UNIVERSITY OF CINCINNATI
07/12/2005
Date:___________________
Xingguo Xiong
I, _________________________________________________________,
hereby submit this work as part of the requirements for the degree of:
Ph.D.
in:
Computer Engineering
It is entitled:
Built-in Self-test and Self-repair for Capacitive MEMS Devices
This work and its defense approved by:
Dr. Wen-Ben Jone
Chair: _______________________________
Dr. Chong Ahn
_______________________________
Dr. Fred Beyette
_______________________________
Dr. Frank Gerner
_______________________________
Dr. Carla Purdy
_______________________________
Built-in Self-Test and Self-Repair
for Capacitive MEMS Devices
A dissertation submitted to the
Division of Research and Advanced Studies
of the University of Cincinnati
in partial fulfillment of the
requirements for the degree of
DOCTOR OF PHILOSOPHY
in the Department of
Electrical and Computer Engineering and Computer Science
of the College of Engineering
July, 2005
by
Xingguo Xiong
B.S.(Physics) Wuhan University, Wuhan, China, July 1994
Ph.D (E.E.) Shanghai Institute of Microsystem and IT, Shanghai, China, July 1999
Thesis Advisor and Committee Chair: Dr. Wen-Ben Jone
Abstract
With the rapid development of MEMS (microelectromechanical system) and its increasing applications
to safety-critical applications, MEMS testing and fault-tolerant MEMS design are becoming more and more
important. A robust and efficient MEMS testing solution is in urgent need for MEMS commercialization,
and yield and reliability are also very important issues for MEMS devices. Unfortunately, research in these
fields still remains in its infancy. In this thesis both built-in self-test (BIST) and built-in self-repair (BISR)
of capacitive MEMS devices are studied.
First, we propose a dual-mode built-in self-test (BIST) technique for capacitive MEMS devices. The
BIST technique partitions the fixed (instead of movable) capacitance plates. Due to this partition, the BIST
technique can be extended to bulk micromachining and other MEMS technologies. Based on the partition,
both sensitivity and symmetry BIST modes can be implemented. Since each of both modes has its own
fault coverage, a combination of them ensures a more robust test solution. Three typical capacitive MEMS
devices are used as examples to demonstrate the effectiveness of the dual-mode BIST method. Different
defects are simulated for these devices. Simulation results prove the effectiveness of the dual-mode BIST
technique.
Based on the dual-mode BIST technique, a built-in self-repair (BISR) technique for comb accelerometer
devices is proposed. The BISR technique uses modularized design for each device. The device consists
of several identical modules. Among them, some are connected as the main device, while others act as
redundancy. If any of the working module is found faulty during BIST, the control circuit will replace it
with a good module. In this way, the device can be self-repaired. Performance analysis shows that the
device suffers sensitivity loss due to the modularized design. This can be compensated by adjusting the
device parameters such as shrinking the beam width, etc. Electrostatic force can also be used as a powerful
tool to compensate the sensitivity back to normal. The BISR scheme introduces 50% of area overhead.
However, with this paid price, we gain great improvement in device yield as well as its reliability.
In order to evaluate the effectiveness of the BISR scheme on yield improvement, a yield model for
MEMS redundancy repair is developed. The result demonstrates that a significant yield increase can be
achieved for moderate initial yield. The defect fatal rates are also considered in the yield model. Monte
Carlo simulation is performed and its result demonstrates an effective yield increase due to redundancy
repair. The control circuit for the BISR implementation is also discussed, and the parasitic effects of the
control and the BISR device are analyzed. In order to evaluate the reliability enhancement due to redundancy repair, a MEMS reliability model is also developed. Based on the reliability model, we evaluate the
MEMS reliability in three different failure mechanisms: fatigue, shock and stiction. Analysis results prove
that the BISR design leads to effective reliability enhancement for various failure mechanisms.
2
Acknowledgements
First I would like to express my sincere gratitude to my advisor, Dr. Wen-Ben Jone, for all his guidance,
support and constructive inputs throughout this work. He was always ready to provide his valuable help
in solving various problems and guiding the direction of my research. This work would have never been
possible without his encouragement and expertise. I am greatly impressed by his profound knowledge,
creative thoughts and careful research style. What I learned from Dr. Jone is not only valuable knowledge
and skills, but also the principles and styles for research. These will surely be very beneficial for me in my
future career. As an advisor, Dr. Jone also develops good friendship with his students. He always offers
considerate help in their lives. I would be very thankful to him for all his help to me and my family during
my entire graduate study. He taught me the positive attitude toward academic research as well as life. I am
very grateful for all I learned from him, both in knowledge and in life attitude.
I would like to express my deep appreciation to my thesis committee members for their valuable instructions and help in guiding my thesis research. They are listed alphabetically as follows: Dr. Chong Ahn, Dr.
Fred Beyette, Dr. Frank Gerner, Dr. Wen-Ben Jone and Dr. Carla Purdy. I am sincerely grateful to them for
spending their valuable time reviewing this work, raising important issues and guiding the directions of my
research. I would also like to thank my friends and colleagues in the VLSI Testing and Low Power VLSI
Design group for their valuable discussions and help. They are S. Shenoy, F. Lu, M. Li, W. Mao, S. Gosh,
K. Srivatsa, K. Maddi, S. Hariharan, etc. Special thanks are also due to my close friends Hui Yang, Maojun
Gong and their parents for their friendship and valuable help during the difficult time in my life.
Finally, I would express my great thanks to my wife, Hongli Lu and my little daughter, Grace Cindy
Xiong for their wonderful love in my life and their support to my study and research. I am also very thankful
to my parents, brothers and sisters for their everlasting love and support ever since I was a child. The love
from my family is the strong drive for me to continuously strive in my future career.
3
Contents
1
Introduction
1.1
1.2
1.3
1.4
1.5
1
Introduction to MEMS . . . .
Built-in Self-test of MEMS . .
Built-in Self-repair of MEMS .
Our Work . . . . . . . . . . .
Contents of the Thesis . . . .
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1
5
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10
2
Background
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Non-BISR MEMS Comb Accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
11
13
3
Dual-Mode Built-In Self-Test for Capacitive MEMS Devices
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Basic Concepts of the Proposed BIST technique . . . . . . . . . . . . . . . . . . . . . . .
17
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3.2.1 Basic Knowledge for Capacitive MEMS Devices
3.2.2 The Symmetry BIST Scheme . . . . . . . . . .
3.2.3 The Dual-Mode BIST Technique . . . . . . . . .
The Dual-Mode BIST for Different Technologies . . . .
3.3.1 Surface-Micromachined Comb Accelerometer .
3.3.2 Bulk-Micromachined Capacitive Accelerometer .
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4.2
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29
29
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Simulation Results of Dual-Mode Built-In Self-Test
4.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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27
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Poly-Si Surface-Micromachined Microresonator . . . . . . . . . . . . . . . . . .
4.1.1 Surface-Micromachined Comb Accelerometer
4.1.2 Bulk-Micromachined Accelerometer . . . . . .
4.1.3 Poly-Si Surface-Micromachined Resonator . .
4.1.4 Testing of Defects in Fixed Capacitance Plates
Discussions . . . . . . . . . . . . . . . . . . . . . . .
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19
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3.3.3
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3.3
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29
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35
5
6
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35
4.2.3
Built-in Self Repair of MEMS Devices . . . . . . . . . . . . . . . . . . . . . . .
36
39
5.1
BISR MEMS Comb Accelerometer Design . . . . . . . . . . . . . . . . . . . . . . . . .
39
5.2
Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
42
5.2.2
Sensitivity Compensation through Electrostatic Force . . . . . . . . . . . . . . . .
45
5.2.3
Frequency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
Yield Analysis of MEMS Redundancy Repair
53
6.1
Yield analysis of the BISR comb accelerometer . . . . . . . . . . . . . . . . . . . . . . .
53
6.1.1
6.1.2
Yield Model for MEMS Redundancy Repair . . . . . . . . . . . . . . . . . . . .
Yield Increase Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
57
6.1.3
Yield Model Revision Considering Defect Fatal Rate . . . . . . . . . . . . . . . .
60
Monte Carlo Simulation of Point-stiction Defects . . . . . . . . . . . . . . . . . . . . . .
64
Circuit Support and Simulation Results of BISR Accelerometer
73
7.1
Circuit Support for MEMS BISR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.1 Differential Capacitance Sensing Circuit . . . . . . . . . . . . . . . . . . . . . . .
73
73
7.1.2
BISR Control Circuit Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
7.1.3
Parasitic Capacitance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
7.1.4
Analog Multiplexers and Signal Strength Analysis . . . . . . . . . . . . . . . . .
77
7.1.5
7.1.6
Defective Module Isolation and Load Effect Analysis . . . . . . . . . . . . . . . .
BIST Circuit for BISR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
78
Design and Simulation of BISR Accelerometer . . . . . . . . . . . . . . . . . . . . . . .
79
7.2
8
Capacitance and Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calibration and Sensitivity BIST . . . . . . . . . . . . . . . . . . . . . . . . . . .
Built-in Self-repair of MEMS Comb Accelerometers and Performance Analysis
6.2
7
4.2.1
4.2.2
Reliability Analysis of MEMS Comb Accelerometer
83
8.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
8.2
8.3
Basic Concepts of Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reliability Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
85
8.4
MEMS Failure Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
8.4.1
Material Fatigue and Aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
8.4.2
Mechanical Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
8.4.3
8.4.4
Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
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8.4.5
Wear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
6
8.5
8.6
8.7
8.8
9
8.4.6 Delamination . . . . . . . . . . . . . . . . . . . . . . .
8.4.7 Residual Stress . . . . . . . . . . . . . . . . . . . . . .
8.4.8 Vibration . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.9 Humidity . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.10 Particle Contamination . . . . . . . . . . . . . . . . . .
8.4.11 Electrostatic Discharge . . . . . . . . . . . . . . . . . .
Reliability Analysis of MEMS Comb Accelerometer . . . . . .
8.5.1 Reliability Model of Non-BISR MEMS Device . . . . .
8.5.2 Reliability model of BISR MEMS device . . . . . . . .
8.5.3 Reliability Enhancement and Reliability Analysis . . . .
Reliability of MEMS accelerometer in Material Fatigue . . . . .
8.6.1 Fatigue Analysis and Cycles to Failure . . . . . . . . . .
8.6.2 Reliability Analysis by Cycles to Failure . . . . . . . .
8.6.3 Maximum Stress and Beams . . . . . . . . . . . . . . .
8.6.4 Reliability Comparisons . . . . . . . . . . . . . . . . .
8.6.5 Weak Device and Material Fatigue . . . . . . . . . . . .
The Reliability of MEMS Accelerometer in Shock Survival . . .
8.7.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . .
8.7.2 Reliablity Analysis for Shock . . . . . . . . . . . . . .
8.7.3 Mean Fracture Acceleration and Safety Factor Analysis
8.7.4 Simulation Results . . . . . . . . . . . . . . . . . . . .
Reliability of MEMS Accelerometer in Stiction . . . . . . . . .
8.8.1 MEMS Reliability in Resisting Contact . . . . . . . . .
8.8.2 Stiction Survival Probability after Contact . . . . . . . .
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93
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120
Conclusions and Future Work
127
9.1 Contributions of This Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
9.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7
List of Figures
1.1
Photo of ADXL250 accelerometer by Analog Devices Inc. . . . . . . . . . . . . . . . . .
3
1.2
Photo of digital micromirror device by Texas Instruments Inc. . . . . . . . . . . . . . . .
