MECHANICALLY AMPLIFIED CAPACITIVE
STRAIN SENSOR
by
JUN GUO
Submitted in partial fulfillment of the requirements
For the degree of Doctor of Philosophy
Dissertation Adviser: Dr. Wen H. Ko
Department of Electrical Engineering and Computer Science
CASE WESTERN RESERVE UNIVERSITY
May, 2007 CASE WESTERN RESERVE UNIVERSITY
SCHOOL OF GRADUATE STUDIES
We hereby approve the dissertation of
______
candidate for the Ph.D. degree *.
(signed)______(chair of the committee)
______
______
______
______
______
(date) ______
*We also certify that written approval has been obtained for any proprietary material contained therein.
Dedication
To my parents.
TABLE OF CONTENTS
Mechanical Amplified Capacitive Strain Sensor
Table of Contents ...... 1
List of Tables ...... 7
List of Figures ...... 9
Acknowledgements ...... 15
Abstract ...... 16
Chapter 1 Introduction to Strain Measurement ...... 18
1.1. Introduction and motivation...... 18
1.2. Thesis organization ...... 23
Chapter 2 A Review of Micromachined Strain Gages ...... 25
2.1. Introduction...... 25
2.2. Metallic-foil strain gage...... 25
2.2.1 Principle ...... 26
2.2.2 Gage factor and sensitivity...... 28
2.2.3 Instrumentation ...... 29
2.2.4 Resolution of a typical metallic-foil strain gage...... 30
2.3. Piezoresistive strain gage...... 32
2.3.1 Piezoresistive effect ...... 32
1 2.3.2 Longitudinal and transverse piezoresistive coefficients ...... 33
2.3.3 Impurity concentration (doping level) and temperature effect on
piezoresistance coefficients...... 35
2.3.4 Resolution and issues...... 36
2.4. Resonant silicon micromachined strain gage...... 37
2.4.1 Principle ...... 38
2.4.2 Excitation and detection scheme...... 39
2.4.3 Resolution: minimum detectable frequency ...... 40
2.5. Capacitive strain gage...... 42
2.5.1 Principle and configurations ...... 42
2.5.2 Performance of capacitive strain gage ...... 44
2.6. Summary...... 46
Chapter 3 Capacitive Strain Sensor Employing a Buckled Beam Mechanical
Amplifier ...... 48
3.1. Introduction...... 48
3.2. Mechanical amplifier ...... 49
3.2.1 Background...... 50
3.2.2 Principle ...... 51
3.3. Analytical modeling...... 52
3.3.1 Derivation of the curvature function under an applied force...... 53
2 3.3.2 Calculate the mechanical gain ...... 56
3.3.3 Small input force simplification and the structure nominal mechanical gain 59
3.3.4 Large deflection compensation ...... 60
3.3.5 Calculate the buckling beam equivalent stiffness in x-direction ...... 60
3.4. Finite-Element-Analysis verification...... 61
3.4.1 FEA verification of nominal mechanical ...... 61
3.4.2 FEA verification of equivalent stiffness ...... 62
3.4.3 FEA verification of mechanical gain corresponding to large displacement input
63
3.5. Nonlinearity of the buckled beam amplifier ...... 65
3.6. Fundamental noise analysis ...... 67
3.6.1 Resistive thermal noise estimation...... 68
3.6.2 Mechanical thermal noise (Brownian motion noise) estimation ...... 70
3.7. The maximum stress of the structure and locations...... 73
3.8. Device fabrication...... 75
3.9. Fabricated device ...... 77
3.10. Testing...... 80
3.10.1 Test fixture...... 80
3.10.2 Device functionality test under microscope...... 83
3.10.3 Capacitive readout circuitry...... 84
3 3.10.3.1 C/V characteristics using MS3110®...... 84
3.10.3.2 C/V characteristics using developed interface electronics...... 88
3.11. Conclusions...... 93
Chapter 4 Sensor Backing Design ...... 95
4.1. Strain transmission ratio of a solid backing...... 95
4.1.1 Transmission ratio definition ...... 95
4.1.2 Strain transmission ratio of a solid backing...... 96
4.2. Transmission ratio of a folded-spring backing...... 102
4.2.1 Folded-spring backing structure analysis...... 103
4.2.2 Anchor location and package gain...... 106
4.2.3 Package gain discussions ...... 109
4.2.3.1 The maximum package gain for piezoresistive sensor design...... 109
4.2.3.2 The maximum displacement for capacitive sensor design ...... 111
4.3. Overall sensor module backing design ...... 116
4.4. Summary and conclusions ...... 119
Chapter 5 Sensing Structure Design ...... 120
5.1. Structure modification ...... 120
5.2. Capacitive output sensitivity calculation ...... 122
5.3. Fringe capacitance effect on comb drive finger design ...... 124
5.4. Nominal capacitance and parasitic capacitance...... 126
4 5.5. Series resistance consideration ...... 127
5.6. Structure mechanical property ...... 128
5.6.1 Nonlinearity of the buckled beam amplifier ...... 128
5.6.2 Resonance modes and frequencies...... 128
5.6.3 Mechanical thermal noise analysis ...... 130
5.7. Fabrication process ...... 131
5.8. Fabricated device ...... 135
5.9. Summary and conclusions ...... 136
Chapter 6 Testing Fixtures and Device Testing ...... 137
6.1. Test fixtures for capacitance output characteristics ...... 137
6.1.1 Four-point strain testing fixture ...... 137
6.1.2 Bent cantilever beam testing fixture ...... 141
6.1.3 Capacitive strain sensor applying procedure ...... 144
6.2. Package gain & mechanical gain measurement under microscope ...... 148
6.3. Capacitive output characteristics ...... 151
6.3.1 Interfaced to an off-the-shelf C/V converter...... 151
6.3.2 Interfaced to a developed low noise C/V converter...... 153
6.4. Temperature behavior ...... 155
6.5. Step response ...... 159
6.6. Frequency response testing...... 160
5 6.6.1 Test fixture...... 161
6.6.2 Resonant cantilever beam resonant frequency calculation ...... 161
6.6.3 Actuation displacement versus generated strain...... 163
6.6.4 Test fixture behavior...... 163
6.6.5 Sensor Dynamic behavior...... 165
6.7. Conclusions...... 166
Bibliography ...... 168
6 LIST OF TABLES
Table 1.1. Major requirements for the MEMS strain gage module used for a
rolling-element bearing. 22
Table 2.1. Properties of strain gage conductors. 29
Table 2.2. Longitudinal and transverse piezoresistance coefficients for various
combination of direction in cubic crystals. 35
Table 2.3. Adiabatic piezoresistance coefficients at room temperature. 35
Table 2.4. Summary of four types of micromachined strain gages. 47
Table 3.1. Properties of hyperbolic cosine and sine function. 59
Table 3.2. Comparison of nominal mechanical gains obtained analytically and from FEA.
62
Table 3.3. Comparison of structure stiffness obtained analytically and from FEA. 63
Table 3.4. Design values of the buckled beam strain sensor. 79
Table 3.5. Parameters of the test fixture. 81
Table 4.1. Calculated transmission ratio with various bonding adhesive thickness. The
structure is with thickness of 480 µm and geometrical dimensions is illustrated in
Figure 4.6. 105
Table 5.1. Device parameters. 123
Table 5.2. Resonant frequency and shape of each part of the sensing structure. 129
Table 6.1. Parameters of the developed four-point test fixture. 141
7 Table 6.2. Parameters of the developed bent cantilever beam test fixture. 143
Table 6.3. Beam parameters. 162
Table 6.4. Calculated fundamental resonant frequencies of various beam lengths. 162
8 LIST OF FIGURES
Figure 1.1. Strain definition. 18
Figure 1.2. Strain-stress relationship for common materials. 19
Figure 2.1. Metallic-foil strain gage. 25
Figure 2.2. The resistance change of a cylindrical conductive wire under strain. 26
Figure 2.3. Wheatstone bridge configuration. 30
Figure 2.4. Piezoresistor under a) longitudinal and b) transverse stress. The small red
arrow on the resistor points in the direction of the exciting current. 33
Figure 2.5. Variation of the coefficient π 11 in n-type layers with temperature and surface
concentration. 36
Figure 2.6. A resonant beam subjected to a strain load, ε. 38
Figure 2.7. Resonator excitation and detection diagram. 39
Figure 2.8. Electrostatic excitation and capacitive detection. 40
Figure 2.9. The diagram of a capacitive strain sensor. 42
Figure 2.10. Capacitive strain sensor configuration. 44
Figure 2.11. High sensitivity of a capacitive strain sensor. 45
Figure 3.1. The diagram of a capacitive strain sensor. 50
Figure 3.2. Principle of the buckled beam amplification scheme. 52
Figure 3.3. A buckled beam strain sensor with differential output. 52
9 Figure 3.4. (a) Simplified structure. (b) Forces acting on the beam. 54
Figure 3.5. Beam modeling for calculating Δx. 57
Figure 3.6. Comparison of mechanical gain obtained analytically and by FEA with
l=300 μm , w=1 μm , and h=30 μm with the maximum applied strain of ± 2000 με .
65
Figure 3.7. Finite Element Analysis result of relationship between nonlinearity, mechanical
gain, and the maximum strain. 67
Figure 3.8. Device electrical model. 68
Figure 3.9. The diagram of the front amplifier architecture of the interface electronics. 69
Figure 3.10. Device resonant behavior simulation. 72
Figure 3.11. FEA analysis of the structure under the maximum strain indicates that the
location of maximum stress is near the anchor. 74
Figure 3.12. Major fabrication steps. 75
Figure 3.13. An SEM picture of a buckled beam capacitive strain sensor. 78
Figure 3.14. Description of the test strip and bent-beam test fixture. (a) Top view of the
4-inch device wafer and a test strip. (b) Cross-section view a test strip mounted on
a bent-beam test fixture. 81
Figure 3.15. A picture of the bent-beam test fixture. 82
Figure 3.16. Microphotographs of sensor comb drive fingers before and after an applied
tensile strain. 83
10 Figure 3.17. A computer illustration of the bent-beam test fixture. 85
Figure 3.18. Coupled-field finite-element analysis of the structure under applied strain. 86
Figure 3.19. Measured capacitance change as a function of applied strain. 87
Figure 3.20. Strain sensor electronic interface architecture. 89
Figure 3.21. Sensing electronics die photo [40]. 90
Figure 3.22. System testing board. 92
Figure 3.23. Output voltage vs. input strain. 92
Figure 3.24. Output noise spectral density. 93
Figure 4.1. Strain transfer mechanism. 96
Figure 4.2. Sensor substrate geometric dimensions. 98
Figure 4.3. FEA results on bonding adhesive thickness effect on transmission ratio. 100
Figure 4.4. Measured capacitance change as function of the applied strain using the
four-point test fixture 101
Figure 4.5.The structure of a folded-spring backing for the capacitive strain sensor. 102
Figure 4.6. Top view of a folded-spring backing and the geometrical dimensions used.
The structure has a thickness of 480 μm . 103
Figure 4.7. Structure deformation under an applied strain. 104
Figure 4.8. Package gain. 106
Figure 4.9. Transmission ratio analysis. 107
Figure 4.10. Package gain as a function of bond pad length Lp . The sensor structure is
11 with the following geometrical parameters: Gap Lgap =1000 µm, bonding pad
thickness h =480 µm, adhesive layer thickness ha =20 µm. 111
Figure 4.11. Top view of a capacitive strain sensor module including a folded-spring
backing. The annotated dimensions reflects the real sensor. 113
Figure 4.12. 3-D FEA modeling of a folded-beam backing bonded on a strained specimen
for transmission ratio analysis. 114
Figure 4.13. FEA results of transmission ratio of a folded-beam backing as a function of
varied bonding epoxy thickness. 115
Figure 4.14. Overall sensor backing design. 117
Figure 4.15. Strain transmission ration examined by the finite-element-analysis. 118
Figure 5.1. Improved sensing structure. 120
Figure 5.2. Sensing beam affects the structure deformation as it bends when the structure is
under strain 122
Figure 5.3.Finite-element modeling of capacitance of a comb finger. (a) Top view of a
section of comb drive sensing fingers. (b) Top view of a section of comb drive
sensing fingers in the dashed red line. 124
Figure 5.4. Finite-element analysis results about capacitance output as a function of
applied displacement with two variables: (1) gclr =21µ m; (2) gclr =24 µm. 125
Figure 5.5. Top view of the sensing structure. 129
Figure 5.6. Major fabrication processing steps. 134
12 Figure 5.7. Fabricated device and an SEM picture of the sensing structure. 135
Figure 6.1. Analysis of a four-point bending test fixture for strain sensor evaluation. 138
Figure 6.2. Picture of the developed four-point bending fixture for strain sensor evaluation.
140
Figure 6.3. Diagram of a bent-cantilever beam bending test fixture. 142
Figure 6.4. Picture of a capacitive strain sensor under testing using a bent cantilever test
fixture. 142
Figure 6.5. Finite element analysis results of the strain distribution along the bent beam
surface under a displacement load. The circled area indicates a distorted strain
distribution around the clamped edge. 144
Figure 6.6. Two capacitive strain sensors bonded on a stainless cantilever beam. Elevated
temperature adhesive curing process results in an unbalanced capacitive output
which can be observed under microscope. 147
Figure 6.7. Test setup for package gain & mechanical gain measurement. 148
Figure 6.8. Captured images of comb drive fingers under microscope. The overall gain is
measured to be 8.95. 150
Figure 6.9. Measured C/V converter (MS3110) sensitivity. 151
Figure 6.10. Measured capacitance output characteristics using the off-the-shelf C/V
converter MS3110®. 152
Figure 6.11. Comparison of measured capacitance output characteristics after offset adjust
and FEA results. 153
13 Figure 6.12. Measured noise floor of the developed C/V Converter powered by a 3-V
battery. The measured noise floor is 233 nV Hz nV/Hz. 154
Figure 6.13. Measured capacitance output characteristics using the developed low-noise
C/V converter. 155
Figure 6.14. Voltage output dependency on environmental temperature variation. 156
Figure 6.15. Diagram of the modified test setup for sensor evaluation at elevated
temperature. 157
Figure 6.16. Measured voltage output characteristics at different temperature. 158
Figure 6.17. Measured voltage output sensitivity change of the overall system including
the sensor the developed interface electronics at different temperature (referenced to
sensitivity at 24.6ºC). 158
Figure 6.18. The step response. 160
Figure 6.19. Frequency response testing test setup. 161
Figure 6.20. Diagram of the double-end-clamped beam under displacement load at center. 163
Figure 6.21. Measured PZT displacement without load and beam displacement driven by
the PZT actuator. The driven voltage level is 10 V Vpp. 164
Figure 6.23. Test setup for system dynamic behavior. 165
Figure 6.24. Received power spectrum of strain sensor under a dynamic 10 µεpp signal @ 1
kHz. 166
14 ACKNOWLEDGEMENTS
I would like to express my sincere thanks for my advisor, Professor Wen H. Ko, for his support, encouragement, and guidance through the past four years of my Ph.D. studies.
His technical expertise and constant help have broadened my professional perspective and view of research. I would also like to express sincere thanks to Professor Darrin J.
Young for his constant guidance and advice. I am very grateful to Professor Steven
Garverick, Christian A. Zorman, Thomas P. Kicher, and Alexis R. Abramson for being my committee and contributing their precious time.
Many thanks to my friend, Dr. Hung-I Kuo for his help building lab fixtures and precious technical advices. Many thanks to MFL staffs (Ed Jahnke, Alex Smith, and
David Greer) for their support in device fabrication. I also would like to thank Michael
Suster, Peng Cong, Ping Huang, Rui Zhang, Dona Duan, for their friendship and help.
I would like to thank my family for their constant encouragement and support throughout my research years, especially my mother, who had been supportive and very close to me through my whole lift. I would like to express my deepest thanks to my wife who brought me the most precious gift: a family with two healthy sons.
15
Mechanically Amplified Capacitive Strain Sensor
Abstract
by
Jun Guo
A capacitive strain sensor module has been developed to fulfill the desired requirements such as high sensitivity, high accuracy, extended temperature range, and high signal bandwidth for advanced industrial applications.
The capacitive strain sensor employs a novel mechanical amplifier using buckled beam suspensions to mechanically amplify the applied displacement signal, thus resulting in a 20 dB increment of the differential capacitance output. The detailed principle description and the analysis of the mechanical amplifier are provided, including the full analytical modeling and the Finite-Element-Analysis modeling. The modeling accuracy has been verified by the measurement results of a fabricated test structure with the maximum discrepancy less than 5%. The fundamental noise analysis indicates that the fundamental Brownian motion noise of the mechanical structure has an input-referred noise power spectral density of 3x10-7 micro-strain/sqrtHz. Therefore, the sensor can be
operated in ambient without requiring a vacuum packaging, thus substantially reducing the
system packaging complexity and cost.
16 Understanding and predicting the transfer behavior of strain from the test substrate to
the sensing element is another critical step toward achieving the high accuracy requirement.
The thesis provides the detailed analysis on the strain transfer behavior and parametrically
establishes the relationship between the strain transfer behavior and device material and
geometrical parameters. Based on the analysis, a novel folded-spring silicon backing
structure is designed to reduce the strain transmission loss, yet still providing the
microfabrication batch process capability to reduce the device fabrication cost. The
analysis shows the structure stiffness has been reduced close to three orders in magnitude,
thus substantially improve the transmission ratio from 60% to 99.2%. Additionally, the
transmission ratio is less sensitive to bonding adhesive thickness, a key advantage to
achieve high measurement consistency.
The overall capacitive strain sensor module has a sensor dimension of 2mmx3mm and delivers a nominal capacitance of 0.23 pF, and a measured capacitance output sensitivity of
280 aF/micro-strain. Integrated with the developed low-noise C/V converter interface electronics, the overall system is able to detect a minimum strain of 0.033 micro-strain with the maximum range of ± 1000 micro-strain, indicating an equivalent dynamic range of 89
dB has been achieved. Tested from room temperature to 112ºC, the integrated system shows a maximum variation of 2.9% in sensitivity.
17 1. CHAPTER ONE
INTRODUCTION TO STRAIN MEASUREMENT
1.1. INTRODUCTION AND MOTIVATION
Strain is a fundamental engineering phenomenon that describes an object’s deformation under stress. It is defined as the elongation per unit length, denoted by the
Greek letter ε (epsilon), as illustrated in Figure 1.1 and given by the equation [1],
ΔL ε = . (1-1) L
Force (F) Force (F)
L ΔL
Figure 1.1. Strain definition.
If the object is under tension, the strain is called a tensile strain, representing the elongation or stretching of the material. If the object is in compression, the strain is a compressive strain as the object shortens. Tensile strain is taken as positive (+) and compressive strain as negative (-) [2]. Because the strain is the ratio of two lengths, it is a dimensionless quantity with no units. The strain of an object is usually small; therefore the unit micro-strain is more often used. One micro-strain equals to 10-6 strain, and is denoted by “ με ”.
18 The importance of strain measurement has been well established since the seventeenth
century when Hooke pointed out that for many common materials, there was a constant
ratio between stress and strain [1]. The constant of proportionality between stress and
strain (Figure 1.2) implied in Hooke’s law is known as the modulus of elasticity of the
material, or, after the man who is credited with defining it, Young’s modulus.
Mathematically, this is expressed as
σ E = , (1-2) ε
where E, σ, ε denote the material’s Young’s modulus, the stress, and the strain,
respectively.
Linear range
σ Δσ = E
Stress, Δε Δσ
Δε E: Young’s modulus
Strain, ε
Figure 1.2. Strain-stress relationship for common materials.
With the relationship between stress and strain established, it becomes apparent that
we can determine the stress in a body under a given external load by measuring the strain
and multiplying by the modulus of elasticity. This feature is widely used in experimental
stress analysis, where experimental stress analysis uses the strain values measured on the
19 surface of a specimen or structural part to state the stress in the material and also to predict its safety and endurance. It is also worth mentioning that for many material, the Young’s modulus may not be constant, but a function of applied stress and strain [1].
A strain gage (alternatively: strain gauge) is a device used to measure the strain of an
object. Fundamentally, all strain sensors are designed to convert mechanical deformation
(strain) into other measurable signals, often electrical signals, by generating a change in
resistance, resonant frequency, or capacitance that are proportional to the strain
experienced by the sensor.
There are many types of strain sensors (gages) based on different operating principles.
Among them, the oldest and the most common type of strain gauge is metallic-foil gage,
consisting of an insulating flexible backing (typically made of polyimide) which supports
a metallic-foil pattern. Metallic-foil gage is the most widely used gage due to its
easiness to handle and use, and well established availability to different applications [3].
However, this type of strain gage suffers from low sensitivity as the gage factors are
typically around 2-5. The development of the Microelectromechanical Systems (MEMS)
technology has spawned a new era in strain gage development. Micromachined
piezoresistive (also called semiconductor strain gage), resonant, capacitive, optical strain
gages have been developed; and achieved much higher performance in terms of sensitivity,
dynamic range, and operating temperature range. The established batch fabrication
process enables the sensors to be produced in mass quantities with consistent performance
and low unit cost. As a result, besides being used for machinery components safety
monitoring, the micromachined strain gages are widely used in sensing a variety of
physical and chemical quantities including acceleration, angular rate, chemical
20 concentration, etch rate, fluid density, fluid viscosity, force, humidity, mass, mass flow,
pressure, proximity, surface profiles, vibration, and many others. For example, a silicon
resonant strain gage fabricated on a silicon membrane has been used in a pressure sensor to
deliver a guaranteed accuracy of ± 0.075% and ± 0.1% stability for 24 months operation
[4].
