Temperature Compensated Radio-Frequency Free-Free Beam MEMS

Resonators using a Commercial Process

By George Xereas

Department of Electrical & Computer Engineering

McGill University

Montreal, Quebec, Canada

April 2013

A thesis submitted to McGill University in partial fulfillment of the

requirements for the degree of Master of Engineering.

Copyright 2013 George Xereas

iii

ABSTRACT

Frequency references are needed in every modern electronic device including mobile

phones, personal computers, and medical instrumentation. With modern consumer

mobile devices imposing stringent requirements of low cost, low complexity, compact

system integration and low power consumption, there has been significant interest in

developing batch-manufactured MicroElectroMechanical Systems (MEMS) Radio-

Frequency (RF) resonators. An important challenge for MEMS resonators is to match the

temperature stability of quartz resonators.

The first part of this Thesis focuses on developing Free-Free beam MEMS resonators

using PolyMUMPs, a commercial multi-user process available from MEMSCAP. An

analytical method is derived in order to enable the design of resonators operating at 1

MHz, 5 MHz, 10 MHz, 20 MHz and 30 MHz. Their performance is evaluated using the

CoventorWare Finite Element Method (FEM) software suite.

In the second part of this Thesis, we present a 20 MHz temperature compensated Free-

Free beam MEMS resonator that is also developed using PolyMUMPS. We introduce a

novel temperature compensation technique that enables high frequency stability over an

industrial grade temperature range. The approach is based on two strategic principles,

passive compensation by using a structural gold layer on the resonator, and a Free-Free

beam design that minimizes the effects of thermal mismatch between the vibrating

structure and the substrate. Numerical simulation results showed that in the passively

compensated design, a strategic deposition of gold on top of the resonator induces a

positive frequency shift that counterbalances the characteristic negative Temperature

iv

Coefficient of Frequency (TCf) of polysilicon resonators. Temperature stability increased

from -8.48 ppm/℃ to -1.66ppm/℃. The performance of the proposed designs is

comparable to resonators fabricated through complicated custom processes that are

characterized from low fabrication yields and high costs.

v

ABRÉGÉ

Les références de fréquence sont nécessaires dans tous les appareils électroniques

modernes, incluant téléphones mobiles, ordinateurs personnels et instruments médicaux.

Les applications mobiles requièrent des oscillateurs à coûts réduits, faible complexité et

grande capacité d’intégration pour une consommation faible. Ceci a induit, en particulier,

un intérêt considérable pour les résonateurs radiofréquences (RF)

'MicroElectroMechanical Systems (MEMS)' produits à grande échelle. Un défi important

pour les résonateurs MEMS est d'avoir une stabilité en température comparable avec les

résonateurs quartz.

La première partie de cette Thèse se concentre sur le développement des résonateurs

MEMS du type 'Free-Free beam' avec PolyMUMPS, un processus commercial à

utilisateurs multiples disponible par MEMSCAP. Une méthode d'analyse est présentée

pour permettre la conception de résonateurs qui fonctionnent à 1 MHz, 5 MHz, 10 MHz,

20 MHz et 30 MHz. Leur performance est évaluée en utilisant la suite logicielle 'Finite

Element Method (FEM)' de CoventorWare.

Dans la deuxième partie de cette Thèse, nous présentons un résonateur MEMS du type

'Free-Free beam' fonctionnant à 20 MHz avec une compensation de température, qui est

développé avec PolyMUMPS. Nous introduisons une nouvelle technique de

compensation de la température qui permet une stabilité à haute fréquence sur une

gamme de températures compatibles avec les normes industrielles. L’approche est

basée sur deux principes stratégiques: compensation passive en utilisant une couche

d’or structurel sur le résonateur, et une conception ‘Free-Free beam’ qui minimise les

vi

effets de non-concordance thermique entre la structure vibrante et le substrat. Les

simulations numériques ont montré que par compensation passive, un dépôt stratégique

de l’or sur le dessus du résonateur induit un déplacement de fréquence positif qui

contrebalance le coefficient de température de la fréquence (TCf) négatif caractéristique

des résonateurs polysiliciums. La stabilité thermique a augmenté de -8.48 ppm/℃ à

1.66ppm/℃. La performance des résonateurs proposés est comparable à celle des

résonateurs fabriqués par des processus complexes qui sont caractérisés par des coûts

élevés et des rendements de fabrication faibles.

vii

ACKNOWLEDGMENTS

First and foremost, I would like to express my sincerest gratitude to my advisor Professor

Vamsy P. Chodavarapu. He gave me his support and believed in me at a time where I

was near the stages of despair. Ever since I joined the lab, he has been as good a role

model as I could have hoped for, constantly challenging me with new research ideas for

innovation. I would also like to thank my engineering colleagues, Adel Merdassi, Maurice

Cheung, Yucai Wang for all the feedback they’ve given me for my work and helping me

whenever I encountered difficulties.

I was fortunate to make some wonderful friends during my time at McGill, each inspiring

in their own way. My gratitude goes to my fellow engineering colleagues and friends,

Stefanos Koskinas, Kishor Ramaswamy and Andrianos Andrianopoulos for always being

there for me, in good and bad times. Of course, I would like to thank Rose LaRiviere for

supporting me in every decision I’ve made, pushing me to pursue my dreams and putting

up with me and my unpredictable schedule.

Last but not least, I would to like to thank my parents, Antony and Catherine, my sister

Vivian and of course my uncle Elias for enabling me to do everything I’ve done so far.

Their unconditional love and support means the world me, none of this would have

happened if they hadn’t given me the courage to believe in myself.

viii

TABLE OF CONTENTS

ABSTRACT ......................................................................................................................... iii

TABLE OF CONTENTS .................................................................................................... viii

LIST OF TABLES ................................................................................................................ x

LIST OF FIGURES ..............................................................................................................xi

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

1.1 Quartz Oscillators - Motivation ................................................................................. 2

1.2 MEMS Devices ......................................................................................................... 4

1.3 MEMS Resonators.................................................................................................... 4

1.4 Transduction Mechanisms in MEMS Resonators .................................................... 7

1.4.1 Piezoelectric Transduction ................................................................................ 8

1.4.2 Electrostatic Transduction ................................................................................. 9

1.4.3 Alternative Transduction Mechanisms ............................................................ 13

1.5 Quality Factor and Energy Loss ............................................................................. 13

1.5.1 Anchor Damping .............................................................................................. 15

1.5.2 Gas Damping ................................................................................................... 16

1.5.3 Thermoelastic Dissipation (TED) ..................................................................... 17

1.5.4 Surface Losses ................................................................................................ 18

1.6 MEMS Resonator Stability ...................................................................................... 18

1.6.1 Short Term Stability ......................................................................................... 19

1.6.2 Medium Term Stability ..................................................................................... 19

1.6.3 Long Term Stability.......................................................................................... 21

Chapter 2 MEMS-CMOS Integration and Standard Processes ........................................ 23

2.1 RF MEMS Integration with CMOS .......................................................................... 23

2.1.1 Pre-CMOS Approach ...................................................................................... 24

2.1.2 Intra-CMOS Approach ..................................................................................... 26

2.1.3 Post-CMOS Approach ..................................................................................... 27

2.2 Standard Processes for RF Resonators ................................................................. 29

2.2.1 UW-MEMS ....................................................................................................... 29

2.2.2 SOIMUMPS ..................................................................................................... 32

2.2.3 CMOS .............................................................................................................. 34

2.2.4 PolyMUMPs ..................................................................................................... 38

Chapter 3 Free-Free (FF) Beam MEMS Resonator ......................................................... 42

3.1 Free-Free Beam Resonator Structure .................................................................... 43

3.1.1 Resonant Frequency Design ........................................................................... 44

3.1.2 Mechanical Body Model of Free-Free Vibration .............................................. 45

ix

3.1.2 Electrostatic Spring Stiffness ........................................................................... 50

3.1.2 Support Structure Design ................................................................................ 52

3.2 Layout and Fabrication ........................................................................................... 58

3.2.1 PolyMUMPs Process Flow .............................................................................. 59

3.2.2 PolyMUMPs Process Limitations .................................................................... 61

3.2.3 Temperature Compensation of FF beam Resonators ..................................... 64

3.3 Free-Free Beam Design Characteristics ................................................................ 67

Chapter 4 Finite Element Modeling (FEM) Simulations, Analysis, and Results ............... 70

4.1 MEMS Resonator Evaluation Methods .................................................................. 70

4.2 Model Initialization .................................................................................................. 72

4.3 Modal Analysis ........................................................................................................ 74

4.4 Electrostatic Spring Stiffness .................................................................................. 78

4.5 Harmonic Electromechanical Analysis ................................................................... 79

4.6 Temperature Compensation Evaluation ................................................................. 84

Chapter 5 Conclusion and Future work ............................................................................ 87

5.1 Conclusion .............................................................................................................. 87

5.2 Future Work ............................................................................................................ 88

References ........................................................................................................................ 90

x

LIST OF TABLES

Table 1: Stability characterization ..................................................................................... 18

Table 2: Comparison of the three different approaches for CMOS-MEMS integration [33]

.......................................................................................................................................... 24

Table 3: CMOS-MEMS performance prediction based on CMOS technology used [49]. 35

Table 4: Eigenvalues β for the first 10 modes of vibration a FF beam [59]. ..................... 49

Table 5: Torsion coefficient α value depending on the cross-sectional geometry of the

beam [59] .......................................................................................................................... 58

Table 6: Set of mandatory design rules for PolyMUMPs, relative to this work [50]. ......... 61

Table 7: Design parameters of uncompensated FF beam resonators. ............................ 68

Table 8: Design parameters of temperature compensated FF beam resonators. ............ 68

Table 9: Material properties used in FEM simulations. ..................................................... 72

Table 10: Material properties used in FEM simulations. ................................................... 76

Table 11: Quality factors for the uncompensated FF beam resonators. .......................... 83

Table 12: Quality factors for the uncompensated FF beam resonators. .......................... 86

xi

LIST OF FIGURES

Fig. 1.1Typical oscillator design .......................................................................................... 2

Fig. 1.2 Intel’s Roadmap showing the Technology Pipeline [3] .......................................... 3

Fig. 1.3 Comb-Drive MEMS resonator integrated with the CMOS controlling circuit [7] .... 5

Fig. 1.4 A SEM (Scanning Electron Microscope) image of a commercial MEMS oscillator

(courtesy of Discera) [10] .................................................................................................... 6

Fig. 1.5 Number of publications on MEMS resonators versus year published ................... 6

1.3 MEMS Resonators Basics ............................................................................................ 7

Fig. 1.6 Basic mechanical lumped model for a MEMS resonator ....................................... 7

Fig. 1.7 A processed SEM photo of an FBAR resonator [12] ............................................. 9

Fig. 1.8 Clamped-Clamped beam MEMS resonator [13] .................................................. 10

Fig. 1.9 A simple model of an electrostatically driven MEMS resonator [11] .................... 10

Fig. 1.10 SEM photos of electrostatic resonators. a) An electrostatic comb drive is used

to reduce the gap b) A 74 MHz wine-glass mode resonator with 4 perimeter supports.

Close-up shows the magnitude of the gap........................................................................ 12

Fig. 1.11 Impedance matching techniques. a) Wine-Glass Disk [19], quarter-wavelength

supports are located at radial nodal point locations. b) Contour-Mode disk [20], supported

by polysilicon stem that is impedance mismatched to a diamond disk. ............................ 15

Fig. 1.12 Effect of anchor dissipation on the operation of a MEMS resonator as the

frequency increases. a) Clamped-Clamped beam resonator with an original quality factor

of 8,000 at 7MHz. b) Free-Free beam resonator where the support beams are

strategically placed at nodal points. [13] ........................................................................... 16

Fig. 2.1 A cross-sectional schematic of the CMOS-MEMS integration technology by

Sandia National Laboratories [34] ..................................................................................... 26

Fig. 2.2 A cross-sectional schematic of the intra-CMOS integrated pressure sensor,

developed by Infineon Technologies ................................................................................ 27

Fig. 2.3 A schematic design of the Texas Instruments post-CMOS integrated DMD [38].

