geometry and dimensions of the device, not to scale (dimensions in µm).

using standard micromachining technology and optical lithography.

Since the device is expected to modulate the carrier signals through variable

electrical resistance, the dc electrical resistance of the device is designed to be fairly

large, in this case it is about ~ 40 Ω at room temperature.

4.4.2 Microfabrication

The device was fabricated using two masks and by means of standard

microfabrication techniques. The wafer level layout of the device in a three inch glass

wafer is shown in Figure 4-6. The die level layout of the device is shown in Figure 4-7.


Page 82

Figure 4-6 The wafer-level layout of the device

To further view the layout of the device, Figure 4-7 shows the layout of the device as

well as the PAD, and the layout of the device is shown in Figure 4-8. The pink color

represents tungsten structure and the light brown color is for gold pads.

Figure 4-7 The die-level layout of the device

Figure 4-8 The layout of the device


Page 83

The first mask is to define the device by dry etching tungsten using a mixture of

SF6 and Argon in an inductively coupled plasma (ICP) etcher. The gold pads are then

defined by the second mask and fabricated by means of a lift-off process. The substrate

we used in this study is Pyrex glass. Figure 4-9 shows the fabrication process, and the

process parameters can be found in Appendix A.

Figure 4-9 The process flow of the fabrication of the device

The physical properties of both tungsten and Pyrex glass are summarized in Table

4-1. Figure 4-10 shows the SEM micrograph together with the cross-sectional view of the

device.


Page 84

Table 4-1 Physical Properties of Device Materials Used in the Model

Density (kgm-3, ×103)

Thermal conductivity (Wm-1K-1)

Specific heat

(Jkg-1K-1)

Electrical resistivity

(Ωm, ×10-7)

Temp. coefficient of

resistivity (K-1, ×10-4)

Tungsten 19.3 173 133 3.1 9.0

Pyrex glass 2.23 1.4 835 > 10P

14P

N/A

Figure 4-10 (a) SEM micrograph of the fabricated device. (b) Cross-sectional view of a device, not to

scale, where W1 = W3 = 160 µm, W2 = 12 µm, H1 = 1062 Å, H2 = 500 µm and H3 = 1 µm.

4.4.3 Mathematical Analysis

In order to predict the intermodulation distortion generated by the variable

electrical resistance, we build a device as described above. Before we develop model to

predict the intermodulation, we need to establish a thermal model to determine the device

temperature in terms of materials properties, geometry and signal frequencies. We will

first develop the thermal model for the device we fabricated. The three-dimensional view

of the device is shown in Figure 4-11.

20µm

Substrate

Al

(a)

(b)

W2

H1

H2

H3

W1 W3G


Page 85

Figure 4-11 The three-dimensional view of the device on a pryex glass substrate

4.4.3.1 Thermal Modeling

As mentioned above, we need to develop a thermal model which can

quantitatively determine the temperature variation for an alternating input signal. In order

to build such a model system, we make the following assumptions: 1) we neglect the heat

transfer effect due to the CPW ground electrodes which can be seen in Figure 1; 2) we

assume the heat is generated only by the Ohmic heating of the device; 3) the temperature

across the whole device is uniform and the bottom aluminum electrode is maintained at

room temperature; and 4) we neglect the electrical skin effect at high frequencies and

assume a uniform current distribution over the cross section. This latter assumption is

verified by the calculation of the skin depth of tungsten at a frequency of 10 MHz, which

yields the skin depth of about 89 µm which is much larger than the width and the

thickness of the device. Figure 4-12 shows the schematic representation of the device for

thermal analysis.

Au

W

Substrate

Contact Pad


Page 86

Figure 4-12 The cross-sectional device-on-substrate schematic showing the heat generated by tungsten as uniformly distributed over a semicircle with a radius of half the width of the device, i.e. r1

= W2/2, and is transferred to the ambient through conduction. The arrows illustrate the isotropic nature of heat conduction, r2 = H2 + H3, not to scale.

The modes of heat transfer from the device to the ambient may be conduction

through the substrate, convection through the air, or radiation. To evaluate the relative

importance of the three different modes of heat transfer, we calculated the equivalent heat

transfer coefficient, h, of each mode. Since the dependencies of the heat transfer

coefficients of tungsten on temperature for different heat transfer modes vary

significantly, we need to choose a temperature or a range of temperatures at which we

calculate the heat transfer coefficient. In addition, thermal convection and thermal

radiation become more important than thermal conduction at high temperatures, so we

assume the device has a high temperature of about 2273 oK and the ambient temperature

is 300 oK. These assumptions give an upper-bound estimation of the importance of

radiation and convection relative to conduction.

According to the Stefan-Boltzmann law, the radiative heat transfer coefficient, hBr B,

can be written as 39

))(( 212

22

12,1 TTTTFhr ++= εσ (4-45)

Substrate

T = 300 K

r2

DeviceT

r1


Page 87

where ε is the emissivity of the radiating surface, σ is Stefan-Boltzmann constant (5.67 ×

10P

-8P WmP

-2PKP

-4P), FB1,2 B is the shape factor between surface area of body 1 and body 2, TB1 B and

TB2 B are the absolute temperatures of the device and ambient, respectively, ε = 0.23 for

tungsten at 2000 oC 40, and FB1,2 B = 0.5. Substituting the values of these constants in Eqn (4-

45) yields hBrB = 88.19 WmP

-2PKP

-1P.