An “ideal” strain sensor should have the following listed properties:
1. Little or no effect on the strain distribution of the specimen under measurement
2. Small in size
3. High sensitivity
4. Wide signal bandwidth
5. Easy transporting, handing, and low applying cost
6. High temperature stability
7. Low fabrication cost
To date, there is no one instrument that fully meets all above specifications and different applications would desire different performance requirements. Therefore, the practical gage design is really application requirement defined. This study is orientated to design a type of strain gage which can be used to monitor the strain field of a rolling-element bearing under load, with the main requirements list in Table 1.1. To fulfill
the requirements, a strain gage with resolution of 0.1 με over DC to 10 kHz bandwidth,
and a dynamic range over 80 dB has to be developed, which serves as the motivation of
this study.
21 Table 1.1. Major requirements for the MEMS strain gage module used for a rolling-element bearing.
Parameter Goal
Operating ranges ± 1000 με *
Accurate to within 0.5 με
Resolution 0.1 με
Operating temperature range -40 D C – 150 D C
Frequency bandwidth DC – 10 kHz
* 1 με = 1 micro-strain = 10-6 strain.
To achieve this goal, many types of micromachined strain gages have been investigated including: metallic-foil, piezoresistive, resonant, and capacitive types.
Among them, the capacitive type strain gage has excellent temperature stability, low
long-term drift, and high sensitivity, which make it a promising candidate for this study.
However, a typical micromachined capacitive strain gage has extremely small capacitance output at resolution level due to its miniature size, thus requiring either a nearby high-sensitivity interface circuit or large device size/area. To overcome this problem, a mechanical amplifier employing buckled beam suspension has been developed. The mechanical amplifier amplifies the small displacement induced by the strain; as a result, much greater capacitance output can be achieved and the requirements on the interface circuit and the sensor size can be reduced.
This thesis presents the mechanical amplification mechanism concept, structure modeling, and sensor design/fabrication/test based on this concept. Tested with an in-house designed interface circuit, the designed strain demonstrated a strain resolution of
22 0.033 με over 10 kHz bandwidth with a maximum range of ±1000 με .
1.2. THESIS ORGANIZATION
Chapter 1 gives an overall introduction.
Chapter 2 gives a review of four types of micromachined strain gages such as metallic-foil, piezoresistive, resonant, and capacitive strain gages. The chapter addresses the principle and performances of those sensors.
Chapter 3 introduces the concept of a novel mechanical amplification mechanism employing the buckled-beam suspensions, and provides the mechanical amplifier structure
analysis. The research efforts focus on revealing the structure geometrical effects on the
amplifier’s performances in terms of gain, linearity, and bandwidth. As a result, the optimal design can be obtained according to application requirements.
Chapter 4 investigates the mechanism of strain transferring from the specimen to the sensing structure - a critical necessity in sensor’s backing design to accurately and
efficiently transfer the strain from the specimen to the sensing structure. Based on the
analysis, a spring-folded sensor backing is designed to achieve the overall sensor system
performance.
Chapter 5 presents the device sensing element design considerations to fulfill the
specification requirements such as sensitivity, resolution, etc. Additionally, nonlinearity,
device fundamental noise sources including electrical thermal noise and mechanical
Brownian motion noise will be also addressed.
Chapter 6 presents the test results of the fabricated mechanically amplified capacitive
23 strain sensor and its associated test fixture development. The tests include: 1) capacitive
output characteristics; 2) the voltage output characteristics when integrated with the
developed low-noise capacitance-voltage (C/V) converter and the sensitivity evaluation; 3) the overall system’s behavior under elevated temperature; 4) the system’s step response and frequency response.
24 2. CHAPTER TWO
A REVIEW OF MICROMACHINED STRAIN GAGES
2.1. INTRODUCTION
Resistive (metallic-foil and piezoresistive gages), resonant, capacitive types are the
common types of strain gages based on micromachining technology. The principle and
performance of each type will be reviewed in this chapter.
2.2. METALLIC-FOIL STRAIN GAGE
A
Foil grid A’
Backing A – A’
Figure 2.1. Metallic-foil strain gage.
It was Lord Kelvin [5] who first reported in 1856 that metallic conductors subjected to
mechanical strain exhibit a change in their electrical resistance. The metallic-foil gage is
folded metallic wire or trace mounted on a supporting backing, as illustrated in Figure 2.1.
The gauge is attached to the object by a suitable adhesive such as Vishay® M-bond610.
As the object is deformed, the foil is deformed, inducing a mechanical strain, and resulting
25 in the electrical resistance change.
The metallic gage is made from thin strain-sensitive metallic foil patterned by firstly
printing with acid-resist ink and then followed by a chemical etching process [2].
Backing materials are needed to provide the mechanical support for easy handling and the
electrical insulation. The backing is usually made of a dielectric material such as
polyimide, epoxy, and glass fiber reinforced epoxy orientated for different application
requirements. For example, the glass-fiber-reinforced epoxy can operate at temperature
range from -269 D C to +290 D C , which is much wider than a typical polyimide backing
(-195 D C ~ 175 D C )[6]. With decades of engineering efforts, there are a variety of
combinations between the strain-sensing metallic alloys and backing materials readily
available in the market, which makes the metallic-foil strain gage currently the most commonly used strain gages on market.
2.2.1 PRINCIPLE
r rdr−
L dL L + dL = ε L
Figure 2.2. The resistance change of a cylindrical conductive wire under strain.
The metallic-foil strain gage is basically a resistive sensor with resistance changing
according to the applied strain. This principle can be described by analyzing the
26 resistance change of a cylindrical conductive wire with electric resistivity of ρ , length of L , and cross section area of A . The resistance R is a function given by,
L R = ρ . (2-1) A
The resistance change under strain ε is a combination effect of changes in length,
cross-section area, and resistivity,
ρ ρLL dR=− dL dA + dρ , (2-2) AA2 A
and,
dR dL dA dρ =−+dρ . (2-3) RLAρ
dA The cross-section area change, , can be expressed as, A
dA 2dr ==−2νε , (2-4) Ar
where ν is the Poisson’s ratio of the material.
Therefore, the resistance change can be expressed as:
dR dρ = ()1 + 2ν ε + . (2-5) R ρ
The ratio of resistance change described in (2-5) consists of two terms, ()1+ 2ν ε , and
dρ , respectively. The former term describes the resistance change due to the object ρ
geometrical changes, and the second term describes the resistance change due to the
27 piezoresistive effect, which describes the resistivity change of a material under mechanical stress.
dρ For metals, a fraction change in resistivity, , does occur with mechanical stress ρ
dρ through the piezoresistive effect. Mathematically, the fraction change, , may be ρ
dV related to the fraction change in volume that takes place with the longitudinal V mechanical stress given by:
dρ dV ⎛ dL dA⎞ dL = C = C⎜ + ⎟ = C ()()1 − 2ν = Cε 1 − 2ν , (2-6) ρ V ⎝ L A ⎠ L
where C is known as the Bridgman constant, ranging from 1.13 to 1.15 for most strain gage alloys [2].
Combining (2-5) and (2-6) gives,
dR =++⎡121CCν () −⎤ε . (2-7) R ⎣ ⎦
2.2.2 GAGE FACTOR AND SENSITIVITY
The gage factor, G , is used to evaluate the sensitivity of a strain gage. It is defined as the ratio between the gage resistance change and the applied strain,
ΔR / R G = . (2-8) Strain(ε )
Combining with (2-7), the gage factor, G, is given by,
G = 1+ C + 2ν (1− C) . (2-9)
28 The modern metallic-foil pattern is made from chemical etching, result in a rectangular cross section shape, which is claimed to have a 10% increment of gage factor compared to a round wire [2].
Obviously, the higher the gage factor, the more sensitive the gage and the greater the electrical output if other variables remaining the same. Table 2.1 lists the properties of the typical materials used for metallic-foil strain gage. A metallic-foil strain gage typically has gage factor around 2 to 5.
Table 2.1. Properties of strain gage conductors [2].
Common name Composition Gage factor Nichrome Ni-80%, Cr-20% +2.0 Manganin Ni-4%, Mn-12%, Cu-84% +0.47 Advance Ni-45%, Cu-55% +2.1 Constantan Ni-45%, Cu-55%* +2.4 Platinum Pt-100% +6.1 Platinum-Tungsten Pt-92%, W-8% +4.0 Karma Ni-74%, Cr-20%, Al-3%, Fe-3%) +2.0 *: Different cold working conditions.
2.2.3 INSTRUMENTATION
The resistance change of a strain gage is rather small (typically only a fraction of 1% over the full range), and usually measured using a Wheatstone bridge, which allows a small resistance change relative to the initial value to be measured. In many applications, a full-bridge configuration is used because it provides the highest sensitivity, but it requires four strain gages. In other applications, half and quarter bridge circuits can be used. A
quarter-Wheatstone bridge configuration is illustrated in Figure 2.3. R1 denotes the
resistance of the strain gage with nominal resistance of Ro . The ‘bridge’ consists of two
29 half bridges ABC and ADC , and the values of R2 , R3 , R4 are normally selected to be
Ro . The bridge has a terminal voltage of Vin either excited by a DC (or AC ) voltage
source or a current source. Taken a voltage excitation Vin for example, for a small
resistance change ΔR1 , the bridge output voltage Vout can be derived as [7],
Vin ΔR1 Vin Vout = = Gε (2-10) 4 R1 4
A
+ R1 R3
B V in V To instrumentation amplifier D out
- R2 R4
C
Figure 2.3. Wheatstone bridge configuration.
Thus the output voltage Vout is seen in (2-10) to be directly proportional to the excitation voltage. Greater excitation voltage produces greater sensitivity of the bridge output voltage to strain. However, it also increases the system power consumption and generates heat on the sensing element, which may result in thermal instability and even over-heating problems.
2.2.4 RESOLUTION OF A TYPICAL METALLIC-FOIL STRAIN GAGE
The resolution of a metallic-foil strain gage is determined by the summation of noise
Vn , contributed by the sensor VnS− and the input-referred interface electronics noise VnA− , which is,
30 22 VVnnSnA=+−− V. (2-11)
According to (2-10), the input referred strain at resolution level is simply determined by,
4Vn ε res = . (2-12) VinG
The sensor noise is mainly Johnson noise (thermal noise) induced by the thermal agitation of the charge carriers (the electrons) inside an electrical conductor in equilibrium, which happens regardless of any applied voltage [8], and theoretically given by,
Vthermal = 4K BTRB , (2-13)
where KB, T, R, B denote the Boltzman constant, absolute temperature, sensor nominal resistance, and frequency bandwidth, respectively.
For a metallic-foil strain gage with a nominal resistance of 120Ω , the calculated
Johnson noise power spectral density is 1.4 nV Hz , and the integrated noise voltage amplitude over frequency bandwidth from DC to 10 kHz is 0.14 μV .
The input referred noise power spectral density of an instrumentation amplifier varies, usually on the order of nV Hz . For example, the low-noise instrumentation operational amplifier AD8221 from Analog Device® has an input-referred noise of
8 nV Hz @1 kHz [9], and a total of 0.8 μV over DC to 10 kHz bandwidth. (The interface electronics 1 f noise is ignored by assuming employing advanced AC exciting scheme.)
Therefore, the overall input referred noise voltage amplitude calculated from (2-11) is
31 approximately 0.8 μV over DC to 10 kHz bandwidth, and the minimum detectable strain of a metallic-foil strain gage is calculated to be approximately 1.6 με for a 1V excitation voltage, according to (2-12). If, an ultra-low noise amplifier such as AD797 with input referred noise power spectral density typically around 0.9 nV Hz @1 kHz and a
50 nVpp− over 0.1 Hz to 10 Hz is used and the noise of overall sensor system is dominated by the sensor’s Johnson noise, the theoretically calculated sensor minimum detectable strain is 0.28 με and sensor itself consumes a exciting current of 8.3 mA.
Generally, a metallic-foil strain gage can achieve a resolution on the order of micro-strain, not good enough to meet this project’s requirements.
2.3. PIEZORESISTIVE STRAIN GAGE
2.3.1 PIEZORESISTIVE EFFECT
In 1953, Dr. Charles S. Smith published the paper on the measured piezoresistive coefficients in Germanium (Ge) and Silicon (Si) in 1954 [10]. The piezoresistive effect describes the electrical resistivity change of a material due to an applied mechanical stress.
The resistance change of a semiconductor material such as silicon still can be expressed by
dρ (2-5). However, the piezoresistive effect term is much greater than the geometrical ρ effect term ()1+ 2ν ε ; therefore, the resistance change of a semiconductor material is dominated by the stress-dependent resistivity change.
The general equation predicting the resistivity change under applied stress for Si, Ge
like cubic crystalline material with stress applied along major crystalline axis, is given by
32 [11],
dρ = π ⋅σ + π ⋅σ , (2-14) ρ t t l l
where, π l and π t denote the effective values for the longitudinal and transverse
piezoresistive coefficients, respectively; σ t and σ l are stress components in longitudinal and transverse directions. The definition of longitudinal and transverse stress components,
σ t and σ l , is illustrated in Figure 2.4. The longitudinal direction describes the change of the resistance if the stress is applied in the same geometrical direction as current flowing direction; while the transverse direction describes the change to a load applied perpendicular to current flowing direction.
Resistor Resistor Contacts Contacts
------
++++++++++++ ++++++++++++ Force
Electric field direction Electric field direction Force
Figure 2.4. Piezoresistor under a) longitudinal and b) transverse stress. The small red arrow on the resistor points in the direction of the exciting current.
2.3.2 LONGITUDINAL AND TRANSVERSE PIEZORESISTIVE COEFFICIENTS
longitudinal and transverse piezoresistive coefficients, π l and π t ,can be calculated
33 from the piezoresistive coefficient tensorπ ij [12],
6 dρi = ∑π ijσ j (2-15) ρ j=1
Because of the symmetry property of the cubic crystalline structure such as silicon, the
tensor π ij only has three independent terms: π11 , π12 , and π 44 . The resistivity change can be expressed in the following form,
⎡⎤Δρ11⎡⎤πππ11 12 12 ⎡⎤σ ⎢⎥Δρ ⎢⎥πππ 0 ⎢⎥σ ⎢⎥22⎢⎥12 11 12 ⎢⎥
1 ⎢⎥Δρ33⎢⎥πππ12 12 11 ⎢⎥σ ⎢⎥=⋅⎢⎥ ⎢⎥ ρ ⎢⎥Δρ41⎢⎥π 44 ⎢⎥τ ⎢⎥Δρ ⎢⎥0 π ⎢⎥τ ⎢⎥52⎢⎥44 ⎢⎥ Δρ π τ ⎣⎦⎢⎥63⎣⎦⎢⎥44 ⎣⎦⎢⎥ (2-16)
Table 2.2 lists the longitudinal and transverse piezoresistance coefficients for various combination of direction in cubic crystals. Table 2.3 lists the measured results that Smith
[10] published in 1954. Combining Table 2.2 and Table 2.3, π l and π t can be calculated numerically. The results show that, the maximum longitudinal piezoresistance
coefficient π l occurs in the <111> direction for p-type lightly doped silicon, or in the
<100> direction for n-type lightly doped silicon with values of 93.5×10-11Pa-1, and
-102.2×10-11Pa-1, respectively. The Gage factor of the piezoresistor for either cases can be expressed as,
dρ ΔR / R ρ π l ⋅σ l π l Elε l G = ≈ = = = π l El . (2-17) Strain(ε ) ε l ε l ε l
Combining with the measured single-crystal silicon material’s Young’s modulus, 166
34 GPa in <111> direction, and 130 GPa in <100> direction [13], the maximum achievable gage factors are 155 for p-type silicon, and 133 for n-type silicon, respectively. Obviously, these gage factors are much higher than those of metallic-foil gages, thus resulting in a much higher sensitivity.
Table 2.2. Longitudinal and transverse piezoresistance coefficients for various combination of direction in cubic crystals [11].
Longitudinal π l Transverse π t Direction Direction
(100) π11 (010) π12
(001) π11 (110) π12
(111) 1 3()π11 + 2π12 + 2π 44 (110) 1 3()π11 + 2π12 − 2π 44 (111) (110) 1 2(π11 + π12 + π 44 ) 1 3()π11 + 2π12 − 2π 44 (001) (110) 1 2(π11 + π12 + π 44 ) π12 (110) 1 2(π11 + π12 + π 44 ) (110) 1 2(π11 + π12 − π 44 )
Table 2.3. Adiabatic piezoresistance coefficients at room temperature [10].
Material Resistivity, π11 π 12 π 44 ρ (10-12 cm2/10-11Pa-1) (Ω-cm) Silicon (p-type) 7.8 +6.6 -1.1 +138.1 Silicon (n-type) 11.7 -102.2 +53.4 -13.6
2.3.3 IMPURITY CONCENTRATION (DOPING LEVEL) AND TEMPERATURE
EFFECT ON PIEZORESISTANCE COEFFICIENTS
Impurity concentration (doping level) and temperature are major factors affecting piezoresistance coefficients. O. N. Tufte studied the piezoresistance coefficients of single-crystal silicon with impurity concentrations from approximately 1×1015 to 1×1020 cm-3 and tested over various temperature range [14, 15].
35 Figure 2.5 shows the results of measured coefficient π11 variation in n-type layers with temperature and surface concentration. It shows that the coefficients decrease when
D -11 -1 doping increases. For example, tested at 300 K , π11 is -90 (10 Pa ) at doping level of
1.8×1016 cm−3 and decreased to -35 (10-11Pa-1) at doping level 2.1×1020 cm−3 . Figure 2.5 also shows the coefficients decrease with elevated temperatures, especially when the doping is light, indicating the piezoresistive devices are highly sensitive to temperature change, which results in a long-term sensitivity stability problem.
120 16 N =1.8×10 110 s
100
18 90 N =8.8×10 s 80 ) -1 70 Pa (
11 60
× 10 × 50 11
π 20 - N =2.1×10 40 s
30
21 N =1×10 20 s 10 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 o Temperature ( C)
Figure 2.5. Variation of the coefficient π 11 in n-type layers with temperature and surface concentration [15].
2.3.4 RESOLUTION AND ISSUES
Piezoresistive sensors have two main noise sources, 1/f noise and Johnson noise [16].
The origins of 1/f noise is caused mainly by conductance fluctuation associated with
36 contaminations and crystal defects in the active layer [8]. Its power spectral density has a
1/f frequency dependence; therefore it is most significant at low frequencies. The 1/f fluctuations of piezoresistive device element are shown to vary inversely with the total number of carriers in the piezoresisitors, as formulated by Hooge in 1969 [16]. Therefore, while 1/f noise is reduced for large heavily doped piezoresisitors, gage factor considerations favor lightly doped piezoresisitors. A balancing between these conflicting constraints requires optimal device parameters.
With the optimal design, a piezoresistive strain gage can achieve much higher sensitivity and the minimum detectable strain is approximately two orders of magnitude lower than a metallic-foil gage. However, as mentioned previously, the temperature and long-term sensitivity stability problem limits its usage in advanced industrial applications.
2.4. RESONANT SILICON MICROMACHINED STRAIN GAGE
In a resonant strain sensor, the measurand, strain, affects the behavior of the oscillation of a solid structure, e.g., the natural resonance frequency; thus, by measuring the frequency change, the strain information can be extracted. The use of resonant micromachined silicon structures is a well-known way to realize sensitive transducers due to the excellent mechanical properties of the single-crystal silicon such as very high intrinsic mechanical
Q-factor, good resistance to fatigue, and low thermal expansion coefficient [17]. The resonant micromachined strain gages have high sensitivity and temperature stability compared to the metallic-foil and piezoresistive gages. Moreover, the well established fabrication techniques allow cost-effective batch fabrication of sensors for mass markets such as the automotive and consumer electronics industry [18].
37 2.4.1 PRINCIPLE
The basic structure of micromachined resonant structure for strain sensing is a double-clamped vibrating beam. Let us consider a beam with length A , width b, thickness h << b (and b << A ), as illustrated in Figure 2.6.
ε
b A h
ε
Figure 2.6. A resonant beam subjected to a strain load, ε.
The fundamental resonance frequency fo is given by:
Eh f= 1.028 , (2-18) o ρ A2
and becomes,
2 Eh ⎛⎞A f=+ 1.0282 1 0.295⎜⎟ε , (2-19) ρ A ⎝⎠h
when a force induced strain ε is applied [19]. E , ρ , h , and A denote the material
Young’s modulus, mass density, beam thickness, and length beam, respectively. Eq.
(2-19) is only valid as the beam is unbuckled. The critical length at buckling is given by
[20]:
38 π h lc = (2-20) 3 − ε
2.4.2 EXCITATION AND DETECTION SCHEME
The resonant sensor requires components to excite and detect the mechanical vibration.
The complete sensor solution consists of four parts: the resonator, the excitation unit, the detection unit, and the feedback circuitry, as illustrated in Figure 2.7. An amplifying feedback loop circuitry makes sure that the resonator is maintained in the desired resonance mode, and also the resonance frequency is changing as a result of a change in the measured quantity [17]. This principle takes advantage of the bandpass characteristic of the resonant structure itself and requires relatively large Q-factors to obtain sufficient frequency stability [18].
Measurand
Output signal Excitation Detection Resonator Mechanism Mechanism
Amplifier
Figure 2.7. Resonator excitation and detection diagram.