.......................................................................................................................................... 29

Fig. 2.4 UW-MEMS fabrication process – Part A [39] ....................................................... 30

Fig. 2.5 UW-MEMS fabrication process – Part B [39] ....................................................... 31

Fig. 2.6 SEM photo of a UW-MEMS Clamped-Clamped beam resonator [40]................. 31

Fig. 2.7 Cross sectional view of the layers used in the SOIMUMPS process [42]. .......... 32

Fig. 2.8 Micrograph of the square resonator reported by Khine et al. [45] and a large-

span view of its frequency response. ................................................................................ 34

Fig. 2.9 Cross-section view of three typical FF-Beam CMOS-MEMS resonators. (I) Metal-

Oxide composite, (II) Metal-Via composite, (III) Single metal layer [47] ........................... 36

xii

Fig. 2.10 a) Schematic view of the CMOS-MEMS Free-Free beam resonator by Li et al. b)

Cross sectional view of the layers used in the fabrication of the Free-Free beam

resonator [49] .................................................................................................................... 36

Fig. 2.11 Comparison of the temperature sensitivity of a Mere-Metal FF-Beam and a

SiO2-Metal composite FF-Beam [49]. ............................................................................... 38

Fig. 2.12 Cross sectional view of the layers used in the PolyMUMPs process [50]. ........ 39

Fig. 2.13 A SEM image of a disk resonator fabricated through PolyMUMPs [56]. ........... 40

Fig. 3.1 Schematic view of the FF beam resonator along with the required readout

circuitry. ............................................................................................................................. 44

Fig. 3.2 Forces and strain applied on a differential element of a uniform vibrating beam.46

Fig. 3.3 Normalised vibration shape function for the first 5 modes of the FF beam

resonator. .......................................................................................................................... 53

Fig. 3.4 FEM simulations illustrating the first 6 modes of vibration of a FF beam. The blue

areas denote the nodal points. .......................................................................................... 53

Fig. 3.5 Top-down view of the FF beam resonator structure. ........................................... 54

Fig. 3.6 (a) λ/4 torsional beam; (b) Acoustic π network model for a λ/4 torsional beam; (c)

Equivalent acoustic network when the beam is anchored at port B. ................................ 55

Fig. 3.7 Process flow used for the fabrication of the FF beam resonators. ...................... 60

Fig. 3.8 FF beam length Lr versus the resonant frequency. ............................................. 62

Fig. 3.9 Supporting beam length Ls versus the resonant frequency. ................................ 62

Fig. 3.10 Layout of a 40 MHz FF beam resonator. ........................................................... 63

Fig. 3.11 20 MHz compensated FF beam resonator. ....................................................... 65

Fig. 3.12 Chip layout of the 9 resonators that were fabricated in PolyMUMPs. ............... 69

Fig. 4.1 a) Layout of the 1 MHz resonator, cyan and white areas denotes a Poly-2 and

Poly-1 layer respectively. b) Fabrication process used in the PolyMUMPs run. .............. 73

Fig. 4.2 3D meshed model of the 1 MHz FF beam resonator........................................... 74

Fig. 4.3 3D modal shapes for the uncompensated designs operating at a) 1MHz, b) 5MHz,

c) 10MHz, d) 20MHz, e) 30MHz. ...................................................................................... 77

Fig. 4.4 3D modal shapes for the compensated designs operating at a) 1MHz, b) 5MHz, c)

10MHz, d) 20MHz. ............................................................................................................ 78

Fig. 4.4 Plot of resonant frequency, for the 20 MHz design, versus applied bias voltage

VDC. .................................................................................................................................... 79

Fig. 4.5 Frequency response at 20V VDC, for: a) 30 MHz resonator, b) 20 MHZ resonator.

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

Fig. 4.6 Frequency response at 20V VDC for: a) 10 MHz resonator, b) 5 MHz resonator. 82

Fig. 4.7 Frequency response at 20V VDC for the 1 MHz resonator. .................................. 83

xiii

Fig. 4.8 Fractional frequency versus temperature for the 20MHz uncompensated, gold

compensated and SiO2 compensated design. .................................................................. 85

1

Chapter 1 Introduction

Clocks or frequency references are the fundamental building block of every modern electronic

device including Global Positioning Systems (GPS), mobile phones, personal computers, medical

diagnostic equipment and televisions. A frequency reference is responsible for many tasks

including synchronizing the operation of a digital logic circuit, keeping track of real time, setting

the clock frequency for a digital transmission, and frequency conversion and filtering in RF

transceivers. In fact, multiple frequency references are required for any advanced electronic

systems. Thus, each year approximately 10 billion frequency reference devices are fabricated ,

representing a $6.3 billion commercial market [1].

In general, a frequency reference device consists of a resonator and an amplifier connected in a

feedback loop as can be seen in Fig. 1.1., a configuration also known as an “oscillator”. Ideally

the output signal should have a high spectral purity, and most importantly, high frequency

stability. While the concept is simple, its implementation is rather complicated. A series of

external factors can affect the performance of frequency reference devices. Phase noise,

temperature stability and material aging are some of the most important parameters that need to

be addressed before successful implementation can be achieved.

2

Fig. 1.1Typical oscillator design

1.1 Quartz Oscillators - Motivation

For over 50 years, quartz oscillators have dominated the timing device market because they are

able to offer a unique combination of high precision and low cost frequency references [2].

However, the continuing evolution of electronics has increased the requirements for frequency

references. Silicon compatibility and ever shrinking components are the main challenges that

quartz technology needs to address. Micro-fabrication foundries have remained on track with

Moore’s Law and are expected to do so for another decade [3]. As can be seen in Fig. 1.2, 22nm

technology is currently commercially available, while 14nm is expected to arrive in the near

future. What has remained largely the same for the past few decades, and can be characterized

as the bottleneck of the industry, is the quartz oscillator. An ordinary quartz reference system can

take up nearly half the space required by a digital chip [4]. Furthermore, while shrinking digital

components have lowered the power consumption of chips; crystal oscillators have largely

maintained the same power levels for decades.

3

Fig. 1.2 Intel’s Roadmap showing the Technology Pipeline [3]

Perhaps the largest and most important drawback of quartz reference systems is that they cannot

take advantage of the developments in silicon manufacturing technology such as for the past four

decades, billions of dollars were invested every year into research and development of

Complementary Metal Oxide Semiconductor (CMOS) and MicroElectroMechanical Systems

(MEMS) processes. Batch microfabrication allows the production of thousands of devices at very

low cost. Furthermore, monolithically integrated chips with frequency reference devices and

integrated circuitry (IC) can be produced in CMOS or any other batch fabrication method leading

towards compact and low power electronic systems. While repeated attempts have been made to

integrate quartz resonator technology with silicon or III-V electronics, the results are usually

associated with high costs or high fabrication complexity [5].

The introduction of a series of portables devices such as tablets and smartphones has only

served to outline the limitations of quartz technology. Their large size and high power

consumption, high cost and low silicon compatibility has prompted researchers to investigate

4

other technologies for frequency references [6]. The most promising of those, is the silicon based

MEMS resonators.

1.2 MEMS Devices

MEMS technology has existed for 4 decades but it has become more relevant in the last decade,

with the development of miniaturized electronic devices. It is responsible for integrating

mechanical systems at micrometer or few tens of nanometer dimensions. MEMS applications

range from gyroscopes and accelerometers, pressure, touch and airflow sensors to fluidic

systems and displays. They have revolutionized pre-existing industries and have also enabled

the creation of entirely new markets. Smart-phones, tablets and automobile airbag systems were

prohibitively expensive or simply impractical to create before MEMS technology arrived.

One of the fundamental advantages of MEMS is that it has been built upon the well-established

semiconductor industry. Thus, technological advancements in the silicon industry have a direct

impact on MEMS. MEMS resonators are an example of one of the most prominent and useful

MEMS devices.

1.3 MEMS Resonators

With modern consumer mobile devices imposing stringent requirements of low cost, low

complexity, compact system integration and low power consumption, there has been significant

interest to develop MEMS resonators that can be batch-manufactured. Thereby reducing the

production cost, and more importantly, allowing resonator devices to be on-chip with integrated

circuits (IC) [6]. Fig. 1.3 shows an example of IC system closely integrated with a MEMS

resonator.

5

Fig. 1.3 Comb-Drive MEMS resonator integrated with the CMOS controlling circuit [7]

The first MEMS resonator was developed in 1967 by Nathenson et al. [8]. Since then, a

tremendous amount of research has been put into this area. A series of designs have been

suggested, using different structure geometries and materials. An initial wave of optimism was

quickly dismissed when the complexities of realizing MEMS resonators were fully understood. A

series of breakthroughs had to be made in the fields of anchor damping, gas damping,

thermoelastic dissipation, thermal expansion and the thermal coefficient of elasticity before

researchers could design effective, stable, and efficient MEMS resonators.

All of the suggested devices were confined to research laboratories until 2003. In 2003, Discera

released the MRO-100 (Fig. 1.4), the first commercial silicon micro-oscillator targeting the RF

market. A series of startup companies including SiTime, Silicon Clocks and Harmonic Devices

have since entered the market [9]. The interest in MEMS resonators is increasing rapidly (Fig.

1.5) as their commercial potential is being recognized by major semiconductor industries such as

Bosch and STMicroelectronics.

6

Fig. 1.4 A SEM (Scanning Electron Microscope) image of a commercial MEMS oscillator

(courtesy of Discera) [10]

Fig. 1.5 Number of publications on MEMS resonators versus year published

7

1.3 MEMS Resonators Basics

In general a MEMS resonator is represented mechanically as a mass-spring-damper second

order system as shown in Fig. 1.6.

Fig. 1.6 Basic mechanical lumped model for a MEMS resonator

.

The spring and the mass are used to describe the oscillation while the damper is used to describe

the energy losses. The resonant frequency (fo) of such a system can be found from Equation 1.1,

m

kfo

2

1 (1.1)

where, k represents the spring constant and m the mass of the resonating structure. This is a

very simplified perspective view but the basic principle always holds. Equation 1.1 shows that the

frequency of operation can be controlled by changing the spring constant or mass of the

resonating structure. As will be demonstrated later in this thesis, this is an essential principle

regarding the design of MEMS resonators.

1.4 Transduction Mechanisms in MEMS Resonators

As the name suggests MEMS devices are based in both the mechanical and electrical domains.

The coupling efficiency of the two domains is usually a good indicator of the potential success of

the device. A MEMS resonator is a mechanical structure, but in order to operate it requires an

electrical input. Its output is a mechanical vibration which is converted into an electrical signal in

8

order to be “sensed” and subsequently utilized. There are many transduction mechanisms that

convert mechanical energy into electrical energy. The choice of the transduction mechanism is an

important deciding factor in the MEMS resonator design. Electrostatic and piezoelectric

transduction mechanisms are most commonly used due to the ease of fabrication and excellent

performance of the designs. However, alternative methods based on optical and magnetomotive

transduction do exist, and have met success in niche markets.

1.4.1 Piezoelectric Transduction

Piezoelectric materials such as quartz have the ability to couple together the mechanical and

electrical domains very efficiently. When a voltage is applied across a piezoelectric material it

causes the creation of mechanical stress across the structure. If this voltage is alternated at the

natural frequency of the structure, the material will vibrate with ideally no energy loss. In turn, this

vibration will cause an electrical field to be generated across it, thus creating a perfect physical

resonator.

Piezoelectric devices, by nature, have very high transduction efficiencies and are relatively simple

to fabricate. Because of this, quartz, Surface Acoustic Wave (SAW) and Thin-Film Bulk Acoustic

Wave (FBAR) resonators have been very successfully commercialized.

While quartz is an exceptional piezoelectric material, it is difficult to fabricate and control it in the

“micro” domain. In order to address this problem, researchers turned to piezoelectric materials

such as aluminum nitride (AlN), zinc oxide (ZnO) and zirconium titanate (PZT) [11]. These are

much more likely to be found in MEMS devices such as SAW and FBAR filters.

9

Fig. 1.7 A processed SEM photo of an FBAR resonator [12]

In theory, the quality factor of piezoelectric MEMS resonators is unparalleled compared to other

transduction methods. However, the practical results deviate greatly from the theoretical

possibilities. One of the main cause of losses are the metal electrodes which are placed in direct

contact with the piezoelectric material in order to transfer the output. The metal-piezoelectric

interface introduces non-linearities that diminish the quality factor of the devices.

1.4.2 Electrostatic Transduction

In general MEMS resonators based on electrostatic transduction provide excellent characteristics,

such as, high quality factor, low phase noise, low power consumption and very low space

requirements. It is widely believed that they will bring a revolution in the area of radio frequency

(RF) electronics and timing devices.

10

MEMS resonators that are based on electrostatic transduction are usually composed of a semi-

free moving structure, an actuating electrode and a sensing electrode. A typical example is the

Clamped-Clamped beam resonator as shown in Fig. 1.8. A sinusoidal signal at the actuating

electrode is used to drive the resonating structure into its natural frequency of vibration. The

mechanical displacement causes a change in the capacitance of the space between the sensing

electrode and the vibrating structure. The constant variation of the capacitance causes an output

current at the sensing electrode [13]. The electrical signal that is produced has the same

frequency as the natural mechanical frequency of the vibrating structure.

Fig. 1.8 Clamped-Clamped beam MEMS resonator [13]

Fig. 1.9 A simple model of an electrostatically driven MEMS resonator [11]

11

The electrostatic coupling coefficient, η, defines the efficiency of the electrostatic transduction.

For a simple electrostatically driven resonator, as shown in Fig. 1.9., η can be defined by

Equation 1.2,

2d

AV

x

CV P

rP

(1.2)

where, C is the capacitance between the resonating structure and the sensing electrode, Vp is the

DC potential difference between the electrodes and the resonating structure, A is the electrode

overlap area, εr is the relative permittivity of the material between the structures and d is the

transducer gap.

A number of important points can be extracted from Equation 1.2. The first to note is that η is

inversely proportional to the square of the transducer gap, d. Because of the exponential nature

of the relationship, the transducer gap is designed to be as small as physically possible. Thus, a

series of attempts have been made to reduce it to nanometer levels. The limitations of etching

and photolithography have lead researchers to employ innovative fabrication methods. In

particular, the most common technique involves depositing an exceptionally thin film of silicon

dioxide between the structures. The film is then etched away with dry or wet etching, leaving in

place a ~100-200 nm gap [14]. Other techniques include using an electrostatic comb drive to

reduce the transducer gap [15], as shown in Fig. 1.10, or employing atomic layer deposition to

place the silicon dioxide layer [16].

12

a) b)

Fig. 1.10 SEM photos of electrostatic resonators. a) An electrostatic comb drive is used to reduce

the gap b) A 74 MHz wine-glass mode resonator with 4 perimeter supports. Close-up shows the

magnitude of the gap.