For a free, or natural, heat convection in the air and the device facing up, the heat

transfer coefficient for a turbulent flow is given as 41

333.022.0 Thc ∆= (4-46)

where ∆T is the temperature difference between the device and the ambient air in degrees

Fahrenheit. A simple calculation shows that the heat transfer coefficient for free

convection is about hc = 2.75 WmP

-2PKP

-1.

For calculation of the heat transfer via thermal conduction through the substrate,

we need to take into account the spreading effect. The spreading angle for a thin and

infinite long plate is given as42

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛=

6.0

180355.0tanh90 K

sπθ (4-47)

where K is the thermal conductivity of the substrate. In our case, the spreading angles are

θS = 2.98P

o for a glass substrate. A simple calculation shows that the equivalent coefficient

of heat transfer for the conduction mode can be written as 42

))tan(1(wH

Khcondθ

+= (4-48)

For the geometry shown in Figure 4-12, hBcondB = 1.36 × 10P5 PWmP-2 PKP

-1 P


Page 88

From the preceding discussion, it can be seen that heat transfer by means of the

conduction mode is dominant, compared with those through natural convection and

thermal radiation. Thus, in the following derivation of the temperature variation of the

device, we neglect the heat loss due to convection and thermal radiation.

Since the dimensions of the tungsten device are much smaller than that of the

substrate and the length of the device is much larger than its width, the heat transfer

problem of the devices is analyzed in a cylindrical coordinate system to include the effect

of heat spreading in the lateral direction, as shown in Figure 4-12. When the signal

current flows through the tungsten device, the temperature of the device will increase due

to Ohmic heating. To simplify the analysis, we physically removed the device from the

substrate and assume that the heat generated by the device is uniformly distributed in a

half cylinder of radius of r1 which is equal to half the width of the device, and the

temperature of the device is equal to the center temperature of the cylinder. The heat

generated in the cylinder is transferred only through thermal conduction.

Heat energy generation due to Ohmic heating of two sinusoidal signals with equal

power delivery, i.e. [ ])sin()sin()( 210 ttItI ωω += , is

[ ] [ ]221011

21

221

20 )sin()sin(

2)sin()sin(2)( ttg

HrL

LrttItg e ωωρ

πωω

+=+

= (4-49)

where1

31

20

0 HrI

g e

πρ

= , ρBeB is the resistivity of the device, rB1 B and H1 are quantities as defined

in Figure 4-10, Figure 4-11 and Figure 4-12, the factor of 2 is used to double the power of

a full cylindrical configuration of the system.

The problem can then be formulated as a two-dimensional heat transfer boundary-

value problem together with boundary conditions, i.e.


Page 89

Let 0),(),( TtrTtr −=θ :

t

trrrHktg

rtr

rrtr

∂∂

=−+∂

∂+

∂∂ ),(1)()(),(1),(

12θ

αθθ (4-50)

00),(2

≥== ttr rrθ

00),( 20 ≥≥== rrtr tθ

where α = k/(ρBd BCBp B) is the thermal diffusivity of the substrate, r2 is taken to be equal to H2.

H(r1-r) is the unit step function as defined below

⎪⎩

⎪⎨

⎧

<−

≥−=−

)0(0

)0(1)(

1

1

1

rr

rrrrH (4-51)

Using the integral transfer method43, the kernel and eigenvalues for Eqn (4-50) in

the cylindrical coordinate system are

0)()()(2),( 20

21

0

2

=−= rJandrJrJ

rrK

m

mmm β

ββ

β (4-52)

where J B0B and J B1 B are the Bessel functions of the first kind of order 0 and order 1,

respectively, and β BmB are the eigenvalues of 0)( 20 =rJ β .

The solution to Eqn (4-50) can then be written as 43

∫∑ ∫∞

=

−= 222

0 01 02

21

02

2

)()'()()(2),(

r

nn

t

n

nt drtgrrJdterJrJ

ekr

tr nn βββαθ αβαβ (4-53)

As assumed above, the heat generation is only confined in a volume of a cylinder with a

radius of rB1 B, so Eqn (4-53) can then be reduced to

∫∑ ∫∞

=

−= 122

0 01 02

21

02

2

)'()()()(2),(

r

nn

t

n

nt drrrJdttgerJrJ

ekr

tr nn βββαθ αβαβ (4-54)


Page 90

Since )()( 10 rJrdrrrJ nn

n ββ

β =∫ 44, Eqn (4-54) can be further reduced to

∑ ∫∞

=

−=1 02

21

11012

2

)()(

)()(2),(22

n

t

nn

nnt dttgerJ

rJrJe

kr

rtr nn αβαβ

ββββα

θ (4-55)

Substituting Eqn (4-49) into Eqn (4-55) yields

[ ]∑ ∫∞

=

− +=1 0

221

221

11012

2

)sin()sin()(

)()(2),(22

n

t

nn

nnt dttterJ

rJrJe

kr

rtr nn ωω

ββββα

θ αβαβ (4-56)