During the last few years, many different excitation and detection mechanism for mechanical micro resonators based on silicon technology have been investigated. Six of them are frequently employed including: 1) Electrostatic excitation and capacitive detection [21-25]; 2) Dielectric excitation and detection [26]; 3) Piezoelectric excitation and detection [27]; 4) Resistive heating excitation and piezoresistive detection [28]; 5)
39 Optical heating and detection [29, 30]; 6) Magnetic excitation and detection [4, 31]. Use electrostatic excitation and capacitive detection as an example. The technique requires two electrodes arranged in close proximity, where one of the electrodes is a part of the vibrating structure. An alternating voltage across the electrodes results in an alternating electrostatic pulling effect on the resonator element, as illustrated in Figure 2.8. This is an effective vibration excitation technique especially in configurations where the resonator oscillates in vacuum, thus avoiding the damping of the air in the varying gap between the electrodes. This electrostatic excitation structure also constitutes a capacitor with varying electrode distance, which can be used to measure the vibration. The use of the capacitive excitation and detection principle requires comparatively large electrode areas and small electrode distances, about a few microns, in order to have reasonably low excitation and detection voltage levels.
Exciting voltage
Force
Resonant beam
Figure 2.8. Electrostatic excitation and capacitive detection.
2.4.3 RESOLUTION: MINIMUM DETECTABLE FREQUENCY
Neglecting the noise generated in the displacement sensor, in the oscillation control amplifier, and other interface electronics, the minimum detectable frequency shift Δω was estimated based on the noise generated by the thermal vibration of the beam [32],
40 ω k TB Δω = o B , (2-21) kQ A2
where B , Q , k , kB , and T denote bandwidth, mechanical quality factor, spring constant of the beam, Boltzman constant, and absolute temperature, respectively. A2 is the mean square amplitude of the self-resonating beam.
The high Q-factor is important to achieve a high resolution and a low unwanted mechanical coupling to the external world, a key feature leading to high accuracy and long-term stability [17]. The Q-factor can be defined as the total energy stored in the structure divided by the sum of energy losses from the vibrating element per cycle. The
Q-factor is dependent on damping. For most of the micromachined resonators, the internal-material-loss damping and the air damping are the two major damping sources.
Single-crystal material such as silicon has extremely low internal-material-loss damping, and material energy loss usually caused by the vibration coupling loss at the structure supports, which can be minimized by a balanced resonant design, e.g. a double ended tuning fork (DETF) [33]. To reduce the air damping, the device typically requires a vacuum package. If, however, the resonator is to be operated in air it is crucial to have a design of the electrodes with holes to reduce the “squeeze-film damping” [34].
The resonant strains sensor is highly sensitive. Test results [23, 24] demonstrated an equivalent gage factor of 2327, much higher sensitivity than even the piezoresistive strain gage. Recent research working [35] utilizing DETF resonator combining with a 4th order sigma delta phase lock loop achieved a 33 nano-strain (0.033 με ) resolution over 10 kHz bandwidth.
41 In addition, this structure has small gauge length, which allows measurements of small strain field with high accuracy. Its frequency output feature also allows a digital interface, provides low susceptibility to electrical interferences. It also offers great stability over temperature. However, it suffers from bucking problem and requires a sophisticated excitation and detection interface electronics and a vacuum packaging in most of the cases to achieve the competitive resolution performance.
2.5. CAPACITIVE STRAIN GAGE
2.5.1 PRINCIPLE AND CONFIGURATIONS
Aε C = e g
Change g
Strain ε Change A Vout
Change εe
Capacitive strain Interface electronics sensor module
Figure 2.9. The diagram of a capacitive strain sensor.
The basic diagram for a capacitive strain sensor is illustrated in Figure 2.9. For a capacitor, which is simplified as two parallel plates, the capacitance C is mathematically expressed by (neglecting the fringe effect of the plates),
Aε C = e , (2-22) d
where A, εe, d denote the overlapping area, medium electrical permittivity, and separation distance (gap).
42 In a capacitive strain sensor, the measurand, strain ε, induces a geometrical change or medium permittivity change; hence results in a proportional change in capacitance, which can then be converted to an electrical output signal through an interface circuit. K. I.
Arshak [36] developed a capacitive strain sensor based on PZT and PVDF dielectrics, whose permittivity exhibits a large change under an applied strain, and achieved a performance comparable to piezoresistive gages in terms of gage factor, nonlinearity, and hysteresis. However, for most of other cases, geometrical change is used for capacitive sensing applications. Figure 2.10 illustrates two basic configurations of capacitive strain sensing: lateral sensing (Figure 2.10.a) and transverse sensing (Figure 2.10.b). In the lateral sensing configuration, when undergone an applied strain, the fingers moves along the finger length direction, thus results in a change of overlapping area; whereas in the transverse sensing configuration, the fingers moves perpendicularly to the finger length, and results in a gap change. Generally, the lateral sensing configuration has less sensitivity, but can achieve better linearity and range; whereas transverse sensing configuration can achieve better sensitivity but poor in linearity.
43 Finger length, Lf Anchor A Gage length, Lg Anchor A Anchor B
Finger length, Lf
Gap, g0
Gap, g0
Anchor B Gage length, Lg ε Strain Direction ε Strain Direction (a): Lateral sensing (b): Transverse sensing
Figure 2.10. Capacitive strain sensor configuration.
2.5.2 PERFORMANCE OF CAPACITIVE STRAIN GAGE
A capacitive strain gage, along with other capacitive sensors, offers two major advantages: 1) high sensitivity; 2) excellent high temperature performance which provides a long term stability and feasibility in some extreme high temperature applications.
High sensitivity
Considering a capacitive strain sensor illustrated in Figure 2.11 (only one pair of comb drive finger is illustrate as an example), if the gage length Lg is 1 mm , the initial finger length Lf is 1 μm , and the applied strain ε is 1000 με , the relative capacitance change of the
ΔC L ε ΔC structure, = g = 1, which results in an equivalent gage factor ( ε ) of 1000, C L f C approximately one order of magnitude greater than the piezoresistive strain gages. J.
Aebersold [37] developed a MEMS capacitive strain sensor utilizing a transverse comb
44 drive for diagnosis of spinal fusion. The structure has 150 μm sidewalls with 25 μm gap.
At the unstrained state, the capacitive output was 7.56 pF and increased to 17.04 pF at
1571 micro-strain input bending strain, indicating an equivalent gage factor of 794.
Finger length, Lf
Gage length, Lg
Figure 2.11. High sensitivity of a capacitive strain sensor.
Excellent high temperature performance
The capacitance output of a capacitive strain gage given in (2-22) shows that the capacitance change is only determined by the geometrical dimension variation under applied strain (assuming air is used for gap material). Because of the material’s low linear thermal expansion coefficient (Si: 2.6×10-6ºC-1@300 D K ), the geometrical dimension change over required operating range (-40º - 150 D C ) is approximately 0.05%, which is negligible.
Single-crystal silicon is also a mechanically stable material over wide temperature range. David W. Schindel et al [38], studied the single-crystal silicon material Young’s modulus property at high temperature utilizing miniaturized resonant tuning forks and demonstrated that the Young’s modulus of silicon had only a 2.7% drop at 700 D C . Other research work [39] also concluded that the silicon was a mechanically and chemically
45 stable material for temperature up to 500 D C . Silicon material’s excellent high temperature performance offers silicon capacitive strain sensor excellent temperature stability. Therefore, long term sensitivity and offset stability can be expected for a capacitive strain gage.
Disadvantages
The major disadvantages of micromachined capacitive strain gages are: 1) Although a capacitive strain sensor has a high gage factor, the absolute capacitance change at resolution level is very small, typically on the order of atto-Farad ( aF ), due to the device’s miniature size. Therefore, a highly sensitive interface electronics is desired to achieve the required resolution. 2) A capacitive strain sensor contains movable structure, which is susceptible to contaminations such as dust particles; therefore properly designed encapsulation is required to provide sufficient protection which complicates the design and increases the cost; 3) The device also requires additional design concerns about the mechanical resonant frequency and modes, fatigue problems, shock problems because of the movable structures.
2.6. SUMMARY
Four types of micromachined strain gages including metallic-foil, piezoresistive, resonant, and capacitive are addressed in this chapter. The comparison is summarized in
Table 2.4.
46 Table 2.4. Summary of four types of micromachined strain gages.
Strain sensor Advantages Disadvantages technology Easy handling Low sensitivity Metallic-foil Many geometrics readily Large size available Small Long-term stability Piezoresistive High sensitivity Temperature stability Small High packaging cost High sensitivity Sophisticated electronics Resonant Temperature stability Buckling problem Long-term stability High sensitivity Size Capacitive Temperature stability Highly-sensitive Long-term stability interface electronics
47 3. CHAPTER THREE
CAPACITIVE STRAIN SENSOR EMPLOYING A BUCKLED BEAM
MECHANICAL AMPLIFIER
3.1. INTRODUCTION
Capacitive strain gage is highly sensitive and superb stable in terms of sensitivity and offset over large temperature range. However, a MEMS capacitive strain gage has small absolute capacitance change at resolution level due to its miniature size, thus requiring highly-sensitive interface electronics or enlarging device size to accommodate more sensing area, a key drawback limiting its usage in advanced industrial applications.
To overcome this disadvantage, a mechanical amplification mechanism employing a set of or sets of buckled beam suspensions is investigated. The mechanical amplifier based on the bending of a buckled beam suspension provides increased device sensitivity, thus attractive in reducing the sensitivity requirement and power dissipation in the interface circuits or the device area.
This chapter describes amplifier’s basic principle and presents the structure analysis of the buckled beam suspension which is employed in the mechanical amplifier. Efforts focus on revealing the relationship between the mechanical gain and structure geometrical parameters, as well as the device non-linear characteristics. The analytical model was firstly built and analyzed using the simple beam theory. The FEA method is then employed to verify the analytical results. Finally, based on the analysis, a capacitive strain
48 sensor is designed and fabricated using a single-mask micro-fabrication process. The device consists of four parallel MEMS capacitive strain sensors with a nominal capacitance value of 440 fF , converting an input strain to a capacitance change with a sensitivity of 281 aF per micro-strain ( με ). Interfaced with a low-noise integrated sensing electronics [40], which employs a continuous time synchronous detection architecture converting the capacitive signal to an output voltage for further signal process, the sensor module achieves a strain resolution of 0.0009 με Hz and consumes 1.5 mA from a 3V supply.
3.2. MECHANICAL AMPLIFIER
The mechanical amplified capacitive strain gage diagram is illustrated in Figure 3.1.
The capacitive strain sensor is basically a displacement sensor. An applied strain,ε ,
induces a displacement change, Lgε , which is amplified by a mechanical amplifier; the amplified displacement altered the capacitor overlapping area, which results in a linear capacitance change. An interface circuit is then utilized to detect the capacitance change and convert it to electrical voltage output.
49 Lgε AmLgε Capacitive ΔC Strain, ε Gage length Am Sensor Vout
Mechanical Gain
Mechanically amplified capacitive strain Interface electronics sensor module
Figure 3.1. The diagram of a capacitive strain sensor.
3.2.1 BACKGROUND
Many mechanical amplifiers have been developed to accurately measure the residual strain of micromachined thin films by amplifying the small strain displacement to a level which can be identified by naked eyes under an optical microscope. Y. B. Gianchandani et al [41] developed a bent-beam amplification mechanism utilizing a pair of narrow bent beams with an apex at their middle points. The narrow beams amplify and transform deformations caused by residual strain into opposing displacements of the apices, where vernier gauges are positioned to quantify the deformation. A resolution of 60 με is achieved by a pair of beams with 2 μm in width, 200 μm in length, and a bent angle of
2.86°. L. Lin et al [42] developed a mechanical amplification mechanism consisting of three beams, a test beam, a slope beam, and an indicator beam. The small displacement caused by the residual strain induces an angle at the slope beam, and results in a large displacement at the end of the indicator beam as the indicator beam is much longer than the slope beam. A resolution of 10 με has been achieved for a 500 μm long indicator beam.
However, no mechanical amplification mechanism has been employed in stand alone strain sensors. It is thus of interest to develop a capacitive strain sensor employing the
50 mechanical amplification mechanism, so that the device can overcome the previously described disadvantages: size and capacitance output to interface circuit.
3.2.2 PRINCIPLE
The principle of the mechanical amplification mechanism employed in this study is illustrated in Figure 3.2. The structure consists of two buckled beams with a small buckling angle, α , and one sensing beam at center. When a strain, ε, is applied, it
causes a lateral displacement, Δx = Lgε , where Lg is the gauge length measured by the distance between anchors A and B. For a small buckling angle, the center deflection of the sensing beam, Δw, is greater than Δx , thus resulting in a mechanical amplification
gain, Awxmech = ΔΔ. A linear differential capacitive output can be obtained by designing comb drive fingers at the center sensing beam, as shown in Figure 3.3. Figure
3.3 illustrates the concept of a buckled beam strain sensor: when under a compressive strain, the top and bottom beams will move up and the center beam will move down,
+ − causing an increased Cs and a decreased Cs , thus resulting in a differential capacitance change, which is proportional to the displacement, Δx .
51 Lg
L Sensing Beam L s Δw
Anchor A α Anchor B
Δx Buckled Beams
Figure 3.2. Principle of the buckled beam amplification scheme.
Buckled beam suspensions Comb drive sensing Δw fingers
− Cs C Δw com + Cs
A Δw B Δx Buckling direction
Figure 3.3. A buckled beam strain sensor with differential output.
3.3. ANALYTICAL MODELING
The analytical modeling procedure is described as follows. 1) The beam curvature function under an applied force is derived by employing the beam governing equations and boundary conditions according to the simple beam theory. 2) The x-direction
52 displacement, Δx , and y-direction displacement, Δy , are obtained based on the derived beam curvature function and the assumption that the overall length of the buckled beam remains constant according to the simple beam theory. The mechanical gain is then calculated by ΔΔyx. 3) Beam equivalent stiffness, an important mechanical property, is also obtained using the similar manner.
3.3.1 DERIVATION OF THE CURVATURE FUNCTION UNDER AN APPLIED
FORCE
The buckled beam structure analysis employs the simple beam theory [43]. The derivations in this section are based on the following assumptions and facts: (1) The beam made of single-crystal silicon is of homogeneous material that has the same modulus of elasticity in tension and compression. (2) The beam is straight or nearly so. (3) The beam is long in proportion to its depth. (4) The beam is not disproportionately wide. (5)
The maximum stress does not exceed the proportional limit.
Based on the assumption, the beam has the following behaviors: (1) As the beam bends, fibers on the convex side lengthen, and fibers on the concave size shorten. The neutral surface is normal to the plane of the loads and contains the centroids of all sections; hence the neutral axis of any section is the longitudinal central axis. 2) Plane sections remain plane, and hence unit fiber strains and stresses are proportional to distance from the neutral surface. 3) Longitudinal displacements of points on the neutral surface are negligible [20].
53 Δw Guided left end Buckled beams Fixed Right End F Δx
(a)
Deformed beam shape Arbitrary Location (x,y) Original beam shape M
y −F h M o α F x L
(b)
Figure 3.4. (a) Simplified structure. (b) Forces acting on the beam.
The structure is simplified as a beam structure constrained as shown in Figure 3.4.a with the right end fixed and the left end guided along the x-axis. Because of the symmetrical behavior, only the left half of the bean is under analysis. The forces acting on the left half buckled beam are shown in Figure 3.4.b, where F is the equivalent force
applied by the external strain, M o is the moment from the guild end, M is the moment in the beam at an arbitrary location (x, y), and the governing equation is given by [43] according to the simply beam theory and behavior,
d 2 y EI = M = M − Fy, (3-1) dx 2 o
where E, I denote the material Young’s modulus and the moment of inertia, respectively.
54 D = EI , u , and k are introduced for convenience, where
F L2 = u 2 , (3-2) EI
F u 2 = = k 2 . (3-3) EI L2
The general solution for equation (3-1), if F < 0 (tensile), can be expressed as,
ux ux y = C sinh + C cosh + C . (3-4) 1 L 2 L 3
The boundary conditions are:
y x=0 = 0 , (3-5)
dy = tanα , (3-6) dx x=0
dy = tanα . (3-7) dx x=L
C1 , C2 , and M can be solved as,
L C = tanα . (3-8) 1 u
L ML2 C = − tanα tanh(u ) = − . (3-9) 2 u 2 u2EI
C3 = −C2 (3-10)
u M = D tanα tanh(u ) . (3-11) L 2
55 Therefore, y can be solved as,
tanα kL ytens = [tanh( )()()1− cosh()kx + sinh kx ]. (tensile) (3-12) k 2
yapex represents the y coordinate at the joint location,
tanα kL yapex _ tens = 2 tanh( ). (tensile) (3-13) k 2
Similarly, for an applied compressive force (F>0), the solutions are:
tanα kL ycomp = []tan( )()()1− cos()kx + sin kx . (compressive) (3-14) k 2
tanα kL yapex _ comp = 2 tan( ). (compressive) (3-15) k 2
3.3.2 CALCULATE THE MECHANICAL GAIN
Equation set (3-12, 3-14) represents the curvature function of the beam under an
applied force. According to the simple beam theory, the length of the curved beam
remains the same under an applied force. Therefore, the x-direction displacement,
Δx , can be calculated as the difference of the beam projections on x-axis with and
without the applied force; and the y-direction displacement, Δy , can be calculated as,
yapex − h , where yapex represents the y coordinate at the joint location, which are
given in (3-13) and (3-15). The mechanical gain is then obtained by calculation of
ΔΔyx.
56 Deformed beam under an applied force
Beam without an applied force dl dy dx y h α x Δx
Figure 3.5. Beam modeling for calculating Δx.