4
1.3
Photo of LambdaRouter optical switch by Lucent Technologies Inc. . . . . . . . . . . . .
4
1.4
Photo of MEMS gene chip by Affymetrix Inc. . . . . . . . . . . . . . . . . . . . . . . . .
5
2.1
The general design of MEMS comb accelerometer. . . . . . . . . . . . . . . . . . . . . .
13
2.2
The schematic diagram of differential capacitance.
. . . . . . . . . . . . . . . . . . . . .
14
3.1
Schematic diagram of a capacitive MEMS device. . . . . . . . . . . . . . . . . . . . . . .
19
3.2
MEMS capacitance structure for our symmetry test scheme. . . . . . . . . . . . . . . . .
21
3.3
Fixed capacitance plates partition for MEMS device. . . . . . . . . . . . . . . . . . . . .
22
3.4
Voltage biasing schemes for the three modes of MEMS device. . . . . . . . . . . . . . . .
23
3.5
The control circuit for dual BIST technique. . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.6
Structural diagram of a comb accelerometer. . . . . . . . . . . . . . . . . . . . . . . . . .
24
3.7
Structural diagram of bulk-micromachined accelerometer. . . . . . . . . . . . . . . . . . .
25
3.8
Revised design of bulk-micromachined accelerometer. . . . . . . . . . . . . . . . . . . .
26
3.9
Structural diagram of a comb microresonator. . . . . . . . . . . . . . . . . . . . . . . . .
27
5.1
SEM photo of a bulk-micromachined comb accelerometer device[39]. . . . . . . . . . . .
40
5.2
Modularized comb accelerometer structure. . . . . . . . . . . . . . . . . . . . . . . . . .
40
5.3
The relationship between displacement sensitivity of accelerometer and beam/mass width.
43
5.4
The sensitivity increase of accelerometer due to DC voltage biasing. . . . . . . . . . . . .
46
5.5
The sensitivity increase of accelerometer due to DC voltage biasing. . . . . . . . . . . . .
49
6.1
Fault distribution among the modules of BISR accelerometer. . . . . . . . . . . . . . . . .
55
6.2
The yield increase vs initial yield for different m numbers. . . . . . . . . . . . . . . . . .
59
6.3
The yield after repair vs initial yield for different m numbers. . . . . . . . . . . . . . . . .
59
6.4
The yield increase vs initial yield for different n numbers. . . . . . . . . . . . . . . . . . .
60
6.5
The yield increase for different m and n numbers when Y0=0.698. . . . . . . . . . . . . .
61
6.6
The yield increase vs initial yield for different m numbers when defect fatal rate is considered. 63
9
The yield increase vs initial yield with/without considering defect fatal rate (m = 2, n = 4,
k = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
6.8
Point-stiction and its formation.
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65
6.9
The random scatter of a point-stiction defect for 1000 samples (two defects) in Monte Carlo
simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
6.10 The sensitivity history for 1000 samples (two defects) in Monte Carlo simulation. . . . . .
67
6.11 The sensitivity distribution of 1000 non-BISR device samples (two defects). . . . . . . . .
68
6.12 The good-module-number distribution of 1000 BISR device samples (three defects).
. . .
68
6.13 The yield comparison between non-BISR and BISR devices. . . . . . . . . . . . . . . . .
70
6.14 The yield increase due to redundancy repair for six cases. . . . . . . . . . . . . . . . . . .
70
6.15 The yield comparison between non-BISR and BISR devices with more defects. . . . . . .
72
6.16 The yield increase due to redundancy repair with more defects. . . . . . . . . . . . . . . .
72
6.17 The comparison between theoretical prediction and Monte Carlo simulation result. . . . .
72
7.1
The differential capacitance sensing circuit. . . . . . . . . . . . . . . . . . . . . . . . . .
73
7.2
Switching circuit for redundancy repair. . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
7.3
Parasitic capacitance analysis for one BISR module. . . . . . . . . . . . . . . . . . . . . .
76
7.4
Simplified parasitic analysis circuit in phase 1 and 2. . . . . . . . . . . . . . . . . . . . .
76
7.5
Sensitivity simulation results for the BISR accelerometer design. . . . . . . . . . . . . . .
80
7.6
Frequency simulation results for the BISR accelerometer design. . . . . . . . . . . . . . .
80
7.7
ANSYS simulation results for sensitivity of BISR/non-BISR accelerometers. . . . . . . .
82
8.1
The bathtub curve of failure rate [60]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
8.2
The exponential reliability distribution function. . . . . . . . . . . . . . . . . . . . . . . .
86
8.3
The block diagram of series reliability model. . . . . . . . . . . . . . . . . . . . . . . . .
88
8.4
The block diagram of parallel reliability model. . . . . . . . . . . . . . . . . . . . . . . .
88
8.5
The block diagram of k-out-of-n redundancy reliability model. . . . . . . . . . . . . . . .
89
8.6
A typical S-N curve of a ductile material. . . . . . . . . . . . . . . . . . . . . . . . . . .
90
8.7
Weibull distribution of the probability of fracture versus stress. . . . . . . . . . . . . . . .
92
8.8
The structural diagram of a non-BISR MEMS device. . . . . . . . . . . . . . . . . . . . .
96
8.9
The reliability model for the non-BISR MEMS device. . . . . . . . . . . . . . . . . . . .
96
8.10 The structural diagram of the BISR MEMS accelerometer. . . . . . . . . . . . . . . . . .
98
8.11 Reliability model for BISR device. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
6.7
8.12 The S-N curve of poly-silicon tensile specimens with cyclic loading [93]. . . . . . . . . . 102
8.13 ANSYS stress analysis of one BWC BISR module. . . . . . . . . . . . . . . . . . . . . . 104
8.14 The reliability curves for BISR and non-BISR device (t = 10x ). . . . . . . . . . . . . . . 106
8.15 The accelerometer with combined defects (stiction + narrowed-beam). . . . . . . . . . . . 108
10
8.16
8.17
8.18
8.19
8.20
8.21
8.22
8.23
8.24
8.25
8.26
8.27
8.28
8.29
8.30
8.31
Deflection of accelerometer under Z-axis shock. . . . . . . . . . . . . . . .
Weibull fracture probabilities for thick/thin film beams in [66]. . . . . . . .
The maximum displacement versus input shock acclerations. . . . . . . . .
The maximum stress versus input shock accelerations. . . . . . . . . . . .
The Weibull fracture probabilities for non-BISR device and BISR modules.
The z-axis shock survival probability for non-BISR/BISR devices. . . . . .
The shock survival probability increase of BWC/EFC BISR devices. . . . .
The mean fracture accelerations of BWC/EFC BISR devices. . . . . . . . .
The safety factors of BWC/EFC BISR devices at 4000g. . . . . . . . . . .
The threshold acceleration for Z-axis contact. . . . . . . . . . . . . . . . .
The threshold acceleration for X-axis contact. . . . . . . . . . . . . . . . .
SEM photo showing the stiction of one comb finger to substrate [54]. . . . .
Stiction survival probabilities vs pf for non-BISR/BISR devices (N = 60).
Stiction survival probability increase versus pf for different N numbers. . .
Stiction of movable microstructure to substrate. . . . . . . . . . . . . . . .
The anti-stiction numbers of non-BISR/BISR devices. . . . . . . . . . . . .
11
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109
112
112
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114
115
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118
119
121
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123
124
126
List of Tables
3.1
Diversity of the three MEMS device examples. . . . . . . . . . . . . . . . . . . . . . . .
18
3.2
Voltage biasing scheme for comb accelerometer. . . . . . . . . . . . . . . . . . . . . . . .
25
3.3
Voltage biasing for bulk-micromachined accelerometer. . . . . . . . . . . . . . . . . . . .
26
3.4
Voltage biasing scheme for microresonator. . . . . . . . . . . . . . . . . . . . . . . . . .
28
4.1
Stiction defect simulation results for comb accelerometer. . . . . . . . . . . . . . . . . . .
30
4.2
Finger height mismatch simulation of comb accelerometer. . . . . . . . . . . . . . . . . .
30
4.3
Stiction simulation results for bulk accelerometer. . . . . . . . . . . . . . . . . . . . . . .
31
4.4
Etch variation simulation results for bulk accelerometer. . . . . . . . . . . . . . . . . . . .
31
4.5
Stiction defect simulation results for microresonator. . . . . . . . . . . . . . . . . . . . .
32
4.6
Finger height mismatch simulation of microresonator. . . . . . . . . . . . . . . . . . . . .
32
4.7
Broken-via defect simulation results of comb accelerometer. . . . . . . . . . . . . . . . .
33
4.8
Gap variation defect simulation results of bulk accelerometer. . . . . . . . . . . . . . . . .
34
4.9
Side-etch variance defect simulation results for microresonator. . . . . . . . . . . . . . . .
34
6.1
Monte Carlo simulation results for point-stiction defects in non-BISR device. . . . . . . .
69
6.2
Monte Carlo simulation results for point-stiction defects in BISR device. . . . . . . . . . .
69
6.3
Comparison of Monte Carlo simulation results between non-BISR and BISR devices. . . .
69
6.4
Monte Carlo simulation results for more point-stiction defects. . . . . . . . . . . . . . . .
71
7.1
Design of BISR/non-BISR accelerometers.
. . . . . . . . . . . . . . . . . . . . . . . . .
79
7.2
Simulation results of BISR/non-BISR accelerometers. . . . . . . . . . . . . . . . . . . . .
81
8.1
Mechanical and electrical parameters degradation of p+ silicon cantilever beams due to aging. 91
8.2
Non-BISR/BISR accelerometers design parameters for reliability evaluation. . . . . . . . . 100
8.3
The ANSYS stress analysis result for one beam of both BISR and non-BISR devices. . . . 104
8.4
The reliability analysis for accelerometers with combined defects. . . . . . . . . . . . . . 108
8.5
The measurement results of [66]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
8.6
The mean fracture stress for non-BISR/BISR devices. . . . . . . . . . . . . . . . . . . . . 115
8.7
The Z-axis acceleration required for contact of movable microstructure and substrate. . . . 118
13
8.8
The simulation results of X-axis shock.
. . . . . . . . . . . . . . . . . . . . . . . . . . . 119
14
Chapter 1
Introduction
In 1987, the world’s first micromachined electrostatic motor [1] was developed in Berkeley Sensor and Actuator Center, University of California. The micromotor was fabricated with an IC-compatible processing
technology. The rotor diameters in all motors are between 60 and 120 microns. This exciting innovation
triggered the naissance of a brand-new area: Micro Electro Mechanical System (MEMS). People began to
realize that mechanical structures can also be scaled down in size to the micron range. These microfabricated structures can be integrated with VLSI circuits on the same chip to ensure a better compatibility
for the interface between the mechanical and electro parts. During last decades, MEMS devices based
upon various working principles for various applications have been conceived [2]-[6]. The rapid progress
in MEMS microfabrication technologies has helped turn these novel designs into reality. Combined with
nano-technology, MEMS is believed to be the technology to trigger the next wave of technology revolution.