Another point to be noted from Equation 1.2 is that the electrostatic efficiency depends on the

potential voltage, VP, across the structures. This provides designers with a tool to control the

operation of the resonator. A 0V bias voltage will switch off the device completely and reduce the

power consumption to effectively zero watts. This could prove vital in portable electronic

applications where power efficiency is of the outermost importance.

Setting VP to a very high value can lead to increase in the electrostatic efficiency. However, there

are a number of limitations that restrict using a high value for VP. First, most electronic devices

operate at a voltage range of 1.8V to 15V, anything beyond that is largely overlooked by

commercial manufacturers. Second, high bias voltages cause the devices to exhibit high non-

linearities, leading to increased phase noise and a reduced quality factor.

Equation 1.2 provides more information regarding the operation of MEMS resonators. It shows

that the electrostatic transduction efficiency is directly proportional to the electrode overlap area,

A. A closer investigation will show a “weak” point of simple parallel plate resonators, such as the

clamped-clamped design. In order to operate at high frequencies the dimensions of the

13

resonators are reduced exponentially. Consequently the electrostatic efficiency drops

exponentially as the devices shrink, leading to reduced quality factors.

1.4.3 Alternative Transduction Mechanisms

A series of transduction mechanisms exist in nature. Only some are applicable to MEMS. While

several techniques have been demonstrated, only piezoelectric and electrostatic transduction

methods have seen commercial success. Other transduction techniques such as optical and

magnetomotive techniques have produced exceptional results in MEMS resonators. However, the

complexity of the optical systems [17] and the large size and power consumption of the

magnetomotive devices [18] restricts them to niche applications.

1.5 Quality Factor and Energy Loss

A resonating system can be defined to a large degree by two parameters, the resonant

frequency, fo and the quality factor, Q. The resonance frequency is the frequency at which the

device exhibits higher amplitudes of vibration compared to other frequencies. The quality factor is

defined by Equation 1.3.

ycleipatedPerCEnergyDiss

edEnergyStorQ 2 (1.3)

As can be seen, Q is a dimensionless parameter. It characterizes a resonators operation from

two perspectives. First, it provides information regarding its efficiency. A high Q indicates that the

oscillations will die slowly and that there is little energy lost during periodic operation. Second, it

provides information regarding the relation of the resonator’s bandwidth and center frequency.

Equation 1.3 can be rewritten as Equation 1.4.

14

f

fQ o

(1.4)

where, Δf is the half-power bandwidth. A high Q is associated with very high selectivity, an

attribute which is very desirable in RF filters.

A final note to make is that the Q of a resonator is highly dependent on the frequency of

operation. In general, it is more difficult to achieve high-Q at very high frequencies (VHF) as

anchor damping, thermoelastic dissipation and material non-linearities become increasingly

harder to counter. As such, the Q-f product is very commonly seen in the literature because it

incorporates the frequency dependency and allows for a more “fair” comparison of the various

resonator designs.

As is the case for a physical resonator, a MEMS resonator slowly loses energy to the surrounding

environment. The usual macroscopic energy loss mechanisms such as air damping and anchor

damping remain. However, the micro dimensions and the small energies involved emphasize the

importance of energy loss mechanisms that were previously overlooked, such as, the

thermoelastic dissipation and surface losses. The most important energy loss mechanisms in

MEMS resonators are summarized in the following section. The final quality factor of the device

can be found by adding the individual contributions of each factor according to Equation 1.5.

surfaceTEDgasanchortotal QQQQQ

11111 (1.5)

15

1.5.1 Anchor Damping

Anchor damping describes the transfer of kinetic energy from the resonator to its support

structure. Mechanical vibrations travel at the speed of sound, with what is called acoustic or

elastic waves. When these waves hit the resonator/anchor interface they either get reflected back

or they transverse through to the substrate. Elastic waves that manage to transverse to the

substrate are lost forever and so is the energy that they contain.

Anchor damping imposes considerable losses in MEMS resonators. Unfortunately the losses

increase exponentially as the resonant frequency increases, therefore reducing the Q factor

dramatically [13]. A series of techniques have been developed to address this form of energy

loss. Some of the most notable attempts include impedance mismatch techniques shown in Fig.

1.11 [19], [20]; as well as, quarter-wave impedance matching [13] as shown in Fig. 1.22. The

latter is extensively investigated in Chapter 3 of this dissertation.

a) b)

Fig. 1.11 Impedance matching techniques. a) Wine-Glass Disk [19], quarter-wavelength supports

are located at radial nodal point locations. b) Contour-Mode disk [20], supported by polysilicon

stem that is impedance mismatched to a diamond disk.

16

a) b)

Fig. 1.12 Effect of anchor dissipation on the operation of a MEMS resonator as the frequency

increases. a) Clamped-Clamped beam resonator with an original quality factor of 8,000 at 7MHz.

b) Free-Free beam resonator where the support beams are strategically placed at nodal points.

[13]

1.5.2 Gas Damping

Gas damping describes the transfer of kinetic energy from the resonator to the surrounding air. In

electrostatic transduction resonators air damping is the dominant energy loss mechanism after

anchor damping, when operated near atmospheric level pressures. The minute gap between the

input-output electrodes and the resonator gap, as well as the high vibration speeds involved,

introduces considerable Couette flow damping [11]. Additionally, more complicated mechanisms

such as squeeze film damping cause further drop of the Q of the resonators [13].

Because of this, electrostatic MEMS resonators are operated in vacuum or very low pressure

environments. Under these conditions the air behaves as kinetic particles rather than a

continuous medium. Furthermore, the air damping is now easier to calculate as there is only a

17

single energy loss mechanism. That is, the collision of air molecules with the vibrating structure.

The Q resulting from air dumping can usually be expressed in the form as given below,

P

TkCQ b

gas (1.6)

here, P is the pressure, kb is the Boltzman constant, T is the operating temperature and C is a

constant. The constant, C, depends on several design parameters such as the mode of vibration

and the shape of the resonator. Two notable points can be extracted from Equation 1.6. First, the

energy losses decrease when the operating pressure is reduced. Second, the energy losses

increase when the environment temperature rises. Both relations can be attributed to the fact that

air collisions are proportional to the temperature and inversely proportional to the pressure.

1.5.3 Thermoelastic Dissipation (TED)

Thermoelastic dissipation describes the transfer of energy from the resonator to thermal energy

due to temperature non-uniformities. During the operation of the device the resonator constantly

changes its shape depending on the mode of vibration. As it moves, parts of the structure are

compressed while other parts are tensile stressed. Due to thermodynamics, the region that is

compressed assumes a higher temperature, while the region that is tensile stressed assumes a

lower temperature. The result is a temperature difference across the structure. In order to balance

it, a heat flow between the “hot” part and “cold” part of the resonator is created. However, all the

energy that is converted to heat is forever lost.

In 1938, Zener C. [21] was the first who analyzed this phenomenon. He derived Equation 1.7 that

can calculate the TED losses for simple rectangular beam structures.

18

2

22

p

TM

TMC

ff

ffQTED (1.7)

where, fM is the mechanical frequency of the resonator, fT is the frequency of heat transport

across the beam, ρ is the density, α is the thermal expansion coefficient, Cp the specific heat, E

the Young’s modulus and T is the temperature of the resonator.

1.5.4 Surface Losses

Surface losses are related to the defects found at the surface of most materials. Discontinuities,

lattice defects, impurities and contamination from elements such as hydrogen and oxygen

introduce non-linearities that eventually lead to energy dissipation. The magnitude of the energy

loss is not considerable at the microscale. However, it becomes increasingly important at sub-

micron levels where the surface to volume ratio is high [20], [22].

1.6 MEMS Resonator Stability

One of the most important parameters of a frequency reference, although commonly overlooked,

is their signal stability. A reference as the name suggests should maintain a constant value

through time. There are numerous and complicated effects that affect the stability of a resonator.

In order to understand and address them, researchers have categorized them in three types

depending on their duration, as summarized in Table 1.

Table 1: Stability characterization

Stability Type Duration Dominant Effect

Short Term Less than second Phase noise

Medium Term Seconds to hours Temperature

Long Term Days to months Material Aging

19

Stability values are reported using the normalized frequency deviation from the target, fo, and

measured in parts-per-million (ppm). This deviation is equivalent to the fractional change of

resonant frequency Δf/fo, defined as by Equation 1.8.

o

o

o f

ff

f

f

(1.8)

A variation of 100 ppm corresponds to a 0.01% deviation from the target frequency. While this

seems negligible it actually corresponds to 9 seconds a day or 4.5 minutes a month on a clock.

As a rule of thumb, general consumer electronics require a deviation that is less than 200 ppm in

a temperature range of 0 oC to 70 oC, while mobile phones and GPS devices can require

deviations as little as 0.1 ppm [23].

1.6.1 Short Term Stability

Short term stability is involved with frequency deviations that occur in a period of less than a

second. They are usually characterized in the frequency domain as phase noise, or as jitter in the

time domain. In MEMS resonators they are usually attributed to non-linearities of the material.

While they have been extensively characterized [47], little can be done to reduce them as they

are directly linked to the physical properties of the material. Short term stability is usually not an

issue if the device’s power handling capabilities are respected. Exceeding them however, can

cause a dramatic increase in the non-linearities present and introduce noise.

1.6.2 Medium Term Stability

The dominant parameter in medium term stability is temperature sensitivity. Commercial grade

electronics are required to operate in the temperature range of 0 oC to 70 oC. While, industrial

grade and military grade applications require ranges of -40 oC to 85 oC and -55 oC to 125 oC,

20

respectively. Unfortunately, physical effects such as thermal expansion and the dependence of

the Young’s modulus on temperature have a big impact on the stability of MEMS resonators.

Temperature sensitivity is affected at large by two parameters, the coefficient of thermal

expansion, α, and the temperature coefficient of elasticity, TCE [23]. In order to calculate their

combined effects on frequency stability, the scientific community has introduced the temperature

coefficient of frequency, TCF. For a simple Free-Free beam resonator the TCF is defined by

Equation 1.8.

2

TCE

TCF (1.8)

An important point that can be extracted from Equation 1.8 is that the TCF depends only on the

properties of the material. Unfortunately, the highly negative TCE of silicon (-60ppm/oC) [24]

leads uncompensated MEMS resonator to have TCF’s in the order of -20ppm/oC to -30ppm/oC

[6]. In contrast, quartz resonators can achieve TCF’s in the order of ~0.01ppm/oC due to the

crystallographic properties of the material.

Quartz resonators are created by cutting a crystal. The physical properties of the material depend

heavily on the precise angle the crystal was cut. For example an “X” cut will yield an extensional

mode of vibration, while an “AT” cut will yield a shear mode of vibration with different TCF’s. This

crystal lattice property has proven to be invaluable in the design of modern quartz resonators. It is

possible to produce essentially zero-TCF resonators by precisely cutting and combining quartz

crystal planes.

In fact one of the toughest challenges that MEMS resonators have to address is matching the

temperature stability of quartz resonator systems. Numerous attempts have been made to nullify

21

the temperature sensitivity of MEMS resonators. They are in general categorized as passive or

active approaches depending on if there is power consumption or not. Active techniques usually

employ a technique of a “micro-oven” where a high current is responsible for direct heating of the

resonating structure [25]. Another approach is electrical stiffness compensation through

electrostatic tuning [26]. The passive method is a low power and simple approach that uses a

resonator structure composed of composite materials [27]. The compensation material is chosen

such that a high mismatch exists in the temperature coefficient of thermal expansion.

Temperature compensation remains one of the toughest and complicated challenges that MEMS

resonators need to face before they can be successfully commercialized.

1.6.3 Long Term Stability

Surface contamination is the dominant factor in the long term stability of MEMS resonators. The

resonating frequency of a MEMS device depends on the resonating mass, according to Equation

1.1. When the surface absorbs or desorbs gas molecules, the vibrating structure’s mass changes

and thus the resonating frequency shifts. This effect becomes increasingly important as

dimensions of MEMS resonators decrease.

The most successful approach is to isolate the resonator from the external environment through

hermetic seal packaging. It is important to note that a standard hermetic packaging is not enough.

The fabrication needs to be done in a vacuum environment. This is because any gas molecules

that remain in the seal package will significantly affect its performance. Extensive research is

being performed in this area. In 2006, Kaajakari reported 4-500ppm/month stability using anodic

bonding encapsulation [28]. However the enclosed gasses did not allow for further improvement.

In 2007, Kim et al. reported for the first time ppm level stability of a year long period [29]. The

22

group first encapsulated the MEMS resonators in an epitaxially deposited wafer-level packaging

and then performed high temperature (1,000 oC) annealing to remove any gasses left in the

resonator’s chamber. The long term stability results of the epi-seal packaged resonator can be

seen in Fig. 1.13.

Fig. 1.13 Long-term stability results of an epi-seal encapsulated MEMS resonator. This was first

time a ppm level stability was reported for a year long period of operation [29].

23

Chapter 2 MEMS-CMOS Integration and Standard Processes

A MEMS resonator is a mechanical vibrating structure. However, it is of limited usefulness on its

own. CMOS circuitry is required in order to control it and highlight its strengths. As will be

demonstrated, the fabrication process that is selected will to a large extent determine the

integration method. The first part of this chapter discusses the most prominent technologies by

which CMOS-MEMS integration can take place. While the second part discusses standard

processes that can be used in the fabrication of MEMS resonators.