To simplify Eqn (4-56) , let [ ]∫ +=t

n dttteF n

0

221 )sin()sin()(

2

ωωβ αβ , and 2nb αβ=

[ ]

[ ]

[ ]

[ ]⎭⎬⎫

⎩⎨⎧

++++++

−

⎭⎬⎫

⎩⎨⎧

−−+−+−

+

⎭⎬⎫

⎩⎨⎧

++

−

⎭⎬⎫

⎩⎨⎧

++

−

⎥⎦

⎤⎢⎣

⎡

+++

+−−

++

++−=

ttbb

e

ttbb

e

ttbb

e

ttbb

e

bbbbb

bbeF

bt

bt

bt

bt

bt

n

)sin()()cos()(

)sin()()cos()(

)2sin(2)2cos(42

1

)2sin(2)2cos(42

1

)(1

)(1

41

411)(

2121212221

2121212221

222222

111221

221

221

222

221

ωωωωωωωω

ωωωωωωωω

ωωωω

ωωωω

ωωωωωωβ

(4-57)

After making substitution of Eqn (4-57) into Eqn (4-56), the distribution of temperature

at location r and time t, i. e. T(r, t) is

432100),( θθθθθ ++++=−TtrT (4-58)

Where

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+++

+−−

++

++−

= ∑∞

=

−

42221

2

42221

2

4222

2

4221

2

22

1 22

1

1102

2

010

)()(

441

)()()(2

2

2

n

n

n

n

n

n

n

n

nn

t

n nn

nnt

n

n

e

rJrJrJe

krgr

βαωωαβ

βαωωαβ

βαωαβ

βαωαβ

αβαβββββαθ

αβ

αβ (4-59)


Page 91

[ ])2sin(2)2cos(4

1)(

)()(111

2

1422

122

1

1102

2

011 tt

rJrJrJ

krgr

nn nnn

nn ωωωαββαωββ

ββαθ ++

−= ∑∞

=

(4-60)

[ ])2sin(2)2cos(4

1)(

)()(222

2

1422

222

1

1102

2

012 tt

rJrJrJ

krgr

nn nnn

nn ωωωαββαωββ

ββαθ ++

−= ∑∞

=

(4-61)

⎥⎦

⎤⎢⎣

⎡

−−+−

+−= ∑

∞

= tt

rJrJrJ

krgr n

n nnn

nn

)sin()()cos(

)(1

)()()(2

2121

212

142

2222

1

1102

2

013 ωωωω

ωωαββαωωββ

ββαθ (4-62)

⎥⎦

⎤⎢⎣

⎡

++++

++−= ∑

∞

= tt

rJrJrJ

krgr n

n nnn

nn

)sin()()cos(

)(1

)()()(2

2121

212

142

2222

1

1102

2

014 ωωωω

ωωαββαωωββ

ββαθ (4-63)

As discussed above, it is assumed that the temperature at r = 0 is equal to the

temperature of the device, so substitute r = 0, i.e. 1)( 00 ==rn rJ β , ∞=t or steady state,

into Eqn (4-60), we have:

tDtDtDtD

tDtDtDtDTTtrT st

)sin()cos()sin()cos(

)2sin()2cos()2sin()2cos(),(

214215

214215

242312110

ωωωωωωωω

ωωωω

+−+−−−−+

−−−−=− (4-64)

where TBstB is the temperature for cases where frequency dependent terms disappear or

average temperature, D1 – D8 are constants which are determined by the material

properties, dimensions of the device and substrate, and levels of input power, as shown in

Eqns (4-59) - (4-63).

∑∞

=

=1 2

21

311

22

01

)()(2

n n

nst rJ

rJkrgrT

nββ

β (4-65)

∑∞

= +=

1422

1

2

22

1

112

2

02

11 4)(

)(n n

n

nn

n

rJrJ

krgrD

βαωβ

βββα

(4-66)

∑∞

= +=

1422

1

1

22

1

112

2

012 4

2)(

)(n nnn

n

rJrJ

krgrD

βαωω

βββα (4-67)


Page 92

∑∞

= +=

1422

2

2

22

1

112

2

02

13 4)(

)(n n

n

nn

n

rJrJ

krgrD

βαωβ

βββα

(4-68)

∑∞

= +=

1422

2

2

22

1

112

2

014 4

2)(

)(n nnn

n

rJrJ

krgrD

βαωω

βββα (4-69)

∑∞

= +−=

1422

21

2

22

1

112

2

02

15 )()(

)(2n n

n

nn

n

rJrJ

krgrD

βαωωβ

βββα

(4-70)

∑∞

= +−−

=1

42221

21

22

1

112

2

016 )(

)()(

)(2n nnn

n

rJrJ

krgrD

βαωωωω

βββα (4-71)

∑∞

= ++=

1422

21

2

221

112

2

02

17 )()(

)(2n n

n

nn

n

rJrJ

krgrD

βαωωβ

βββα

(4-72)

∑∞

= +++

=1

42221

21

22

1

112

2

018 )(

)()(

)(2n nnn

n

rJrJ

krgrD

βαωωωω

βββα (4-73)

4.4.3.2 Third-Order Intermodulation Products

From Eqn (4-64), it can be seen that the steady-state temperature of the device varies with

a frequency which is twice that of the input signal, i.e. 2ω, for a single input signal. For

two-tone signals, i.e. ωB1 B, ωB2 B, the steady-state time-varying temperature has components

with frequencies equal to the sums and differences of integer multiples of the input

signals. The electrical resistance of metals varies with temperature and is generally

correlated with the temperature coefficients of resistivity as45

...])()()(1[ 30

2000 +−+−+−+= TTTTTTe ηαγρρ (4-74)

In Eqn (4-74), γ, α, η, are the temperature coefficients of the electrical resistivity of the

material for the first second, and third powers of the temperature, respectively.