1.1 Introduction to MEMS
MEMS is the acronym of Micro Electro Mechanical System. The word ”micro” indicates the most important feature of MEMS: its size is extremely small. The typical size of MEMS components is in the range
between 1 micron (m) to 1 millimeter (mm). This means the key feature size of a MEMS device is usually
smaller than the diameter of human hair. For a microstructure in this scale, it is beyond the capability of
conventional precision fabrication technologies. Thus, special MEMS microfabrication technology must
be developed. The conventional precision fabrication technology can reach the domain of 1 millimeter and
above. For feature size below 1 micron, the quantum effect cannot be ignored. It belongs to the recently
emerged concept of NEMS (Nano Electro Mechanical System). Thus, MEMS devices mostly concentrate
on the feature size in 1 1000m. Further, the electro and mechanical parts of a MEMS device interact
with each other, so that it can be called a ”system”. For example, in a MEMS system, the signals in a
mechanical sensor can be sensed by the electro circuit, while the actuation instructions from the electro
circuit can be implemented by a mechanical actuator. Thus, MEMS can incorporate the environment data
1
collection, signal processing and actuation in the same ”smart” system.
When compared with conventional electro-mechanical products, MEMS has the following specific features and corresponding advantages:
Small volume, low weight and high resolution. Since the size of a MEMS device is very small,
its weight can fall to nano gram or below. This ensures high function density within a tiny space.
Further, high resolution can be achieved since the sensing structure is very small.
High reliability. Since MEMS devices are very small, the influence from environments (for example,
heat expansion, noise and electromagnetic interference, etc) can be minimized. Thus, more stable
performance can be expected for MEMS devices. They can be used in some harsh working environments.
Low energy consumption and high efficiency. To perform the same function, the energy consumption
of a MEMS device is usually tenth that of a conventional mechanical device. However, its working
speed can be ten times faster. Further, a MEMS device has a better compatibility in its mechanical
and electrical interface, so the signal delay between these two portions can be minimized. Thus,
MEMS devices are more suitable for high-speed operations.
Multi-function and intelligentized. Since MEMS sensors and actuators can be fabricated on the same
chip with VLSI circuits, MEMS devices can realize more functions and become ”smart”.
Low cost. Since MEMS products can be fabricated in batches like VLSI circuits, one fabrication flow
can produce a large number of MEMS devices without incurring any additional cost. The microfabrication process is mostly compatible with well-developed VLSI fabrication technologies. Further, the
material used in one individual MEMS device is tremendously reduced due to its tiny size. Thus, the
cost of MEMS is much lower than the conventional devices.
Due to the above advantages, MEMS has attracted tremendous interests from scientists and researchers
in the world. More and more people from various backgrounds are working jointly to explore this exciting
field. Many countries in the world have realized the importance of MEMS and given it a very high priority
in funding the corresponding research. Initially, governments and universities are the major supporters
for MEMS researches. After MEMS has been found broad applications, industry began to join this area.
Some commercial products have been developed successfully, and achieved great market successes. Several
typical examples for commercial MEMS devices are listed as follows.
ADXL series accelerometers [7]. Analog Devices Inc. (ADI) has developed the world’s first MEMS
accelerometer ADXL50 [8] [9] in 1991. The device is shown in Figure 1.1. It is a monolithically
integrated accelerometer with on-chip signal detection circuitry fabricated by the poly-silicon surface
micromachining technology. With a 50g acceleration measurement range, it is used in automobiles
2
for airbag deployment during crash accident. Each movable comb finger constitutes differential capacitances with the fixed fingers in its left and right sides. Acceleration along its sensitive direction
will lead to the deflection of all central movable fingers, hence the differential capacitances change.
By sensing the change of the differential capacitances, the amount of acceleration can be measured.
The ADXL50 accelerometer has now been replaced by newer generation products ADXL150 (single axis) and ADXL250 (dual axis) accelerometers [7]. The ADXL series accelerometers have been
widely used in the world’s automobile market.
Figure 1.1: Photo of ADXL250 accelerometer by Analog Devices Inc.
Digital micromirror device (DMD). Texas Instruments Inc. (TI) developed DMD [10] devices for
digital projection display systems. The device is shown in Figure 1.2. It is a micromechanical spatial
light modulator fabricated on silicon substrate using the surface micromachining sacrificial layer
technology. The device structure is an array of electrostatistically deflected aluminum micromirrors
which are suspended above the silicon substrate by very thin torsional hinges. When the driving
voltage is applied between the plate and the bottom deflection electrodes of a mirror, the mirror will
rotate and the incident light on the mirror surface will change its propagation direction. The DMD
devices have been successfully used in light projectors to produce high-resolution and high-contrast
images.
LambdaRouter optical switch. Lucent Technologies Inc. has developed LambdaRouter optical switches
[11] for steering the light from one fiber to another. The device is shown in Figure 1.3. As an alloptical lightwave-routing device, it is actually an array of microscopic mirrors. Each of them can tilt
in various directions to steer the light propagation. The micro-mirrors route information in the form
of photons to and from any of 256 input/output optical fibers. In contrast to conventional devices,
such an optical device need not convert the light into electrical signal during the switching process.
It has been widely used in the all-optical communication networks.
3
Figure 1.2: Photo of digital micromirror device by Texas Instruments Inc.
Figure 1.3: Photo of LambdaRouter optical switch by Lucent Technologies Inc.
4
GeneChip DNA chip. In 1996, Affymetrix Inc. in Santa Clara, Calif. introduced the world’s first
commercial DNA chip, GeneChip [12]. The device is shown in Figure 1.4. The DNA chip is actually
a DNA microarray which contains microscopic groups of thousands of DNA molecules of known
sequences attached to a solid surface. The array is exposed to labeled sample DNA and hybridized in
order to determine the identity and abundance of its complementary sequences. Such DNA chips can
be used for biomedical applications such as illness diagnosis, gene analysis, etc. It has the advantages
of fast diagnosis with high accuracy and small sample volume requirement. It is expected to be the
main diagnostic tool in future medical applications.
Figure 1.4: Photo of MEMS gene chip by Affymetrix Inc.
MEMS has found broad applications in areas such as automobile, communication, medical health care,
aerospace, consumer products, etc. The world MEMS market is increasing steadily with a fast pace. It is
expected to be 40 billion USD in 2005, and will increase to more than 60 billion USD in 2010 [13]. The
rapid growth in the MEMS market is a strong impulse for the research and development of MEMS.
1.2 Built-in Self-test of MEMS
Built-in self-test (BIST), is the technique of designing additional hardware features into integrated circuits
to allow them to perform self-testing. That is, each circuit can test its own operation using its own circuits.
As a result, this greatly reduces the dependence on an external automated test equipment (ATE). Advantages
of implementing BIST include: 1) lower cost of test, since the need for external electrical testing using an
ATE is avoided; 2) better fault coverage, since special test structures can be incorporated into the chips; 3)
shorter test times, if the BIST method can be designed to perform BIST in parallel mode; 4) capability to
perform tests during in-field usage, which allows the consumers themselves to test chips prior to mounting
or even after these are in the application boards.
According to the International Technology Roadmap for Semiconductors, 2002 Updates, MEMS will
5
begin to be integrated into system-on-Chip (SoC) designs during 2003 and 2004 [14]. Thus, we can expect
that MEMS devices will be fabricated on the same chip with digital, analog, memory, and FPGA circuit
technologies very soon. For this purpose, a thorough and effective testing solution for MEMS devices is in
an emergent need to ensure reliability. However, the great diversity of MEMS structures and their working
principles, various defect sources, multiple field coupling, as well as the essential analog features, all make
MEMS testing very challenging [15]-[21]. Built-in self-test is also believed to be the promising solution
for MEMS testing [21]. Furthermore, BIST is also the prerequisite for our research on MEMS built-in selfrepair (BISR). In order for the BISR implementation of MEMS devices through redundancy repair, each
MEMS module must be tested to decide whether it is faulty. The test and diagnosis result of each module
will be fed to the BISR control circuit. If any of the working modules in the main device is found faulty, the
BISR control circuit will separate out the faulty module and replace it with a good redundant module. The
most efficient solution for such a diagnosis of each module is built-in self-test (BIST). The BISR function
relies on the output of the BIST circuit. Thus, an efficient and robust BIST solution is the prerequisite for
MEMS BISR.
Sensitivity BIST has been proposed by researchers [17]-[19], and the major application is for in-field
testing. A testing stimulus (e.g., electrostatic force) is used to activate the device. Such a testing stimulus
is also calibrated before releasing. If the device failed to exhibit an output response of the calibrated value,
this indicates that the device is faulty. Further, Symmetry BIST suitable for CMOS MEMS devices has also
been proposed [21]. It partitions the central mass (movable capacitance plate) into two separate portions
connected by an insulated layer. By comparing the test responses between symmetric parts of the device,
the symmetry test approach can detect any left-right asymmetry caused by local defects. The symmetry
BIST aims at local defects which alter the device symmetry and no test stimulus calibration is needed. It
can be used in manufacturing test as well as in-field test.
1.3 Built-in Self-repair of MEMS
With the rapid growth in the complexity of VLSI chips, it is necessary to have built-in self-repair for VLSI
circuit. Built-in self-repair techniques have been successfully applied to VLSI, especially for memory chips
[22]. Without redundancy repair, no functional memory chip would have been possible. Redundancy repair
can greatly enhance the yield and reliability of VLSI chips. Similarly, it is reasonable to expect a fast growth
in the complexity of MEMS devices. Many MEMS devices will be integrated on a single chip to realize
more functions. Each MEMS device may also contain many more components. Especially, nanotechnology
is going to be combined with MEMS, and together they will create a new field of Nanoelectromechanical
System (NEMS). This may enable hundreds of MEMS devices to be integrated in a single chip, with each
MEMS device containing thousands of finger and beams, etc. For such large scale integration of MEMS
components and devices, it is necessary to implement built-in self-repair features into the chip to improve
the yield as well as the reliability.
6
Therefore, fault-tolerant MEMS design is extremely crucial with the following three reasons. First,
with increasing applications of MEMS to safety-critical fields, such as aerospace, automobile and medical
applications, MEMS reliability is becoming a very important issue. Especially, many MEMS devices have
movable parts and their repeated movements (vibrations, etc.) may lead to different kinds of structural
material fatigues. For example, the friction between the contacting surfaces of movable and fixed parts of
a micromotor may wear out the device structure. Thus, even if a MEMS device is tested as fault-free, it
still may fail after serving for a certain lifetime. Such a failure during in-field usage is a potential threat
to human lives especially for safety-critical applications. Second, due to the involvement of multiple fields
in MEMS design and fabrication, in contrast to the well-developed VLSI technology, MEMS fabrication
is vulnerable to more defect sources. Currently, MEMS device fabrication yield is much lower than that
of VLSI circuits. Third, there is increased tendency that MEMS is going to be integrated into systemon-chip (SoC) designs using a standard CMOS process [14]. That is, MEMS devices will be fabricated
on the same chip with digital, analog, memory, and FPGA circuit technologies. Fault-tolerant design for
traditional CMOS circuits have been proposed and many available techniques are existent [23][24][25].
However, fault-tolerant design for MEMS devices has never been studied. It will be uneconomical to get
rid of the entire SoC chip, if there exist minor MEMS defects. Thus, it is emergent to find a solution to
have a defective MEMS device fix itself, whenever the test process (in-field or manufacturing) finds defects
existent. By implanting the built-in self-repair (BISR) feature into MEMS devices, the reliability as well as
the yield can be greatly improved.
Efforts on trimming the device geometry parameters with certain physical or chemical processes have
been reported. In [26], a highly focalized laser beam is used to precisely trim the geometry parameter of
the thin film resistor, so the resistance can be accurately adjusted to meet the design expectation. If the
dimension of a thin film resistor is larger than the designed value, the laser trimming can repair the deviated
resistance back to the good value. In [27], the ion milling and isotropic RIE etching techniques are used to
reduce the beam width and the beam height of a microgyroscope. Hence, the resonant frequencies in the
driving and detection modes can be precisely matched to ensure the proper function of the microgyroscope.