The length of the beam can be calculated as,
2 2 2 L L ⎛ dy ⎞ L ⎡ 1 ⎛ dy ⎞ ⎤ L 1 ⎛ dy ⎞ l = dl = 1+ ⎜ ⎟ dx ≅ ⎢1+ ⎜ ⎟ ⎥dx = L + ⎜ ⎟ dx . (3-16) ∫0 ∫0 ∫0 ∫0 ⎝ dx ⎠ ⎣⎢ 2 ⎝ dx ⎠ ⎦⎥ 2 ⎝ dx ⎠
The length of the beam corresponding to the condition of F = 0 is express as,
2 L 1 ⎛ dy ⎞ 1 2 lo = L + ⎜ ⎟ dx = L +L tan α . (3-17) ∫0 2 ⎝ dx ⎠ 2
Therefore,
2 2 2 2 Δx ≅ l − h − l0 − h . (3-18)
Using Taylor expansion and the first-order approximation, assuming h< 2 L 1 ⎛ dy ⎞ 1 2 Δx ≅ l − lo = ⎜ ⎟ dx −L tan α . (3-19) ∫0 2 ⎝ dx ⎠ 2 dy According to (3-12), can be expressed as (for tensile force input), dx 57 dy tanα ' = [tanh(kL )()()1− cosh kx + sinh kx ] dx k 2 (3-20) = tanα − tanh kL sinh kx + cosh kx []()2 () Therefore, 2 ⎡tanh 2 kL +1 tanh 2 kL −1 ⎤ ⎛ dy ⎞ 2 ( 2) ( ( ) ) . (3-21) ⎜ ⎟ = tanα ⎢ cosh()2kx − tanh()kL sinh(2kx) − 2 ⎥ ⎝ dx ⎠ ⎢ 2 2 2 ⎥ ⎣ ⎦ 2 ⎛ dy ⎞ The integration of ⎜ ⎟ can then be expressed as (for tensile force input), ⎝ dx ⎠ 2 ⎡ 2 kL kL ⎤ l ⎛ dy ⎞ tanh ( ) sinh()2kL + 2kL tanh( ) ⎜ ⎟ dx = tan 2 α ⎢ 2 ()sinh()2kL − 2kL + − 2 ()cosh()2kL −1 ⎥ , (3-22) ∫0 ⎝ dx ⎠ ⎢ 4k 4k 2k ⎥ ⎣ ⎦ 2 kL kL 1 ⎡tanh () sinh()2kL − 2kL tanh( ) ⎤ Δx = tan 2 α ⎢ 2 ()sinh()2kL − 2kL + − 2 ()cosh()2kL −1 ⎥ . (3-23) tens 2 ⎢ 4k 4k 2k ⎥ ⎣ ⎦ Similarly, when F is compressive, 2 kL kL 1 ⎡tan () sin()2kL − 2kL tan( ) ⎤ Δx = tan 2 α ⎢ 2 ()2kL − sin()2kL + − 2 ()cos()2kL −1 ⎥ . (3-24) comp 2 ⎢ 4k 4k 2k ⎥ ⎣ ⎦ Δy is calculated as, 2tanα kL Δy = yapex − h = tanh( )− h . (3-25) k 2 Combining (3-23)-(3-25), we obtain mechanical gain for an applied tensile force, tanα 2 tanh kL − h Δy ( 2) ; (3-26) A = ≅ k mech_ tens 2 Δx 2 ⎡tanh ()kL/ 2 sinh()2kL − 2kL tanh()kL/ 2 ⎤ tan α⎢ ()sinh()2kL − 2kL + − ()cosh()2kL −1 ⎥ ⎣ 4k 4k 2k ⎦ 58 And for an applied compressive force, we obtain, tanα 2 tan kL − h Δy ( 2) . (3-27) A = ≅ k mech_ comp 2 Δx 2 ⎡tan ()kL/ 2 sin()2kL − 2kL tan()kL/ 2 ⎤ tan α ⎢ ()− sin()2kL + 2kL + − ()cos()2kL −1 ⎥ ⎣ 4k 4k 2k ⎦ Table 3.1 lists some the properties of hyperbolic cosine and sine functions used in the derivation. Table 3.1. Properties of hyperbolic cosine and sine function e x + e−x e x − e −x cosh(x) = sinh(x) = 2 2 cosh(−x) = cosh(x) sinh(−x) = sinh(x) cosh(x) + sinh(x) = e x cosh(x) − sinh(x) = e −x cosh 2 (x) − sinh 2 (x) = 1 cosh(2x) = 2cosh 2 (x) −1 = cosh 2 (x) + sinh 2 (x) = 1+ 2sinh 2 (x) 3.3.3 SMALL INPUT FORCE SIMPLIFICATION AND THE STRUCTURE NOMINAL MECHANICAL GAIN When the applied force is small, the above equations can be simplified based on the following approximations: x3 x3 1 sinh(x) − x ≅ ,tanh(x) − x = −2 ,cosh(x) −1 = x2 , when x is small. 6 6 2 (3-22) can be simplified as, 2 l ⎛ dy ⎞ ⎛ sinh(2kL)+ 2kL 1 ⎞ ⎜ ⎟ dx = tan 2 α⎜ − k 2 L3 ⎟ (3-28) ∫0 ⎝ dx ⎠ ⎝ 4k 2 ⎠ 59 Therefore, (3-23) can then be reorganized as, 1 2 ⎛ sinh(2kL) − 2kL 1 2 3 ⎞ Δxtens ≅ tan α⎜ − k L ⎟ . (3-29) 2 ⎝ 4k 2 ⎠ The mechanical gain corresponding to a small tensile input force or strain can then be calculated as, 2tanα ⎡⎤tanh kL− kL Δy k ⎣⎦( 22) 1 Amech == ≅ . (3-30) Δαx 11223⎡⎤⎛⎞sinh() 2kL− 2kL 2tan tanα ⎢⎥⎜⎟− k L 24k2⎣⎦⎢⎥⎝⎠ Similarly, for compressive small force or strain, the same result about mechanical gain expressed in Eq. (3-30) can be derived. Eq. (3-30) is defined as the nominal mechanical gain of the structure. 3.3.4 LARGE DEFLECTION COMPENSATION Assuming the buckling angle α is small, a modified equation can be derived by replacing α with α + Δα and employing the Taylor expansion, where Δα equals to 1 Δx , representing the buckling angle change with an applied force or displacement. 2tanα L 1 ⎡ 1 Δx⎤ A* = 1 − . (3-31) mech ⎢ 2 ⎥ 2tanα ⎣ 4tan α L ⎦ 3.3.5 CALCULATE THE BUCKLING BEAM EQUIVALENT STIFFNESS IN X-DIRECTION The structure stiffness in x-direction can be calculated using equation, 60 1 3 2 E tw 3 F F k EI 12 Et ⎛ w ⎞ , (3-32) k x = = = = = 2 ⎜ ⎟ Δx 1 2 ⎛ sinh()2kL − 2kL 1 2 3 ⎞ 1 2 2 3 1 2 3 tan α ⎝ L ⎠ tan α⎜ − k L ⎟ tan αk L tan αL 2 ⎝ 4k 2 ⎠ 12 12 where, kx, E, h, w, L denote the structure equivalent stiffness in x-direction, material Young’s modulus, thickness of the beam (in direction perpendicular to the paper surface), the beam width, and beam length, respectively. 3.4. FINITE-ELEMENT-ANALYSIS VERIFICATION Finite-element-Analysis using ANSYS software package has been employed to verify the validation of the analytical results. 3.4.1 FEA VERIFICATION OF NOMINAL MECHANICAL Structures of different geometric dimensions are studied using FEA analysis. The buckling beam suspension length, L , is fixed to be 300 μm and height, h , varies from 15 μm to 60 μm with calculated nominal mechanical gains of 10, 5, and 2.5 according to (3-30), respectively. In FEA analysis, a small displacement load of 0.0001 μm in x direction is applied, serving as the small input signal Δx . Δy is obtained by getting the displacement reading at the element located at the beam joint position in post data processing. The mechanical gain is calculated by ΔyxΔ . Different beam widths (1 μm , 3 μm , and 6 μm ) are used to examine the beam width effect on modeling accuracy. Table 3.2 lists the comparison of results obtained analytically and by FEA, corresponding to different geometrical structures. It shows the analytical results match closely with the FEA results for small beam width, e.g., beam width less than 3 μm . 61 With greater beam width, the buckled beam behavior does not exactly follow the simple beam theory, resulting in an increased discrepancy. For instance, if hm= 15μ and wm= 6μ , the discrepancy between the two analysis methods is 14%. Never the less, the analytical analysis results of the nominal mechanical gain give sufficient accuracy in most cases. Table 3.2. Comparison of nominal mechanical gains obtained analytically and from FEA. L ( μm ) h ( μm ) Beam width Analytical ANSYS Ratio w ( μm ) results 1 10 9.95 0.99 15 3 10 9.60 0.96 6 10 8.62 0.86 1 5.0 4.99 1.00 300 30 3 5.0 4.95 0.99 6 5.0 4.80 0.96 1 2.5 2.50 1.00 60 3 2.5 2.49 0.99 6 2.5 2.48 0.99 3.4.2 FEA VERIFICATION OF EQUIVALENT STIFFNESS Similar FEA analysis is used to obtain the structure stiffness in x-direction. In FEA analysis, an x-directional force F.= 0 0001μ Nis applied on the structure’s left guided end. The structure is analyzed and the left guided end displacement, Δx , is obtained, and the stiffness is calculated as, F Δx . Table 3.3 lists the results obtained analytically according to (3-32) and by FEA, corresponding to different geometrical structures. Similarly, it shows the analytical results match closely with the FEA results with discrepancy less than 5% when the buckled beam width is small. The discrepancy increases with greater beam width. 62 Table 3.3. Comparison of structure stiffness obtained analytically and from FEA. L ( μm ) h ( μm ) Beam width Analytical ANSYS Ratio w ( μm ) results ( Nm) ( Nm) 15 3 520 501 0.96 6 4160 3607 0.86 30 3 130 129 0.99 6 1040 1010 0.97 300 3 32.5 31.8 0.98 60 6 260 252 0.97 3.4.3 FEA VERIFICATION OF MECHANICAL GAIN CORRESPONDING TO LARGE DISPLACEMENT INPUT The FEA analysis of the structure follows the same procedure discussed in section 3.4.1. The structure has geometrical parameters of L=300 μm , h=30 μm . Two beam width, 1 μm and 3 μm , are used in simulation to examine the beam width effect. The material has Young’s modulus of 130GPa with Poisson’s ratio of 0.3, corresponding to the properties of single-crystal silicon material in <100> direction. The structure is applied with a maximum load equivalent to ± 2000 με and analyzed with large displacement modeling method which enables the nonlinear structure analysis to achieve more accurate results when the structure is under large load. Figure 3.6 shows the comparison of mechanical gain obtained analytically by (3-30), (3-31), (3-26), (3-27), and by FEA. Eq. (3-30) gives the function of the structure’s nominal gain and provides the first order approximation of the mechanical gain, which is represented by the flat solid line in the figure, indicating that it is constant with various applied load. Using FEA results as 63 reference, the maximum discrepancies between them at ±2000με are 16% and 10% for tensile and compressive load, respectively. Eq. (3-31) has considered the angle changing effect and shows the mechanical gain varies with the applied load. For instance, with an applied tensile (positive) strain, the buckling angle decreases with the load, resulting in an increased gain and represented by a tilted up solid straight line in the tensile strain region. Similar manner happens for the compressive load. As (3-31) provides the second order approximation of the mechanical gain, thus resulting in higher accuracy than (3-30). Using FEA results as reference, the maximum discrepancies between them at ±2000με are 5% and 1% for tensile and compressive load, respectively. Shown in Figure 3.6, the FEA results with structure of w=1 μm match perfectly with the analytical results obtained by (3-26) and (3-27), indicating that equation set (3-26) and (3-27) provides the most accurate results if using the FEA results as reference. The FEA simulation results also show that the beam width does affect the mechanical gain. Greater beam width decreases the mechanical gain. However, with typical device geometrical dimensions such as L is several hundreds of µm, w is several micros, and the mechanical gain less than 10, the beam width effect is negligible. For instance, with L=300 μm and the nominal mechanical gain of 5, the structure with buckled beam width of 1 μm demonstrates a mechanical gain of 4.99 with small input strain around zero. When beam width increases to 3 μm , the simulated mechanical gain decreases to 4.95, a 1% gain drop in amplitude, which is rather small and negligible in this study. It is interesting to mention that the gain variation is asymmetric with the strain sign. 64 Compressive strain tends to have smaller gain error compared to the nominal gain. For instance, for a -2000 με (compressive) input strain, the simulated mechanical gain is 4.5, a 10% gain drop (gain error) in amplitude. However, the simulated mechanical gain for a +2000 με (tensile) input is 5.8, indicating a +16% gain error. In other words, the tensile strain tends to have greater nonlinearity. 6.0 Analytical results of (3-26) and (3-27) 5.8 Analytical results of (3-31) 5.6 FEA (w=1µm) results 5.4 FEA (w=3µm) results 5.2 Analytical results of (3-30) 5.0 4.8 Mechanical gain gain Mechanical 4.6 Compressive Strain Tensile Strain 4.4 -2000 -1600 -1200 -800 -400 0 400 800 1200 1600 2000 Applied strain (micro-strain) Figure 3.6. Comparison of mechanical gain obtained analytically and by FEA with l=300 μm , w=1 μm , and h=30 μm with the maximum applied strain of ± 2000 με . 3.5. NONLINEARITY OF THE BUCKLED BEAM AMPLIFIER It is obvious that an improved sensitivity can be achieved by employing greater mechanical gain, or reducing the buckling angle, according to (3-30), where mechanical 65 gain is reversely proportional to tangent of buckling angle, α . However, with large applied strains, the change of the buckling angle induced by the beam deformation must be taken into account; therefore, a modified mechanical gain for large deflection, (3-31), has to be used. In other words, the mechanical gain is rather a varying parameter depending on the input strain, thus resulting in a nonlinear characteristic. The 1 nonlinearity can be estimated by the term ε in (3-31), where ε represents the 4 tan 2 α applied strain on the sensing structure. Higher mechanical gain favors sensitivity, which, on the other hand, has larger nonlinearity. Therefore, there is obviously a tradeoff between the sensitivity (mechanical gain) and the linearity. The optimal device parameters are a matter of application requirements. For example, for applications focusing on resolution, a higher mechanical gain is favored and more sophisticated calibration procedure is required. On the other hand, a smaller mechanical gain has more linear characteristic, requires simpler calibration procedure, and is more suitable for cost-sensitive applications. However, in all cases, it is of interest to know the quantitative inter-relationship between the nonlinearity, mechanical gain, and the operating strain range. Using FEA tool, the buckled beam suspension structures has buckle length L= 500μ m , w6m= μ with various buckling height, h , are under investigation. The nominal mechanical gains vary from 3 to 7 and the maximum strains range from 750 με to 2250 με . The nonlinearity is defined as deviation from a best fit straight line at full scale. Figure 3.7 illustrates the Finite Element Analysis results of relationship between 66 nonlinearity, mechanical gain, and the maximum strain. It shows that higher mechanical gain tends to have greater nonlinearity and it is also true with greater applied strain. In this study, the device is designed to have mechanical gain of 5 and an expected full-scale (2000 με ) nonlinearity around 7%. 16.0% Gain=7 14.0% Gain=6 12.0% 10.0% Gain=5 8.0% Nonlinearity 6.0% Gain=4 4.0% Gain=3 2.0% 0.0% 500 1000 1500 2000 2500 Maximum applied strain (micro) Figure 3.7. Finite Element Analysis result of relationship between nonlinearity, mechanical gain, and the maximum strain. 3.6. FUNDAMENTAL NOISE ANALYSIS Resistive thermal noise and structure Brownian noise are the two major noise sources. Resistive thermal noise is introduced by the device series resistance; whereas structure 67 Brownian noise comes from the molecular agitation. Here, the 1f noise of semi-conductive capacitor is neglected because it can be reduced for large heavily doped structure which has large carrier number in the device [16]. 3.6.1 RESISTIVE THERMAL NOISE ESTIMATION R1 Cx-p + Cs R2 Ccom - Cs R3 Cx-n Cp1 Cp0 Cp2 Csub Figure 3.8. Device electrical model. Figure 3.8 shows the device electrical model. The device has three sensing terminals named Cx− p , Ccom , and Cx−n , constructing a differential capacitance output. Terminal Csub represents the device substrate and is usually connected to the circuit + − + ground. CS and CS represent the device’s capacitive outputs, where CS increases − with an applied tensile strain and CS decreases with an applied tensile strain. The + − nominal capacitance of CS and CS are designed to be around 0.5 pF. R1 , R2 , R3 are the series resistances of the sensing capacitors. Cp0 , Cp1 , Cp2 are the parasitic 68 capacitance between the sensing terminals and the substrate with typical values around 3pF each, mainly contributed by the anchors and bonding pads. CI 2 2 Vn-s Vn-i ~ ~ V + s Cs To next stage 0 - Cs CI Figure 3.9. The diagram of the front amplifier architecture of the interface electronics. The analysis of the noise contribution of the series resistors to the overall sensor module with the integrated interface electronics can be explained using Figure 3.9, which illustrates the architecture diagram of the differential charge amplifier of a proposed continuous time synchronous detection capacitance to voltage converter [40]. The + − MEMS sensors, modeled as differential capacitors, CS and CS , are driven by a clock signal with an amplitude of Vs and are interfaced by a differential charge amplifier, with integrating capacitor CI , which converts the change of differential sensor capacitance to 2 an output voltage. Vn−i Δf represents the amplifier input referred voltage noise power spectral density, where Δf is the signal bandwidth of interest. The amplifier has a noise voltage of 20μ V over bandwidth DC− 10kHz if referred to the output of the front-end amplifier [40]. 69 2 Vn−s Δf represents the sensor resistive thermal noise power spectral density, which is determined by, 2 Vn−s Δf = 4K BT( R1 + R3 ), (3-33) where KB is the Boltzman constant and T is the absolute temperature. The noise induced by R2 is cancelled out because of the differential architecture. Therefore the noise voltage of the sensor referred to the output of the amplifier is expressed as, Cs Vo,n−s = 4K BT( R1 + R3 )Δf . (3-34) CI The typical values for R1 and R2 are less than 1000Ω ; CS and CI are of 0.5 pF and 1.6 pF , respectively. The calculated resistive thermal noise voltage referred to the amplifier’s output over DC-10kHz bandwidth is 0.18μ V , indicating it is negligible when compared with the previously calculated interface electronics noise of 20μ V . 3.6.2 MECHANICAL THERMAL NOISE (BROWNIAN MOTION NOISE) ESTIMATION Mechanical-thermal noise, commonly referred to as Brownian motion noise, can be the limiting noise component for a MEMS device, as the device geometrical dimension getting smaller and smaller, at normal pressure or in liquids, the small moving parts are especially susceptible to mechanical noise resulting from molecular agitation and random collision of air molecules in the ambient with the suspended structures [44]. Therefore, mechanical thermal noise is a critical consideration for the sensor design in order to 70 achieve the overall stringent sensor performance requirements. This mechanical Brownian noise referred to displacement fluctuation can be expressed as [44], 4K Tω m 4K T Δx =⋅=⋅Bo BWB BW (3-35) 2 2 ω 3mQ ()ωres mQ res where KB is the Boltzman constant; T is the absolute temperature with unit of Kelvin; ωres is the structure resonant frequency; m is the mass; Q is the mechanical quality factor; BW is the required signal bandwidth. Therefore, the input strain referred mechanical thermal noise power spectral density of each suspended structure for frequencies below the mechanical resonant frequency of the structure can be expressed as, 4KB T εΔnn==KxK n 3 (3-36) ωres mQ where εn is input strain referred mechanical thermal noise power spectral density; Kn is the conversion factor between the noise displacement and the input-referred strain, which is a function of mechanical gain and gauge length, and equals to 200 for the designed sensor. Each device component (top, center, and bottom suspensions) contributes non-correlated mechanical noise. However, the Brownian motion noise of the center suspension is cancelled due to the differential mode. Therefore the mechanical noise is 71 the summation of the noises from top and bottom suspensions. Finite element simulations have been performed to examine the device resonant behavior. The frequency and shape of the resonant modes will determine the noise contribution in the signal band. Figure 3.10.a shows the top suspension primary resonant mode at 103 kHz, which indicates that the beam oscillates with a tilt angle with constant averaged capacitance to the first-order approximation. Therefore, this mode does not contribute significant noise. However, the second resonant behavior at 139 kHz, as shown in Figure 3.10.b, will become the dominant mechanical thermal noise source as the structure oscillates in a symmetric configuration with respect to a reference. Other higher frequency modes will contribute negligible amounts of noise. f = 139 KHz fres = 103 KHz res (a) (b) Figure 3.10. Device resonant behavior simulation. The designed device structure exhibits a proof mass of 650 ng for each suspended structure. The resulting mechanical thermal noise power spectral density is estimated in air, with a Q of unity (typical for MEMS structures operated in air [45]), to be 2 x 10-6 Å Hz along the vertical axis, which corresponds to an equivalent input strain noise 72 power spectral density of 3 x 10-7 με Hz , thus negligible compared to the minimum design requirement of 0.001 με Hz . Therefore, the sensor can be operated in ambient without requiring a vacuum packaging, thus substantially reducing the system packaging complexity and cost. 3.7. THE MAXIMUM STRESS OF THE STRUCTURE AND LOCATIONS The maximum stress of the structure should be far below the fracture stress referring to the pre-defined limit to which the device can be stressed. Investigations in [46, 47] show that the surface or edge roughness greatly affects the fracture stress and for structures such as cantilever beams obtained by RIE and wet etching, the fracture stress of silicon micro cantilevers ranges from 1.1GPa to 3.4GPa dependent on the load direction. FEA results show that for the mechanical amplifier structure, the maximums tress happens at the locations close to the anchor, as shown in the Figure 3.11. The maximum stress is less than 100MPa if the structure is under 1000με , far below the silicon fracture stress. It is therefore concluded that the device is safe to use under the maximum strain 1000 µε. 73 Maximum stress Stress (Unit: MPa) Figure 3.11. FEA analysis of the structure under the maximum strain indicates that the location of maximum stress is near the anchor. 74 3.8. DEVICE FABRICATION Device layer (Si) Silicon oxide Substrate (Si) (a) Starting with SOI wafer. Comb finger gap Photoresist Device layer (Si) Silicon oxide Substrate (Si) (b) Photoresist patterning and deep-reactive-ion-etch (DRIE). Comb finger gap Device layer (Si) Silicon oxide Substrate (Si) (c) Dicing and wafer cleaning up. Comb finger gap Undercut Device layer (Si) Silicon oxide Substrate (Si) (d) Time controlled BHF release. Metal (Al) Comb finger gap Undercut Device layer (Si) Silicon oxide Substrate (Si) (e) Metal (Al) sputtering. Figure 3.12. Major fabrication steps. A single mask fabrication process, as shown in Figure 3.12, was developed to fabricate the device. The process started with a silicon-on-insulator (SOI) wafer with 75 device/oxide/handle layer of 20/2/500 μm (Figure 3.12.a), respectively. The device layer has resistivity of 0.5-10 ohm− cm with a measured sheet resistance around 3 kΩ per square. The wafer was firstly patterned using photoresist Shipley 1813®, and etched using a deep-reactive-ion-etch (DRIE) process to generate the device structure such as the comb drive sensing finger (Figure 3.12.b). The oxide layer of the SOI wafer serves as the etch stop layer. The device has minimum feature size of 3 μm and the DRIE etching aspect ratio is around 7:1. The wafer device side is then coated with a 2- μm thick photoresist for protection and diced using SiC saw, followed by a cleaning processing using piranha bath (H2SO4:H2O2 4:1) to clean up the photoresist residuals and contaminations from dicing process (Figure 3.12.c). The devices are released in BHF (1:7) solution for approximately 30 minutes (Figure 3.12.d). Finally, a layer of 200 Å-thick aluminum was sputtered on the device to reduce the device series resistance to be around 300 Ω (Figure 3.12.e). Fabrication outstanding issues: 1. Undercut problem in releasing process The undercut etching rates observed in device release process varied from wafer to wafer. The undercut measured from the edge of the pattern to lines where the undercut stops sometimes exceeded 30 micros to release device structure with 2 µm-thick oxide underneath. Therefore, enough anchor areas are required to avoid device being peeled off after the release. It is believed the undercut etch rate is related to the bonding strength between the device layer and the oxide layer. Experimental results showed that the undercut length could be reduced to half if the device is after a 1000ºC, 1 hour annealing 76 process before release. 