Page 93

For a first-order approximation and simplicity in the following derivation, we

included only linear terms in (T-T0). Thus the resistance, Rsw, of the device can be written

as

)](1[ 00 TTRR swsw −+= γ (4-75)

where Rsw0 is the resistance of the device at temperature T0.

When the device with a resistance represented by Eqn (4-75) is used as an

element in a circuit, as shown in Figure 4-13, the voltage across the load can be written as

)( 000 TTRRRR

RVVswswLS

LinL −+++

=γ

(4-76)

Assuming that 000 )( swLSsw RRRTTR ++<<−γ and using a Taylor series expression,

Eqn.(4-76) can be approximated as

)](1[ 00

0

0

TTRRR

RRRR

RVVswLS

sw

swLS

LinL −

++−

++≈ γ (4-77)

where RL, RS, and Vs are the load resistance, source resistance and source voltage,

respectively.

Figure 4-13 The circuit configuration in which the microstructure is in series with a load where RS and RL are for source resistance and load resistance, respectively. RSW represents the resistance of the

device that is variable with input power.

Substitute a two-tone input voltage given by

[ ])sin()sin()( 210 ttVtV sin ωω += (4-78)

RL Vin

-

+

VL

RS RSW


Page 94

where Vs0 is the magnitude of the two-tone input voltage, and Eqns (4-49) & (4-64) into

Eqn (4-77) one can get the voltage for a two-tone signal including the thermally-

modulated electrical resistance across the load. If we define the voltage components with

frequency 212 ωω − and 122 ωω − as the third-order sideband voltages, after simple

mathematical manipulation, we can get the third-order sideband power across the load

with respect to the input power to the load as the following

(i) 212 ωω − component

( ) [ ]⎭⎬⎫

⎩⎨⎧

++++++

−=2/1

622

512/100

02/1

103

int )()()(

log20 DDDDRRRRR

RRP

LswswLS

swLrd γ (4-79)

(ii) 122 ωω − component

( ) [ ]⎭⎬⎫

⎩⎨⎧

−+++++

−=2/1

462

532/100

02/1

103

int )()()(

log20 DDDDRRRRR

RRP

LswswLS

swLrd γ(4-80)

From Eqns (4-79) and (4-80), it can be seen that the third-order intermodulation

distortion power is proportional to the square of the temperature coefficient of the

electrical resistivity. This agrees with the result for coaxial waveguides [21]. If the

frequencies of the two-tone signal are sufficiently high, e.g. ωB1 B and ωB2 B are on the order of

magnitude of a megahertz or greater, and the difference frequency (ωB1 B-ωB2 B) is sufficiently

small, then D1, D2, D3, D4, D6, D7 and D8 become zero and D5 becomes Tst. Therefore, the

intermodulation distortion rdP3int at frequencies of 212 ωω − and 122 ωω − can be further

reduced to

( ) ⎭⎬⎫

⎩⎨⎧

+++−= st

LswswLS

swLrd TRRRRR

RRP 2/1

00

02/1

103

int )(log20

γ (4-81)


Page 95

Based on Eqns (4-65) and (4-81) and the expression of )/()( 13

1200 HrIg e πρ= , it is found

that the power of the third-order intermodulation distortion is proportional to the square

of the electrical resistivity and inversely proportional to the square of the thermal

conductivity of the substrate. This behavior can be understood because large device

resistivity and small thermal conductivity of the substrate allow the device to have a

temperature variation, thus large electrical resistance variation and large signal

modulation for a given input power, leading to large intermodulation in the low

frequency limit. In contrast, it is worthwhile to notice that when the difference frequency

is comparable with, or larger than, the inverse of the thermal time constant of the device,

the device with smaller thermal conductivity will have larger intermodulation for a given

input power. This is because the modulation of the signals is limited by the finite heat

transfer rate, which is proportional to the thermal conductivity. This result can be seen by

looking at the dependence of the intermodulation on the thermal conductivity of the

device, as expressed in Eqns (4-65) - (4-73) and Eqns (4-79) - (4-80). It is interesting to

notice that the intermodulation distortion increases with room-temperature electrical

resistance Rsw0, if Rsw0 is smaller than RL and RS. This suggests that decreasing the room-

temperature electrical resistance of the device is an effective way to reduce the thermally-

induced intermodulation distortion.