2.1 RF MEMS Integration with CMOS

While countless RF MEMS resonators have been suggested, only a handful of them have been

commercialized. The SiTime vacuum encapsulated resonator [30], the Vectron International high-

shock oscillator [31] and the Discera multiple output oscillator [32] are among the most

successful yet. A common characteristic among these products is that they are fabricated through

CMOS-MEMS integration. In fact, this connection can be extended to the general field of RF

MEMS. The STMicroelectronics accelerometers and gyroscopes, the Texas Instruments digital

micro-mirror and the Analog Devices RF MEMS switch are all devices based on a monolithic

integration of CMOS and MEMS.

24

There are multiple hybrid ways of integrating MEMS with CMOS, including wafer-to-wafer

bonding and chip-to-wafer bonding. This section will only discuss methods that lead to a single

chip integration. In general they can be categorized in three types of approaches according to

when the micromachining process steps are added to the CMOS circuits. The micromachining

process steps can be performed before the CMOS process sequence (pre-CMOS), along with the

CMOS sequence (intra-CMOS) or after the CMOS sequence is completed (post-CMOS). Each

approach has its own benefits and limitations. Thus, the integration decision lies on the specific

application at hand. A quick comparison between the three approaches can be seen in Table 2, a

more detailed discussion follows.

Table 2: Comparison of the three different approaches for CMOS-MEMS integration [33]

Pre-CMOS Intra-CMOS Post-CMOS

Thermal constrain on MEMS No Yes Yes

Constrain on MEMS layer thickness No No Yes

Effective use of real estate Yes No Yes

Residual stress annealing Yes Yes No

Planarization required Yes No Yes

Foundry options Yes Limited Yes

Interruption of CMOS process No Yes No

2.1.1 Pre-CMOS Approach

In pre-CMOS or “MEMS-first” fabrication, the micromachining takes place before the CMOS

sequence. This is particularly useful for MEMS technologies that require a high temperature

annealing. For example, thick polysilicon structures are typically deposited using low-pressure

chemical vapor deposition (LPCVD) at approximately 600 ℃, followed by a thermal anneal at

25

1100 ℃ in order to reduce residual stress. Exposing CMOS circuitry to such high temperatures

would irreparably damage the metallization layers. It is essential that the annealing takes place

prior to the CMOS sequence.

Once the MEMS micro-structures are fabricated, they are typically covered and sealed. The

wafer’s surface is then planarized using chemical mechanical polishing (CMP). The wafers with

the embedded microstructures are subsequently used as a starting point for the CMOS process.

An important design challenge of the pre-CMOS approach is the formation of interconnects

between the MEMS and the CMOS circuit.

The “MEMS-first” integration principle was initially demonstrated by Sandia National Laboratories

[34]. In their approach, the M3EMS (Modular, Monolithic, MEMS), trenches are first etched into a

silicon wafer using an anisotropic wet etchant. Following that, a multi-layer polysilicon MEMS is

built inside the trench. After annealing the trench is filled with LPCVD oxide, the wafer is

planarized and sealed using a nitride layer. A few holes are opened in the nitride layer in order to

allow access to polysilicon studs in the trench. The CMOS circuitry is fabricated on top of the

processed wafers and the metallization layers are used for the MEMS-CMOS interconnections.

Finally, the CMOS area is covered by an HF resistant passivation layer in order to protect the

circuits during the MEMS release process. A cross section of the post-CMOS integration

technology is shown in Fig. 2.1.

26

Fig. 2.1 A cross-sectional schematic of the CMOS-MEMS integration technology by Sandia

National Laboratories [34]

Since Sandia National Laboratories first demonstrated the principle a plethora of post-CMOS

methods have been demonstrated. Some of these include the Analog Devices, Palo Alto

Research Center and UC Berkeley “Mod MEMS” [35], the UC Berkeley “SOIMEMS” [36] and

recently the Analog Devices “iMEMS” [37]. While there are significant improvements over the

original concept, the difficulty in interconnecting the MEMS with the CMOS area remains.

2.1.2 Intra-CMOS Approach

Intermediate micromachining is typically used to integrate polysilicon microstructures in CMOS

processes. The MEMS structure is constructed after the CMOS doping has been performed but

before the interconnect metallization. The key point here is that the doping profiles of CMOS can

withstand temperatures up to 900 ℃ without being affected. Polysilicon can thus be deposited

27

and thermally annealed without any considerable degradation of the CMOS circuits. The

metallization layer is then used to establish the MEMS-CMOS interconnects.

An example of an intra-CMOS integrated device is the “KP100” pressure sensor developed by

Infineon Technologies. The mechanical microstructures are constructed during the standard

BiCMOS sequence, using the 0.8 μm capacitor polysilicon layer. The electronic circuits are

constructed in parallel with the pressure membrane. However, the standard BiCMOS process is

interrupted at the interconnect metallization step in order to perform surface micromaching of the

polysilicon. When the microstructures are formed, the back-end metallization steps are

performed, followed by a passivation layer. A schematic cross section of an intra-CMOS

fabricated by Infineon Technologies is shown in Fig. 2.2.

Fig. 2.2 A cross-sectional schematic of the intra-CMOS integrated pressure sensor, developed by

Infineon Technologies

2.1.3 Post-CMOS Approach

The main advantage of post-CMOS micromachining is that all the fabrication steps can be

outsourced. The CMOS process sequence can be performed at any CMOS foundry and the

micromachining steps can be later performed at a MEMS specific foundry. This flexibility however

comes at a great price. The metallization layers of the CMOS area cannot withstand high

temperatures. Because of this, post-CMOS micromachining is restricted to maximum

28

temperatures of 450 ℃. Processes such as depositing polysilicon in a LPCVD furnace and

thermal annealing are precluded from the fabrication sequence, as they require temperatures up

to 900 ℃. MEMS structures can thus only be formed using low temperature methods such as,

plasma enhanced chemical vapor deposition (PECVD), E-beam evaporation, electroplating and

sputtering.

Besides the thermal budget constraints, the post-CMOS approach has two main challenges. First,

the CMOS substrate should have a high degree of planarity as the MEMS will be built on top.

Second, the sacrificial layers of the microstructures have to be chosen carefully. Typical sacrificial

materials such as silicon dioxide (SiO2) are etched using hydrofluoric acid (HF) which can

damage the CMOS passivation layer.

In general, post-CMOS can be further categorized in two methods. In the first method the

microstructures are formed by machining the CMOS layers themselves. In this case, the

microstructures are usually pre-defined during the CMOS sequence. As such, the

micromachining steps are limited to releasing the structures or performing a final deposition step.

In the second method, the microstructures are built from scratch on top of the CMOS substrate.

While the latter case is more complicated, it comes with a strong advantage. Building the MEMS

on top of the CMOS circuits provides substantial real estate savings.

An example of a post-CMOS integrated device is Texas Instruments digital micro-mirror device

(DMD) [38]. The MEMS are built on top of the CMOS circuitry. Aluminum is used as the structural

material, as it can be deposited and processed at low temperatures, ensuring CMOS

compatibility. An expanded view of the DMD can be seen in Fig. 2.3.

29

Fig. 2.3 A schematic design of the Texas Instruments post-CMOS integrated DMD [38].

2.2 Standard Processes for RF Resonators

The work presented in this thesis was performed using PolyMUMPs, which is a commercial multi-

user process available from MEMSCAP. This section starts with an introduction to other

compelling standard processes, such as UW-MEMS, SOIMUMPS and CMOS. The fabrication

process flow, along with how it can be used to fabricate MEMS resonators is discussed. The key

performance characteristics of each technology are presented. The section ends with a detailed

discussion of PolyMUMPs and why it was chosen over the other fabrications processes.

2.2.1 UW-MEMS

The UW-MEMS is a cost effective, multi-user MEMS fabrication process that is offered by the

University of Waterloo in Ontario, Canada. UW-MEMS is a seven mask process that is based on

gold surface micromachining. It was derived by work performed at the Center for Integrated RF

Engineering at the University of Waterloo.

30

The process starts with a polished 0.025” thick Alumina substrate. A thin Cr layer is first

deposited on the wafer and patterned using the lift-off technique. PECVD is used to deposit a 0.5

μm layer of SiO2, followed by reactive ion etching (RIE) in order to pattern the oxide. A 400 Å

Cromium (Cr) and 100 nm gold (Au) bilayer is deposited as a seed base, in order to electroplate 1

μm of Au. Cr is used here as an adhesive material between layers. The process continues with

sputtering another 300 Å Cr adhesion layer and depositing 0.7 μm SiO2 using PECVD.

Fig. 2.4 UW-MEMS fabrication process – Part A [39]

A 2.5 μm polyimide layer is spin coated on the wafer; it serves as a sacrificial material for the

fabrication of the microstructures. At this point, anchor holes or dimples can be created in the first

Au layer using RIE. The next deposition step involves sputtering a Au seed layer, followed by

1.25 μm electroplated Au. This final layer is typically used as the structural material for the MEMS

devices. The wafers are then diced and the sacrificial SiO2 layer is removed in a plasma dry etch

process in order to release the structures.

31

Fig. 2.5 UW-MEMS fabrication process – Part B [39]

At first glance the UW-MEMS process could be considered ideal for the fabrication of MEMS

laterally driven resonators. The reason is that it can form a suspended Au structure above

another Au layer. The use of metal as the structural material would mean that the resonator

would yield a very high electrostatic transduction efficiency. Several MEMS resonators have been

demonstrated using UW-MEMS technology, including Clamped-Clamped [40] and Free-Free [41]

beam designs. Typically, the first gold layer is used as an actuation electrode and the second

gold layer is used as the vibrating mechanical structure. A SEM photo of a UW-MEMS Clamped-

Clamped beam resonator can be seen in Fig. 2.6.

Fig. 2.6 SEM photo of a UW-MEMS Clamped-Clamped beam resonator [40].

While the technology seems ideal, there are several limitations that need to be accounted for

during the design process. First, the minimum resolution that can be achieved using UW-MEMS

32

is 10 μm. As was discussed in Chapter 1, the frequency of operation of a MEMS resonator is

inversely proportional to the square root of its mass according to Equation 1.1. Effectively, MEMS

resonators that are fabricated using UW-MEMS are limited to operating frequencies below 1MHz.

Of course applications do exist in this range, particularly in simple timing devices that currently

use crystals operating at 32.768 kHz. Second, the transducer gap between the first and second

Au layer is 2.5 μm. According to Equation 1.2, a DC bias voltage of more than 50V is required to

reach acceptable electrostatic transducer efficiency. Finally, another limitation results from the

material used for the vibrating structure. While gold offers high conductivity, its mechanical

properties are very sensitive to temperature variations. A post-processing step would thus be

required in order to produce a temperature insensitive and stable resonator.

2.2.2 SOIMUMPS

SOIMUMPs is a multi-user MEMS process, offered by MESCAP Inc., which is based on silicon-

on-insulator (SOI) micromachining. A cross section that shows the layers of the process can be

seen in Fig. 2.7., a short description follows.

Fig. 2.7 Cross sectional view of the layers used in the SOIMUMPS process [42].

33

The designer can choose between two different types of wafers. As such, the process starts with

a SOI wafer that has a device layer thickness of 10μm or 25 μm and an oxide layer thickness of

1μm or 2μm. The silicon is then doped, patterned and etched down to the oxide layer. This is the

layer that is typically used for mechanical structures such as resonators. Optional steps include

patterning and etching the bottom side of the substrate to expose the oxide layer, thus allowing

through-hole structures. While a backside etch might seem unnecessary for the fabrication of a

resonator, it can actually be very useful, as it reduces substrate parasitics and capacitive feed-

through signals. The overall effect is a considerably improved noise performance. The process

ends with two metallic layers. The first is typically used for electrical routing and bond pads, while

the second is deposited using E-Beam evaporation and allows for finer metal features.

The SOIMUMPS process cannot deliver laterally driven resonators, as there are no overlapping

conductive surfaces separated by a transduction gap. However, it has a low minimum feature

size of 2 μm and it can be used to create longitudinally driven structures. Because of this, several

MEMS resonators have been implemented using the SOIMUMPS technology, including Free-

Free beam [43], and Lame-mode differential resonators [44] [45].

In 2007, Khine et al. demonstrated 6.3MHz square Lame-mode resonators with Q values

exceeding 1 million [45]. The anchor beams were positioned at the nodal points of the excited

mode. The 25 μm thick Si layer was used to construct the vibrating mechanical structure, the

actuating and sensing electrodes, while the 50nm SiO2 layer was used as the sacrificial material.

A micrograph of the square resonator and its corresponding frequency response can be seen in

Fig. 2.8. The transmission curve shown was recorded at a vacuum level pressure of 36 μTorr

under a 50V DC bias voltage.

34

Fig. 2.8 Micrograph of the square resonator reported by Khine et al. [45] and a large-span view

of its frequency response.

2.2.3 CMOS

UW-MEMS, SOIMUMPs and PolyMUMPs are all standard processes capable of delivering

competitive MEMS resonators, but they can only provide two-chip oscillation systems. The

controlling ICs need to be packaged and wire-bonded to the MEMS structures. While this is not

an issue in scientific research, it becomes an important obstacle if the devices are to be

successfully commercialized. Two-chip solutions are typically associated with higher costs and

larger space requirements. Because of this, a single-chip solution is generally preferred, even if it

slightly lacks in performance.