It is possible that the concepts of geometry trimming and ion milling techniques be applied to MEMS
device repairing. However, the methods of [26][27] have the following disadvantages. First, these efforts
are not ”self-repair” and they require an extremely precise control over the process to avoid any overtrimming or over-milling. Second, the repairing process may have to be performed for each individual
device separately with different adjustment, because each defective device may have its own geometry
deviation. This leads to extremely high cost for the device repairing process through trimming, milling or
etching, so it is not suitable for batch fabrication processes. Third, since specific processing equipments are
required, such a repair cannot be in-field. Fourth, the defects that can be repaired are very limited, because
only defects involving deviation in geometry parameters can be dealt with. For other defects such as stiction
and broken beams, they cannot be applied. Finally, since the repairing processes are essentially removing
away structural materials, such schemes are only one-way repair, i.e., they can only reduce the geometry
7
parameters but cannot enlarge them. Thus, if a geometry parameter is smaller than the designed value, such
schemes cannot repair a defective device into a good one.
Self-repair techniques through redundancy for VLSI circuits have been well developed [28][29]. Similarly, the idea of redundancy for repairing may also be a promising solution for MEMS devices. However,
in reality, the application of BISR implemented by redundancy to MEMS devices is much more difficult
than that to VLSI circuits. The reason is that in a VLSI circuit, the replacement of a faulty circuitry by a
redundant component can be easily realized by using switches to isolate the defective one from contributing
its signal. While in a MEMS device, all the parts are mechanically connected as a whole, which makes it
very difficult to physically separate a faulty portion from the main device. For example, it is impossible to
use a good moving finger to replace a defective one by multiplexer switching. The necessity for fault tolerant MEMS has been predicted in [15], however, due to the above difficulty, to the best of our knowledge,
no research on BISR of MEMS devices through redundancy has ever been conducted before.
1.4 Our Work
In our MEMS BIST research, a dual-mode built-in self-test technique for capacitive MEMS devices is
proposed [30]. Our BIST technique partitions the fixed (instead of movable) capacitance plates at each side
of the movable microstructure into three portions: one for electrostatic activation and the other two equal
portions for capacitive sensing. Due to such a partitioning method, the BIST technique can be extended
to bulk-micromachining MEMS devices and other technologies. Both sensitivity and symmetry dual BIST
modes can be implemented based on this partitioning technique. Each of the sensitivity and symmetry
BIST has its own fault coverage. A combination of both BIST modes ensures a better fault coverage for
the testing. The proposed BIST technique has been verified with three typical capacitive MEMS devices:
surface-micromachined comb accelerometer, bulk-micromachined capacitive accelerometer, and surfacemicromachined comb resonator. Results obtained by ANSYS [31] simulation show the effectiveness of
the BIST technique. The voltage biasing schemes for the dual mode BIST on these example devices are
discussed. The control circuit for the dual-mode BIST implementation is also studied. Various defects such
as stiction, etch variation and finger height mismatch are simulated with ANSYS and HSPICE to verify the
effectiveness of the BIST method. The defects on fixed capacitance plates are also simulated. Simulation
results show that the proposed dual-mode BIST technique is an effective test solution for various capacitive
MEMS devices. The proposed dual-mode BIST technique can also be extended to other MEMS devices in
a similar way.
Based on the BIST research, a built-in self-repair technique for comb accelerometer devices is proposed
[32]. Modularized design is used for the BISR accelerometer device. In this research, the device consists
of six identical modules. Among them, four modules work jointly as the main device, and the other two
modules act as redundancy. During the in-field usage, the dual-mode BIST technique will be performed
on each module. If any of the working modules in the main device is found faulty, the control circuit will
8
separate it out and replace it with a good module. In this way, the entire device can be self-repaired into a
good one given the number of faulty modules is less than the redundancy. Performance analysis is carried
out on the BISR device which suffers sensitivity loss due to modularized design. The sensitivity loss can be
compensated by adjusting the design parameters such as shrinking the beam width, etc. Further, electrostatic
force can also be used as a powerful tool to compensate the sensitivity loss. The BISR technique also
introduces an area overhead of about 50%. With this paid price, we do achieve a great improvement in the
device yield and reliability. Thus, the technique is intended for high-reliability applications such as systemon-chip or aerospace, etc. The BISR technique can also be applied to the case where yield is a problem,
and cannot be improved by other ways. For a moderate initial yield, the BISR technique demonstrates a
significant yield increase after redundancy repair.
In order to evaluate the effectiveness of the BISR technique in yield improvement, a yield model for
MEMS redundancy repair is developed. Based on statistical analysis, the yield after repair and yield increase of BISR devices are derived. The relationship between the yield increase and initial yield for different m (redundancy) and n (the number of main working modules) values is also analyzed. The simulation
results demonstrate that for a moderate initial yield of 0.698 (for demonstration purpose only), a yield increase of 35:7% can be achieved after redundancy repair when m = 2 and n = 4. The proposed BISR
technique is most efficient in yield increase for moderate initial yield. If the initial yield is too large or too
small, the yield increase will not be so significant. The yield model further considers different defect fatal
rates by introducing the concept of critical area, and the yield with different defect fatal rates is even better.
In order to verify the developed yield model, ANSYS Monte Carlo simulation is performed. Point-stiction
defects with random size from 0:1 to 6m are distributed randomly within the device area. A group of 1000
devices is simulated for different defect densities. The Monte Carlo simulation results clearly indicate the
effectiveness of the BISR scheme in yield increase for different initial yields.
The BISR control circuit is also studied. The control circuit takes the BIST result as input and makes
the necessary switch among the modules to replace the faulty module with a good one. Analog switches
(transmission gates) are used for the switching between the modules. A current sensing detection circuit
for differential capacitances is used, because it ensures better separation of the faulty modules. The control
circuit coordinates four different modes of the device: normal working mode, BIST mode, BISC (built-in
self-calibration) mode and BISR mode. The loading effect of each faulty module is analyzed. Parasitic
effects of the sensing circuit are also studied in detail. Based on the analysis, optimized non-BISR and
BISR devices are designed to demonstrate the concept.
Further, in order to quantitatively evaluate the effectiveness of the BISR scheme in increasing the device
reliability, a reliability model for MEMS has been developed. A series model is used to describe the reliability of a non-BISR device. A combined series and k-out-of-n redundancy model is used for the reliability
evaluation of the BISR MEMS device. Based on the reliability models, the reliabilities of both non-BISR
and BISR devices for different failure mechanisms are discussed. The failure mechanisms include material
fatigue, shock, and stiction. For the failure mechanism due to material fatigue, the result turns out that
9
the reliability of BISR devices are equal to or better than that of the non-BISR device. The analysis also
shows that material fatigue is not a major threat to a comb accelerometer device, since its displacement in
normal operation is very small. The MEMS reliability analysis under shock environment demonstrates a
significant reliability increase of the BISR devices. When compared with non-BISR devices, the average
fracture acceleration of BISR devices is greatly expanded. This leads to a significant increase in the shock
survival probability for the BISR design. Further, we analyze two cases of a stiction defect. First, we evaluate the threshold acceleration to bring movable fingers into contact with fixed fingers or the substrate. The
simulation results turn out that the threshold accelerations of the BISR devices are improved, which leads to
an increased device reliability in stiction defects. Second, we also evaluate the stiction survival probability
after such a contact is made. The results also demonstrate an effective stiction survival probability increase
for the BISR design, since the number of comb fingers in each module is reduced. The results prove that
the BISR devices lead to a significant reliability enhancement in various failure mechanisms.
1.5 Contents of the Thesis
This thesis is organized as follows. First, we have given a brief introduction to MEMS, MEMS built-in
self-test and built-in self-repair in Chapter 1. In Chapter 2, we will introduce some background information
about the MEMS comb accelerometer and its working principles. In Chapter 3, a dual-mode built-in selftest technique for capacitive MEMS devices is proposed. In Chapter 4, ANSYS and HSICE simulation
results for the dual-mode BIST method on three typical capacitive MEMS devices are described. The result
demonstrates the effectiveness of the proposed dual-mode BIST technique. Chapter 5 introduces our builtin self-repair technique for the MEMS comb accelerometer. Performance analysis on the sensitivity and
resonant frequency of the BISR device is also discussed. Chapter 6 develops a yield model for MEMS
redundancy repair and performs yield analysis on the MEMS redundancy repair technique. Chapter 7
illustrates the circuit support and simulation results of the designed BISR devices. Chapter 8 gives reliability
analysis for the MEMS BISR method. A reliability model is developed to quantitatively evaluate the MEMS
device reliability for three failure mechanisms: material fatigue, shock and stiction. Finally, Chapter 9
comes to some conclusions and suggests the future work of the research.
10
Chapter 2
Background
2.1 Introduction
Micro Electro Mechanical System (MEMS) has achieved tremendous progress in recent decades. Various
MEMS devices based upon different working principles have been developed [3]-[2]. MEMS has also
found broad applications in various areas. With the rapid development of MEMS technology, and its integration into system-on-chip (SoC) designs, MEMS testing is becoming a more and more important issue.
An efficient and robust test solution is in urgent need for MEMS. However, due to the great diversity of
MEMS structures and their working principles, various defect sources, multiple field coupling, as well as
the essential analog features, MEMS testing remains a very challenging work. Various efforts have been
made in this area [16]-[21]. However, a thorough testing solution like that in digital VLSI has not been
achieved. MEMS testing is still in its early stage.
Further, as more and more MEMS devices are utilized, fault-tolerant MEMS design becomes extremely
crucial with the following three reasons. First, with increasing applications of MEMS to safety-critical
fields, such as aerospace, automobile and medical apparatus, the reliability of MEMS products becomes a
very important issue. Especially, many MEMS devices have movable parts and their repeated movements
(vibrations, etc.) may lead to different kinds of structural material fatigues. For example, the friction
between the contacting surfaces of movable and fixed parts of a micromotor may wear out the device
structure. Thus, even if a MEMS device is tested as fault-free, it still may fail after serving for a certain
lifetime. Such a failure during in-field usage is a potential threat especially for safety-critical applications.
Second, there is increased tendency that MEMS is going to be integrated into system-on-chip (SoC) designs
using a standard CMOS process [14]. That is, MEMS devices will be fabricated on the same chip with
digital, analog, memory, and FPGA circuit technologies. Fault-tolerant design for traditional CMOS circuits
have been proposed and many available techniques are existent [23][24][25]. However, fault-tolerant design
for MEMS devices has never been well studied. It will be uneconomical to get rid of the entire SoC chip, if
there exist minor MEMS defects. Thus, it is emergent to find a solution to have a defective MEMS device
11
fix itself, whenever the test process (in-field or manufacturing) finds defects existent. By implementing the
built-in self-repair (BISR) feature into MEMS devices, the reliability as well as the yield rate of MEMSbased products can be greatly improved.
Fault-tolerant (or self-repairable) MEMS are in no doubt an urgent need. However, the realization of
MEMS self-repair remains extremely challenging. Due to the very tiny size of MEMS devices (in the
range of microns), the direct manipulation of MEMS parts proves to be very difficult. If one portion of the
movable part is faulty, it is not feasible to physically remove the faulty portion and replace it with a good
part, as the way macro machines are repaired. A few efforts on the structural repair of MEMS devices with
certain physical or chemical processes have been reported. In [26], the highly focalized laser beam is used
to precisely trim the geometry parameter of a thin film resistor, and thus the resistance can be accurately
adjusted to meet the design expectation. If the dimension of the thin film resistor is larger than the designed
value, the laser trimming can repair the deviated resistance back to the good value. In [27], the ion milling
and isotropic RIE etching techniques are used to reduce the beam width and height of a microgyroscope.