2. Short circuit problem in thin aluminum film deposition Thicker aluminum film increases the conductivity of the device structure, thus reducing the structure series resistance. However, any layer thicker than 300 Å may result in a short circuit problem of the device. 3.9. FABRICATED DEVICE The SEM picture in Figure 3.13 shows a fabricated strain sensor. It consists of four sets of individual sensors in parallel with a total number of 552 fingers. The size of the sensor excluding the anchors and wire bonding pads is 1000×700 microns with the detailed dimensions listed in Table 3.4 . The buckling angle, α, is designed to be 5.7º, which has a nominal mechanical amplification gain of 5. 77 4 sets of sensors Figure 3.13. An SEM picture of a buckled beam capacitive strain sensor. 78 Table 3.4. Design values of the buckled beam strain sensor. Lg Gauge length 1000 μm Lb Buckled beam Length 300 μm w Beam width 6 μm h Buckling beam height 30 μm α Buckling angle 5.7º N Total number of sensing fingers 552 g Measured gap between fingers 3.6 μm Amech_nom Nominal mechanical amplification gain 5 Co Nominal capacitance 0.42 pF C , C , + ~ 3 pF p0 p1 Parasitic capacitance to substrate of Cs , Cp2 − Cs , Ccom 79 3.10. TESTING 3.10.1 TEST FIXTURE In dicing procedure, the 4-inch device wafer is diced into rectangular test strips. Each test strip is with size of 40 × 5 mm and the center of the strain sensor is located a distance of 11.5 mm from the edge, as shown in Figure 3.14.a. A bent-beam test fixture is developed to examine the device capacitive output characteristics as a function of applied strain. The test fixture consists of a vise to fix one end of the test strip, and a manipulator to apply the displacement load, as shown in Figure 3.14.b. When a displacement is applied, the test strip bends, thus resulting in a strain generated on the strip surface where the sensor is located. The sensor is bonded on the test strip substrate through the strong oxide layer as part of the structure of the SOI wafer, the strain generated on the test strip surface can be efficiently transferred to the sensor with negligible loss. Detailed discussion on the strain transferring mechanism will be presented in Chapter 4. The relationship between strain generated on sensor and the applied displacement is given by, 3t( L− x ) ε = s ΔZ , (3-37) s 2L3 where εs, t, L, xs, Δz denote the strain generated on sensor, test strip thickness, distance between the clamped edge and the edge where displacement is applied, distance between the sensor center to the clamped edge, and applied displacement, respectively. 80 4-inch device wafer Displacement applied by Sensor a manipulator Dicing edge Vise clamp Sensor 0.48 mm Silicon test strip Test strip xsensor= 1.5 mm L= 30 mm 10-mm clamp area (a) (b) Figure 3.14. Description of the test strip and bent-beam test fixture. (a) Top view of the 4-inch device wafer and a test strip. (b) Cross-section view a test strip mounted on a bent-beam test fixture. Table 3.5. Parameters of the test fixture. Symbol Description Value t Test strip thickness (SOI wafer substrate thickness) 480 μm L Distance between the clamped edge and the edge where 30 mm displacement is applied xs Distance between the sensor center to the clamped edge 1.5 mm Δzmax Applied maximum displacement 1.2 mm Table 3.5 lists the parameters involved in the test fixture. It is calculated that the test fixture can deliver a sensitivity of 0.76 με for every micron displacement (Δz). It is also 81 found experimentally that the applied displacement, Δz, can be up to 1.2 mm in either directions without breaking the test strip, indicating the maximum strain been able to be generated by this test fixture is ± 910 micro-strains according to parameter listed in Table 3.5. The actuating manipulator has a displacement resolution around 2.5 μm , limiting the test fixture’s resolution and accuracy to be around 2 με . A picture of the test fixture is shown in Figure 3.15. Input Displacement Sensor + Readout Circuitry Silicon Test Strip Vise Clamp Manipulator Figure 3.15. A picture of the bent-beam test fixture. 82 3.10.2 DEVICE FUNCTIONALITY TEST UNDER MICROSCOPE + Cs Ccom L1 L1 L2 L2 − Cs Figure 3.16. Microphotographs of sensor comb drive fingers before and after an applied tensile strain. The device is firstly tested by observing the fingers movements under an optical microscope. Figure 3.16 shows the micrographics of the structure before and after an applied tensile strain. The left figure in Figure 3.16 shows the initial finger positions before an applied strain. The finger overlapping lengths L1 , L2 are the same, indicating + − the capacitive outputs Cs and Cs equals. After an applied tensile strain, as shown in the right figure, the overlapping length L1 increases and L2 decreases, indicating a differential capacitive output. This test method lacks the required resolution due to the resolution limitation of the optical microscope; but it is convenient for device functionality test. 83 3.10.3 CAPACITIVE READOUT CIRCUITRY Higher-resolution test results are achieved by employing a capacitance-to-voltage (C/V) conversion circuitry. Two readout circuits are used to examine the device’s capacitive output characteristics. The device is firstly tested by an off-the-shelf C/V conversion chip MS3110® from Microsensors and then tested using a developed C/V conversion chip [40] to achieve higher resolution. 3.10.3.1 C/V CHARACTERISTICS USING MS3110® The test strip is mounted on a PCB circuit board using epoxy and clamped at edge by a vise, as shown in Figure 3.17. The strain sensor is connected to the circuit board through 2-mil aluminum bonding wires. The key component of the C/V conversion circuitry is an off-the-shelf capacitive readout chip MS3110® from Microsensors. The chip is capable of interfacing a differential capacitor pair with a typical resolution of 4.0 aF Hz . It also provides an on-chip EEPROM to store the trim and program settings. The C/V circuitry provides a voltage output, which is measured by a Keithley 6 1 digital multi-meter, 2 Model 2000. Before applying the strains, the MS3110® chip C/V characteristics in terms of linearity and sensitivity is examined by its internal calibration capacitors, which is claimed to have 10% accuracy according to the chip’s datasheet. 84 Figure 3.17. A computer illustration of the bent-beam test fixture. The measurements results are compared to the ANSYS FEA results, which were developed to simulate the capacitance change as a function of the strain load. The FEA results are achieved utilizing a coupled-filed analysis method of the full structure, as illustrated in Figure 3.18. The modeling analysis starts by building the mechanical full structure (Figure 3.18.a). Then the mechanical boundary constraints are applied (Figure 3.18.b) and solved to obtain the structure deformation with an applied displacement load using a nonlinear static mechanical solver (Figure 3.18.c), a key approach to solve the deformation if the structure is under large load. Then the mesh is updated, electrical elements and boundary conditions are added, followed by an electrical field solver to obtain the electrical fields densities (Figure 3.18.d). Capacitance can be determined by the calculation of the total energy stored in the elements. By this approach, the fringe capacitance of the comb fingers are taken into account, thus resulting in more accurate results. 85 (a) Define the structure of a capacitive strain sensor. Boundary constraints (b) Apply the boundary constraints. Deformed structure Contour plot of Y-component of displacement of the structure after applying a displacement (c) Obtain the mechanical deformation of the structure under strain. 0 V 1 V Contour plot of electrical field density after applying voltages on the deformed structure (d) Obtain the electrical field density after remesh the structure and apply electrical boundary conditions. Figure 3.18. Coupled-field finite-element analysis of the structure under applied strain. 86 FEA simulation results Measurement results 0.4 . 0.3 0.2 0.1 0.0 -0.1 -0.2 Compressive Tensile Differential Capacitance Out (pF) -0.3 -900 -600 -300 0 300 600 900 Applied strain (micro-strain) Figure 3.19. Measured capacitance change as a function of applied strain. Sensitivity: με 0.4 281 aF/ 0.2 Measurement Results (pF) out 0.0 -0.2 Finite-Element Simulation Results Capacitance Out C Capacitance Out -0.4 Compressive Tensile -1000 -500 0 500 1000 Applied Strain (με) Figure 3.19 shows a measured capacitive output characteristics as a function of the applied strain, and the comparison between the measurements and the FEA results. The 87 maximum applied strain is around 820 µε in tensile and compressive. The measurement results agree well with the FEA results with slightly (less than 2%) lowered sensitivity due to fabrication error. The device achieved an overall sensitivity of 281 aF/µε, while the sensitivity of tensile strain (300 aF/µε) is higher than that of compressive strain (268 aF/µε), which aggress with the analytical analysis and FEA results. 3.10.3.2 C/V CHARACTERISTICS USING DEVELOPED INTERFACE ELECTRONICS A capacitor-to-voltage conversion interface circuit has been developed by M. Suster [40, 48] to achieve the overall sensitivity performance of the strain sensor system. The circuit employs a continuous time synchronous detection architecture due to its low noise performance [49, 50] compared to other capacitive sensor interface configurations such as a switched-capacitor amplifier which exhibits an increased noise contribution due to aliasing of the high frequency amplifier thermal noise into the signal band. Figure 3.20 shows the continuous time synchronous detection capacitance to voltage converter architecture for the strain sensor. The MEMS capacitive strain sensor, modeled as differential capacitors, are driven by a clock signal with an amplitude of Vs and are interfaced by a differential charge amplifier, with integrating capacitorCI , which converts the change of differential sensor capacitance to an output voltage. The fully differential architecture is employed to eliminate common-mode noise sources such as substrate coupling, power supply noise, and noise from Vs . A clock frequency of 1 MHz is chosen to modulate the sensor information away from the 1/f noise of the amplifier, a critical means to achieve a high sensitivity. An input common-mode feedback (ICMFB) circuit is 88 incorporated with the charge amplifier, suppressing any offset signal due to the parasitic capacitance mismatch. The charge amplifier output is then mixed by the same clock signal and low-pass filtered to obtain the desired strain information. This technique is commonly referred as Chopper Stabilization, a powerful method to eliminate input DC offset voltage and 1f noise. MEMS CI Sensor + Cs 2 Mixer Low Pass Filter Vn C C fb vs P ICMFB OCMFB V V 0 o,amp out Cfb - CP Cs CLK CI Figure 3.20. Strain sensor electronic interface architecture. The differential output voltage of the charge amplifier can be determined as ΔCs VVoamp, = s, (3-38) CI where ΔCs is the sensor differential capacitance change due to an input strain, CI is the charge amplifier integrating capacitor, and Vs is the clock amplitude. With chosen Vs of 1.2V, CI of 1.6 pF , the circuit has output voltage signal range from 20 μV to 200 mV , corresponding to the minimum and maximum input strain of 0.1 με and 1000 με (80 dB dynamic range), respectively. The circuit has an calculated input referred voltage noise power spectral density of 5 nV Hz . The amplifier exhibits 89 a DC voltage gain of 73 dB with a unity gain frequency of 50 MHz [40]. The sensing electronics are fabricated using a 1.5 μm CMOS process and consume 1.5 mA of DC current from a 3V supply. 2.2 mm Clock Bandgap Reference Charge Low-Pass Filter Mixer Amplifier Figure 3.21. Sensing electronics die photo [40]. Figure 3.21 shows the chip photo with the core electronics occupying an area of approximately 0.6 mm x 1.7 mm . The MEMS capacitive strain sensor chip is wire bonded to the sensing electronics to form the prototype system as shown in Figure 3.22. The four parallel sensors presented in Figure 3.13 are enclosed in the white box as illustrated in this figure. The sensor chip is then subjected to a bent-beam testing process for system characterization as described in Section 3.10.1. Figure 3.23 presents the measured output voltage versus an applied input strain, indicating that the prototype system can achieve a maximum input signal of 1000 με , 90 corresponding to an output voltage of 420 mV , with a nonlinearity of 1.5% of full scale (including all measurement instrument errors). Figure 3.24 shows the measured output voltage noise power spectral density, demonstrating that the microsystem can achieve a low noise level of 375 nV Hz , corresponding to an input referred capacitance noise and displacement noise of 0.24 aF Hz and 0.009Å/ Hz , respectively. Considering the mixer noise conversion gain and noise contribution from the mixer, low-pass filter, and measurement equipment, the predicted noise performance is within 1 dB of the measurement results. A further reduced electronic noise floor is expected with improved CMOS technologies. The low frequency tone near 500 Hz and the 1f noise exhibited in Figure 3.24 have been verified as artifacts from the measurement equipment. 91 Four Parallel Sensors MEMS Chip Sensing Electronics Figure 3.22. System testing board. 0.45 0.4 0.35 0.3 0.25 0.2 Voltage (V) 0.15 0.1 0.05 0 0 100 200 300 400 500 600 700 800 900 1000 Strain (με) Figure 3.23. Output voltage vs. input strain. 92 Figure 3.24. Output noise spectral density. 3.11. CONCLUSIONS The structure of the mechanical amplifier employing the buckled beam suspension has been analyzed analytically and by FEA. The mechanical gain of the structure is determined by the buckling angle for small input strains. Large input strains alter the buckling angle, thus resulting in mechanical gain variations, a key source for the device’s nonlinear characteristics. The nonlinearity of the mechanical amplifier has been analyzed and test results agree well with the analysis with less than 2% discrepancy. A strain sensor employing this mechanical mechanism has been designed, fabricated, and tested through a bent-beam test fixture by bending a silicon test strip consisting of the sensor. The comb finger movement under applied strains can be 93 observed under a microscope, while more accurate measurement results were achieved by interfacing a capacitive readout circuitry. Interfaced with the developed circuits employing the continuous time synchronous detection architecture, the sensor prototype demonstrates a strain sensing resolution of 0.0009 με Hz with a maximum input strain of ± 1000 με . 94 4. CHAPTER FOUR SENSOR BACKING DESIGN The stress/strain packaged in micro-sensor induced during sensor installation or operating under various thermal conditions may cause degradation in device performance and long-term stability [51]. On the other hand, accurately transmitting substrate strain/stress to the strain/stress sensor is crucial to achieve high performance in high accuracy applications. Therefore, understanding and predicting the transfer behavior of stress from the substrate to the sensing module is important for the overall system performance. This chapter presents a detailed analysis on the strain transfer behavior and parametrically establishes the relationship between strain transfer behavior, device material, and geometrical parameters. The analysis was verified experimentally using the developed capacitive strain gages. 4.1. STRAIN TRANSMISSION RATIO OF A SOLID BACKING 4.1.1 TRANSMISSION RATIO DEFINITION For a typical strain gage application, the gage is bonded on the specimen surface through a layer of adhesive. The strains measured by the sensor are the strains transferred from the specimen surface to the sensor through the layer of adhesive. Therefore, the accuracy and efficiency of this transfer mechanism is critical in order to achieve the required accuracy and sensitivity of the overall system. To quantitatively analyze this phenomenon, the concept of transmission ratio, T , is 95 introduced. The transmission ratio T is defined as the ratio between strain seen by the sensor ε s and the strain ε applied on the specimen under test, and given by: ε T = s . (4-1) ε 4.1.2 STRAIN TRANSMISSION RATIO OF A SOLID BACKING Sensing structure Lg Sensor substrate ha Specimen under test Adhesive layer (a) Sensing structure Lg+ΔLs Sensor substrate Adhesive layer Specimen under test γ Lg+ΔL (b) Figure 4.1. Strain transfer mechanism. Figure 4.1 illustrates the cross-section view of a strain sensor bonded on a specimen through an adhesive layer for strain measurement before and after the strain is applied. The sensor module consists of the sensing structure anchored on the sensor substrate (backing). As the sensing structure is typically much more flexible than the backing, the overall structure mechanical stiffness is basically determined by the backing. When the specimen is under strain ε, its length changes from Lg to Lg + ΔL = Lg + Lgε . The 96 sensor responds with length change of ΔLs . Assuming the sensor substrate has a mechanical stiffness of ks, the associated shear force Fs generating length change of ΔLs is expressed by, FLks = Δ ss. (4-2) This force is provided by the adhesive layer through a deformation with a shear strain of γ, as shown in Figure 4.1. The averaged shear stress τ of the adhesive layer is governed by: FLLss1 ⎛⎞ΔΔ− G * τγγavrg==GG avrg ≈ =⎜⎟, (4-3) A2 2⎝⎠ 4 ha where Fs is the total shear force, A is the area of the adhesive layer; G is the shear modulus of elasticity of the adhesive layer, which is a function of material elastic modulus E E (Young’s modulus) and Poisson ratio, ν, expressed by G = [52]; ha is the 2(1 +ν ) adhesive layer thickness. *: The derivation of (4-3) assumes that the shear stress τ in the adhesive layer is linearly distributed from zero at the center to the maximum at the edge, LLg + Δ . Combining (4-1), (4-2), and (4-3), the transmission ratio between ΔLs and ΔL can be derived as: ε ΔL 1 T ==ss = . (4-4) kh εΔL 18+ s a GA A uniform solid backing with length of L , width of w , and height of h is illustrated 97 in Figure 4.2. the mechanical stiffness in (4-4) is given by: Ewh k = (4-5) s L where E, w, h, L denote the sensor substrate Young’s modulus, substrate width, thickness, and length, respectively. Strain direction Strain direction w z h x L y Figure 4.2. Sensor substrate geometric dimensions. Therefore, the transmission ratio is given by: 11 T == . (4-6) kh Ehh 18++s aa 18 GA GL2 It is shown that the transmission ratio is a function of bonding area, adhesive shear modulus, adhesive thickness, and the sensor substrate (backing) properties such as dimension and modulus of elasticity. The transmission ratio can be improved by the following ways: Employing strong adhesive which has large shear modulus. Typical epoxy based adhesives have shear modulus on the order of GPa. Metallic bonding using copper, gold, or other alloys can achieve much higher shear modulus [53], thus resulting in high transmission ratio. 98 Increasing the bonding area length, L. However this approach may also increase the device’s size. Reducing adhesive layer thickness by applying a very thin layer of adhesive ( ha ). Reducing the sensor substrate stiffness by employing thinner substrate or employing a more flexible substrate which will be discussed later. Using the capacitive strain sensor discussed in the previous chapter as an example: The device has a backing substrate (silicon) thickness of 480 μm and material Young’s modulus of 166 GPa [13]. The device is with width of 5 mm and length of 3 mm . The adhesive is epoxy Vishay® M-bond 610 (Young’s modulus: 3.3GPa , shear modulus: 1.3 GPa ). The sensor has an equivalent stiffness of 133×106 Nm. The calculated transmission ratio is 0.65 for adhesive thickness of 10 μm , indicating a significant sensitivity drop if this device is bonded on the specimen using the regular M-bond 610 epoxy adhesive. 99 1.0 0.8 Calculated results using Eq. (4-6) Measured data 0.6 Transmission Ratio 0.4 FEA results 0 10203040 Adhesive Layer Thickness (micro) Figure 4.3. FEA results on bonding adhesive thickness effect on transmission ratio. Figure 4.3 shows the Finite-Element-Analysis analysis results of transmission ratio of the above structure with various adhesive thicknesses. The results show that the transmission ratio changes from 0.57 to 0.37 when the adhesive thickness varies from 10 μm to 30 μm, agreeing with (4-6) with a maximum discrepancy of 16%. Further testing has been performed using a fabricated capacitive strain sensor described in Chapter 3[54]. The device is diced with final size of 5 mm ×3 mm , bonded on a piece of stainless steel through an adhesive layer MB610® with estimated thickness of 10 μm , and tested on a four-point test fixture developed in our laboratory. (The detailed description of the four-point test fixture can be found in Chapter 6). The measured sensitivity, as shown in Figure 4.4, is 160 aF/micro-strain, which is approximately 60% of measured sensitivity (280 aF/micro-strain) using the test strip described in Chapter 3, indicating a significant 100 transmission ratio drop because of the rigidness of the sensor backing. The test results also agree well with the predicted data illustrated in Figure 4.3 with a discrepancy of 5% comparing with the FEA result and a discrepancy of 8% comparing with the analytical result. It is also observed that a hysteresis around 10 με (2.2%FS) was found a 450 με tensile stress sweep due to the adhesive layer plastic deformation when under stress beyond its elastic limit. 60 Measured sensitivity: 160 aF/micro-strain 50 40 Strain increasing 30 Strain decreasing 20 10 Differential Capacitance Output (fF) Output Capacitance Differential Max. hysteresis: 10 micro-strain 0 0 100 200 300 400 500 Applied Tensile Strain (micro-strain) Figure 4.4. Measured capacitance change as function of the applied strain using the four-point test fixture This example demonstrated that the transmission ratio had to be improved for the desired sensor. This objective can be achieved by employing a more flexible backing structure such as a folded-spring backing instead of a uniform solid silicon backing, which will be discussed in the following sections. 