As was discussed in the beginning of this chapter (Section - 2.1.2 – Intra-CMOS), the most direct

approach of integrating ICs with MEMS is to fabricate both in the same standard CMOS process.

In the past decade, a few research groups [46], [47], [48] have followed this path and have made

significant progress in fabricating MEMS resonators using standard CMOS processes. However,

their initial efforts were hindered by high motional impedances (Rm), caused by large electrostatic

gaps [46], [48]. Fortunately, recent advancements in CMOS microfabrication technologies have

35

allowed for dramatic improvements in the performance of CMOS-MEMS resonators. The effects

of the CMOS technology used on the performance of resonators are summarized in Table 3. A

clear trend can be observed, in which increased resolution can create reduced gaps and thus

higher efficiency electrostatic resonators.

Table 3: CMOS-MEMS performance prediction based on CMOS technology used [49].

CMOS Technology Layers Used

Initial Gap (nm)

Height (μm)

Area Overlap (μm2) Rm(Ω)

0.18 μm 1P6M M1-M4 280 4.67 84.1 1.32Μ 90 nm 1P9M M1-M7 140 4.02 72.36 151k 65nm 1P10M M1-M7 100 2.74 49.32 53.2k 45 nm 1P10M M1-M5 70 1.9 36 13.3k

In the past, a series of exotic designs have been suggested in order to counter the limitations

imposed by CMOS technology. For the purposes of this thesis, only the most common

configurations will be described and a focus will be given in the fabrication process flow rather

than design.

While there are countless ways to fabricate a CMOS-MEMS resonator using pre-processing (pre-

CMOS) or post-processing (post-CMOS), the approaches to fabricate it using intra-processing is

limited to the three types as shown in Fig. 2.9. In the first approach the resonating structure is

composed of alternating layers of metal and SiO2, with vias being used to connect the structure

and protect the enclosed oxide. In the second method, the resonating structure is composed of

metal and via layers. The last approach follows a simpler implementation and only uses a single

metal layer for the mechanical structure.

36

Fig. 2.9 Cross-section view of three typical FF-Beam CMOS-MEMS resonators. (I) Metal-Oxide

composite, (II) Metal-Via composite, (III) Single metal layer [47]

A typical example of a CMOS-MEMS is the Free-Free Beam resonator by Li et al. [49], shown in

Fig. 2.10a. The device is based on a TSMC standard 0.18 μm 1P6M process. The resonating

structure is a composite, consisting of five metal layers and four SiO2 layers as shown in Fig.

2.10b. The chips are fabricated at TSMC along with other IC components. Then, the resonators

are released using a commercial SiO2 etchant (Silox Vapox III, from Transene Inc.). The vias that

run down the resonating structure protect the enclosed SiO2 during this wet-release process and

also help to form smooth sidewalls. The ICs that form the trans-impedance amplifier are protected

by the passivation layer during the wet-release process.

Fig. 2.10 a) Schematic view of the CMOS-MEMS Free-Free beam resonator by Li et al. b) Cross

sectional view of the layers used in the fabrication of the Free-Free beam resonator [49]

37

A high performance MEMS resonator directly integrated with CMOS ICs could have a

tremendous impact in the timing industry. While numerous attempts have been made to achieve

this, none have met commercial success. The reason is that the CMOS technology was

specifically developed to create ICs, not MEMS devices. A series of design challenges need to be

addressed before CMOS-MEMS resonators can compete with conventional silicon or quartz

resonators.

The first issue arises from the materials of the resonating structure. As can be seen in Fig. 2.10

the CMOS process involves alternating layers of metal and oxide. The mechanical structure can

thus be composed of either a) metal, b) a composite of metal-oxide, c) oxide. The main problem

here is the TCf of the fabricated devices. Metal has a highly negative TCE and a highly positive α,

as such, resulting devices will have a highly negative TCf. Oxide has a highly positive TCE and a

very small α, thus leading to a highly positive TCE. Finally, in a metal-oxide composite, the TCE

of metal dominates and the devices are characterized by a negative TCf [49]. A comparison

between a composite FF-Beam resonator and a mere-metal FF-Beam can be seen in Fig. 2.11.

While the composite resonator has a vastly improved performance, a TCf of -102 ppm/℃makes it

impractical for commercial applications. As was noted in the first Chapter of this thesis (Section

1.62), TCf’s in the order of ~1 ppm/℃ or less are the standard in the timing market. Further

research in this field is required in order to decrease the sensitivity of MEMS resonators

produced.

38

Fig. 2.11 Comparison of the temperature sensitivity of a Mere-Metal FF-Beam and a SiO2-Metal

composite FF-Beam [49].

2.2.4 PolyMUMPs

PolyMUMPs is a low cost, multi-user MEMS process offered by MESCAP Inc. It is based on a

three-layer polysilicon surface micromachining process flow. The polysilicon layers are used as

the structural material, the phosphosilicate glass (PSG) is used as the sacrificial layer, and the

silicon nitride (Si3Ni4) provides electrical isolation between the MEMS structure and the substrate.

The process flow has been optimized to accommodate multiple different designs on the same

wafer, while maintaining very high yields. The thicknesses of the polysilicon and oxide layers

were chosen so that they could meet the requirements of multiple projects. While this

standardization allows for a reduction in cost and makes the process available to many

developers, it introduces several limitations. A cross section that shows all seven layers of the

process can be seen in Fig. 2.12., a short description follows, a detailed process overview can be

found in the “PolyMUMPs Design Handbook” [50].

39

Fig. 2.12 Cross sectional view of the layers used in the PolyMUMPs process [50].

The process starts with n-type silicon wafers that are heavily doped with phosphorus. The doping

aids in isolating the substrate from the devices on the substrate as it blocks charge feedthrough

between the two. In order to reinforce this electrical isolation, a 600nm layer of Si3Ni4 is deposited

using LPCVD on the surface of the wafer. The first conductive polysilicon layer (Poly-0) is then

deposited using LPCVD and patterned using photolithography. This is followed by the first (2.0

μm) sacrificial layer of PSG that is deposited using LPCVD. The PSG layer is then patterned and

the first structural layer of polysilicon (Poly-1) is deposited on top at a thickness of 2.0 μm.

After annealing, the polysilicon layer is lithographically patterned in order to form the first

structural layer of the devices. The second sacrificial PSG layer is then deposited and annealed

at a thickness of 750 nm. Two distinct masks that serve different purposes are used to pattern

and etch the PSG layers. The first masks allows for the creation of VIA’s between the Poly-1 and

Poly-2 layers. The second mask allows for etching both the first and second oxide layer, thus

providing the capability of independently anchoring the subsequent Poly-2 layer to the substrate.

After the oxide layers are patterned, the second polysilicon layer (Poly-2) is deposited at a

thickness of 1.5 μm. This is the final structural layer that can be released in the PolyMUMPs

40

process. Lithography and RIE are then used to pattern the polysilicon and oxide layers. The

fabrication flow finishes with a metal layer deposition at a thickness of 0.5 μm. This final, highly

conductive layer is typically used for the electrical interconnections and bond pads of the MEMS

devices. After the metal is patterned using photolithography, the MEMS structures are released

by immersing the chips in a 49% HF solution.

PolyMUMPs has been repeatedly used for the fabrication of MEMS filters [51], [52] and

resonators [53], [54], [55], [56]. Fig. 2.13 shows an example of a PolyMUMPs made disk micro-

resonator. It operates at a radial mode of vibration with a resonant frequency of 920 MHz. The

DC bias and RF interconnects were implemented in the Poly-0 layer, while the disk structure and

the side actuating and sensing electrodes were implemented in the Poly-1 layer [56]. As will be

shown later this is a typical configuration for the fabrication of MEMS resonators using

PolyMUMPs.

Fig. 2.13 A SEM image of a disk resonator fabricated through PolyMUMPs [56].

In general there are two common ways to fabricate a MEMS resonator in PolyMUMPs. In the first

method, the Poly-0 is used to provide the actuating and sensing electrodes, while the Poly-1 is

used for the vibrating mechanical structure. The second method differs from the first in that it

41

adds a structural Poly-2 layer on top of Poly-1. This causes an increase in the thickness of the

resonator which results in an increase of the resonating frequency.

The configuration described, allows for a simple and direct implementation, but it introduces a

serious limitation to the reported designs. The problem is that the first sacrificial oxide layer has a

thickness of 2 μm, meaning that the electrostatic transducer gap will also be 2 μm. This value

however, is impractically high and results in devices that need very high DC bias voltages in order

to operate, a fact that can be confirmed by Equation 1.2.

An alternative design that bypasses these limitations is introduced in this thesis [55]. In this

approach the Poly-0 layer is only used for fabrication of the interconnections. The first sacrificial

oxide layer is then completely etched away so that the Poly-1 layer can be anchored directly to

the substrate. Poly-1 is subsequently used for the ground plate and the sensing and actuating

electrodes. Finally, the Poly-2 layer is used for the fabrication of the resonator’s vibrating

structure. The main advantage that comes from this process flow is that the electrostatic gap

between Poly-2 and Poly-1 is only 750 nm. This reduces the required DC bias voltage by a factor

of three. In addition, the suggested design introduces for the first time a novel and effective

temperature compensation technique. A strategic deposition of the final Au layer of PolyMUMPs

allows for a creation of tensile stresses that cancel out the negative TCE of polysilicon.

Temperature insensitive resonators with very low TCfs and high Q values are demonstrated. A

detailed description can be found in Chapter 3.

42

Chapter 3 Free-Free (FF) Beam MEMS Resonator

To date, a myriad of MEMS resonator designs have been suggested. One of the toughest

challenges a designer has to get through is to short-list the best type of resonator for the

application at hand. As was discussed in Chapter 1, Clamped-Clamped MEMS resonators have

been repeatedly used for many applications. Their main advantages are having a simple design

and fabrication process. However, they are unable to perform well at HF and VHF. Anchor

dissipation and a high 3rd order intermodulation distortion dominate their response.

Alternative and more modern designs, such as the Wine-Glass disk [19], Contour-Mode disk [57]

and Spoke-Supported ring [58] resonators have solved these problems. They are able to operate

at HF, VHF and UHF with minimal effects on their performance. However, they require

exceptionally small transducer gaps (on the order of 50nm). Effectively, this requirement imposes

the use of complicated fabrication processes that are in turn characterized with low yields and

high cost.

The FF beam resonator is a design that stands in the middle ground between increasingly

complex and overly simplistic designs. It alleviates the problems found in Clamped-Clamped and

cantilever beam resonators, while managing to maintain a simple structure that can be batch

43

produced. The design, first introduced by Wang et al. in 2001 [13]. This design has been modified

in the current research work with the aim to prove that a FF beam resonator can be fabricated in

a commercial, standard, multi-user process, such as PolyMUMPS, and with little detrimental

effects to its performance. The following chapter attempts to discuss the key parameters of the

FF beam design, and the process by which it was adapted to meet the fabrication constraints set

by PolyMUMPS.

3.1 Free-Free Beam Resonator Structure

A device schematic of the FF beam resonator, along with the recommended readout

configuration, is shown in Fig. 3.1. The device consists of a main resonating structure (known as

the free-free beam) and four supporting torsional beams placed at the flexural nodal points of the

structure. The supports are suspended over the ground plate and subsequently connected to rigid

contact anchors. An electrode under the main structure is responsible for providing the required

electrostatic excitation via an applied AC voltage (VAC). A DC bias voltage (VDC) applied on the

resonating structure is responsible for amplifying the weak electrostatic force that is created by

VAC.

The mechanical vibration of the FF beam creates a time-varying capacitance C(z,t) in the

electrode-resonator gap. This in turn causes the generation of an output current on the FF beam

according to [6],

t

z

z

CV

t

tzCVi DCDCout

),( (3.1)

where, z is the vertical displacement. A design challenge arises here as the current is generated

on the same structure as VDC is applied. Contrary to a typical two-port MEMS resonator designs,

44

the FF beam resonator is a one-port configuration. In order to allow sensing of the output current

a DC bias tee is required, as shown in Fig. 3.1.

Fig. 3.1 Schematic view of the FF beam resonator along with the required readout circuitry.

3.1.1 Resonant Frequency Design

As was discussed in Section 1.3, the simplest way to model a MEMS resonator is to use a

second order mass-spring-damper system that has a resonant frequency (fo) given by Equation

1.1. While this method is commonly used as a starting point to give a general approximation of fo,

it is incapable of providing accurate results at high frequencies. The reason is that resonator

geometry, anchor damping and material stiffness are not included in the model. Thus, more

advanced theories are used in the design of a MEMS resonator; the choices are in the mode,

type and frequency of vibration.

45

For the case of the FF beam resonator, two theories are typically applied. These theories are

commonly known as the Euler-Bernoulli and the Timoshenko methods. The Euler-Bernoulli theory

holds as long as the length (Lr) of the beam is much larger than its width (Wr) and thickness (h).

For low frequency designs (<30MHz), the ratios of Lr to Wr and Lr to h tend to be large, in which

case the Euler-Bernoulli theory is very accurate. However, as the resonant frequency increases,

the length of the devices shrinks and the beam lengths approach the dimensions of h and Wr.

The effects of shear displacements and rotatory inertias become increasingly important in the

physical behavior of the device. The problem is that the Euler-Bernoulli method does not take

these parameters into consideration. As such, at VHF the theory is unable to provide accurate

results. At frequencies higher than 30MHz, the Timoshenko method is more appropriate [13].