Hence, the resonant frequencies in the driving and detection modes can be precisely matched to ensure the
proper function of the gyroscope. The structural repair of a MEMS device is very expensive in terms of both
the repairing expenses and time, and special apparatus is needed during the structure repair process. It also
has to be individually performed for each device, since the structural variance is different for each of them.
A precise control on the repair process is also required to avoid any over-repairing. Thus, structural repair
of MEMS is extremely costly, and is not suitable for batch fabrication. The defects that the structural repair
method can deal with are also limited. For example, some defects such as stiction and broken beams cannot
be repaired by this scheme. Further, the structural repairing process essentially removes away structural
materials, as a result, it is one-way repair. That is, it can only reduce the geometry parameters but cannot
enlarge them. Thus, if a faulty device has a geometry parameter which is smaller than the designed value,
structural repair is not applicable.
Self-repair techniques through redundancy for VLSI circuits have been well developed [23][24][25].
Intuitively, it is possible that the concept of redundancy repair can be a promising solution for MEMS selfrepair. However, in reality, the application of MEMS BISR through redundancy is much more difficult than
that of VLSI circuits. In a VLSI circuit, the replacement of a faulty circuitry by a redundant component
can be easily realized by using switches to isolate the defective one from contributing its signal. While
in a MEMS device, all the movable parts are mechanically connected as a whole, which makes it very
difficult to physically separate a faulty portion from the main device. For example, it is impossible to use
a good movable finger to replace a defective one in a comb accelerometer by multiplexer switching. The
need and necessity for fault-tolerant MEMS has been predicted in [15]. It also suggested the possibility of
using modular redundancy repair for array-structured MEMS devices, such as a microvalve array for fluid
flow control. However, to the best of our knowledge, no built-in redundancy repair method for non-arraystructured MEMS devices has ever been reported so far.
In this work, built-in self-repair through modular redundancy of a MEMS comb accelerometer is ex12
plored. We revise the design by partitioning it into identical modules such that healthy modules can work
jointly to guarantee the normal function. The partitioning cannot affect the expected sensitivity of the
MEMS device. Thus, several techniques are proposed to compensate the sensitivity loss due to modularized design. In the BISR mode, the device first performs the built-in self-test (BIST) process to check the
function of each individual module. The BISR control circuit will then restructure some good modules
as a main device for normal operation. The remaining good modules will serve as redundant devices for
reliability enhancement. Further, based on the yield model for MEMS redundancy repair, the optimal ratio
between the main device area and redundant device area will be derived such that the yield can be maximized with moderate area overhead. A BIST technique based on dual-mode (sensitivity and symmetry)
testing [30] will be applied to guarantee the success of identifying defective modules. The results obtained
based on computer simulation demonstrate that, with a tolerable device area overhead, the proposed BISR
technique can improve the manufacturing yield of a comb accelerometer from 0.698 to 0.947 without sacrificing the sensitivity. This technique is especially powerful for a manufacturing process that has moderate
yield where a sufficient number of modules are healthy. Further, this technique can be efficiently applied
to safety-critical applications, since the reliability of a BISR MEMS device for in-field applications can be
enhanced. Though a comb accelerometer is used to present the key idea, the redundancy repair concept can
also be extended to other capacitive MEMS devices such as bulk-micromachined accelerometer, RF-MEMS
tunable capacitor, and humidity sensor, etc.
2.2 Non-BISR MEMS Comb Accelerometer
fixed
finger
(left)
fixed
finger
(right)
anchor
beam
central movable mass
movable finger
Figure 2.1: The general design of MEMS comb accelerometer.
A typical surface-micromachined comb accelerometer [8] is shown in Figure 2.1. The comb accelerom13
eter is made of a thin layer of poly-silicon on the top of a silicon substrate. The thickness of the poly-Si
structure layer is about 2m. The fixed portion of the device includes four anchors and many left and right
fixed fingers. The movable portion of the MEMS device includes four tether beams, central movable mass
and all the movable fingers extruding out of the mass. The entire movable portion is floating about 1.5m
above the substrate. As shown in Figure 2.1, the central movable mass is connected to the four anchors
through four flexible beams. The movable fingers extrude from both sides of the central mass, and can
move together with it. There is a pair of fixed fingers around the left and right sides of each movable finger.
Each movable finger and its left and right fixed fingers constitute a differential capacitance pair 1 and 2
separately as shown in Fig. 2.2. In the static state, each movable finger stays in the middle position between
the left and right fixed fingers, and the capacitance gaps of both 1 and 2 are equal to d0 . Assume there are
v pairs of finger groups in the MEMS device, and let C1 (C2 ) represent the sum of all 1 (2 ) capacitances.
We have
C1 = C2 =
nf "0 (Lf )h
:
d0
where nf is the total number of differential capacitance groups, "0 is the dielectric constant of air, Lf is the
length of each movable finger, is the non-overlapped length at the root of each movable finger, and h is
the thickness of the device.
Assume the mass of both the central movable mass and all the movable fingers as M . If there is an
acceleration a in perpendicular to the beams while in parallel to the device plane, the central mass will
experience an inertial force M a. This will result in a certain amount of beam deflection along the
direction of the inertial force, hence the equivalent amount of displacement of the central mass and the
movable fingers. Thus, each capacitance gap will be changed accordingly which leads to the change of
corresponding capacitances (Fig. 2.2).
movable finger
fixed
finger
F1
c
1
c
fixed finger F2
2
d0 −x
d0 +x
X
0x
Figure 2.2: The schematic diagram of differential capacitance.
As shown in Figure 2.2, the inertial force results in a deflection of the beams and a certain displacement
x of movable fingers along the X direction. Given x d0 , we have C1 and C2 changed to [2]
C1 =
nf "0 (Lf )h
(d0 + x)
nf "0 (Ldf )h (1 dx );
0
0
14
x
nf "0 (Lf )h nf "0 (Lf )h
(1 + ):
(d0 x)
d0
d0
change of C1 and C2 , we know the displacement x,
C2 =
By sensing the capacitance
acceleration. This is the working principle of a MEMS comb accelerometer.
15
hence the experienced
16
Chapter 3
Dual-Mode Built-In Self-Test for Capacitive
MEMS Devices
3.1 Introduction
According to the International Technology Roadmap for Semiconductors, 2002 Updates, MEMS will begin
to be integrated into system-on-Chip (SoC) designs recently [14]. Thus, we can expect that MEMS devices
will be fabricated on the same chip with digital, analog, memory, and FPGA circuit technologies very soon.
For this purpose, a thorough and effective testing solution for MEMS devices is in an emergent need to
ensure reliability. However, the great diversity of MEMS structures and their working principles, various
defect sources, multiple field coupling, as well as the essential analog features, all these factors make MEMS
testing very challenging [16]-[21].
MEMS devices are calibrated before shipping. However, new defects may be developed during infield usage. Calibration is not convenient after MEMS devices are released. Sensitivity BIST has been
proposed by researchers [17]-[19], and the major application is for in-field testing. Its basic concept is
simple: a testing stimulus (e.g., electrostatic force) is applied to activate the device to its full working
range, and failure to demonstrate a full-range output within some tolerance level means the device is faulty.
Sensitivity testing is an easy way to check whether the device is free to move according to the design
expectation. However, it requires the electrostatic force (or other test stimulus) to be calibrated before it
can be applied. Also, it is not effective in identifying some hard-to-detect defects, such as the capacitance
asymmetry caused by local defects. In order to solve these problems, Symmetry BIST suitable for CMOS
MEMS devices has also been proposed [21]. The basic idea of symmetry BIST is to partition the central
mass (movable capacitance plate) into two separate portions connected by an insulated layer. By comparing
the test response between symmetric parts of the device, the symmetry test approach can detect any leftright asymmetry caused by local defects. The symmetry BIST method aims at local defects which alter the
device symmetry, and no test stimulus calibration is needed. It can be used in manufacturing test as well
17
as in-field test. Since most capacitive MEMS devices have some extent of structure symmetry, this method
can be applied to many different kinds of capacitive MEMS devices. Compared to sensitivity test, it has
the advantage that no test stimulus calibration is needed. It aims at local hard-to-detect defects which alter
the symmetry of the device. However, for global defects which change both sides of the device in the same
amount, the symmetry test approach cannot be used. Since each of sensitivity test and symmetry test has its
own defect coverage, by combining them together, a more robust test for MEMS device can be expected.
In this chapter, a dual-mode built-in self-test technique which partitions the fixed (instead of the movable) capacitance plates is introduced. Due to this partitioning, the movable capacitance plate is not divided,
so the BIST scheme is not limited to CMOS MEMS devices, and it can be easily extended to bulk micromachined MEMS devices and other technologies. Another major contribution of this work is that both
sensitivity and symmetry BIST modes based on this partitioning are also implemented. Since each of sensitivity and symmetry BIST methods has its own defect coverage, by combining them together, a more robust
test can be expected. The proposed BIST technique has been verified using three typical capacitive MEMS
devices: surface-micromachined comb accelerometer, bulk-micromachined capacitive accelerometer, and
surface-micromachined comb resonator. The criterion for selecting these three devices as examples is to
ensure the diversity of technologies for demonstrating the versatility of the BIST technique, as shown in
Table 3.1. Results obtained by ANSYS [31] fault simulation show the effectiveness of the BIST technique.
Table 3.1: Diversity of the three MEMS device examples.
comb
micro
bulk
accelerometer
resonator
accelerometer
fabrication
technology
surface-
surface
bulk-
micromachine
micromachine
micromachine
working
principle
sensor
sensor
actuator
driving
perpendicular
side-driving
perpendicular
3.2 Basic Concepts of the Proposed BIST technique
In order to make the following discussion clearer, we define some biasing voltages first.
Vmp(Vmn): the modulation voltage in positive (negative) phase (high frequency, e.g., 1MHz),
Vd: the test driving voltage (usually DC) applied to the fixed driving plates to activate the device in
the BIST modes.
Vnom: the nominal voltage on the movable plate, usually the time average value of modulation voltage
Vmp .
18
Vdp; Vdn: the complementary driving voltages applied to the fixed driving plates to activate an actuator
in normal mode.
3.2.1 Basic Knowledge for Capacitive MEMS Devices
F1
C1
d0
B1
B2
M
C2
d0
F2
Figure 3.1: Schematic diagram of a capacitive MEMS device.