According to Eqn (4-75), one can get γTst = (Rsw-Rsw0)/Rsw0, which when

substituted into Eqn (4-81) gives

( )( ) ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+++

−−= 2/1

00

2/10

103

int)(

log20swLswLS

Lswrd

RRRRRRRR

P (4-82)


Page 96

By using Eqn (4-82), one can conveniently derive the intermodulation distortion in the

low frequency limit of both the frequencies of the two-tone signal and the difference

frequency.

4.5 Results and Discussion

4.5.1 Model Predictions

Figure 4-14 (a) shows the variation of the electrical resistance of the device for an input

signal of )sin()( 0 tVtV ω= [see Figure 4-14 (b)]. It can be seen that the resistance varies

with a frequency of twice that of the exciting signal, i.e. 2ω, due to the fact that the

dissipated power is proportional to the square of the input signal and the resistance

increases with thermal heating. The variation of the electrical resistance of the device is

caused by the thermal effect which is associated with the material properties of the

device. This suggests that the magnitude of the resistance variation depends on the

frequency of the input signal. For an input signal )sin(0 tII ω= , the resistance of the

device can be represented as )]2sin(1[0 ∆++= tRR ωδ , where δ is a measure of the

resistance variation due to Ohmic heating and ∆ is the phase shift caused by the finite

thermal time constant of the device.


Page 97

Figure 4-14 (a) The electrical resistance variation showing a sinusoidal-type variation with a frequency of 2ω, i.e. R = sin(4πft+∆). (b) The input sinusoidal signal with a frequency of f = 3.2 kHz,

i.e. I = I0sin(2πft).

Figure 4-15 shows the value of δ as a function of frequency on a log scale with an

input power of 40 mW. It is clearly seen that the resistance of the device is allowed to

have a noticeable variation, and the variation decreases monotonously with frequency

until 105 Hz. For frequency larger than 105 Hz, the variation of the resistance is

negligible, which suggests that the intermodulation effect in the device could be

neglected if the frequency, or difference frequency for RF signals, is larger than 105 Hz.

In order to understand the frequency dependence of the intermodulation distortion, the

intermodulation distortion for a two-tone signal has been calculated based on Eqns (4-79)

- (4-80). The frequency of one tone is 10 MHz and the input power for a 50 ohm load is

40 mW. The sideband power of the 3rd order intermodulation distortion as a function of

the difference frequency is shown in Figure 4-16. It can be seen that the intermodulation

exhibits a maximum value in the low frequency limit and starts decreasing at about 1 Hz

and is less than -60 dBc for difference frequencies greater than 40 kHz. This result


Page 98

suggests that the intermodulation may become significant for frequencies which are close

to or smaller than the inverse of the thermal time constant (~ 27.9 µs) of the device.

Figure 4-15 Variation of the resistance of the device as a function of the frequency. The input power for a 50 ohm load is 40 mW.

Figure 4-16 The third-order intermodulation distortion of the device as a function of difference frequency ∆f = f2 - f1, f2 = 10 MHz. The input power for a 50 ohm load is 40 mW

4.5.2 Static and Transient Electrical Resistance

As mentioned above, the variable electrical resistance, which is caused by the Ohmic

heating of the input signal, is responsible for generation of the intermodulation of the


Page 99

device. By measuring the static electrical resistance of the device versus the current of the

input signal, one can determine the variation of the electrical resistance, and thus the

maximum intermodulation distortion during operation of the device. The electrical

resistance measurement is carried out using a standard four point method. The sheet

resistance of the tungsten film with a thickness of 0.11 µm has been measured using the

four probe method and is found to be 2.8 Ω per square. The corresponding electrical

resistivity of the tungsten film is calculated to be 3.1 × 10-7 Ωm. Compared with the bulk

value of 5.6 × 10-8 Ωm, the resistivity of the tungsten film is about a factor of 5.5 times

larger. This result could be attributed to the two-dimensional constraint effect as well as

the effect due to the presence of defects or impurities of the film.

Figure 4-17 shows the electrical resistance variation of the device as a function of

the measuring current. The maximum resistance which corresponds to the point where the

device is failed by being burned out is recorded. The temperature coefficient of the

electrical resistivity and melting temperature for bulk tungsten are 4.8 × 10-3 K-1 and

3400 K, respectively.

Figure 4-17 The electrical resistance of the device as a function of the measuring current using a four point probe test setup


Page 100

From the resistance variation in Figure 4-17, the temperature coefficient of the

electrical resistance is calculated as 6.1 × 10-5 K-1 assuming the device has a uniform

temperature and the maximum temperature is the melting temperature. The calculated

value is much smaller than the theoretical value of tungsten. This suggests that the

temperature of the device is not uniform and/or the temperature at which the device fails

may not be the melting temperature of tungsten.