However, the tradeoff is that calculations involved are much more complicated, even in simple

structures. As the complexity of MEMS resonator devices increases, Finite Element Methods

(FEM) become increasingly important in the design process.

The resonators presented in this thesis operate at 1MHz, 5MHz, 10MHz, 20MHz and 30MHz.

Since none of the designs operate at VHF, the Euler-Bernoulli theory was used to determine the

dimensions of each device. The goal of this section is to discuss how the Euler-Bernoulli method

was used in the designing process of the resonators, and how their dimensions were determined.

3.1.2 Mechanical Body Model of Free-Free Vibration

The mechanical vibration of a flexural beam can be described by solving the equations of motion

for a differential element of the beam. An overview of the derivation is presented here, while a

more detailed discussion can be found in [59]. Let us consider a differential element of a uniform

flexural beam as shown in Fig. 3.2. The mechanical structure has a Young’s modulus, E, density,

46

ρ, cross-sectional area, A, moment of inertia, I, and length, L. It is subjected to an axial load, P,

and acted on by a shear force, V and moment, M. The goal is to find an expression that describes

the displacement of the beam during vibration in the form of v(x,t). It is expected that the

expression will heavily depend on the mode of vibration.

Fig. 3.2 Forces and strain applied on a differential element of a uniform vibrating beam.

According to the Euler-Bernoulli theory, the vibration of a uniform beam that has a constant

flexural rigidity, EI, and is not subjected to a load, P, can be described by a fourth order

differential equation in the form of Equation 3.2,

(3.2)

As was discussed in the previous section, the effects of shear displacements and rotary inertia

can be neglected. The Euler-Bernoulli equation can be solved using separation of variables by

expressing v(x,t) as Equation 3.3

(3.3)

Equation 3.2 can then be re-written as Equation 3.4

47

(3.4)

This can be rearranged so that the left side is a function of distance and the right side a function

of time as given by Equation 3.5

(3.5)

The only way this equality can hold is if both sides are equal to a constant. It can be shown that

should a solution exist, this constant must be positive. The derivation is out of the scope of this

thesis so it is omitted - a detailed explanation can be found in [59]. Assuming the constant to be

ω2, Equation 3.5 can be re-written as Equation 3.6,

(3.6)

(3.7)

where,

(3.8)

Since, both Equation 3.6 and Equation 3.7 are homogeneous ordinary differential equations; their

solutions can be easily expressed as Equation 3.9,

(3.9)

and,

(3.10)

where, α is a phase angle that depends on the initial conditions. In order to facilitate the

calculations, it is easier to work with non-dimensional variables. As such, instead of working with

distance x, the fraction ξ will be used as,

(3.11)

48

The value of β is determined by using the four boundary conditions that govern Equation 3.6. For

a Free-Free beam, these are governed by the shear strain, V, and bending moment, M. At ξ=0,

V=M=0, thus,

(3.12)

(3.13)

Similarly, at ξ=1, V=M=0, thus,

(3.14)

(3.15)

Equation 3.9 can now be re-written as Equation 3.16,

(3.16)

where,

(3.17)

The above equation is also known as the vibration mode shape function. In order to obtain non-

trivial solutions, β can only take specific values that depend on the boundary conditions and the

mode of vibration. Table 4 summarizes the eigenvalues β for the first 10 modes of vibration of a

uniform Free-Free beam. Since the resonators of this work operate on the fundamental mode of

vibration, β is equal to 4.73.

49

Table 4: Eigenvalues β for the first 10 modes of vibration a FF beam [59].

Mode of Vibration Eigenvalue β

1 0 (Rigid Body)

2 4.730

3 7.853

4 10.996

5 14.173

6 17.274

7 20.420

8 23.562

9 26.703

10 29.845

A final expression for the vibration mode shape function can now be determined by substituting

the appropriate value of β into Equation 3.16,

(3.18)

Similarly, it can be expressed using the distance variable x, as,

(3.19)

The resonators presented in this thesis operate on the fundamental mode of vibration; as such

two nodal points are expected to be present. An analytical solution of Equation 3.19 yields the

first nodal point at,

(3.20)

and the second nodal point at,

(3.21)

In order to complete the mechanical characterization of the beam, an expression for the resonant

frequency must be derived. The process starts with rearranging Equation 3.8 as,

50

√

(3.22)

For a uniform beam with a large length-to-thickness ratio, the moment of inertia can be found

from Equation 3.23

(3.23)

The resonant frequency can then be simplified in the form of Equation 3.24

√

√

√

(3.24)

A final expression for the resonant frequency can now be derived by substituting the appropriate

eigenvalue β from Table 4. For a FF beam operating at the fundamental mode of vibration, β is

equal to 4.73, thus fo can be expressed as Equation 3.25,

√

(3.25)

The above expression works exceptionally well for low-frequency designs that are defined by high

Lr to h ratios. It is important to note here that the thickness, h, is typically determined by process

constraints. As such, the parameter that controls the frequency is the length, Lr, of the FF beam.

3.1.2 Electrostatic Spring Stiffness

A closed form expression for the resonant frequency of an ideal FF beam was derived in the

previous section. A key point in the analysis was that the beam is not subjected to a load, P.

However, this is not the case for a FF beam MEMS resonator. As was previously discussed, a

bias voltage, VDC, is applied on the resonating structure in order to improve the transduction

efficiency. The FF beam is subjected to an electrostatic force, Fe, that consists of two

components, the first originates from VAC while the second from VDC. The introduction of Fe

51

makes Equation 3.25 no longer valid. The Euler-Bernoulli theory must be revisited in order to

derive an expression that accounts for electromechanical coupling. Assuming excitation signal

VAC in the form of Equation 3.26,

(3.26)

The electrostatic force Fe which the beam is subjected to can then be expressed as Equation

3.27 [60],

| |

(3.27)

where, do and Co are the initial transducer gap and initial transduction capacitance, respectively.

The resulting displacement of the beam is in the form of Equation 3.28 [60],

| | (3.28)

A key point to note here is that the beam displacement is 90° out of phase with Fe. This means

that the restoring spring force, Fk, which attempts to bring the beam back to equilibrium, is not in

phase with Fe. A comparison between Equation 3.27 and Equation 3.28 shows that the first

component of Fe reinforces Fk, while the second component reduces it. The latter however

dominates the response, thus the overall effect is a reduction in the value of Fk. In turn, this

causes the beam to act as if it has a reduced spring constant, effectively reducing the resonant

frequency of vibration. In order to model this “electrostatic spring softening” effect, the electrical

stiffness, ke, of the beam is combined with the mechanical stiffness km. The derivation is out of

the scope of this thesis; only a description of the approach is given here. A detailed derivation can

be found in [60]. When electromechanical coupling is taken into consideration, the resonant

frequency fres of a FF beam can be expressed as Equation 3.29 [61],

52

√ ⟨

⟩ (3.29)

where, fo is the resonant frequency of a FF beam when VDC=0V, given by Equation 3.25, and

<ke/km> is the ratio of electrical to mechanical stiffness integrated over the actuating electrode

width, We, such that it satisfies Equation 3.30 [61],

⟨

⟩ ∫

(3.30)

where, do is the transducer gap and εo is the relative permittivity of free space. Substituting

Equations 3.25 and 3.30 into Equation 3.29 yields a final expression for the resonant frequency of

a FF beam resonator under electrostatic excitation as Equation 3.31,

√

( ∫

)

(3.31)

A close inspection of Equation 3.31 reveals that the resonant frequency depends on the bias

voltage. While at first this dependency can be seen as a stability problem it is actually of great

interest to designers. In a technique commonly known as “bias tuning” or “frequency pulling”,

designers can control the resonant frequency of operation by carefully changing the bias voltage.

3.1.2 Support Structure Design

An important consideration of the FF beam resonator is the suppression of anchor dissipation

through strategic support design and placement. Its introduction led to a series of designs that

utilized the idea and eventually allowed for MEMS resonators capable of operating in the GHz

range. The goal of this section is to discuss the approaches of “strategic support placement” and

“strategic support design” and their effects in the performance of the resonator.

The strategic support placement idea originates from the vibrational shape function described by

Equation 3.19. The expression shows that the shape function of a FF beam depends on the

53

mode of vibration. MATLAB and FEM simulations were performed to evaluate the shape function

for the first 5 modes of vibration; the results can be seen in Fig. 3.3 and Fig. 3.4. The points that

are located on the x-axis of the graph, also known as nodal points, are the solutions of Equation

3.19. During vibration they sustain no translational movement. As the shape function changes

depending on the mode of vibration, different modes have different nodal point configurations.

Fig. 3.3 Normalised vibration shape function for the first 5 modes of the FF beam resonator.

Fig. 3.4 FEM simulations illustrating the first 6 modes of vibration of a FF beam. The blue areas

denote the nodal points.

54

As shown in Fig. 3.5, the FF beam resonator is supported by four torsional beams that are

attached symmetrically at its two fundamental nodal points. The idea is that since the nodal points

remain steady, the supports will sustain minimal translational movements during vibration. Anchor

dissipation due to translational movements will thus be minimized. The vibration of the beam is

now only hindered by torsional impedance introduced by the supporting beams.

Fig. 3.5 Top-down view of the FF beam resonator structure.

Strategic support design concept aims to minimize the torsional losses. The idea originates from

the fact that at high frequencies torsional beams behave as acoustic transmission lines. By

designing the supporting beam to have a length, Ls, effectively equal to a quarter-wavelength of

the operating frequency of the resonator, it is possible to present virtually zero torsional

impedance to the resonating beam.

The reason is that an acoustic quarter-wavelength beam transforms the fixed boundary condition

found at the rigid anchor to a free end condition at the resonator end. The equivalent acoustic π

55

network model for a quarter-wavelength torsional beam can be seen in Fig. 3.6b. In this Figure,

the boundary conditions have yet to be established. The variables Z and -Z represent the series

and shunt impedances of the beam and are complementary to each other.

(a)

(b)

(c)

Fig. 3.6 (a) λ/4 torsional beam; (b) Acoustic π network model for a λ/4 torsional beam; (c)

Equivalent acoustic network when the beam is anchored at port B.

Assuming that the supporting beam is attached to the FF beam at port A and to a rigid anchor at

port B, the boundary conditions indicate port B will be open-circuited. The equivalent acoustic

network model is shown in Fig. 3.6c. The impedance ZA as seen from port A is equal to Equation

3.32,

(3.32)

56

Effectively, the FF beam “sees” a zero torsional resistance during its vibration. Anchor dissipation

is minimized and the Q of the FF beam resonator increases. The advantages of quarter-

wavelength supports are significant. However, an expression for the length, Ls, of the supporting

beam has yet to be derived. The following section describes the derivation process. An overview

is presented here, while a more detailed discussion can be found in [61]. The process starts by

solving an acoustic wave equation for a beam of length, Ls.

For a torsional beam, the acoustic wave equation takes the form of a second order differential

equation as given below,

(3.33)

where, γ is the torsional constant, J is the polar moment of inertia, G is the shear Young’s

modulus and ρ the density of the beam. Equation 3.33 can be solved using separation of

variables by expressing θ(x,t) as Equation 3.34,

(3.34)

where, Θ(x) is the torsional mode shape function. Substituting Equation 3.34 into Equation 3.33

yields a homogenous second order differential equation as given below,

(3.35)

At this point it should be noted that the second term of Equation 3.35 includes the wave

propagation constant k, that is defined as Equation 3.36,

√

(3.36)

Thus, Equation 3.35 can be re-written in the form of Equation 3.37,

(3.37)

57

A general solution can now be obtained in the form of Equation 3.38

(3.38)

or similarly as,

√

√

(3.39)

Boundary conditions can now be used to obtain specific solutions for the supporting beam. At the

end where the beam is fixed, the angular displacement should be zero thus,

(3.40)

At the end where the beam is free to move, the longitudinal stress should be zero thus,

(3.41)

Further analysis as described in [61] indicates that for a torsional beam to behave like an acoustic

quarter-wavelength transmission line it should satisfy the following condition as given below,

√

(3.42)

where, Js, G and the torsional constant γ are given by,

(3.43)

(3.44)

(3.45)

Here, v is the Poisson ratio, h is the height and Ws is the width of the beam. The torsion

coefficient α is dimensionless and is determined by the h to Ws ratio, as shown in Table 5.

58

Table 5: Torsion coefficient α value depending on the cross-sectional geometry of the beam [59]

h/Ws Α

1 0.141

1.5 0.196

1.75 0.214

2 0.229

2.5 0.249

3 0.263

4 0.281

6 0.299

8 0.307

10 0.313

∞ 0.333

Several points can be extracted from Equation 3.42. First, it should be noted that Ls is inversely

proportional to the frequency of operation of the resonator. It can be shown that at low

frequencies (e.g. <1MHz), the length of the supporting beams is impractically large. As such,

quarter-wavelength transformations are typically limited to HF and VHF designs. Furthermore,

Equation 3.42 shows that Ls is proportional to the square root of γ. The beam thickness is

typically defined by process constraints, which leaves Ws as the only parameter that can be

controlled. Equation 3.45 and Table 5 also show that in order to minimize γ, the beam cross-

section should be a square. While this is not always possible (because of fabrication constraints),

it is advisable to keep low h to Ws ratios.