A typical MEMS differential capacitance structure is shown in Figure 3.1 where M represents the movable plate, F1 and F2 denote fixed plates, while B1 and B2 are both beams of the MEMS device. As shown
in Figure 3.1, the movable plate M is anchored to the substrate through two flexible beams B1 and B2. It
constitutes differential capacitances C1 and C2 with the top and bottom fixed plates (F1 and F2). In the
static mode, the movable plate M is located in the center between F1 and F2, thus
"S
C1 = C2 = 0 :
d0
where "0 is the dielectric constant of air, S is the overlap area between M and F1/F2, and d0 represents the
static capacitance gap between M and F1/F2. A vertical stimuli (such as acceleration etc.) will result in the
deflection of beams and a certain displacement of movable plate M along the vertical direction. Assume the
central movable mass moves upward with a displacement of x. Given x << d0 , C1 and C2 under the test
stimuli can be derived by
C1 =
C2 =
In order to sense the displacement
applied to F1 and F2 separately
"0 S
(d0
"0 S
x
(1 + );
x) d
d
"0 S
(d0 + x)
0
0
"d0 S (1 dx ):
0
0
x of the movable plate M, modulation
voltage Vmp and Vmn are
VF 1 = Vmp = V0 sqr(!t);
VF 2 = Vmn = V0 sqr(!t):
where V0 represents the modulation voltage amplitude, ! denotes the frequency of the modulation voltage,
and t gives the time for operation. According to the charge conservation law, the charge in capacitances C1
19
and C2 must be equal, so we have
C1 (VF 1 VM ) = C2 (VM
VF 2 ):
where VM is the voltage level sensed by the movable plate M. Solving the above equations, we have
VM = (x=d0 )V0 sqr(!t):
It can be observed from this result that under the above modulation voltage biasing, the central movable
plate M acts just as a voltage divider between the top and bottom fixed plates F1 and F2 respectively. By
measuring the voltage level on central movable electrode VM , we can find the displacement x of the central
movable plate M, which in turn is directly proportional to the physical stimuli. Thus, we can derive the
value of the applied physical stimuli. This is the working principle for most differential capacitive MEMS
devices.
In the sensitivity BIST mode, a certain amount of driving voltage Vd can be applied to the driving plate
to mimic the action of a physical stimulus with electrostatic force. In the above example, if voltage Vd is
applied to the fixed plate F1 and nominal voltage Vnom is applied to M, an electrostatic attractive force Fd
will be experienced by the central movable mass
" SV 2
Fd = 0 2 d :
2d
The electrostatic force is used to mimic the input stimuli during the BIST mode, and the device response
to the electrostatic force is measured and compared with the good device response to check whether the
device is faulty. This is the basic idea for the sensitivity test mode of a capacitive MEMS device. For
vertical electrostatic driving, the driving voltage cannot exceed a threshold value by which the deflection
exceeds 1/3 of the capacitance gap d0 . Otherwise, the movable plate will be stuck to the fixed plate through
a positive feedback, and a short-circuit will occur.
3.2.2 The Symmetry BIST Scheme
Now, we introduce our symmetry test scheme. A simplified MEMS capacitance structure is given in Figure
3.2 where S1-S4 are fixed plates. As shown, each of the top and bottom fixed capacitance plates is divided
into two equal portions. For simplification, here we omit the capacitance for electrostatic actuation which is
necessary for BIST implementations. The basic idea of our symmetry test scheme is to check whether the
two symmetric capacitances (e.g., C1 and C2 in Figure 3.2) on the same side of the movable microstructure
remain equal all the time, after activation.
In Figure 3.2, fixed plates S1 and S2 lie at the same side of the movable plate M. The capacitance
between M and S1 (S2) is defined as C1 (C2 ). The modulation voltage Vmp and Vmn are applied to S1 and
20
Vmn
Vmp
S1
S2
C1
B1
C2
B2
M Vm=0?
C3
C4
S4
S3
Figure 3.2: MEMS capacitance structure for our symmetry test scheme.
S2 separately. If the device is fault-free, regardless whether the movable plate is in rest or moving a certain
displacement along the vertical direction, the values of C1 and C2 should always remain equal. Take the
voltage level on central movable plate M as VM , according to the charge conservation law, charge Q1 and
Q2 in capacitances C1 and C2 must remain equal
C1 (Vmp VM ) = C2 (VM
Since Vmp
Vmn ):
= Vmn , from the above equation we have
VM = Vmp (C1 C2 )=(C1 + C2 ):
If C1 equals C2 , we have: VM = 0. Under the above voltage biasing scheme, the voltage level on the
central movable plate is always zero for good devices. However, if there is any local defect which alters the
symmetry of the device, the movable plate will tilt and C1 will not be equal to C2 . In this way, the output
voltage VM will not be zero anymore. Thus, by checking the voltage output on the movable plate, we can
find any defect which alters the symmetry of the device. Furthermore, according to the phase polarity of
VM , we can know whether the defect lies at the left or right side of the device. For example, if a stiction
defect in the right side (which introduces C2 in Fig.3.2) of the mass causes C2 to be smaller than C1 , VM
will have he same phase polarity as Vmp , and vice versa.
The above analysis is for checking both capacitances in the top side of the device. However, the verification for both bottom capacitances (C3 and C4 ) can be easily performed in a similar way, and they should
have the same result.
3.2.3 The Dual-Mode BIST Technique
As discussed above, our BIST technique for capacitive MEMS devices is to divide the fixed capacitance
plate(s) at each side of the movable microstructure into three portions: one for electrostatic activation and
the other two equal portions for capacitance sensing, as shown in Figure 3.3. Note that M is the movable
plate, D1 and D2 are the fixed driving plates, while {S1, S2, S3, S4} are the fixed sensing plates. As shown
in Figure 3.3, after capacitance partitioning, two BIST modes (sensitivity test and symmetry test) can be
easily implemented on the device. During normal operation, we have Test Enable signal TE=0. If the device
21
S1
D1
S2
M
S3
D2
S4
Figure 3.3: Fixed capacitance plates partition for MEMS device.
is a sensor, the modulation voltage Vmp is applied to {S1, D1, S2}, and the complementary modulation
voltage Vmn is applied to {S3, D2, S4} (Figure 3.4(a)). The voltage on central mass M is sensed as the
output voltage Vout indicating the device sensitivity. If the device is an actuator (e.g., microresonator), the
driving voltage Vdp is applied to {S1, D1, S2}, and the complementary driving voltage Vdn is applied to
{S3, D2, S4} separately to implement the electrostatic actuation in normal operation. In a word, the driving
capacitance plates (D1 and D2) for BIST will also participate the normal operation, so there is no loss of
capacitance area due to the BIST implementation.
In the BIST mode (TE=1), the Test Selection (TS) signal can select one of the two BIST modes. When
TS=0, the device is in the sensitivity test mode. Test driving voltage Vd is applied to D1 to activate the
device, modulation voltage Vmp is applied to {S1, S2}, and Vmn is applied to {S3, S4}(Figure 3.4(b)).
The voltage level on the movable electrode (i.e., plate) M is measured for the device sensitivity analysis.
Voltage VM is compared with the expected value (calibrated) within a tolerance level to find whether the
device is faulty.
When TS=1, the device is in the symmetry test mode, and the proposed new symmetry test scheme
is used here. In this case, test driving voltage Vd is applied to D1 to activate the device. The modulation
voltage Vmp is applied to S1, and Vmn is applied to S2 separately (Figure 3.4(c)). The voltage level VM of
the movable electrode is measured to see whether it is a constant zero. If there is a non-zero voltage output
on movable electrode M, it indicates there is a local defect which causes the asymmetry of the device. Based
upon the value and polarity of VM , we can also have an idea about the approximate location of the local
defect. The above analysis (and the following examples in Section 3) is for the case where the movable
electrode is driven upward (Vd is applied to D1). However, for the case in which the movable electrode is
driven downward (Vd is applied to D2), the implementation can be easily extended. Note that in the BIST
mode, the device should be driven in both directions for a thorough test. Since each of both sensitivity test
and symmetry test has its own defect coverage, by combining them together, a more robust testing result
can be ensured.
The defect on driving electrodes D1 and D2 can also be detected, if it causes sensitivity change or
left-right asymmetry to the MEMS device. For example, if the left part of D1 is missing due to improper
photoetching, the mass will experience a larger electrostatic force in its right part than its left part in BIST.
22
Hence, the movable mass M will tilt and a symmetry test can detect the defect.
Vmp
Vmp
Vmp
Vmn
Vd
Vd
S1
S2
D1
S1
S2
D1
Vout
S1
Vout
D1
S2
Vout
Vnom
S3
S4
D2
S3
Vmn
(a) normal operation
S4
D2
S3
D2
S4
Vnom
(c) symmetry BIST mode
Vmn
(b) sensitivity BIST mode
Figure 3.4: Voltage biasing schemes for the three modes of MEMS device.
To implement the BIST technique, a control circuit is needed to switch the device among the normal
operation mode and both BIST modes. Such a control circuit is not complex and only contains some
switches made of analog Muxes. Taking the proposed BIST method for capacitive microsensors as example,
the control circuit design is shown in Figure 3.5.
TE TS
TE
0x
10
11
0
Vd
1
Vmp
Vnom
Vmn
D1
S1
TE
0
1
S2
M
TE TS
0x
10
11
D2
S3
TE
0
1
S4
TE TS
0x
10
11
Figure 3.5: The control circuit for dual BIST technique.
In this circuit, totally only six Muxes are needed: three 3-to-1 Muxes and three 2-to-1 Muxes. The
differential capacitance detection circuit for BIST modes can be shared with that of the normal operation
mode. Thus, the circuit overhead for the BIST technique implementation is small. In the following, we
will apply our BIST technique to three typical capacitive MEMS devices: surface-micromachined comb
accelerometer, bulk-micromachined accelerometer and poly-Si microresonator. Through these examples,
we can see how our BIST technique can be applied to various capacitive MEMS devices.
23
3.3 The Dual-Mode BIST for Different Technologies
3.3.1 Surface-Micromachined Comb Accelerometer
Surface-micromachined comb accelerometers have been popular examples for MEMS testing in many papers, because the surface micromachining technology is compatible with the well-developed CMOS VLSI
fabrication technology. A typical comb accelerometer structure [21] is shown in Figure 3.6. The device
prototype comes from ADXL series accelerometers developed by Analog Devices Inc.[8][9]. In Figure
3.6, M1-M8 are movable fingers, Ms is the central mass, D1-D8 are driving fingers, and S1-S8 are sensing
fingers. All beams are connected to the substrate through four anchors. For simplicity, only four groups
of driving/sensing fingers are given here. The fixed portion of the device includes driving fingers D1-D8
and sensing fingers S1-S8. Differential capacitances are constructed between the movable fingers and the
sensing fingers. If the device experiences an acceleration a in the vertical direction which is perpendicular
to the beam, the central mass will experience an inertial force Ms a which will deflect the beam connected
to the central mass. Thus, the movable fingers will also experience the same amount of displacement. This
will change the differential capacitance, which can be detected by the interface circuit so that we know the
value of the acceleration a. Since the fixed capacitance electrodes are separate fingers, partitioning for the
dual-mode BIST technique can be easily realized.
driving fingers
D1
D3
D2
D4
S1
S3
M5
M1
M2
M6
S2
sensing fingers
S5
S4
Ms
S7
M3
driving fingers
M7
S6
S8
D5
D7
D6
D8
M4
M8
Figure 3.6: Structural diagram of a comb accelerometer.
During normal operation, TE=0, modulation voltage Vmp is applied to {S1, S3, S5, S7, D1, D3, D5,
D7}, and Vmn is applied to {S2, S4, S6, S8, D2, D4, D6, D8}. The voltage level in the movable fingers
VMs is measured as the output voltage to determine the acceleration. When TE=1 and TS=0, the device
works in the sensitivity test mode. A certain test driving voltage Vd is applied to {D1, D3, D5, D7} to
activate the device with electrostatic force. The modulation voltage Vmp is applied to {S1, S3, S5, S7},
while Vmn is applied to {S2, S4, S6, S8}. The output voltage on movable mass Ms is measured for the
device sensitivity. This value is compared with the expected good device value within a certain tolerance
level to find whether the device is faulty. When TE=1 and TS=1, the device is in the symmetry test mode.