To clarify this discrepancy between the experiment measurements and the

theoretical values, the static electro-thermal analysis with three-dimensional Solid69

element of the device has been performed using finite element package ANSYS®. In the

simulation, the modeled device has exactly the same geometry and properties as the real

device. The thermal-electric solid element of Solid69 includes the Joule heating. The

lower surface of aluminum film (see Figure 4-12) is the only thermal boundary and set to

be 300 K since it is in intimate contact with a chuck metal stage of the probe station. The

heat loss due to radiation and natural convection is neglected as discussed in Section

4.4.3.1. From the simulation, we found that the temperature of the device are not uniform,

with lowest temperature of 322 K at the pad edges and highest temperature of 537 K at

the center of the device for an direct current of 33 mA. The current of 33 mA corresponds

to a point in which the device starts to get ‘melt’. But in the thermal model presented in

Section 4.4.3.1, we assume that the device has a uniform temperature. To make the

developed thermal model applicable to the analysis of experimental data, we used an

‘averaged’ temperature with respect to the electrical resistance of the device to represent

the ‘uniform’ device temperature in the model. This ‘averaged’ ‘uniform’ temperature of

the device is calculated to be 453 K for a current of 33 mA. Based on the simulated


Page 101

‘averaged’ temperature of 453 K and the corresponding resistance, an equivalent

temperature coefficient of electrical resistivity of 1.8 × 10-3 K-1 was obtained for the

device.

Meanwhile, to obtain the electrical resistivity of tungsten film, we measured its

sheet resistance, Rsh, using a four-probe method and found Rsh = 2.8 Ω/. The thickness

of tungsten film is measured using Zygo NewView 6000 and found to be about 0.11 µm.

The corresponding electrical resistivity of tungsten film is calculated to be 3.1 × 10-7 Ωm.

Compared with the bulk value of 5.6 × 10-8 Ωm, the resistivity of tungsten film is about a

factor of 5.5 times larger. This result could be attributed to the two-dimensional

constraint effect as well as the effect due to the presence of defects or impurities of the

film.

To further understand the transient behavior of the device, we measured the time

dependent electrical resistance of the device. The experimental setup for measuring the

transient electrical resistance of the device is shown in Figure 4-18. This setup is

Figure 4-18 Block diagram of the measurement system for the transient electrical resistance of the microscale devices

composed of a computer-controlled oscilloscope and an arbitrary waveform generator

which is the input to a power amplifier. The oscilloscope recorded the voltage traces

50Ω

DUT

Probe Station

Oscilloscope CPU Controller

Function Generator Amplifier


Page 102

across the device. The current flowing through the device is calculated from the serial 50

ohm resistor.

The recorded data of the device with different voltages across the device are

shown in Figure 4-19. It can be seen that the electrical resistance of the device increases

with time and finally reaches a steady-state value, which corresponds to the thermal

equilibrium of the device for a given current. The time for the device to reach equilibrium

is about 60 µs, which corresponds to a frequency of about 17 kHz. When increasing the

measuring current or voltage, the steady state electrical resistance of the device increases

while the time it takes to reach thermal equilibrium does not change significantly from

that at lower current levels. This result suggests that the device may exhibit a large

intermodulation effect when the difference frequency of a two-tone signal is less than

inverse of the time it takes to reach its thermal equilibrium state, i.e. 17 kHz.

Figure 4-19 The transient electrical resistance of the device with different applied voltages

4.5.3 Comparison Between Experiment and Simulation

As mentioned above, the signals into the load are modulated by the electrical resistance

with a frequency equal to the difference frequency, resulting in third-order


Page 103

intermodulation distortion. In order to verify the analytical model presented above, the

intermodulation distortion caused by the thermal effect has been measured in an

experimental setup as shown in Figure 4-20. Two waveform generators which produce

signals of 10 MHz and 10.064 MHz, respectively, are used as sources of input signals.

The solid-state power amplifiers (SSPA) are used to magnify the input signals. Before the

input power is applied to the system, a mixer and a power attenuator are used both for

mixing the two signals and controlling the input power level. The device is mounted on

the stage of a SUSS Z probe station and is probed with a low loss coaxial cable. The

output spectrum of the power across the load is recorded using a HP 8596E spectrum

analyzer.

Figure 4-20 Block diagram of the experimental setup for the two-tone intermodulation measurement, where f1 and f2 are two tone signals and SSPA is for solid-state power amplifier. This figure is

provided by Professor Elliot Brown from University of California at Santa Barbara.

Considering the physical structure (see Figure 4-10) and its configuration in a

circuit as shown in Figure 4-20, the equivalent electrical circuit can be modeled as a

serial resistor, serial inductor and a parallel capacitor. To calculate the intermodulation, it

can be simplified to a circuit as shown in Figure 4-13. For a total input power of 100mW


Page 104

into the system of the device and the load, the measured power across a 50Ω load is about

40mW. The dissipated power would be 32mW for the device which has a resistance of

40Ω, yielding a reflected power of 28mW. Thus the reflection coefficient for the device

in a 50Ω impedance coaxial cable is 0.53. The equivalent impedance of the device can

then be calculated to be 163Ω, which is much greater than the nominally designed value

of 50Ω.

The spectrum of the output power into a 50Ω load is shown in Figure 4-21. Notice

that the output power has been normalized with respect to the net input power assuming a

perfect match between the device and the transmission line. For instance, in the case of

the attenuation of 0 dB, the recorded output power is normalized with respect to 72 mW

instead of 100 mW. The frequencies of the two-tone signal are 10 MHz and 10.064 MHz,

respectively. It is worth noting that the levels of the third-order sideband power for both

2f1 - f2 and 2f2 - f1 are different, as shown by the scattered dots in Figure 4-22. The solid

line is for the modeled result which is calculated using Eqns (4-79) (4-80). The predicted

level of the third-order sideband power at the frequency of 2f1 - f2 is slightly greater than

that for 2f2 - f1. However, the overall dependence of the intermodulation on the input

power shows good agreement between the experimental and predicted results (see Figure

4-22). In addition, according to Eqn (4-82), it is found that the predicted maximum, i.e.

low frequency limit, sideband power of the third-order intermodulation distortion due to

the thermal effect is about -27.4 dBc for our device.