101 4.2. TRANSMISSION RATIO OF A FOLDED-SPRING BACKING Capacitive sensing structure Bonding Pads Longitudinal direction Wire bonding Ls terminals ws Anchors Bonding adhesive Folded-spring Figure 4.5.The structure of a folded-spring backing for the capacitive strain sensor. Figure 4.5 shows a folded-spring backing employed by a capacitive strain sensor. The sensing structure consisting of sets of capacitive comb driving fingers is anchored on the substrate through the strong silicon-oxide fusion bonding provided by the SOI wafer. The backing structure consists of two bonding pads, connected with two silicon folded springs. The folded spring has length Ls and small width ws, thus resulting in smaller stiffness, ks , in longitudinal direction, a key feature of the sensor module design that makes a more flexible sensor backing. As a result, higher transmission ratio can be expected. 102 4.2.1 FOLDED-SPRING BACKING STRUCTURE ANALYSIS 400 μm 200 μm Bonding pads 1 mm Folded-beam 3 mm springs Longitudinal direction 500 μm Figure 4.6. Top view of a folded-spring backing and the geometrical dimensions used. The structure has a thickness of 480 μm . The mechanical stiffness in longitudinal direction of the folded-spring backing is examined using the FEA method. The structure stiffness in longitudinal direction is obtained by analyzing the structure deformation under a certain longitudinal force through a built 3-D FEA structure model. With the substrate thickness of 480 μm , the simulated structure stiffness in longitudinal direction is 400 kN m with geometrical parameters shown in Figure 4.6. As comparison, a solid rectangular silicon backing structure, as shown in Figure 4.2, with width w of 3 mm , length l of 2 mm , thickness h of 480 μm exhibits a longitudinal stiffness of 120000 kN m according to (4-5). The folded-spring backing structure demonstrates much less stiffness, ks , almost three orders of magnitudes less than that of a rectangular solid backing. 103 Lab Bonding pad, area: Ap A B ks Bonding adhesive C D Specimen Sensor backing equivalent stiffness (a) Lab+ΔLs Bonding pad, area: Ap A B ks Bonding adhesive C D Specimen γ Lbc+ΔL (b) Figure 4.7. Structure deformation under an applied strain. The transmission ratio of the folded-spring backing is analytically analyzed by examining the structure deformation under an applied strain, as illustrated in Figure 4.7. The structure is simplified as two bonding pads connected through a spring with an overall stiffness of 400 kN m , the previously simulated stiffness of the overall structure. Assuming the sensor is anchored at the center of the bonding pads at location A and B. (For cases the sensor is anchored off the center of the bonding pads will be discussed in the next section.) C and D are the projected locations of A and B on the specimen, as illustrated in Figure 4.7.a. The initial distance between A and B is Lab. When the specimen is strained, a displacement ΔL is generated, and the associated displacement ΔLs is generated on the 104 sensor (Figure 4.7.b). The required shear stress τ is: ΔLk τ = s s , (4-7) Ap where Ap is the area of each bonding pad. Assuming the deformation of bonding pads of the sensor backing under strain is negligible (as kkpad s ), the distortion angle γ is governed by: ⎛⎞ΔΔLL− s G τγ=≅G ⎜⎟, (4-8) ⎝⎠2 ha where G is the shear modulus of elasticity of the adhesive layer; ha is the adhesive layer thickness. Combining (4-7) and (4-8) gives: ΔL 1 T =≅s . (4-9) folded kh ΔL 12+ s a GAp Table 4.1. Calculated transmission ratio with various bonding adhesive thickness. The structure is with thickness of 480 µm and geometrical dimensions is illustrated in Figure 4.6. Epoxy MB610® 10 µm 15 µm 20 µm 25 µm 30 µm thickness Transmission Ratio 0.996 0.994 0.992 0.990 0.988 The calculated transmission ratio of a folded-spring backing using (4-9) is listed in Table 4.1. With epoxy adhesive layer thickness varying from 10 µm to 30 µm, the calculation results show that the transmission ratio only slightly drops from 0.996 to 0.988. It demonstrates that the transmission ratio has been greatly improved by employing the 105 folded-spring backing structure. 4.2.2 ANCHOR LOCATION AND PACKAGE GAIN Previously in SECTION 4.2.1, the sensor is assumed to be anchored at the centers of the two bonding pads. However, if the sensor is anchored off-center, e.g., at the edges of bonding pads, as illustrated in Figure 4.8, the transmission ratio differs, and sometimes can be more than 1, indicating a strain amplification nature. This strain amplification phenomenon is called the package gain. It is attractive in applications utilizing strain/stress sensitive elements such as a piezoresistive sensor, thus worth a detailed discussion. Lg A B ks Bonding pad Bonding adhesive Lp Lp Sensor backing Specimen equivalent stiffness (a) Lg A B ks Lp Lp Specimen (b) Figure 4.8. Package gain. 106 Package gain analysis: y xB Center line Lp Sensor structure anchor (right) C B Lg 2 Sensor backing C' Bonding adhesive Specimen ()LL2gp+ (L2Lgp+ ) 2 y' (a) y xB ' C B C' Specimen ()1LL2++ε ( gp) y' (b) Figure 4.9. Transmission ratio analysis. Figure 4.9 illustrates the cross-section view of half the structure before and after the specimen is strained. B represents the right anchor of the sensing structure with distance xB to the center line yy'− . xB can be any value from L2g to ()L2L2gp+ , where Lg is distance between the bonding pads and the Lp is the bonding pads length of the sensor backing. C represents the center of the bonding pads, and C' is its associated 107 projection on the specimen surface. The initial distance between CC'− and center line yy'− is ()LL2gp+ . After the specimen is strained, C' location on the specimen surface moves from ()LL2gp+ to (1LL2++ε )( gp) , where ε is the applied strain. As the sensor structure’s stiffness is small, the projection of C stays at C' to the first-order approximation, according to calculated transmission ratio listed in Table 4.1 (Section 4.2.1). Therefore, ()1LL++ε ( gp) ⎛⎞LLg + p x'B=++−() 1 T bondpadε ⎜⎟ x B , (4-10) 22⎝⎠ where Tbondpad is the transmission ratio of the bonding pad, which is expressed as, 1 TbondPad = , (4-11) Ehha 18+ 2 GLp according to (4-6). The transmission ratio of the overall structure can then be expressed as, ⎛⎞ (LLgp++) ( LL gp) x'BB− x +−Tx⎜⎟ 22bondpad⎜⎟ B xB ⎝⎠ TOverall == . (4-12) ε xB For a short Lp , the transmission ratio of bonding pads TbondPad is very small. For example, if the bonding pad is with Young’s modulus E of 166 GPa, thickness h of 480 µm, length L of 500 µm, and the adhesive layer is with shear modulus G of 1.3 GPa, thickness ha of 10 µm, the calculated TbondPad is only 0.048 according to (4-11). Therefore, the (4-12) can be simplified as, 108 (LLg + p ) 2 TOverall ≅ , for short Lp . (4-13) xB (4-13) shows the simplified transmission ratio of an overall sensor module with short bonding pad length, Lp . It depends on device geometrical dimensions and the location of the sensing structure anchors. If the anchors are located at the center of the bonding pads, as shown in Figure 4.7, the transmission ratio is close to one. However, if the anchor is L located at the inside edge of the bonding pads, x = g , a maximum transmission ratio B 2 can be achieved, which is, LLg + pp L Γ max ≅=+1 . (4-14) LLg g This maximum transmission ratio is greater than one, indicating the strain seen by the sensing structure is greater than the strain applied on the specimen. 4.2.3 PACKAGE GAIN DISCUSSIONS 4.2.3.1 THE MAXIMUM PACKAGE GAIN FOR PIEZORESISTIVE SENSOR DESIGN The package gain is attractive in applications utilizing strain/stress sensitive elements such as a piezoresistive sensor due to the fact that the resistance of a piezoresistive sensing element varies with the applied stress or strain (See discussion in Chapter 2). It is therefore worth discussion to examine the sensor geometrical parameters to achieve a maximum package gain. To achieve a maximum package gain, a long Lp would be 109 employed; therefore the simplified equation (4-13) is no more suitable and (4-12) should be used instead. L Combining (4-11) and (4-12), and letting x = g give, B 2 x'− x ⎛⎞ BB ⎜⎟ x1L Γ ==+−B 11⎜⎟p . (4-15) overall ε ⎜⎟Ehh L 18+ a g ⎜⎟2 ⎝⎠GLp (4-15) shows that the package gain increases initially with the ratio of LLp gap when 1 Lp is small and the term 1− is negligible. However, as the Lp increases, Ehha 18+ 2 GLp 1 the term 1− becomes significant and the package gain will gradually saturate Ehha 18+ 2 GLp and finally reach a peak value. If Lp continues increasing, the package gain will decreases. Figure 4.10 shows the analysis results of the package gain as a function of bonding pad length with the described geometrical parameters and various adhesive layer thicknesses. Taking adhesive layer of 20 μm as an example, the analysis results show that the maximum package gain of 2.55 is achieved for Lp length of 3000 µm, an 8.3 dB sensitivity improvement if this structure is employed by a piezoresistive type strain sensor. Therefore, a piezoresistive strain sensor will be favored by this package gain in terms of sensitivity. 110 3.0 FEA results Analytical results, h =30 μm h =20 μm a a 2.5 Analytical results, h =20 μm a 2.0 Analytical results, h =10 μm a 1.5 Transmission Ratio (Package Gain) Ratio Transmission 1.0 0 1000 2000 3000 4000 5000 Bonding Pad Length, L (μm) p Figure 4.10. Package gain as a function of bond pad length Lp . The sensor structure is with the following geometrical parameters: Gap Lgap =1000 µm, bonding pad thickness h =480 µm, adhesive layer thickness ha =20 µm. 4.2.3.2 THE MAXIMUM DISPLACEMENT FOR CAPACITIVE SENSOR DESIGN For capacitive strain sensor, as discussed early, is fundamentally a displacement sensor. The displacement applied to the sensor, Δx , is the production of the applied strain, ε , and the sensor’s gage length, Lg , that is: Δx = Lgε . As discussed previously, the package gain amplifiers the strain seen by the sensing structure, but not the production of ε Lg , as it also decreases device’s gage length. Therefore, the package gain is not attractive in capacitive strain sensor design. Derived from (4-13), the displacement seen by the sensor is given by: 111 Δεx =≅+2xB T Overall ε( L g L p ) . (4-16) Eq. (4-16) shows that displacement applied on the sensor is only determined by the sensor size (Lg+Lp) as long as using a short Lp, e.g. Lp less than 1mm. In sensor design, the Lp is chosen in such a way that there should be enough space to accommodate other necessary structures such as sensing-structure anchors, wire-bonding pads, device encapsulation space, etc. Figure 4.11 illustrates the top view of capacitive strain sensor module with a folded-spring backing. The sensor module has an overall size of 3mm× 2mm with sensing structure gauge length of 1000 μm . The bonding pad length is 500 μm , which is chosen to leave enough space accommodating wire-bonding pads, anchors, and encapsulation structure for sensing structure protection. 112 2000μ m Encapsulation space Wire-bonding pads 1000μ m 3000μ m Anchors Folded-spring backing thickness: 480 µm Sensing structure L500mp = μ Figure 4.11. Top view of a capacitive strain sensor module including a folded-spring backing. The annotated dimensions reflects the real sensor. The transmission ratio of the structure is examined using a 3-D FEA modeling method, which is illustrated in Figure 4.12. The modeled structure consists of the sensor folded-spring backing, adhesive layer, and a layer of specimen made of stainless steel (17-4-PH) with Young’s modulus of 200 GPa and Poisson ratio 0.3. The sensor sensing element has stiffness much less than that of the sensor backing, so it is omitted in modeling. A strain is applied on the specimen layer in the longitudinal direction, and then transferred to the sensor backing through the adhesive layer, inducing a displacement on the sensor substrate, which can be solved using a nonlinear FEA solver and obtained later through the post processor. 113 Bonding pads A B Longitudinal Folded-beam direction spring Applied strain on specimen Bonded Epoxy Stainless steel specimen (17-4-PH) Figure 4.12. 3-D FEA modeling of a folded-beam backing bonded on a strained specimen for transmission ratio analysis. The FEA modeling examined the transmission ratios of various epoxy thicknesses varying from 5 μm to 35 μm in a 5 μm step with results shown in Figure 4.13. The simulated transmission ratios vary from 1.38 to 1.46 with a fluctuation of 6%. Thin adhesive layer, e.g. ha = 5 μm , has slightly lower transmission ratio, which is consistent with (4-15). If the adhesive layer can be controlled to be within 20 μm to 35 μm , the transmission ratio is near the maximum and the fluctuation can be even lowered to be ±0.2% , at least two orders of magnitude improvement compared to a solid backing illustrated in Figure 4.3. 114 1.50 1.45 1.40 Transmission Ratio (Package Gain) 1.35 0 5 10 15 20 25 30 35 40 Bonding Adhesive Thickness, h (μm) a Figure 4.13. FEA results of transmission ratio of a folded-beam backing as a function of varied bonding epoxy thickness. 115 4.3. OVERALL SENSOR MODULE BACKING DESIGN The overall sensor module backing design is illustrated in Figure 4.14.a, which consists two major parts: the sensor backing and interface electronics backing fabricated using the same SOI substrate and connected through a folded beam. The electronics backing has a recess to accommodate the circuit die for easy alignment. The wiring terminal pads are with width of 500 μm and length of 2000 μm for easy wire soldering. Figure 4.14.b is a computer drawing illustrating the overall sensor module with the encapsulation and the electronics attached. As the capacitive sensing structure contains movable parts such as comb drive fingers, which are prone to dust particles, an encapsulation is necessary to provide the protection. This encapsulation can be made of polyimide material such as SU-8 for its relative flexibility and excellent microfabrication capability of thick structure. The overall structure is with size of 3mm(W)×7mm(L)×500µm(T). The finite-element-analysis modeling method is used to examine the strain transfer characteristic of the overall structure. With results shown in Figure 4.15, the package gain provided by the structure varies from 1.68 to 1.78 with adhesive thickness varying from 10 µm to 30 µm, approximate 6% amplitude fluctuation. The slightly higher transmission ratio compared to data shown in Figure 4.13 is due the electronics backing effect. With a typical adhesive thickness of 15 µm, the package gain is 1.75, which serves as the nominal value guiding the sensing structure design. 116 Wire-bonding pads (To Electronics) Connection pads Sensing structure 7 mm Folded-spring backing 3 mm Electronics and Interface electronics Sensor wiring terminals recess area (a) Bonding wires Encapsulation Interface electronics (b) Figure 4.14. Overall sensor backing design. 117 1.80 1.75 1.70 Transmission (Package Gain) Ratio 1.65 5 101520253035 Bonding Adhesive Thickness, h (μm) a Figure 4.15. Strain transmission ration examined by the finite-element-analysis. 118 4.4. SUMMARY AND CONCLUSIONS This chapter focuses on the sensor module’s backing design by investigating the strain transfer mechanism. The main goal of this investigation is to understand the mechanism and design a backing structure which can transfer the strain from the specimen under test to the sensing structure accurately and efficiently with high transmission ratio. The traditional solid backing would not fulfill this requirement due to its low transmission ratio. Analysis reveals that the transmission ratio can be improved by reducing the backing structure mechanical stiffness. A folded-spring backing is then designed for this purpose. Finite-element-analysis shows that the proposed structure reduces the equivalent stiffness from 120000 kN m to 400 kN m . As a result, the transmission ratio has been greatly improved and it is less sensitivity to adhesive thickness, a key advantage to achieve measurement accuracy in engineering point of view. Further investigation also reveals that the inherited package gain of this structure can amplify the strain; therefore piezoresistive type sensors can be benefited by employing this structure. However the package gain does not benefit the capacitive strain sensor due to the fact that the capacitive strains sensor is measuring the displacement. Finally, the backing structure that also accommodates the interface electronics die is designed according to the analysis and examined by a 3-D FEA modeling performed, which reveals the structure has an inherited nominal package gain of 1.75, indicating that for every micro-strain to be measured on the specimen, the sensor sees 1.75 με , which has to be considered in sensor’s sensing structure design. 119 5. CHAPTER FIVE SENSING STRUCTURE DESIGN This chapter presents the sensing element design to fulfill the specification requirements such as sensitivity, resolution, etc. at a given temperature. Additionally, nonlinearity, device fundamental noise sources including electrical thermal noise and mechanical Brownian noise will be addressed. 5.1. STRUCTURE MODIFICATION Lg Δw Buckled beam suspensions Δx α Applied strain: Anchor A Anchor B Comb drive Sensing beam sensing finger (a) Lg Sensing beams Δw Comb drive sensing finger Δx Applied strain: Anchor A (b) Figure 5.1. Improved sensing structure. 120 The sensing structure described in Chapter 3 (Figure 5.1.a) does not provide the maximum efficiency in sensitivity as the sensing beam occupies part of the gauge length and the buckled beam length is thus limited. The modified structure shown in Figure 5.1.b overcomes this problem by attaching the sensing beams to the joints of the buckle-beam suspensions. This structure has several improvements: 1. Higher sensitivity Apparently, the sensing beams of the modified design are longer with the same total gauge length Lg . Therefore, more sensing fingers for higher sensitivity can be accommodated on the sensing beams. For example, the structure employed in Chapter 3 has a gauge length of 1000 µm and a sensing beam length of 444 µm with a total number of 37 fingers on each sensing beam. The modified structure can has the same gauge length and the sensing beam length has been increased to 800 µm with a total number of 66 fingers on each beam, which is almost doubled. 2. Better linearity As discussed in Chapter 3, the nonlinearity of the structure can be estimated by the 1 term ε , where α is the initial buckling angle and ε max is the maximum applied 4tan2 α max strain. Because of its relatively longer buckled beam length, for the same applied displacement, the modified structure design (Figure 5.1.b) is undertaking less maximum strain ε max on the buckled beam suspensions than that of the previous design (Figure 5..a) Therefore, improved linearity is expected. 3. Sensing beam less prone to bending 121 Bending of sensing beam Compression Tension Δx Figure 5.2. Sensing beam affects the structure deformation as it bends when the structure is under strain As illustrated in Figure 5.2, the sensing beam deforms because of the bending moments from the buckling beams when a displacement is applied. The deformation will affect the sensor capacitance output characteristics, and sometime may generate a short-circuit problem. This problem is avoided in the modified structure. 5.2. CAPACITIVE OUTPUT SENSITIVITY CALCULATION The sensor sensitivity is defined as the differential capacitance change per unit applied strain, given by, ΔΔCC+−− 2NA tε L A S ≡≅mech e g p , (5-1) ΔStrain g where N is the total number of sensing comb fingers; Amech is the nominal mechanical gain; t is the thickness of sensing structure in direction perpendicular to the paper; ε e is the dielectric permittivity; Lg is the gauge length; Ap is the package gain, and g is the gap between fingers. 122 The finger gap g and sensing layer thickness t are chosen according to fabrication process capability, which are 3 μm, and 20 μm, respectively in this project. Sensitivity and device size will benefit from improved fabrication technology which allows smaller feature size and greater Deep-Ion-Reaction-Etch (DIRE) thickness-to-gap aspect ratio. Table 5.1. Device parameters. N Total number of fingers 260 Amech Nominal mechanical gain 5 t Device thickness 20 μm L Sensing structure gage length g 1050 μm Ap Package gain 1.75 g Nominal gap between capacitive sensing fingers 3 μm S Calculated sensitivity (aF/micro-strain) 282 Table 5. lists the parameters used for the sensing structure. A calculated sensitivity of 282 aF με is expected. 123 5.3. FRINGE CAPACITANCE EFFECT ON COMB DRIVE FINGER DESIGN g 6 5 ε 3 4 L 8 7 g Strain clr Direction 1 2 w2 Comb drive FEA modeled sensing fingers structure (a) (b) Figure 5.3.Finite-element modeling of capacitance of a comb finger. (a) Top view of a section of comb drive sensing fingers. (b) Top view of a section of comb drive sensing fingers in the dashed red line. The lateral sensing (Figure 5.3.a) is used for the capacitive sensing for its large operating range and linear output characteristics. In this configuration, when under an applied strain, the fingers moves along the finger length direction, thus resulting in a change of overlapping area which alters the capacitance proportionally. However, when two fingers are moving close, the fringe capacitance at the finger tip area may introduce certain nonlinear fringe effect. To reduce this nonlinear effect, finger tip clearance, gclr , has to be properly selected. Considering the sensor module with package gain of 1.75, the maximum strain seen by the sensing structure is ±1750 με . For the 1 mm sensor length, the maximum displacement movement of the finger from each other is close to 18 μm . Therefore the tip clearance gclr has to be greater than 18 μm . Two values of gclr (21 μm and 24 μm ) are 124 selected to examine the capacitance output characteristics through the finite element analysis. The analysis follows the electrostatic analysis procedure. Once the structure is defined, potential boundary constraints are applied; then the electrostatic analysis is performed to obtain the electrical field contribution. The energy of each element is obtained and the summation is used to obtain the total energy stored in the capacitor; the capacitance is then calculated by: C = 2W V 2 , where C, W, V denote the capacitance, total energy, and voltage potential, respectively. 3 2.6 g =21 μm clr 2.4 2 2.2 g =24 μm clr -1900 -1800 -1700 -1600 -1500 -1400 1 Single Finger Cout (fF) Cout Single Finger 0 -2000 -1000 0 1000 2000 Applied Strain Seen by the Sensor (με) Figure 5.4. Finite-element analysis results about capacitance output as a function of applied displacement with two variables: (1) gclr =21µ m; (2) gclr =24 µm. The simulated capacitance output is illustrated in Figure 5.4. For gclr = 21 μm , the capacitance output shows noticeable nonlinearity when the sensor sees strain exceeding 1400 με . A 24 μm gap clearance can overcome this problem and provide a linear capacitance output. In sensor design, a 30 μm gap clearance has been used to provide 125 enough safety margins, so that the fringe effect can be neglected. 