Test driving voltage Vd is applied to {D1, D3, D5, D7}, modulation voltage Vmp is applied to {S1, S5},
while Vmn is applied to {S3, S7}. The sensing circuit checks whether the output voltage on movable
24
fingers is a constant zero to detect any asymmetry caused by local defects. If there is a non-zero voltage on
the movable electrode Ms, then it indicates there are local defects which alter the symmetry of the device.
Defects on driving electrodes can also be detected if they cause sensitivity change or left-right asymmetry
to the MEMS device. For example, if part of D1 in Fig. 3.6 is missing, the movable mass will experience a
smaller electrostatic force in its left part than its right part during BIST. Hence, the movable mass Ms will
tilt and symmetry test can detect this defect. The voltage biasing scheme for the comb accelerometer in the
normal and both BIST modes is shown in Table 3.2 using a notation similar to [21].
Table 3.2: Voltage biasing scheme for comb accelerometer.
Voltage
biasing
Normal
operation
Sensitivity
BIST
Symmetry
BIST
Vd
Vnom
-
D1,D3,D5,D7
D2,D4,D6,D8
M1,M4,M5,M8
Vmp
S1,S3,S5,S7
D1,D3,D5,D7
S2,S4,S6,S8
D2,D4,D6,D8
S1,S3,S5,S7
D1,D3,D5,D7
D2,D4,D6,D8
M1,M4,M5,M8
S2,S4,S6,S8
S1,S5
S2,S4,S6,S8
S3,S7
Vmn
3.3.2 Bulk-Micromachined Capacitive Accelerometer
frame
Al electrode
Al electrode
glass
beam
A
mass
Si Mass (M)
A’
Al electrode
(a) bulk−micromachined capacitive accelerometer
glass
(b) cross section view (A−A’)
Figure 3.7: Structural diagram of bulk-micromachined accelerometer.
The structure of a silicon symmetric bulk-micromachined capacitive accelerometer [33][34] is shown
in Figure 3.7. The top and bottom glass covers are bonded to the central Si microstructure through the
silicon-glass anodic bonding technique, so the whole device is in the glass-silicon-glass sandwich structure.
The movable Si mass is connected to the frame through four beams. On the top and bottom sides of the
central mass, Al square electrodes are deposited on the corresponding glass surfaces, such that differential
capacitance can be constructed. When there is acceleration along the vertical direction perpendicular to the
device plane, the mass will experience an inertial force, and the beams will bend along the vertical direction.
The capacitance gap between the central mass and the Al electrodes will change. Thus, by measuring the
differential capacitance change with a sensing circuit, we get the value of the experienced acceleration.
For the BIST implementation, we revise the design by partitioning each of the top and bottom Al
25
electrodes into three portions: the central portion for electrostatic activation and the left/right portions for
sensing. The partitioning must ensure equal size for the left and right portions of each Al electrode. As
shown in Figure 3.8, the top (bottom) Al electrode is partitioned to sensing electrodes {S1, S2}({S3, S4})
and driving electrode D1 (D2).
frame
Al electrode
Al electrode
beam
A
S1 D1 S2
glass
Si Mass (M)
A’
Al electrode
(a) bulk−micromachined capacitive accelerometer
S3 D2 S4
glass
(b) cross section view (A−A’)
Figure 3.8: Revised design of bulk-micromachined accelerometer.
During normal operation (TE=0), the modulation voltage Vmp is applied to all top Al electrodes {S1,D1,S2},
and Vmn is applied to all bottom Al electrodes {S3, D2, S4}. The driving electrodes {D1, D2} also participate the normal operation of the device, and there is no capacitance area loss due to the BIST implementation. By measuring the voltage output on the central mass, we obtain the acceleration value. When TE=1,
the device enters either one of both BIST modes. When TS equals 0, the device is in the sensitivity test
mode, and a certain amount of test driving voltage Vd is applied between D1 and central mass M to activate
the device with electrostatic force. Modulation voltage Vmp is applied to {S1, S2} and Vmn is applied to
{S3, S4}. The voltage level in the central mass M indicates the device sensitivity. This value is compared
with the expected good device value within some tolerance level to check whether the device is faulty.
When TS=1, the device is in the symmetry test mode, and the test driving voltage Vd is applied to D1
to activate the device. The modulation voltage Vmp is applied to S1 and Vmn is applied to S2. If the device
is defect-free, based upon the symmetry of the device structure, the capacitance between M and S1 will
always equal to the capacitance between M and S2. Thus, the output voltage on the central mass will be
a constant zero. If there is any local defect which alters the symmetry of the device, the displacement of
the central mass in the left side may not be equal to that in the right side. Thus, the capacitance between
M and S1 will not be equal to the capacitance between M and S2 anymore. This will lead to a non-zero
output voltage at the central movable mass. By checking whether the output voltage on central mass M is a
constant zero, such local defects can be detected. The voltage biasing scheme of the bulk-micromachined
capacitive accelerometer in normal operation and two BIST modes are given in Table 3.3.
Table 3.3: Voltage biasing for bulk-micromachined accelerometer.
Voltage
biasing
Normal
operation
Sensitivity
BIST
Symmetry
BIST
Vd
Vnom
Vmp
Vmn
S1,S2,D1
S3,S4,D2
D1
D2,M
S1,S2
S3,S4
D1
D2,M,S3,S4
S1
S2
26
3.3.3 Poly-Si Surface-Micromachined Microresonator
y axis
pad
D1
S2
S1
movable plate
M
anchor
M
movable fingers
folded beam
fixed fingers
stationary
electrode
S4
S3
D2
(a) microresonator
(b) revised design
Figure 3.9: Structural diagram of a comb microresonator.
Poly-Si comb microresonator has broad applications in sensors, filters and oscillators [35][36]. A typical microresonator structure is shown in Figure 3.9(a) [37]. As an actuator, the normal operation of a
microresonator is to activate the device into oscillation. Several driving schemes are available, among
them, the most popular one is the push-pull driving. In normal operation, driving voltages Vdp and Vdn are
applied to the top and bottom fixed fingers, and nominal voltage Vnom is applied to the central movable
fingers. The driving voltages Vdp and Vdn are high frequency AC voltages with DC biasing. Assume the
following biasing for Vdp and Vdn :
Vdp = Vd0 + V0 sin!t;
Vdn = Vd0 V0 sin!t:
where Vd0 is the DC biasing for driving voltages, V0 is the voltage amplitude for AC component of driving
voltages, ! is the round frequency, and t is the time for operation.
Such a voltage biasing scheme induces a side-driving electrostatic force on the movable fingers. The
electrostatic forces on the central movable fingers generated by the top and bottom fixed fingers are defined
as F1 and F2 separately. We have
F1 = b"0 Vdp2 =(2d0 );
2 =(2d ):
F2 = b"0 Vdn
0
where b is the thickness of the device, d0 is the capacitance gap between movable and fixed fingers, and "0
is the dielectric constant of air. Thus, the total electrostatic force Ftot experienced by the central movable
mass is
Ftot = F1 F2 =
27
2b"0 V0 Vd os!t
:
d0
From this result, we find that the central mass will experience an alternative electrostatic driving force (with
frequency ! ) and oscillate around its equilibrium location.
For the BIST implementation, we revise the device design by partitioning the top (bottom) fixed fingers
into three portions: the central portion D1 (D2) for driving, and the left-right portions {S1, S2} ({S3, S4})
for sensing, as shown in Figure 3.9(b).
During normal operation, TE=0, the normal driving voltage Vdp is applied to the top fixed fingers {S1,
D1, S2}, and Vdn is applied to all the bottom fixed driving fingers {S3, D2, S4}. The central movable
mass will be driven by the electrostatic force in a push-pull mode. The sensing electrodes {S1, S2, S3,
S4} also participate the normal operation of the device, so there is no capacitance area loss due to the BIST
implementation. When TE=1 and TS=0, the device is in the sensitivity test mode, and test driving voltage
Vd is applied between D1 and central mass M to activate the device with electrostatic force. Modulation
voltage Vmp is applied to {S1, S2} and Vmn is applied to {S3, S4}. The voltage level in the central mass M
indicates the device sensitivity to the given test driving voltage. This value is compared with the expected
good device value within a certain tolerance level to check whether the device is faulty. When TE=1 and
TS=1, the device is in the symmetry test mode, and the test driving voltage is applied to D1 to activate the
device. The modulation voltage Vmp is applied to S1 and Vmn is applied to S2. By checking whether the
voltage output on the central movable plate is a constant zero, we can know whether there are local defects.
The voltage biasing scheme of the microresonator in normal operation and two BIST modes is listed in
Table 3.4.
Table 3.4: Voltage biasing scheme for microresonator.
Voltage
biasing
Normal
operation
Sensitivity
BIST
Symmetry
BIST
Vdp
Vdn
Vd
Vnom
Vmp
Vmn
D1,S1,S2
D2,S3,S4
-
D1
D2,M
S1,S2
S3,S4
D1
D2,M,S3,S4
S1
S2
28
Chapter 4
Simulation Results of Dual-Mode Built-In
Self-Test
4.1 Simulation Results
In order to verify the effectiveness of our BIST technique, ANSYS (v6.1) was used for the fault simulation of the above three devices. We simulated three categories of MEMS defects: stiction, finger height
mismatch, and etch variation [21]. Stiction is a very popular defect type in surface micromachined MEMS
devices, and can be caused by molecular force, particle in photolithography or etching, and many other
possible reasons. For a stiction defect, the movable microstructure is stuck to the substrate or the fixed
microstructure at certain locations, and its movement is hindered. A height mismatch defect is one that the
floating microstructure bends upward or downward due to its internal residual stress, and thus the overlap
capacitance between the movable and fixed electrodes will be changed. An etch variation defect is one that
the thickness of the device structure does not meet the design expectation (under-etch or over-etch) due to
the etch variations caused by fluctuation of temperature, etchant concentration and other reasons. In the
following experiments, the amplitude of biasing voltage V0 is 5V, the noise floor is assumed 1mV, and the
output voltage should be larger than 1mV in order to be detected. Any voltage output below 1mV is treated
as 0.
4.1.1 Surface-Micromachined Comb Accelerometer
A stiction defect [21] at different locations of the right central movable finger is simulated. The location of
the stiction defect is expressed in percentage of the total finger length. The 0% point is defined as the finger
tip location, and the 100% point is defined as the finger root location connected to the central mass. The
simulation results are shown in Table 4.1. A finger height mismatch defect [21] is also simulated for the
comb accelerometer. Without the loss of generality, we assume the left side of the device is fault-free, and
a finger height mismatch only occurs on the right side of the device. The device is simulated for a finger
29
height mismatch with changes from 0.1m to 0.4m, and the results are shown in Table 4.2.
Table 4.1: Stiction defect simulation results for comb accelerometer.