Page 105

Figure 4-21 Output spectrum of the intermodulation distortion with respect to the total input power of the device for cases: (a) Pin = 72 mW, (b) Pin = 36 mW, and (c) Pin = 18 mW, where f1 = 10 MHz, ∆f = f2 - f1 = 6.4 kHz. The measurements were conduced by Professor Elliot Brown from University of California at Santa Barbara

Figure 4-22 Comparison of the modeled third-order intermodulation distortion with experimental measurement at different power levels, the frequency of the first tone signal is f1 = 10 MHz, the

difference frequency is ∆f = f2 - f1 = 6.4 kHz. The measurements were conduced by Professor Elliot Brown from University of California at Santa Barbara


Page 106

4.5.4 Prediction of Intermodulation in an RF MEMS

Switch

As mentioned in Section 4.4.3.2, Eqn (4-82) can be conveniently used to calculate the

thermally-induced intermodulation distortion in MEMS/NEMS devices, since only the

device resistances at difference temperature or input power are needed. Here, we will use

it to calculate the third-order intermodulation distortion in an RF MEMS switch. The RF

MEMS switches are fabricated using micromachining technology with gold as the contact

material and silicon as the substrate. These materials exhibit nonlinear temperature-

dependent material properties.

We first simulated the electrical resistance change as a function of current which

passing through the switch. The simulations were performed on the microswitch using

ANSYS®. The switch has the actual three-dimensional geometry. The materials such as

density, specific heat, thermal conductivity, and electrical resistivity are temperature

dependent and the temperature-dependent bulk-material values for these materials are

used 46 . Figure 4-23 shows the solid model of the switch. The simulated electrical

resistance of the microswitch is shown in Figure 4-24. According to Eqn (4-82), if we

know the switch resistances at high current, or heated state, and room-temperature, the

intermodulation distortion at low difference frequency limit can be calculated. The

calculated intermodulation distortion using Eqn (4-82) at low difference frequency limit

as a function of input power is shown in Figure 4-25. It can be seen that the

intermodulation distortion increases with input power and reaches to - 40 dB at 16 W.

This result indicates that the proper selection of materials and geometry optimization of


Page 107

the microswitch must be made to reduce the intermodulation distortion for high power

applications.

299.836301.983

304.13306.277308.424310.571312.718314.865317.012319.159

Figure 4-23 The solid model of a quarter of the Ohmic contact-type RF MEMS switch

Figure 4-24 The simulated electrical resistance of the microswitch as a function of current which flows through the switch.


Page 108

Figure 4-25 Intermodulation sideband power relative to input power as a function of power transmitted by switch

As a comparison, a CREE SiC RF power MESFET transistor has an

intermodulation distortion of -31 dBc at 10 W PEP (Vdd = 48V, Idd= 250mA, f1 =

2000.0MHz, f2 = 2000.1 MHz)47 and a Philips AN10173-01 PIN diode switch48 has a

third order intermodulation intercept point of 39 dBm, which are much larger than that

for RF MEMS switches.


Page 109

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Chapter 5. Summary and Future Work

Page 112


In this chapter, we will present a summary of the thesis which includes the

primary conclusions of the study, the contribution to the literature, and the proposed

future work to improve the present study, thus improving the understanding of the

microswitch dynamics and the intermodulation distortion effect.

5.1 Dynamic Simulation

The dynamic behavior is an important aspect of the microswitch and is relevant to

its overall performance. The dynamic performance of the switch can be characterized by

some figures of merit: 1) switching speed; 2) bounce; 3) impact force; and 4) damping. It

is often desired that that the switch have a fast switching speed, a bounce-free closure, a

minimized impact force, and a controlled damping for its optimum dynamic performance.

In the literature, there exists a lot of work on modeling and simulation of the

dynamics of MEMS devices, and most of them focus on the squeeze-film damping effect

in these devices. Most models and simulations on dynamics of MEMS switches used

lumped spring-mass-damper systems, suggesting that the time-dependent local

characteristics, e.g. the local damping force, nonlinearity, the transient contact force and

the bounces, of the microswitch are neglected, although they are important for the

performance of the microswitch. In this thesis, we present a more comprehensive

dynamic model which includes almost all of the important aspects of the microswitch. It


Page 113

is expected that this model will precisely predict the dynamic response of the switch and

serve as a convenient and reliable design tool for the future design of the microswitch.

In this thesis, a model, which describes the mechanical dynamics of a RF MEMS

switch using finite element analysis for the structure and a finite difference method for

squeeze-film damping, has been developed. The model takes into account the real switch

geometry, electrostatic actuation, squeeze-film damping, nonlinear contact and the

adherence force. These modeled aspects of the switch are believed to be critical for

simulating the dynamic response of the switch both before and after initial contact. The

model has been used to simulate the switching speed, tip displacement and bowing

deflections, impact force, and bounces of the switch. The simulation results are in

excellent agreement with the measurements of a real switch in terms of closing time, and

the number and duration of bounces.