5.4. NOMINAL CAPACITANCE AND PARASITIC CAPACITANCE + − + − Nominal capacitance Co and Co are initial capacitance of Cs and Cs , respectively, when no strain is applied. They are contributed mainly by the capacitance of the sensing fingers. Other capacitances such as the fringe capacitance between bonding pads and bonding wires are omitted in calculation. The nominal capacitance of the sensing fingers is given by, NtL ε CC+−==ovlp e , (5-2) oo 2g where N is the total number of sensing comb fingers; t is the thickness of sensing structure in direction perpendicular to the paper; ε e is the dielectric permittivity; Lovlp is the initial finger overlapping length; and g is the gap between fingers. The overlapping length Lovlp should be chosen to make sure the fingers are still overlapping under the maximum strain, therefore, Lovlp>= 2ε max A p A mech L g 17.5μ m , (5-3) A 30 μm finger overlapping length is chosen to provide enough safety margins. The nominal capacitance can then be calculated: 260××× 20 30 8.85 × 10−6 C+−== C = 0.23pF (5-4) oo 23× + − The parasitic capacitance C p and C p are the capacitance to the sensor substrate, 126 mainly contributed by the bonding pads and anchors. Each terminal consists one wire bonding pad (120μ m× 120μ m ) and two anchors ( 80μ m× 60μ m ) with a total area of 2.4× 1042μ m . The underneath oxide layer has a thickness of 1.5μ m . The calculated parasitic capacitance is: 2.4××× 1046 3.9 8.85 × 10− C+−== C = 0.55 pF (5-5) pp 1.5 5.5. SERIES RESISTANCE CONSIDERATION The series resistance of the capacitive is a serious concern in sensor design because it introduces the resistive thermal noise to the interface electronic as discussed in Chapter 3. The series resistance should be on the order of kΩ or less in order to be negligible in noise point of view (see detailed discussion in Section 3.6.1 of Chapter 3). The series resistance is mainly contributed by the buckled beam suspensions. The design suspension has width of 10 μm and a length of 500 μm with a L W ratio of 50. The series resistance can then be calculated as, ρ R ==50R 50 , (5-6) s t Where Rs is the sheet resistance; ρ is the resistance of the device layer; and t is the device layer thickness, which, in this design, is 20 μm . For a 10 Ω⋅cm , lightly doped silicon material, the calculated sheet resistance is 5 kΩ per square. Therefore, the series resistance is 250 kΩ , which is too high for this application. 127 To reduce the series resistance, either a heavily doped silicon material or applying a layer of highly conductive material such as aluminum on device top can be employed. For example, a heavily doped silicon material with resistance of 0.01 Ω⋅cm can achieve a series resistance of 250 Ω . A layer of 100Å sputtered aluminum (resistivity: 2.8×Ω⋅ 10−6 cm ) can achieve a series resistance of 140 Ω . In this sensor design, a lightly doped silicon material (10 Ω⋅cm ) is used and a layer of 100Å aluminum is sputtered to reduce the series resistance and to avoid the concern of the high built-in stress of the material due to the heavy doping. 5.6. STRUCTURE MECHANICAL PROPERTY 5.6.1 NONLINEARITY OF THE BUCKLED BEAM AMPLIFIER The buckled beam mechanical amplifier has nonlinear characteristics, which has already been examined and discussed in Chapter 3. Given the maximum strain of ±1750με seen by the sensing structure and a mechanical nominal gain of 5, the expected full-scale nonlinearity is around 7% according Figure 3.7 in Chapter 3. 5.6.2 RESONANCE MODES AND FREQUENCIES The required operating bandwidth for the strain sensor is 10 kHz , indicating the resonant frequency of the sensing structure should exceed this number in order to meet the requirement. 128 Top Center Bottom Figure 5.5. Top view of the sensing structure. The sensing structure consists of three parts, namely, top, center, and bottom suspensions, as illustrated in Figure 5.5. The resonance frequency and shape of each part is examined by FEA. The results are shown in Table 5.2 All three parts exhibit tilting resonant shape at the fundamental resonant frequency with the lowest frequency of 17 kHz at the center part, which is much higher than the 10 kHz bandwidth requirement. Table 5.2. Resonant frequency and shape of each part of the sensing structure. Part First mode Second mode Top fo = 32kHz f1 = 103kHz Center 129 fo = 17kHz f1 = 137kHz Bottom fo = 29kHz f1 = 103kHz 5.6.3 MECHANICAL THERMAL NOISE ANALYSIS As discussed previously, the input strain referred mechanical thermal noise power spectral density of each suspended structure for frequencies below the mechanical resonant frequency of the structure can be expressed as (Section 3.6.2, Chapter 3), 4KBT ε n = Kn 3 (5-7) ωo mQ where εn is input referred strain mechanical thermal noise power spectral density; KB is the Boltzman constant; T is the absolute temperature with unit of Kelvin; ωo is the structure resonant frequency; m is the mass; Q is the quality factor; BW is the required signal bandwidth; Kn is the conversion factor between the noise displacement and the input-referred strain, which is a function of mechanical gain and gauge length, and equals to 200 for the designed sensor. Each device component (top, center, and bottom suspensions) contributes non-correlated mechanical noise. However, because of the differential mode, the Brownian motion of the center suspension is cancelled. Therefore the mechanical noise is the summation of the noises from top and bottom suspensions. The frequency and shape of the 130 resonant modes is examined by FEA with results listed in Table 5.2. As discussed previously, the tilting first resonant behavior does not contribute significant noise to the first order due to device capacitive cancellation. So the main mechanical noise sources are the second resonant behavior at 103 kHz of the top and bottom suspension structure. Higher frequency modes will contribute negligible amounts of noise. The design presented in Figure 5.5 exhibits a proof mass of 1.66 μg for the top and the bottom suspended structure. Therefore, the resulting mechanical thermal noise power spectral density is estimated in air, with a Q of unity (typical for MEMS structures operated in ambient [45] to be 1.2× 10−6 με Hz , thus negligible compared to the minimum design requirement of 0.001με Hz . Therefore, the sensor can be operated in ambient without requiring a vacuum packaging, thus substantially reducing the system packaging complexity and cost. 5.7. FABRICATION PROCESS Compared to the fabrication process for the devices discussed in Chapter 3, the fabrication process has two major differences: 1) A backside DRIE process etching through the whole SOI wafer substrate layer (480 µm) is employed to generate the folded-spring backing structure. 2) A backside oxide plasma etching (drying etching) is employed to release the sensing structure instead of using the time-controlled HF wet etching process used previously to avoid the under cut problem caused by the wet HF etching. The developed fabrication process has three photolithograph masks with the details described as follows: (a) Starting with SOI wafer 131 The starting wafer is a silicon-on-oxide (SOI) wafer. The thicknesses for device layer, buried oxide layer (BOX), and handle layer (serving as the device substrate) are 20 μm, 1.5 μm, and 400 μm respectively. The device layer and handle layer have nominal resistivity of 1-10 ohm-cm; both are Boron doped (P-type). The measured sheet resistance of the device layer is approximately 3000 ohm per square. (b) Metal deposition and patterning (Figure 5.6.b, Mask #1: Metal Pad) The first layer in the process is the pad metal. A metal layer of 3000 Å of aluminum is sputtered and patterned through a lift-off process. This metal area must be covered during the subsequent DRIE etch. Hence, it is limited to relatively large areas such as the bonding pads. To accelerate the lift-off process, the wafer is agitated using ultrasonic and wiped with soft clean room tissue (Alpha wiper). After this step, the bonding pads are defined. (c) Sensing structure patterning (Figure 5.6.c, Mask #2: Device) Silicon is lithographically patterned with the second mask and followed by a deep reactive ion etched (DRIE) to generate the sensing structure. This etch is performed using inductively coupled plasma (ICP) technology, and a special SOI recipe is used to virtually eliminated any undercutting of the silicon layer when the etch reaches the BOX layer. (d) Back Patterning (Figure 5.6.d, Mask #3: Substrate) The wafer is reversed and a 10-μm-thick photoresist is spin-coated and patterned on the backside using Mask #3: Substrate. The following DRIE process will etch through the whole substrate (480 µm) with a selectivity of 80:1 between silicon and mask photoresist; as a result, a photoresist layer more than 5 μm thick is required. The 10-μm-thick photoresist should be able to provide enough safety margins. 132 (e) Back DRIE etching Before the back DRIE etching, a layer of photoresist (1.2 μm) is spin-coated on the front surface, and a dummy wafer is glued on the wafer for protection purpose. Then the wafer is completely etched through using the DRIE process to obtain the sensor substrate structure. In this DRIE process, the recipe is optimized for higher etch rate, a higher undercut is expected. The typical ratio between vertical etch and lateral undercut is 40:1. For a 400 μm etch step, a 10 μm undercut can be expected. After this etch step, the dummy wafer and device wafer are separated by a hot acetone bath for several hours. After separated, the device wafer is very fragile and should be handled very carefully. A 30 minutes oxygen plasma etch is used to clean up the photoresist residuals on the wafer. (f) Oxide dry etch (device release) A plasma dry etch is used to etch the oxide layer and release the devices. The recipe is optimized for high oxide etch rate with minimum attack on silicon. Etch time: 5 minutes. (g) Metal deposition A thin layer (200 Å) of metal (Al) is sputtered on the surface to reduce the device series resistance. The sheet resistance can be improved from 3000 ohm per square to 5.6 ohm per square. 133 Device layer (Si) Silicon oxide Substrate (Si) Handle layer Metal (Al) Device layer (Si) Silicon oxide Substrate (Si) (b) Metal deposition and patterning Metal Comb finger gap Device layer (Si) Silicon oxide Substrate (Si) (c) Sensing structure patterning and DRIE etching Comb finger gap Metal (Al) Device layer (Si) Silicon oxide Substrate (Si) (d) Back patterning Thick PR Comb finger gap Metal (Al) Device layer (Si) Silicon oxide Substrate (Si) (e) Back DRIE etching Comb finger gap Metal (Al) Device layer (Si) Silicon oxide Substrate (Si) (f) Oxide dry etching Thin metal (200Å) Comb finger gap Metal (Al) Device layer (Si) Silicon oxide Substrate (Si) (g) Thin metal deposition Figure 5.6. Major fabrication processing steps. 134 5.8. FABRICATED DEVICE Figure 5.7. Fabricated device and an SEM picture of the sensing structure. 135 Figure 5.7 shows a picture of the fabricated device and an SEM picture of the sensing structure. The measured finger gap is 3.64 µm. 5.9. SUMMARY AND CONCLUSIONS This chapter presents a detailed discussion about the capacitive strain sensor’s structure design and fabrication process. Because of the combined effects of package gain and modified sensing structure, the sensor’s sensitivity has been improved. As a result, only a set of sensing fingers is required to provide the same sensitivity. The device provide + − a sensitivity of 280 aF με , a nominal capacitance of 0.23 pF for C o and C o , and a + − parasitic capacitance of 0.55 pF for C p and C p . The nonlinearity performance analysis reveals the sensor has a maximum full-scale (FS) nonlinearity around 7% over the ±1000με operating range. Mechanical Brownian motion noise analysis reveals the mechanical noise of the sensing structure is 1.2× 10−6 με Hz , which is negligible. 136 6. CHAPTER SIX TEST FIXTURES AND DEVICE TESTING The chapter presents the test results of the fabricated mechanically amplified capacitive strain sensor and its associated test fixture development. The tests include: 1) capacitive output characteristics; 2) the voltage output characteristics when integrated with the developed low-noise capacitance-voltage (C/V) converter and the sensitivity evaluation; 3) the overall system’s behavior under elevated temperature; 4) the system’s step response and frequency response. 6.1. TEST FIXTURES FOR CAPACITANCE OUTPUT CHARACTERISTICS 6.1.1 FOUR-POINT STRAIN TESTING FIXTURE Four-point bending (FPB) test fixture is a standard method determining the material flexural properties [55], and can be used for strain sensor evaluation [56]. The four point bending fixture consists of a beam supported at both ends. The beam is with length of L , width of w , and thickness of h , and often the beam is made of elastic material with high yield strength, e.g., the stainless steel 17-4-PH. The diagram of a FPB beam is illustrated in Figure 6.1. 137 P2 P2 a a FrontView A Support h C C' A' L A − A' w SideView h Figure 6.1. Analysis of a four-point bending test fixture for strain sensor evaluation. A pair of forces P2 are applied a distance a from both of the beam supports. The bending moment M is uniform over the central portion (between C and C' ) of the beam, ⎛⎞P M = ⎜⎟⋅a (6-1) ⎝⎠2 The maximum stresses happen at the top and bottom surface of the beam, if the beam remains elastic; the nominal stress σ n can be expressed as, M hPah σ ==, (6-2) n I 24I where I is the moment of inertia, wh3 I = . (6-3) 12 138 Therefore the strain,ε , generated on the top and bottom surface is given by, σ Pah ε ==n . (6-4) E 4EI The displacementδ at CC'− where P 2 force is applied is expressed by [20], Pa δ = (3L2 − 4a 2 ) . (6-5) 48EI Combining (6-4) and (6-5) gives, (3L22− 4a ) δ = ⋅ε (6-6) 12h Therefore, by mounting strain sensor on the top or bottom surfaces of the beam, the sensors can be evaluated by applying precisely controlled displacements through an actuator. 139 Motorized actuator Four-point bending beam (17-4-PH) Sensor mounted on the beam Micrometer for displacement reading Figure 6.2. Picture of the developed four-point bending fixture for strain sensor evaluation. Figure 6.2 illustrates the developed four-point bending fixture for strain sensor evaluation. The displacement is applied by a computer controlled actuator. The displacement is read by a digital micrometer and interfaced to the computer. The parameters of the four-point bending beam apparatus is listed in Table 6.1, the developed four-point test fixture gives, δ = 1.99εε≅ 2 (m). (6-7) The digital micrometer has an accuracy of 0.5 µm. According to (6-7), the developed FPB test fixture provides a sensitivity of 0.5μεμ m and an overall test fixture resolution 140 of 0.25 με . Table 6.1. Parameters of the developed four-point test fixture. Material Length L (mm) Load distance a (mm) Thickness h (mm) Stainless steel 115 20 1.59 (17-4-PH) Interfaced with a computer, the developed test fixture is capable of automatic test. It also has the advantages allowing testing multiple sensors simultaneously, as the strains generated across the beam surface is uniform. However, the test fixture is very sensitive to beam misalignment and defects such as warp and twist, thus resulting in a poor repeatability and high hysteresis measured to be around 10 to 20 με - a major drawback in high precision testing. Additionally, the test fixture is not capable of performing the high-temperature testing due to its limited space for additional heating elements. This test fixture is soon replaced by a bent cantilever beam test fixture, as will be discussed. 6.1.2 BENT CANTILEVER BEAM TESTING FIXTURE The bent cantilever beam bending test fixture consists of a cantilever beam fixed at one end. A controlled displacement δ is applied at the other end. Unlike the FPB test fixture, the strain generated on the beam surface depends on the location, which is given by [54], 3h( L− x ) ε = sensor δ , (6-8) 2L3 141 where ε , h , L , xsensor , δ denote the strain generated on sensor, beam thickness, beam length, sensor center location, and applied displacement, respectively. Displacement applied by a manipulator Vise clamp Sensor δ Bent cantilever beam h xsensor L Figure 6.3. Diagram of a bent-cantilever beam bending test fixture. Manipulator Interface electronics Bent beam MEMS sensor Clamped edge Figure 6.4. Picture of a capacitive strain sensor under testing using a bent cantilever test fixture. 142 Figure 6.4 illustrates the developed bent cantilever beam test fixture for strain sensor evaluation. The displacement is applied by a manipulator with accuracy around 2.5 μm . Using the parameters of the cantilever beam apparatus listed inTable 6.2, (6-8) gives, ε = 0.166δ , (6-9) indicating that the test fixture provides a sensitivity of 0.166μεμ m and an overall test fixture resolution of 0.4 με . Table 6.2. Parameters of the developed bent cantilever beam test fixture. Material Length L (mm) xsensor (mm) Thickness h (mm) Stainless Steel 140 20 2.47 (17-4-PH) When applying this test fixture, special attention should be paid to the strain distribution distortion near the clamped edge, where the strain generated does not follow the (6-8). Finite element analysis results, as illustrated in Figure 6.5, show that there is noticeable strain distortion for location less than 20 mm away from the edge. Therefore the sensor should be mounted away from those locations. 143 Distorted strain distribution 200 Distorted strain distribution at clamp boundary ) με 150 100 Sensor location 50 Clamped edge Strian on ( Strian Beam Surface 0 0 20 40 60 80 100 120 140 x (mm) sensor Figure 6.5. Finite element analysis results of the strain distribution along the bent beam surface under a displacement load. The circled area indicates a distorted strain distribution around the clamped edge. 6.1.3 CAPACITIVE STRAIN SENSOR APPLYING PROCEDURE The procedure to apply a capacitive strain sensor on a stainless steel test beam is 144 described as follows. The goal of this procedure is to develop a layer of strong, stable bond between the sensor and the test beam. The procedure is developed following the recommendations provided by Vishay® (www.vishay.com). 1 Surface preparing – develop a chemical clean surface with proper roughness. 1.1 Mechanically remove rough impurities (scale, rust, etc). 1.2 Sand the surface using a piece of 150-grid SiC sand paper. Use “8” shaped pattern for the best results. 1.3 Degrease the surface using a chemically pure solvent such as RMS1 cleaning agent developed by HBM company (www.hbm.com). 2 Applying the capacitive strain sensor. 2.1 Apply a layer of MB610® using a fine brush. 2.2 Under microscope, gently put the sensor on the surface using a vacuum tweezer. Adjust position if necessary according to the alignment guild. 2.3 Push the sensor gently to make sure the bottom of the sensor fully contact the bonding adhesive layer. 2.4 Reapply a small amount adhesive around the sensor edge. 3 Sensor curing. 3.1 Put the stainless steel beam with the sensor on a hotplate (sensor up) and adjust the hotplate temperature to be 90ºC. 3.2 Apply a piece of Teflon® sheet on the sensor, and put a weight (50 gram) on the sensor. 145 3.3 Cure for at least 24 hours to achieve the maximum bonding strength. Figure 6.6 shows a picture of two capacitive strain sensors bonded on a cantilever beam ready for testing. When observed under high-power optical microscope, the comb driver fingers exhibit unbalanced capacitance output a(unequal overlapping length forC + andC − ) at room temperature, which is caused by the residual strain introduced by the elevated-temperature curing process due to the thermal-expansion-coefficient mismatch between the sensor and stainless steel beam. 146 Figure 6.6. Two capacitive strain sensors bonded on a stainless cantilever beam. Elevated temperature adhesive curing process results in an unbalanced capacitive output which can be observed under microscope. 147 6.2. PACKAGE GAIN & MECHANICAL GAIN MEASUREMENT UNDER MICROSCOPE To computer CCD Optical microscope Displacement applied by a manipulator Clamped edge δ Sensor Stainless steel cantilever beam Figure 6.7. Test setup for package gain & mechanical gain measurement. The overall module gain, Aoverall , which combines the package gain Ap and the mechanical amplifier’s gain Amech , Aoverall= AA p mech , is examined by measuring the finger overlapping length change under applied strains, as illustrated in Figure 6.7. An optical microscope is used to observe the finger movement, which is then captured using a CCD camera and analyzed by a computer. Some sample images are shown in Figure 6.8. The system provides an length measurement accuracy around 0.3 μm and gain measurement accuracy about ±3% . The measured overall gain, Aoverall , is 8.95±0.3, as shown in Figure 6.8. It is difficult to distinguish the package gain Ap and the amplifier’s gain Amech . However, if using the nominal gain 5 for the amplifier’s gain, the calculated package gain 148 is 1.79±0.05, which is a close match to the FEA result: 1.75, as discussed in Chapter 5. 149 Initial finger position Offset is introduced by the residual strain after the high-temperature adhesive curing After 500 µε tensile strain After 1000 µε tensile strain 45 m) μ Linear fitted slope: 17.9 μm/1000με 40 35 30 25 Comb Drive Finger Overlapping Length ( Length Overlapping Comb Drive Finger 0 100 200 300 400 500 600 700 800 Applied Tensile Strain (με) Figure 6.8. Captured images of comb drive fingers under microscope. The overall gain is measured to be 8.95. 150 6.3. CAPACITIVE OUTPUT CHARACTERISTICS 6.3.1 INTERFACED TO AN OFF-THE-SHELF C/V CONVERTER The device is interfaced to an off-the-shelf capacitance-to-voltage converter (C/V converter) MS3110®, which is illustrated in Figure 6.4. The C/V converter chip MS3110® is capable of interfacing a differential capacitor pair with a typical resolution of 4.0 aF Hz . It also provides an on-chip EEPROM to store the trim and program settings. The C/V circuitry provides a voltage output, which is measured by a Keithley 6 1 digital 2 multi-meter, Model 2000. Before applying the strains, the converter chip C/V characteristics in terms of linearity and sensitivity is examined by its internal calibration capacitors, which is claimed to have 10% accuracy according the chip’s datasheet. The measured sensitivity of the circuitry is 1.71 VpF, as shown in Figure 6.9. 3.0 Circuit sensitivity: 1.71 V/pF 2.8 2.6 2.4 (V) out 2.2 V 2.0 1.8 1.6 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 C (pF) s2 Figure 6.9. Measured C/V converter (MS3110) sensitivity. 