3.2 Layout and Fabrication

The resonators presented in this work are targeted to operate at 1MHz, 5MHz, 10MHz, 20MHz

and 30MHz, respectively. They were developed using PolyMUMPs, a commercial multi-user

process available from MEMSCAP. A detailed discussion of the process can be found in Chapter

2. We are grateful to CMC Microsystems which enabled and supported their fabrication.

59

3.2.1 PolyMUMPs Process Flow

As was discussed in Chapter 2, there are two typical methods to fabricate a MEMS resonator in

PolyMUMPs. The first uses the Poly-0 layer for the actuating and sensing electrodes, and the

Poly-1 layer for the vibrating mechanical structure. The second keeps the Poly-0 configuration the

same, but adds a structural Poly-2 layer on top of Poly-1. This causes an increase in the

thickness of the FF beam which, according to equation 3.25, results in an increase in the

resonating frequency. The problem with both approaches is that the transducer gap between the

Poly-0 and Poly-1 layers is approximately 2 μm. In an era where transducer gaps on the order of

100-200nm are common, this value is impractically high. Designers are met with the choice of low

electrostatic transducer efficiency or very high DC bias voltage, both of which forbid any

commercialization attempt.

Here, we present an alternative and novel fabrication method that is free of the limitations

described above. An overview of the process flow can be seen in Fig. 3.7. A detailed discussion

of each step can be found in Chapter 2; this section solely discusses the purpose of each layer.

The Poly-0 layer is used for the formations of the interconnections and bond-pads. In contrast to

previous techniques, the first sacrificial oxide layer is then completely etched away. The Poly-1

layer can now be anchored directly to the substrate. It is used for the ground plate and the

sensing and actuating electrodes. Finally, the Poly-2 layer is used for the formation of the

resonator’s vibrating structure.

The main advantage that comes from this process flow is that the electrostatic gap between Poly-

2 and Poly-1 is only 750 nm. According to Equation 1.2, a reduction from 2 μm to 750 nm leads to

60

a threefold increase in the transducer efficiency, or a threefold decrease in the required DC bias

voltage. In addition, the suggested design introduces, for the first time, a novel and effective

temperature compensation technique. Finally, strategic deposition of the Au layer of PolyMUMPs

allows for a creation of tensile stresses that cancel out the negative TCE of polysilicon.

Temperature insensitive resonators with very low TCfs and high Q values are demonstrated.

Fig. 3.7 Process flow used for the fabrication of the FF beam resonators.

61

3.2.2 PolyMUMPs Process Limitations

As was previously discussed, PolyMUMPs is a commercial, multi-user process. It was not

designed with a particular application in mind; the goal for it was to be as general as possible.

The process had to meet the specifications of as many projects as possible, while maintaining

high yields. Because of this, a set of rather conservative design rules have been imposed to

developers. Table 6 summarizes a set of design rules that are relative to this work [50].

Table 6: Set of mandatory design rules for PolyMUMPs, relative to this work [50].

Level name Feature Min. Line Width Min. Feature Min. Spacing

Poly-0 Poly Line 3 2 2

Anchor-1 Oxide Hole 3 3 2

Poly-1 Poly Line 3 2 2

Anchor-2 Oxide Hole 3 3 2

Poly-2 Poly Line 3 2 2

Metal Metal Line 3 3 3

An additional design rule not shown in Table 6 is that the minimum space between a Poly-2 and a

Poly-1 structure must be 4 μm in order to ensure isolation. The problems that may arise from

these limitations can be understood by looking at Equation 3.31 and Equation 3.42. They show

that in order to increase the frequency of operation of a FF beam resonator, its dimensions must

be scaled down. As can be seen in Fig. 3.8 and Fig. 3.9, the dimensions of the resonators

decrease exponentially with increasing frequencies.

62

Fig. 3.8 FF beam length Lr versus the resonant frequency.

Fig. 3.9 Supporting beam length Ls versus the resonant frequency.

Current process constraints refrain from the fabrication of FF beam resonators above 40 MHz. In

order to illustrate these concerns, the layout of a 40 MHz resonator can be seen in Fig. 3.10. For

a 40 MHz resonator, the FF beam needs to be 18 μm long, with nodal points at 0.2241*Lr and

63

0.7759*Lr. The supporting beams, placed at the nodal points, need to have a fixed width of 3 μm

in order to meet the design rules. For an 18 μm long resonator, this means that they are centered

at 4 μm and 14 μm, effectively leaving 7 μm of free space between them. However, the 5th design

rule of Table 6 states that the minimum space between Poly-1 structures should be no less than 2

μm. Taking this into account, there is only 3 μm left for the actuating electrode. Anything beyond

that leads to impractically low transduction efficiencies, a fact that can be verified by Equation 1.2.

Fig. 3.10 Layout of a 40 MHz FF beam resonator.

The previous discussion established that there is an upper limit to the operating frequency of FF

beam resonators fabricated through PolyMUMPs. However, as will be shown, a lower limit also

exists. The reason can be found in Equation 3.42 and Fig. 3.9. At frequencies below 10 MHz, the

supporting beam length, Ls, becomes exceptionally high. In particular, an Ls value of 399 μm and

1997 μm is required for operating frequencies of 5MHz and 1MHz respectively. At these lengths,

the beams are unable to support the weight of the resonator. In order to meet the exact design of

64

the FF beam resonator, the only solution is to increase the width. However, this in turn

inadvertently affects the quality factor of the resonators. Because of this, the 1 MHz and 5 MHz

resonators presented in this work deviate from the exact FF beam design process. Their Ls was

arbitrarily set at 200 μm - the quarter-wavelength acoustic transformation was not applied here.

3.2.3 Temperature Compensation of FF beam Resonators

As was discussed in Chapter 1, a drawback of silicon MEMS resonators is their relatively high

temperature instability. The cause of the problem lies in the material properties of polysilicon and

silicon, in particular in the temperature coefficient of elasticity (TCE). The TCE of silicon is on the

order of -60ppm/oC [24], which according to Equation 1.8 leads to a resonator TCf on the order of

-30ppm/oC. Commercial timing applications require a TCf on the order of <1ppm/oC. Effectively,

uncompensated silicon resonators are unable to compete with quartz crystals in the open market.

The work discussed in this section entails a passive temperature compensation technique

employed on a 20MHz resonator. Its novel implementation led to a conference paper publication

at the 2013 IEEE Canadian Conference on Electrical and Computer Engineering [55]. Its

performance compared to uncompensated designs and a discussion of its effectiveness can be

found in Chapter 4. The compensation technique is implemented by depositing a SiO2 or Au

structural layer at the center of the resonator (Fig. 3.11). These two materials were carefully

chosen so as to meet a set of different requirements.

SiO2 was chosen because it is one of the few known materials that has a high positive TCE, on

the order of 183ppm/ oC. An increase in the temperature will lead to a corresponding increase in

the Young’s Modulus; effectively the material’s stiffness increases. In contrast, the Youngs

65

modulus of silicon decreases. The idea is that a strategic combination of the two materials can

lead to a composite structure that has a relatively constant Young’s Modulus as the temperature

varies. A detailed description of the design process can be found in [23].

Contrary to SiO2, Au does not have a highly positive TCE. The reason it was chosen was

because it has a much higher thermal expansion coefficient compared to polysilicon. Additionally,

it is included in the standard PolyMUMPs process, eliminating the need for post-processing steps.

The idea is that an increase in the temperature causes the Au layer to expand at a much faster

rate than the polysilicon layer. This imbalance causes the creation of tensile stresses in the

polysilicon structure that cause a positive shift in the Young’s Modulus. Theoretically, a strategic

placement of Au can lead to a virtual cancellation of the highly negative TCE of polysilicon [62].

Fig. 3.11 20 MHz compensated FF beam resonator.

Due to process constraints, the strategy can be applicable for designs up to a maximum

frequency of 30 MHz. The operating frequency of the compensated resonators can be found from

Equation 3.31. However, due to the use of composite materials, the resonator dimensions need

66

to be recalculated. In particular, three design parameters change. First the thickness, h, of the

structure increases from 1.5 μm to 2 μm. Second, the addition of a SiO2 or Au layer has the effect

of “softening” the resonator. The combined elasticity, Ec, of the composite beam can be found

from [55],

(3.46)

(3.47)

(3.48)

where, I is the moment of inertia, Wr is the width of the resonator, tP is the thickness of the

polysilicon and tD is the thickness of the deposited layer. The combined elasticity of the structure

with an Au layer added was calculated to be 114 GPa, while in the case of SiO2 it was found to

be 120 GPa. The final parameter that changes is the density ρ of the FF beam. The combined

density of the structure can be calculated by Equation 3.49,

(3.49)

where, Mpoly is the total mass of polysilicon, MD is the total mass of the compensation material

and V is the volume of the composite resonating beam. The combined density of the structure

was found to be 6497 kg/m3 when the Au layer was added, and 2210 kg/m3 when a SiO2 layer

was added. After these considerations are taken into account, the equations that were derived in

the previous section for the case of uncompensated resonators can be applied in the design

process.

67

3.3 Free-Free Beam Design Characteristics

The first two sections of this chapter discuss in detail the design process of uncompensated and

temperature compensated FF beam resonators. It was shown that a series of design challenges

need to be addressed in order to successfully implement the devices in a commercial process

such as PolyMUMPs. The FF beam resonator design is heavily dependent on an optimal

configuration. Because of this, a small deviation in any of its dimensions can have significant

impact on its performance. A careful analysis of the dimensions is essential to the success of the

resonators.

Five uncompensated resonators and four temperature compensated resonators were fabricated

for this research work. For the uncompensated designs, the operating frequencies were set at

1MHz, 5MHz, 10MHz, 20MHz and 30MHz. For the temperature compensated designs, the

operating frequencies were set at 1MHz, 5MHz, 10MHz, 20MHz. The design parameters,

material properties and analytical values are summarized in Table 7 and Table 8. A complete

layout of the chip sent to CMC Microsystems can be seen in Fig. 3.12. Individual layouts for every

resonator can be found in Appendix A.

68

Table 7: Design parameters of uncompensated FF beam resonators.

Table 8: Design parameters of temperature compensated FF beam resonators.

69

Fig. 3.12 Chip layout of the 9 resonators that were fabricated in PolyMUMPs.

70

Chapter 4 Finite Element Modeling (FEM) Simulations, Analysis, and

Results

The performance of the proposed resonators was evaluated using Finite Element Modeling

(FEM). The goal of this chapter is to explain the simulation and analysis methods that were used

and discuss the results that were obtained.

4.1 MEMS Resonator Evaluation Methods

The performance of a MEMS resonator is typically evaluated using one of the following

approaches,

1) Analytical Methods: The analytical approach follows closely the design process described

in Chapter 3. As we have seen, it involves the solution of high order differential systems

and a series of mathematical manipulations. While accurate, the complexity of the

method increases exponentially as new effects, such as electrostatic spring-softening,

are taken into consideration. Because of this, the analytical approach is typically used

only in the initial stages of the design process.

2) Electrical-Mechanical Analogy Methods: The electrical to mechanical analogy approach

is based on the fact that the equations governing the behavior of a mechanical system

are very similar to the ones governing the behavior of an electrical system. A detailed

71

analysis of the systems [63], shows that there is a direct correlation between mechanical

and electrical parameters. In particular, a displacement x corresponds to the charge q, a

velocity u corresponds to a current i, a rigid mass m corresponds to an inductance L and

a force F is mathematically analogous to a voltage V. By using the above relations a

MEMS resonator can be transformed into an equivalent RLC circuit. The circuit should in

principle behave the same as a mechanical system, thus, electrical simulations can be

performed on it. In order to interpret the physical meaning of the results they need to be

converted back to the mechanical domain. The key advantage of the electrical-

mechanical analogy approach is that the mathematical models behind electronic circuit

simulation are very advanced, effectively leading to a very fast modeling. However, in the

case of MEMS resonators this comes at the cost of reduced accuracy and flexibility. This

is because mechanical effects, such as thermoelastic dissipation (TED), are impossible

to model in the electrical domain and thus are not considered.

3) Finite Element Methods (FEM): The FEM approach is a numerical technique that tries to

find approximate solutions to partial differential equations of physics and other boundary

value problems. In MEMS analysis, the first step of the approach is to discretize the

mechanical structure into a large number of finite elements. Nodes are then created

between elements in order to connect them. The combination of nodes and finite

elements composes the mesh of the structure. As will be shown in the next section, the

success of a FEM implementation depends to a large extent on the selection of an

appropriate type and density of mesh. Compared to the other MEMS evaluation

approaches, the FEM technique has a large overhead during its initialization process.

The reason is that a 3D model must be created and a careful mesh convergence analysis

needs to be performed before simulations can start. While its initialization is time

consuming, the results are significantly more accurate and reliable. The simulation results

72

of a careful FEM analysis are significantly closer to experimental values compared to the

previous two techniques. Additionally, mechanical effects such as “electrostatic spring-

softening” and TED can very easily be included in the model.

Due to the advantages described above, the FEM approach was chosen for the evaluation of the

MEMS resonators in this work. The simulations presented here were performed using the

CoventorWare finite element modeling suite. A discussion of the approach follows below.

4.2 Model Initialization

Before any simulation and modeling takes place, the material properties need to be inserted into

the FEM model. The numerical values that were used are summarized in Table 9.

Table 9: Material properties used in FEM simulations.