In addition, a dual voltage pulse scheme has been developed and applied to the

switch. This tailored actuation waveform method has been shown to be beneficial to the

performance of the switch. In particular, it is shown that the magnitude and number of

bounces of the switch could be reduced. It is expected that this simulation method will

become a design tool for future switch design and development. The simulation and

experimental results obtained provide some useful insights into the operation of a MEMS

switch.

In the model, a link element has been used to represent the contact between the

contact tip and the drain with an assumption that plastic deformation occurs only in the

first contact and subsequent contacts are pure elastic in nature. This may have

oversimplified the real situation. In the future work, one may develop a more realistic


Page 114

model by including characteristics such as plastic deformation, material transfer, and

contact evolution in the model.

It has also been shown that the dual-voltage pulse actuation may help reduce

bounce and control impact force. Large impact force may cause physical damage to the

contact, increase adherence force, facilitate material transfer, etc. Those results are not

desired to occur in MEMS switches. Also, bounce introduces unwanted contact cycles

which equivalently reduce the lifetime of the switch. The experimental measurements

also indicate that the method may not be that effective if the switch has a short close time

since it is very sensitive to parameter changes. One effective way to improve the

robustness of the dual-pulse actuation is to deliberately design a switch which has large

damping force which can both reduce the bounce and impact force. Meanwhile, damping

force may not dramatically change switch closure time.

5.2 Intermodulation Distortion

Intermodulation usually refers to the ability of a device or system to be immune to

the interferences from the environment when a device or system is in operation. For RF

and microwave application of MEMS switches, the intermodulation property is important

since it is critical to the integrity of the carrier signals. Ohmic contact-type RF MEMS

switches are generally believed to be a more linear device than the existing

semiconductor switches due to the direct contact between the two metals, thus having a

negligible intermodulation.

It is recently found that the thermal heating of the MEMS switch by carrier signal

may create intermodulation distortion to the device. This phenomenon becomes serious


Page 115

when devices are in operation at elevated temperatures. In the literature, there is very

little work on the thermally-induced intermodulation distortion, and the experimental

work is missing. The underlying mechanism responsible for generation of the

intermodulation distortion due to Ohmic-heating is not yet well understood.

As part of the thesis, analytical models for calculating the temperature variation as

well as the thermally-induced 3rd order intermodulation distortion of MEMS/NEMS

devices have been developed. It is found that the intermodulation distortion is closely

related to the thermal characteristics of the device and has been quantitatively determined

in terms of the material properties, the frequencies of the two-tone signals, and the

difference frequency. The intermodulation distortion becomes significant for a difference

frequency of a two-tone RF signal in the range of the inverse of the thermal time constant

of the device. In the high frequency limit, the intermodulation distortion is dominated by

the thermal conductivity of the substrate over other parameters. The intermodulation

distortion caused by Ohmic heating increases with input power.

In addition, a closed-form expression for intermodulation distortion in the low

frequency limit has been derived and can be conveniently used to calculate the maximum

thermally-induced intermodulation distortion in a device or a system. The experimental

measurements of the third-order sideband power of the microfabricated device show good

agreement with the calculated results based on the model.

The significance of the intermodulation work is that we developed analytical

models to predict the intermodulation distortion in passive devices such as an RF MEMS

switch. By using this model, we can quantitatively understand the dependency of

materials properties, device geometry and frequencies. Meanwhile, we have


Page 116

experimentally demonstrated the intermodulation distortion which results from Ohmic

heating. In the future, it would be very useful if we have the intermodulation

measurements of the MEMS switches and make comparison with the prediction.

Appendix A

Page 117

Appendix A

1. RCA cleaning of glass wafer

Process of tungsten (W) structure

2. Deposition of W (~0.11µ) using MRC sputtering machine, DC magnetron sputtering,

current 0.4 ampere, at argon gas of 12 mTorr.

3. Spinning of photoresist (PR) 1813

4. Soft baking for 1 minute at 145 oC

5. Photolithography, UV for 6 seconds

6. Hard-baking for 1 minute at 145 oC

7. Develop in 319 developer for 45 seconds

8. Running water rinsing for 5 minutes

9. ICP etching using Argon and SF6: Pressure 4 mTorr, the flow rates for Ar and SF6 are

2 sccm, and 4 sccm, respectively, Power RF1: 70, Power RF2, 300 Watts

Process of gold (Au) contact pads

10. Deposition of gold (~ 2000Å) using MRC: RF deposition, 350 Watts, 12 mTorr argon

gas

11. Lift-off in acetone for 2 minutes with ultrasound at room temperature

12. Running water rinsing for 5 minutes

Process of aluminum (Al) backside electrode

13. Spinning of PR on front side of the wafer

14. Back-side deposition of Al (~2 µm) using Perkin-Elmer sputtering machine: DC

magnetron sputtering, Argon pressure: 12 mTorr, Current: 6.48 Amperes.

15. Dicing the wafer into chips

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