151 Figure 6.10 presents the measured capacitance output versus an applied input strain, indicating the system can achieve a maximum input signal of 1000 με . The results show an initial offset differential capacitance of 0.1 pF , corresponding to a 350 με compressive strain (negative), which is mainly the result of the residual strain introduced in the elevated-temperature curing process due to the thermal-expansion-coefficient (TEC) mismatch between the sensor and stainless steel beam. The adhesive curing temperature is around 80 D C ; when the system is cooled down to room temperature (25 D C ), a temperature drop of 55 D C is introduced. Considering the TEC of silicon, 2.6× 10−−61D C and 17-4-PH stainless steel 10.8× 10−−61D C , the estimated residual strain is -455 με (negative sign means compressive), which agrees with the measurement. 0.4 Residual compressive 0.2 strain: ~ -350με (pF) out 0.0 Measured C -0.2 Compressive Tensile -0.4 -1000 -500 0 500 1000 Strain on Specimen Surface (με) Figure 6.10. Measured capacitance output characteristics using the off-the-shelf C/V converter MS3110®. 152 Figure 6.11 shows the capacitance output characteristics after the offset adjustment. The results show a measured sensitivity of 280 aF με along with an overall non-linearity of 6.6%FS (including all measurement instrument errors). Compared to the FEA results, the measurements agree well with a a maximum discrepancy around 2%. 0.4 Measurements FEA Results 0.2 (pF) out 0.0 Measured C Measured -0.2 -1000 -500 0 500 1000 Strain on Specimen Surface (με) Figure 6.11. Comparison of measured capacitance output characteristics after offset adjust and FEA results. 6.3.2 INTERFACED TO A DEVELOPED LOW NOISE C/V CONVERTER A low-noise C/V converter employing continuous-time synchronous detection architecture is developed to fulfill the sensitivity requirement. The circuit has been discussed in Chapter 3. The measured noise floor is 233 nV Hz , as shown in Figure 6.12. (The noise measured for this version of chip is lower than that used for sensor discussed in Chapter 3 due to design improvement, though the basic architecture remains the same.) 153 B: CH2 Pwr Spec X:4.992 kHz Y:232.638 n* 1 Y* = Vrms/rtHz m* LogMag 6 decades 1 n* 0HzAVG: 32 12.8kHz Figure 6.12. Measured noise floor of the developed C/V Converter powered by a 3-V battery. The measured noise floor is 233 nV Hz nV/Hz. Figure 6.13 presents the measured output voltage versus an applied input strain range of ± 1000 με after offset adjustment. The results demonstrate an overall sensitivity of 0.705 mV με . The sensor responds to tensile strain with slightly higher sensitivity (0.836 mV με ) than that of compressive (0.57 mV με ) and the overall nonlinearity is 6.0%FS. Therefore the overall system demonstrates an overall noise level of 233 nV Hz , corresponding to an input referred strain of 0.033 με and displacement of 0.33 Å over 10 kHz bandwidth. The overall system can achieve a signal noise ratio (S/N) of 8dB at 0.1 με resolution level. 154 0.8 0.6 0.4 0.2 0.0 -0.2 IC Vout (V) -0.4 -0.6 -0.8 Compressive Tensile -1000 -500 0 500 1000 Strain applied on the Sensor Package (microstrain) Figure 6.13. Measured capacitance output characteristics using the developed low-noise C/V converter. 6.4. TEMPERATURE BEHAVIOR The overall strain measurement system after the sensor is bonded on the substrate shows a significant temperature dependency due to the TEC mismatch between the sensor and the specimen – a key concern in long-term stability performance. The strain reading drift testing is performed to examine the overall system stability. In the testing, a fixed compressive strain of -271 με (compressive) is applied on the test beam. The strain reading is recorded in 0.25-second interval through a computer interface. For comparison, a K-type thermal couple is placed very close to the sensor and the 155 temperature reading is recorded simultaneously. The test results illustrated in Figure 6.14 show that the read strain has a drift of 0.25 με for the 300-second period and there is a strong dependency on temperature if compared to the measured temperature variation. There is also a positive sign of the relationship, indicating the read strain increases with an elevated temperature, which is consistent with the behavior of the TEC mismatch. -271.3 23.80 23.78 -271.4 ) C o ( ) 23.76 με ( -271.5 0.25 με 23.74 -271.6 23.72 Strain Reading Reading Strain -271.7 Reading Temperature 23.70 -271.8 23.68 0 50 100 150 200 250 300 Time (second) Figure 6.14. Voltage output dependency on environmental temperature variation. A modified test setup has been developed to test the sensor performance at elevated temperature, as illustrated in Figure 6.15. A thin rectangular tape heater with size of 13mm× 50mm is attached to the stainless steel bent beam on the surface opposite to where the sensor is mounted. With the maximum power of 10 Walts, the heater can quickly heat the sensor to various temperatures up to 120 D C by adjusting the power supply voltages. A thermal couple is placed very close to the sensor for temperature monitoring. 156 Thermal couple Displacement applied by a manipulator Sensor+Interface δ electronics Tape heater Figure 6.15. Diagram of the modified test setup for sensor evaluation at elevated temperature. The measured sensor system voltage output characteristics is illustrated in Figure 6.16. The behavior is examined under three different temperatures: 24.6 ºC, 52.6ºC, and 112ºC, respectively. The test results show a significant offset voltage shift at different temperature with sensitivity around 6.2 με D C , mainly contributed by the TEC mismatch (8.2×10−6 D C ). The voltage output sensitivity also varies slightly with the temperature, as illustrated in Figure 6.17, which is a combined effect of sensor module and the interface electronics. Using the room temperature (24.6 ºC ) sensitivity as reference, the measured sensitivity slightly increase about 1.3% at 81 ºC, and decreases about 2.8% at 112 ºC. Over the temperature range from room temperature to 112 ºC, the maximum sensitivity variation is 2.8%, 157 0.8 24.6 oC 0.6 112 oC 52.6 oC 0.4 0.2 (V) 0.0 out V -0.2 -0.4 -0.6 -1000 -500 0 500 1000 Applied Strain (με) Figure 6.16. Measured voltage output characteristics at different temperature. C) o 0.02 @24.6 out 0.01 0.00 -0.01 -0.02 -0.03 20 40 60 80 100 120 Sensitivity Change (ratio referenced to V to referenced Change (ratio Sensitivity out o V Temperature ( C) (b) Figure 6.17. Measured voltage output sensitivity change of the overall system including the sensor the developed interface electronics at different temperature (referenced to sensitivity at 24.6ºC). 158 6.5. STEP RESPONSE The step response is measured by quickly applying a strain step to the sensor module and recording the strain output reading versus time in a short time interval, e.g., 0.25 second. Figure 6.18 shows the measured strain after a step tensile strain of 707 με is applied to the sensor module a slope of 70 με sec ond . The measured results show the sensor reading quickly reaches the peak. However it will slowly drop and reaches the final stabilized reading in about 100 seconds. The drop is 1.8 με in amplitude, and a 0.25% in ratio with the peak value. Various step amplitudes have been tested ranging from 100 με to 1000 με in tensile or compressive. The results show varied drop in amplitude, but a constant drop in ratio around 0.25%. This phenomenon is believed to be related to the creep property of the bonding adhesive or the test equipment, though no further study has been performed to investigate this problem. 159 800 700 708.0 600 707.5 ) 707.0 με ( 500 706.5 400 706.0 1.8 με 705.5 300 705.0 200 704.5 Strain Reading Reading Strain 704.0 100 0 50 100 150 200 250 300 0 0 50 100 150 200 250 300 Time (second) Figure 6.18. The step response. 6.6. FREQUENCY RESPONSE TESTING The frequency response testing is to examine the sensing system’s frequency response to an alternated strain signal. The input alternated signal is generated by pushing the cantilever beam using a PZT actuator with various frequencies. 160 6.6.1 TEST FIXTURE The test setup is illustrated in Figure 6.19. A stainless beam made of 17-4-PH is clamped at both ends. A PZT actuator is mounted underneath the beam. Once driven by a sinusoidal signal, the PZT generates alternated displacements, which will bend the beam to introduce the alternated strain. Double Clamped Beam Clamps (17-4-PH) PZT Actuator (P-820-10) PZT Actuator Supporting Base Supports Figure 6.19. Frequency response testing test setup. The ordered PZT actuator has a flat frequency response up to 10 kHz ; therefore in order to perform the frequency response with bandwidth up to 10 kHz , the fundamental resonant frequency of double-end-clamped vibrating beam should be high enough to avoid the signal distortion. 6.6.2 RESONANT CANTILEVER BEAM RESONANT FREQUENCY 161 CALCULATION The basic equation to calculate the fundamental resonant frequency of a double-end-clamped beam is given by, E f = 1.03 ( h ) , 1 ρ L2 where f1 , E , ρ , h , L denote the fundamental resonant frequency, Young’s modulus, mass density, thickness, and length, respectively. For a 1/16” stainless steel 17-4-PH beam, the material properties and geometrical parameters are listed in Table 6.3 and the calculated fundamental resonant frequency as a function of length is listed in Table 6.4. Table 6.3. Beam parameters. E 197 GPa ρ 7.8×103 kg/m3 h 1.59 mm w (beam width) 12 mm Table 6.4. Calculated fundamental resonant frequencies of various beam lengths. Length Fundamental resonant FEA Results frequency 25.4 mm 12.75 kHz 12.9 kHz 38.1 mm 5.7 kHz 5.7 kHz 50.8 mm 3.2 kHz 3.2 kHz 63.6 mm 2.04 kHz 2.0 kHz 76.2 mm 1.41 kHz 1.42 kHz The test fixture uses the length of 63.6 mm, which has a resonant frequency of 2.0 kHz 162 according to Table 6.4. 6.6.3 ACTUATION DISPLACEMENT VERSUS GENERATED STRAIN Clamps x Sensor location ε h Δz: generated by a PZT actuator L Figure 6.20. Diagram of the double-end-clamped beam under displacement load at center. The generated strain (at the center, x = L / 2 ): 12 ε = hΔZ , (6-10) L2 where L, h, ΔZ denote the beam length, beam thickness, and applied displacement, respectively. For the test fixture with length of 63.6 mm , the generated strain for an applied 1 μm displacement is 3.5 με . Therefore, for a PZT actuator with a maximum displacement of 4 μm , the maximum strain which can be achieved is 14 με . 6.6.4 TEST FIXTURE BEHAVIOR The test fixture behavior is examined by measuring the generated displacement at beam center driven by a variety of frequencies and comparing to the displacement of the 163 PZT actuator without attaching to the test beam. The beam vibration amplitude at the center is measured by a laser vibrometer. The PZT actuator is driven by a sinusoidal signal with 10Vpp− . When unloaded, the PZT actuator has a relatively flat frequency response with amplitudes around 800 nm . When loaded with the test beam, the displacement has a flat response up to 6 kHz and quickly reaches the peak and drops. Further examining of vibration modes reveals that a single vibration mode only happen at frequencies under 2 kHz and multiple vibration modes exist if the driving frequencies exceed 2 kHz , indicating the maximum test frequency for this test fixture is 2 kHz , which is consistent with the analysis. 1200 Beam displacement 1000 800 600 Displacement (nm) 400 PZT displacement without 200 0 0 2000 4000 6000 8000 10000 Frequency (Hz) Figure 6.21. Measured PZT displacement without load and beam displacement driven by the PZT actuator. The driven voltage level is 10 V Vpp. 164 6.6.5 SENSOR DYNAMIC BEHAVIOR Figure 6.22 illustrates the test setup for examining the system dynamic behavior. The sensor and the interface electronics are mounted on the test bar driven by a PZT actuator to mimic the dynamic input. The overall system, including the sensor and the interface electronics are powered wirelessly through the inductive link consisting of an internal loop, and an external loop with certain separation distance between them. This wireless inductive link also provides the data transmission link. Sensor + IC Wireless link Figure 6.22. Test setup for system dynamic behavior. Figure 6.23 shows the received power spectrum when a 10 µεpp input signal at 1kHz is applied to the sensor, achieving an average noise floor of - 130dBV/√Hz, which corresponds to a minimum detectable strain of 0.05 µεrms over a 10kHz BW. 165 0 -20 10μεpp input signal @ 1kHz /Hz) 2 -40 -60 -80 -100 -120 Output Power Spectral Density (dBV -140 -160 2 4 6 10 10 10 Frequency (Hz) Figure 6.23. Received power spectrum of strain sensor under a dynamic 10 µεpp signal @ 1 kHz. 6.7. CONCLUSIONS Various test fixtures have been developed to evaluate the performance of the developed capacitive strain sensor. Integrated with the developed low-noise C/V converter interface electronics, the overall system is able to detect a minimum strain of 0.033 με with the maximum range of ± 1000 με , indicating an equivalent dynamic range of 89 dB has been achieved. Tested under elevated temperature, the sensor system shows an offset shift around 6.2 με D C , mainly contributed by the TEC mismatch between the sensor (silicon) and testing specimen (stainless steel 17-4-PH), indicating an offset compensation mechanism is desired. The sensitivity also shows slight variation with elevated 166 temperature and a maximum 2.9% sensitivity drop has been observed at temperature 112ºC. The sensor system also responds to step input with a 0.25% drift. 167 BIBLIOGRAPHY [1] J. M. Gere and S. P. Timoshenko, Mechanics of materials, 4th ed. Boston: PWS Pub Co., 1997. [2] C. C. Perry and H. R. Lissner, The strain gage primer, 2d ed. New York: McGraw-Hill, 1962. [3] M. L. Nagy, C. Apanius, and J. W. Siekkinen, "A User Friendly, High-Sensitivity Strain Gauge," Sensors, vol. 18, pp. 20-27, 2001. [4] K. Ikeda, H. Kuwayama, T. Kobayashi, T. Watanabe, T. Nishikawa, T. Yoshida, and K. Harada, "Silicon Pressure Sensor Integrates Resonant Strain Gauge on Diaphragm," Sensors and Actuators, vol. A 21-23, pp. 146-150, 1990. [5] J. G. Webster and R. Pallàs-Areny, Sensors and signal conditioning, 2nd ed. New York: Wiley, 2001. [6] Vishay Inc., "Interactive Guide to Strain Measurement Technology," www.vishay.com. [7] W. M. Murray and P. K. Stein, Strain gage techniques, 1957. [8] P. R. Gray and R. G. Meyer, Analysis and design of analog integrated circuits, 3rd ed. New York: Wiley, 1993. [9] Analog Device Inc., "Datasheet for instrumentation Amplifiers AD8221," www.analog.com. [10] C. S. Smith, "Piezoresistance Effect in Germanium and Silicon," Physical Review, vol. 94 (1), pp. 42-49, 1954. [11] S. M. Sze, Semiconductor sensors. New York: Wiley, 1994. [12] Y. Kanda, "Piezoresistance Effect of Silicon," Sensors and Actuators, vol. A 28 (2), pp. 83-91, 1991. [13] J. J. Wortman and R. A. Evans, "Young's Modulus, Shear Modulus, and Poisson's Ratio in Silicon and Germanium," Journal of Applied Physics, vol. 36 (1), pp. 153-156, 1965. [14] O. N. Tufte and E. L. Stelzer, "Piezoresistive Properties of Heavily Doped n-Type 168 Silicon," Physical Review, vol. 133 (6A), pp. A1705-A1716, 1964. [15] O. N. Tufte and E. L. Stelzer, "Piezoresistive Properties of Silicon Diffused Layers," Journal of Applied Physics, vol. 34 (2), pp. 313-318, 1963. [16] J. A. Harley and T. W. Kenny, "1/f noise considerations for the design and process optimization of piezoresistive cantilevers," J. MEMS, vol. Vol. 9 (2), pp. pp.226-235, 2000. [17] G. Stemme, "Resonant Silicon Sensors," J. Micromech. Microeng, vol. 1, pp. 113-125, 1991. [18] O. Brand, H. Baltes, and E. Zurich, Micromachined Resonant Sensors - an Overview, vol. 4: Willey-Vch, 1999. [19] T. Ikehara, R. Zwijze, and K. Ikeda, "New method for an Accurate Determination of Residual Strain in Polycrystalline Silicon Films by Analysing Resonant Frequencies of Micromachined Beams," J. Micromech. Microeng, vol. 11, pp. 55-60, 2001. [20] R. J. Roark, W. C. Young, and R. G. Budynas, Roark's formulas for stress and strain, 7th ed. New York: McGraw-Hill, 2002. [21] R. T. Howe, "Resonant Microsensors," presented at 4th Internaional Conference on Solid-State Sensors and Actuators (Tranducer '87), Tokyo, Japan, 1987. [22] R. T. Howe, B. E. Boser, and A. P. Pisano, "Polysilicon Integrated Microsystems: Technologies and Applications," Sensors and Actuators, vol. A 56, pp. 167-177, 1996. [23] R. Legtenberg and H. A. C. Tilmans, "Electrostatically driven vacuum-encapsulated polysilicon resonators Part I. Design and fabrication," Sensors and Actuators, vol. A 45, pp. 57-66, 1994. [24] R. Legtenberg and H. A. C. Tilmans, "Electrostatically driven vacuum-encapsulated polysilicon resonators Part II. Theory and performance," Sensors and Actuators, vol. A 45, pp. 67-84, 1994. [25] T. K. Tang, R. C. Gutierrez, C. B. Stell, V. Vorperian, G. A. Arakaki, J. T. Rice, W. J. Li, I. Chakraborty, K. Shcheglov, and J. Z. Wilcox, "A Packaged Silicon MEMS Vibratory Gyroscope for Microspacecraft," presented at The Tenth Annual International Workshop on Micro Eletro Mechanical Systems, Nagoya, Japan, 1997. 169 [26] M. Elwenspoek, S. B. F. R. Blom, T. S. J. Lammerink, F. C. M. v. d. Pol, H. A. C. Tilmans, T. J. A. Popma, and J. H. J. Fluitman, "Transduction mechanisms and their applications in micromechanical devices," presented at Micro Electro Mechanical Systems, 1989 MEMS '1989, Proceedings: 'An Investigation of Micro Structures, Sensors, Actuators, Machines and Robots'. IEEE, 1989. [27] M. J. Vellekoop, G. W. Lubking, P. M. Sarro, and A. Venema, "Integrated-circuit-compatible design and technology of acoustic-wave-based microsensors," Sensors and Actuators, vol. A 44, pp. 249-263, 1994. [28] T. S. J. Lammerink, M. Elwenspoek, and J. H. J. Fluitman, "Frequency Dependence of Thermal Excitation of Micromechanical Resonators," Sensors and Actuators, vol. A 25, pp. 685-689, 1991. [29] D. L. Seibert and B. Bonvalot, "Optical system for pressure and temperature sensing," Sensors and Actuators, vol. A 43, 1994. [30] T. Kvisteroy, O. H. Gusland, and B. Stark, "Optically excited silicon sensor for permanently installed downhole pressure monitoring applications," Sensors and Actuators, vol. A 31, pp. 164-167, 1992. [31] A. Meckes, J. Behrens, and W. Benecke, "Electromagnetically Driven Microvalve Fabricated in Silicon," presented at 1997 International Conference on Solid-State Sensors and Actuators (Transducer '97), Chicago, 1997. [32] T. R. Albrecht, P. Grütter, D. Horne, and D. Rugar, "Frequency Modulation Detection Using High-Q Cantilevers for Enhanced Force Microscope Sensitivity," Journal of Applied Physics, vol. 69 (2), pp. 668-673, 1991. [33] K. E. Wojciechowski, B. E. Boser, and A. P. Pisano, "A MEMS Resonant Strain Sensor Operated in Air," Micro Electro Mechanical Systems, 2004. 17th IEEE International Conference on. (MEMS), pp. 841-845, 2004. [34] M. Andrews, I. Harris, and G. Turner, "A comparison of squeeze-film theory with measurements on a microstructure," Sensors and Actuators, vol. A 36, pp. 79-87, 1993. [35] K. E. Wojciechowski, B. E. Boser, and A. P. Pisano, "A MEMS Resonant Strain Sensor with 33 nano-strain Resolution in a 10 kHz Bandwidth," presented at IEEE SENSOR, 2005. [36] K. I. Arshak, D. McDonagh, and M. A. Duran, "Development of new capacitive strain sensors based on thick film polymer and cermet technologies," Sensors and 170 Actuators, vol. A 79, pp. 102-114, 2000. [37] J. Aebersold, "Design and development of a MEMS capacitive bending strain sensor," J. Micromechanics and Microengineering, vol. 16, pp. 935-942, 2006. [38] D. W. Schindel, D. A. Hutchins, and S. T. Smith, "A study of material at high temperature using miniaturized resonant tuning forks and noncontact capacitance transducers," J. Acoust. Soc. Am., vol. 102 (3), 1997. [39] G. H. Kroetz, M. H. Eickhoff, and H. Moelleer, "Silicon compatible materials for harsh environment sensors," Sensors and Actuators, vol. A 74, pp. 182-189, 1999. [40] M. Suster, J. Guo, N. Chaimanonart, W. H. Ko, and D. J. Young, "Low-Noise CMOS Integrated Sensing Electronics for Capacitive MEMS Strain Sensors," in IEEE Custom Integrated Circuits Conference, 2004. [41] Y. B. Cianchandani and K. Najafi, "Bent-Beam Strain Sensors," JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, vol. 5 (1), pp. 52-58, 1996. [42] L. Lin, A. P. Pisano, and R. T. Howe, "A Micro Strain Gauge with Mechanical Amplifier," in JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, vol. 6, 1997, pp. 313-321. [43] S. Timoshenko and S. Woinowsky-Krieger, Theory of plates and shells, 2d ed. New York: McGraw-Hill, 1959. [44] T. B. Gabrielson, "Mechanical-thermal noise in micromachined acoustic and vibration sensors," Electron Devices, IEEE Transctions on, vol. 40 (5), pp. 903-909, 1993. [45] B. E. Boser and R. T. Howe, "Surface micromachined accelerometers," IEEE Journal of Solid-State Circuits, vol. 31, pp. 366-375, 1996. [46] C. J. Wilson and P. A. Beck, "Fracture testing of bulk silicon microcantilever beams subjected to a side load," J. MEMS, vol. 5 (3), pp. 142-150, 1996. [47] C. J. Wilson, A. Ormeggi, and M. Narbutovskih, "Fracture testing of silicon microcantilver beams," J. Appl. Phys., vol. 79 (5), pp. 2386-2393, 1996. [48] M. Suster, J. Guo, N. Chaimanonart, W. H. Ko, and D. J. Young, "A High-Performance MEMS Capacitive Strain Sensing Microsystem," IEEE Journal of Microelectromechanical Systems, vol. 15 (5), pp. 1069-1077, 2006. 171 [49] M. Tavakoli and R. Sarpeshkar, "An offset-cancelling low-noise lock-in architecture for capacitive sensing," IEEE Journal of Solid-State Circuits, vol. 38, pp. 244-253, 2003. [50] G. K. F. J. Wu and R. Carley, "A low-noise low-offset chopper-stabilized capacitive-readout amplifier for CMOS MEMS accelerometers," IEEE International Solid State Circuits Conference, pp. 428-429, 2002. [51] W. H. Ko, "Packaging of Micro-fabricated Devices and Systems," Materials Chemistry and Physics, vol. 42 (3), pp. 169-175, 1995. [52] H. Kuo, J. Guo, D. J. Young, and W. H. Ko, "Packaging of Micro Strain Sensors," presented at IEEE Sensors, 2004. [53] B. D. Sosnowchik, R. G. Azevedo, A. Cao, L. Lin, and A. P. Pisano, "Silicon-to-steel bonding using rapid thermal annealing," Advanced Packaging, IEEE Transactions on, vol. 28 (4), pp. 626 - 634, Nov. 2005. [54] J. Guo, H. Kuo, D. J. Young, and W. H. Ko, "Buckled Beam Linear Output Capacitive Strain Sensor," in Solid-state Sensor, Actuator and Microsystems Workshop 2004. Hilton Head Island, 2004, pp. 399-403. [55] American Society for Testing of Materials, "D6272-00 Standard Test Method for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials by Four-Point Bending," West Conshohocken, Pennsylvania: American Society of Testing and Materials, 2001. [56] H. W. Ng and P. Wey, "Strain gage evaluation with four-point bending at moderate temperatures," Experimental Mechanics, vol. 37 (3), pp. 237-244, Sept. 1997. 172