Properties Material

Poly-0 Poly-1 Poly-2 Au SiNi4

Young's Modulus (GPa) 158 158 158 57 254

Poisson Ratio 0.22 0.22 0.22 0.35 0.23

Density 2230 2230 2230 1930 2700

TCE (1/K) 2.80E-06 2.80E-06 2.80E-06 1.45E-05 1.60E-06

Thermal Conductivity (pW/umK) 3.20E+07 3.20E+07 3.20E+07 2.97E+08 2.50E+07

Specific Heat (pJ/kgK) 1.00E+14 1.00E+14 1.00E+14 1.29E+14 1.70E+14

Electric Conductivity (pS/um) 6.67E+10 5.00E+10 3.33E+10 3.21E+13 0.00E+00

After the material properties have been set, the next step is to create the layout of the resonators

and define the fabrication process. The dimensions and parameters of the designs were derived

in Chapter 3 and are summarized in Table 7 and Table 8. For illustrative purposes, the layout and

fabrication process used in the 1 MHz resonator can be seen in Fig 4.1a and Fig. 4.1b

respectively.

73

(a)

(b)

Fig. 4.1 a) Layout of the 1 MHz resonator, cyan and white areas denote a Poly-2 and Poly-1 layer

respectively. b) Fabrication process used in the PolyMUMPs run.

In the previous steps, the geometry, material properties and structural composition of the

resonators was defined. An accurate 3D model can now be created using CoventorWare’s

PostProcessor. However, simulations cannot begin yet. The surfaces and faces of interest need

74

to be specified and the model needs to be meshed. A mesh convergence analysis was performed

in order to find a mesh that could accurately simulate the design while keeping the simulation

times to practical values. The results showed that a “Manhattan Bricks” mesh type with finite

element dimensions of (dx, dy, dz) = (2, 2, 2) yielded the optimal solution. The meshed model of

the 1 MHz FF beam resonator can be seen in Fig. 4.2.

Fig. 4.2 3D meshed model of the 1 MHz FF beam resonator.

The procedure described above was performed in the case of each resonator. While the material

properties and fabrication process remained the same, the geometry and dimensions of the

resonators varied. A mesh convergence analysis needed to be performed in each case.

4.3 Modal Analysis

Now that 3D models of the resonators have been created and meshed, the simulation sequence

can be initiated. The first type of simulation that was executed was modal analysis. The goal of a

modal analysis is to identify the natural frequencies of vibration of a mechanical structure under

75

equilibrium. The natural frequencies along with the corresponding modal shapes are invaluable in

understanding the system response of a MEMS resonator. They show exactly when the

mechanical system will have a maximum response and how it will deform during oscillation.

Effectively, a study of the mode shapes can show if the resonator is working as it was intended

to. If the mode shape response is not desired, e.g. spurious modes are present, the system can

be redesigned to minimize the unwanted parameters; similarly if the response is desired the

system can be optimized to enhance the desired parameters.

In CoventorWare a modal analysis can be performed using the MemMech module. Since modal

analysis of mechanical structures needs to be performed under equilibrium, the resonators for

this work were undamped during the simulations. The amplitudes obtained in the modal shapes

were normalized to 1um. The modal shapes for the uncompensated and compensated designs

can be seen in Fig. 4.3 and Fig. 4.4, respectively.

Several observations can be made here. It should be noted that in the figures the blue color

indicates a point that sustains no movement, green indicates large displacement, while red

indicates maximum displacement. The first point to note is that the resonators operate as

expected at the fundamental mode of vibration. The middle of the Free-Free beam is able to

move freely, while the nodal points remain stable during the vibration.

The second point is that the quarter-wavelength impedance transformation was successful in the

designs where it was implemented. As was discussed in Chapter 3, the supporting beam lengths

for the 1 MHz and 5 MHz resonators were set arbitrarily to 200um. A comparison between Fig.

4.3(a,b) and Fig. 4.3 (c,d,e) shows the benefits of a quarter-wavelength impedance

transformation. In the 1 MHz and 5 MHz resonators, shown in Fig. 4.3(a,b), the supporting beams

76

are presented as steady rigid anchors. Because of this, the FF beam will “see” higher torsional

resistance and thus increased anchor dissipation. In contrast to this, the 10 MHz, 20 MHz and 30

MHz designs, shown in Fig. 4.3(c,d,e), do not suffer from this problem. The supporting beams are

able to rotate freely at the FF beam end, effectively presenting virtually zero torsional resistance.

Because of this, it is expected that the quality factors of the 1 MHz and 5 MHz resonators will be

significantly lower compared to the other designs. The modal frequencies for all the designs are

summarized in Table 10.

Table 10: Material properties used in FEM simulations.

Target Frequency Fundamental Mode

Uncompensated Resonant Frequency (MHz)

1 MHz 1.0547E+06 5 MHz 5.0697E+06 10 MHz 9.0841E+06 20 MHz 1.7428E+07 30 MHz 3.4612E+07

Compensated (Gold) Resonant Frequency (MHz)

1 MHz 1.1619E+06 5 MHz 6.2568E+06 10 MHz 1.3134E+07 20 MHz 1.8794E+07

77

(a) (b)

(c) (d)

(e)

Fig. 4.3 3D modal shapes for the uncompensated designs operating at a) 1MHz, b) 5MHz, c)

10MHz, d) 20MHz, e) 30MHz.

78

(a) (b)

(c) (d)

Fig. 4.4 3D modal shapes for the compensated designs operating at a) 1MHz, b) 5MHz, c)

10MHz, d) 20MHz.

4.4 Electrostatic Spring Stiffness

As was discussed in Chapter 3, Section 3.1.2, the operating frequency of a MEMS resonator is

highly dependent on the applied voltage. An analytical expression that describes this

phenomenon was derived in the form of Equation 3.31. However, the analysis cannot be stopped

there as the “softening” of the resonant frequency often leads to stability issues for the resonator.

A numerical analysis is performed in CoventorWare in order to alleviate any concerns.

79

The modal number and resonant frequency from the previously performed modal analysis are

needed here. These values are substituted in the CoSolve module, along with a DC operating

point. A parametric simulation where the DC operating point was swept from 5V to 40V in steps of

5V was performed for each resonator. The results for the 20MHz design can be seen in Fig. 4.4.

As was expected the resonant frequency is inversely proportional to the square of VDC.

Fig. 4.4 Plot of resonant frequency, for the 20 MHz design, versus applied bias voltage VDC.

4.5 Harmonic Electromechanical Analysis

Thus far, two types of simulations have been performed, a modal analysis and an electrostatic

spring stiffness analysis. The modal analysis was used to find the resonant frequency of the FF

beam resonators and to verify that they operate as expected. The spring stiffness analysis was

used to study the effects of VDC on the operating frequency and establish a DC operating point

80

for the resonators. Missing from this evaluation is the performance of the resonators. A frequency

response is required for this purpose.

The information from the previous simulations is used to perform a frequency-domain harmonic

electromechanical simulation on each resonator. The HarmonicEM module from CoventorWare is

used for this purpose. A 100 mV AC signal is supplied to the actuating electrode, while the

structure is set at a DC operating point of 20V. The frequency of the AC signal is swept, in

increments of 100 Hz, from “-500 KHz+fo” to “+500 KHz+fo” where, fo is the resonant frequency

found from the previously performed modal analysis. The simulations are performed at an

atmospheric pressure of 0.1013 MPa with the viscosity of air set at 1.86e-11 kg (μm s)-1 [64].

The frequency responses for the 30 MHz, 20 MHz, 10 MHz, 5 MHz and 1 MHz designs can be

seen in Fig. 4.5, Fig. 4.6 and Fig. 4.7, a few observations can be made here. Firstly, the results

seem to confirm the resonant frequency values obtained through modal analysis. Secondly, a

closer inspection of the graphs confirms that there are no spurious modes near the operating

frequency. After these observations are made, the frequency response plots are used to extract

the quality factor of each resonator. The results are summarized in Table 11.

81

(a)

(b)

Fig. 4.5 Frequency response at 20V VDC, for: a) 30 MHz resonator, b) 20 MHZ resonator.

82

(a)

(b)

Fig. 4.6 Frequency response at 20V VDC for: a) 10 MHz resonator, b) 5 MHz resonator.

83

Fig. 4.7 Frequency response at 20V VDC for the 1 MHz resonator.

Table 11: Quality factors for the uncompensated FF beam resonators.

Target Frequency Quality Factor (Q)

10MHz 1280

20MHz 1170

30MHz 1150

5MHz 339

1MHz 256

A quick glance of Table 11 reveals that there is no clear correlation between the target frequency

and Q value of the resonator. However, as will be shown, this is not the case. As was discussed

in Chapter 3, Section 3.1.2, the supporting beams for the 1 MHz and 5 MHz designs were

designed without quarter-wavelength dimensions. It was expected that this would lead to a large

torsional resistance, which in turn would cause increased anchor dissipation. Indeed, the results

84

shown in Table 11 confirm this prediction. The Q values of 339 and 256, for the 5 MHz and 1

MHz design respectively, are less than half the Q values of the designs that have undergone

quarter-wavelength transformation.

Another important point that can be extracted from Table 11, is that the Q values of the designs

that have quarter-wavelength supporting beams (10 MHz, 20 MHz, 30 MHz), are essentially the

same. A slight decrease in Q with increased frequency is normal, perfect anchor dissipation is

almost impossible to achieve due to fabrication constraints. Effectively, the results show that the

Q of a FF beam resonator is largely independent of the operating frequency, at least at the VHF

domain. At the ultra-high frequency (UHF) domain, other energy loss mechanisms such as

thermoelastic dissipation become dominant. Considering the above we can conclude that the

design process used for the FF beam resonators was successful.

4.6 Temperature Compensation Evaluation

The following section has been adapted from a recently published conference paper [55]. The

simulations described in the previous sections were used to evaluate the performance of the

temperature compensated resonators. This section discusses the method used to evaluate their

temperature characteristics. The numerical analysis was performed in CoventorWare.

After the mechanical characteristics of the resonators were known, a series of parametric

simulations were performed to evaluate their Temperature Coefficient of Frequency (TCF).

Temperature was varied from -50℃ to 120℃ in increments of 10℃ . At every step, an

electrothermomechanical analysis was performed in MemMech in order to acquire the resonating

frequency. The obtained results are shown in Fig. 4.8., the Δf/fo (fractional frequency change) is

85

plotted against the temperature, for the 20 MHz uncompensated, SiO2 compensated and Au

compensated designs. A closer look of the plot reveals that in the case of Au compensated

design, the metal introduces a positive shift that counterbalances the negative trend of the Δf/fo

(fractional frequency change) versus temperature. This causes the resonant frequency to remain

relatively stable as the temperature varies.

The TCFs extracted from Fig. 4.8 are presented in Table 12. For the 20 MHz design, it can be

seen that SiO2 compensation was able to reduce the TCF by about 30%. Using Au layer

compensation the TCF of the FF beam resonator dropped to -1.66ppm/℃, a value comparable to

resonators fabricated through complicated custom processes. The Au layer temperature

compensation is a significant result that is reported for the first time for resonators fabricated

using a commercial MEMS fabrication process. The TCF for the uncompensated design is -

8.48ppm/℃, which is in the range of previous reported values [65].

Fig. 4.8 Fractional frequency versus temperature for the 20MHz uncompensated, gold

compensated and SiO2 compensated design.

86

Table 12: Quality factors for the uncompensated FF beam resonators.

Target Frequency

(MHz)

TCF (ppm/oC)

Uncompensated Metal Comp. SiO2 Comp.

20 -8.49 -1.660 -6.38

87

Chapter 5 Conclusion and Future work

5.1 Conclusion

In this work, we presented the design of several Free-Free beam MEMS resonators that were

developed using PolyMUMPs, a commercial multi-user process. An analytical model was built in

order to study the physics of the resonators. Phenomena and parameters such as, electrostatic

spring-stiffness, vibration mode shape and resonant frequency were discussed, and

mathematical expressions that described their relations were derived. This information was

subsequently used in the design and optimization of Free-Free beam resonators; the

performance of which was evaluated using the Finite-Element-Method (FEM) software

CoventorWare.

In the second part of this work, we presented the design of temperature compensated Free-Free

beam MEMS resonators that were developed using PolyMUMPS. We described a novel and

effective temperature compensation strategy that is achieved using a composite structural layer

of gold. Numerical simulation results showed that in the passively compensated design, a

strategic deposition of gold on top of the resonator induces a positive frequency shift that

counterbalances the characteristic negative TCf of polysilicon resonators. Temperature stability

88

increased from -8.48 ppm/℃ to -1.66ppm/℃, a result comparable to resonators fabricated

through complicated custom processes.

5.2 Future Work

The device designs have been sent to CMC Microsystems for fabrication. Once we receive the

fabricated devices we will perform initial characterization tests to show that there were no issues

during the fabrication process or transportation. A testing setup will be built in order to

experimentally evaluate the performance of the resonators. This setup will consist of a Vector

Network Analyzer (VNA), two voltage supplies in series and a DC bias. We have requested the

resonators to be mounted on DIP40 packages. We will design and fabricate printed circuits

boards (PCB) for complete evaluation of the resonator devices.

The first step in the evaluation process would be to verify that the resonators work as expected. A

frequency response obtained from the VNA would suffice for this. Ideally the resonators should

be tested in a vacuum environment in order to accurately measure their frequency response

without the effects of air damping. Furthermore, a series of stability tests would need to be

performed in order to characterize resonant frequency shifts due to temperature variations and

pressure fluctuations. Ultimately, the tests should be performed using a probe station in order to

eliminate stress effects induced by the packaging.

89

90

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