Defect
location
Frequency
(kHz)
Sensitivity
BIST(mV)
Symmetry
BIST(mV)
defect-free
0%
10%
20%
30%
11.85
28.80
32.75
39.23
46.80
967.6
107.8
82.2
56.4
39.2
0
2.5
2.1
1.6
1.2
Table 4.2: Finger height mismatch simulation of comb accelerometer.
height mismatch
H (m)
Frequency
(kHz)
Sensitivity
BIST(mV)
Symmetry
BIST(mV)
Defect-free
0.1
0.2
0.3
0.4
11.85
11.85
11.85
11.85
11.85
967.6
951.8
937.1
921.8
904.4
0
42.9
83.9
128.2
181.3
From the simulation results, we can see that for the stiction defect on the movable finger, sensitivity test
can detect the defect very effectively. When there is a stiction defect on the movable finger, the sensitivity
drops dramatically from 967.6mV (faulty-free value) to 107.8mV and below. Hence, by comparing the sensitivity BIST output with the good device value, the stiction fault can be easily detected. When the stiction
defect moves from the movable finger tip to the root, the device displacement is increasingly hindered, and
thus the device sensitivity output further reduces. While there is a non-zero voltage for the symmetry BIST
output indicating the existence of a stiction defect, its change is not as apparent as the sensitivity BIST
output. Thus, sensitivity test performs better than symmetry test in detecting stiction defects at movable fingers. However, the simulation results also show that symmetry BIST exceeds sensitivity BIST in detecting
finger height mismatch defects. For a finger height mismatch of 0.4m, the sensitivity output drops only 6%
when compared with the good device response, but the symmetry BIST output changes from 0 to 181.3mV.
4.1.2 Bulk-Micromachined Accelerometer
Both stiction and etch variation defects [21] are simulated for a bulk-micromachined accelerometer. For
a stiction defect, it is assumed to be present on the surface of the central mass, which is very vulnerable
to stiction defect due to the small capacitance gap there. The stiction defect location is expressed as the
percentage of the half mass width. Here, the 0% point is at the right edge of the mass, while the 100% point
is at the center point of the mass. The simulation results are shown in Table 4.3. Etch variance defects are
very popular for bulk-micromachined devices due to the etching speed fluctuation caused by temperature
and etchant concentration changes. Without the loss of generality, we assume an under-etch variation defect
30
and it is only for the right portion of the central mass (the left portion of the mass is fault-free). Assume the
central mass thickness of the good device as 320m. The etching variance defect with changes from 0 to
1m in step of 0.25m is simulated, and the results are shown in Table 4.4.
Table 4.3: Stiction simulation results for bulk accelerometer.
Defect
location
Frequency
(kHz)
Sensitivity
BIST(mV)
Symmetry
BIST(mV)
defect-free
0%
10%
20%
30%
0.73
19.20
19.45
22.75
22.91
20.9
0
0
0
0
0
0
0
0
0
Table 4.4: Etch variation simulation results for bulk accelerometer.
etch
variation(m)
Frequency
(kHz)
Sensitivity
BIST(mV)
Symmetry
BIST(mV)
defect-free
0.25
0.5
0.75
1.0
0.73
0.73
0.73
0.73
0.73
20.9
21.1
21.2
21.4
21.6
0
30.4
74.5
111.4
149.6
From the simulation results, we can see that for a stiction defect on the mass, it will seriously hinder the
mass displacement due to the large rigidity of the mass. Thus, the mass can hardly have any displacement,
and the sensitivity drops to zero. The sensitivity BIST technique is very effective in detecting such a
defect, because the sensitivity BIST output will drop from 20.9mV (good device response) to zero if there
is any stiction on the central mass. However, since the mass almost stay at its static position, each of both
top/bottom sensing capacitances will keep its static value. Thus, the symmetry BIST output remains zero
all the time, which indicates it is not able to detect such a stiction defect for the device. However, for an
etch variation defect, the sensitivity output will change only about 3% (from 20.9mV to 21.6mV) for an
etch variation of 1.0m. But, the symmetry BIST output will change from 0 to 149.6mV, demonstrating its
effectiveness in detecting the etch variation defect.
4.1.3 Poly-Si Surface-Micromachined Resonator
Both stiction and finger height mismatch defects [21] are simulated for a microresonator. For a stiction
defect, it is assumed to occur at the top movable finger of the resonator, and its location is expressed by
the finger number counted from the right edge. There are totally 36 movable fingers in the top side of the
device, and the stiction can occur at one of fingers 1 to 5. The simulation results are shown in Table 4.5. For
a finger height mismatch, we assume the fingers at the left-side (right-side) of the y axis in Figure 3.9(b)
is fault-free (faulty). A finger height mismatch defect with values from 0 to 0.4m in step of 0.1m is
31
simulated and the results are shown in Table 4.6.
Table 4.5: Stiction defect simulation results for microresonator.
Defect
location
Frequency
(kHz)
Sensitivity
BIST(mV)
Symmetry
BIST(mV)
defect-free
right 1st
right 3rd
right 5th
2.45
85.98
96.84
103.93
66.5
0
0
0
0
0
0
0
Table 4.6: Finger height mismatch simulation of microresonator.
height mismatch
H (m)
Frequency
(kHz)
Sensitivity
BIST(mV)
Symmetry
BIST(mV)
defect-free
0.1
0.2
0.3
0.4
2.45
2.45
2.45
2.45
2.45
66.5
65.9
65.4
64.8
64.1
0
42.9
84.0
128.3
181.4
From the simulation results, we can see that if there is any stiction at the movable sensing finger, it will
greatly hinder the movement of the movable fingers and the sensitivity output drops to zero. Thus, sensitivity test is very effective in detecting such kind of defects. However, for symmetry BIST, since the device
remains in its static location after activation, both the left and right sensing capacitance pairs will remain
equal, and the symmetry BIST output will be always zero for both good and faulty devices. Thus, the symmetry BIST method is not effective in detecting stiction defects on the movable fingers of a microresonator.
However, for a finger height mismatch defect, the symmetry BIST method is very effective. The symmetry
BIST output changes from 0 (for fault-free device) to 181.4mV (height mismatch of 0.4m), while the sensitivity output only changes about 3% (from 66.5mV to 64.1mV). Thus, symmetry BIST exceeds sensitivity
BIST in detecting finger height mismatch defect of the microresonator.
4.1.4 Testing of Defects in Fixed Capacitance Plates
The above discussion has demonstrated the effectiveness of the dual-mode BIST technique for defects
in movable microstructures. However, the fixed capacitance plates of a capacitive MEMS device may
also contain some defects. Depending on their locations and extents, such defects may be fatal and lead
to device failure. Thus, an effective BIST solution must also be able to detect all defects in the fixed
capacitance plates. In this sub-section, the dual-mode BIST technique will be applied to deal with typical
defects in the fixed capacitance plates of a MEMS device. The simulation results prove the effectiveness of
the dual-mode BIST method for these defects which lead to either device symmetry or sensitivity change.
Since the resonant frequency of a MEMS device is fully determined by its movable microstructure, defects
in the fixed capacitance plates do not change the device resonant frequency at all. Thus, no frequency test
32
method is able to detect such defects.
Broken-Via Defect of Comb Accelerometer
In a CMOS MEMS comb accelerometer, the fixed driving (sensing) fingers are connected together into a
group by interconnects through vias. The interconnects are in turn connected to the corresponding driving
voltage or the capacitance sensing circuit. If the via between a fixed finger and its corresponding interconnect is broken, as a result, the fixed finger is electronically separated from the driving voltage or the
capacitance sensing circuit. That is, the fixed finger is electronically missing. This is a broken-via defect of
the comb accelerometer. Multiple broken-via defects can occur simultaneously. For simplification, single
and double broken-via defects in driving and sensing fingers are simulated, separately. The simulated single broken-via defect in a driving (sensing) finger is assumed to occur in the right-most (left-most) driving
(sensing) fixed finger of the top plate. The simulated double broken-via defects occur in the right-most
(left-most) driving (sensing) fixed fingers of both top and bottom plates. The ANSYS simulation results are
shown in Table 4.7.
Table 4.7: Broken-via defect simulation results of comb accelerometer.
Defect
location
Frequency
(kHz)
Sensitivity
BIST(mV)
Symmetry
BIST(mV)
defect-free
1 fixed driving finger
2 fixed driving fingers
1 fixed sensing finger
2 fixed sensing fingers
11.85
11.85
11.85
11.85
11.85
968.7
766.1
608.9
1072.9
1180.7
0
0
0
220.7
0
From the table, it can be concluded that each broken-via defect can be effectively detected by sensitivity BIST, since a significant difference in sensitivity between the defect-free and defective devices is
observed. However, symmetry BIST can detect broken-via defects only when they occur in the sensing
fingers asymmetrically. The asymmetric distribution of broken-via defects on driving fingers does not lead
to any asymmetric response due to the twist-resistance effect of the device structure. However, if broken-via
defects occur in the fixed sensing fingers asymmetrically, both sensitivity BIST and symmetry BIST can
detect them effectively.
Gap Variation Defect of Bulk-Accelerometer
For a glass-silicon-glass sandwich bulk-accelerometer, due to variations of the silicon-glass anodic bonding
process, the Al fixed capacitance plates may be farther or closer to the movable plate than the expected
value. This is called a gap variation defect. Without the loss of generality, the gap variation defect is
assumed to occur uniformly only to the top fixed capacitance plates. The bottom fixed capacitance plates
are assumed to be defect-free. If the gap variation is small, the device can still exhibit a linear response to
33
the acceleration in its sensitive direction. However, the zero-point of the device will shift. Thus, the device
needs to be adjusted by a zero-calibration process. The ANSYS simulation result for each different amount
of gap variation is listed in Table 4.8. The capacitance gap of the defect-free device is assumed 15 m.
Table 4.8: Gap variation defect simulation results of bulk accelerometer.
Gap
variation
Frequency
(kHz)
Sensitivity
BIST(mV)
Symmetry
BIST(mV)
defect-free
0.5m
1.0m
1.5m
2.0m
0.73
0.73
0.73
0.73
0.73
24.9
110.6
199.1
291.4
386.8
0
0
0
0
0
The simulation results show that each gap variation defect can be effectively detected by sensitivity
BIST, since there is a significant sensitivity difference between the defect-free and defective devices. However, because the gap variation occurs uniformly to the top plates, the symmetry BIST method cannot detect
it.
Side-Etch Variation of Microresonator
Due to the unavoidable side-etch effect in a dry/wet etching process, the sidewall of a fixed capacitance
plate may shrink by a certain amount of value. This in turn will lead to the increase of the corresponding
capacitance gaps. This is called a side-etch variation defect. Without the loss of generality, we assume that
the side-etch variation defect occurs only in the left portion of a microresonator, and the right portion is
defect-free. That is, all fixed fingers of the left part of the microresonator have thinner sidewalls. In this
way, the capacitance gap of each pair of fixed and moving fingers (in the left part of the microresonator) is
increased. The ANSYS simulation result for each different side-etch variation defect in the fixed fingers of
the left portion of a microresonator is shown in Table 4.9.
Table 4.9: Side-etch variance defect simulation results for microresonator.
side-etch
variation
Frequency
(kHz)
Sensitivity
BIST(mV)
Symmetry
BIST(mV)
defect-free
0.03m
0.06m
0.09m
0.12m
2.45
2.45
2.45
2.45
2.45
105.4
97.6
99.3
100.1
97.7
0
41.0
73.3
106.3
137.3
According to the simulation results, the symmetry BIST output voltage changes from 0V (for defectfree device) to 41.0137.3mV (for defective devices). In fact, the symmetry BIST output voltage increases
with the amount of the side-etch variance. Thus, symmetry BIST is effective to detect side-etch variance
defects of the microresonator. However, the sensitivity BIST output voltage only changes from 105.4mV
34
(for defect-free device) to 97.6mV (for defective devices) which is not significant enough. This indicates
that sensitivity BIST is not effective for side-etch variance defects of the microresonator.
The above simulation results show that all defects discussed in fixed capacitance plates can also be
detected by the dual-mode BIST method, either by sensitivity BIST or symmetry BIST or both.
4.2 Discussions
4.2.1 Capacitance and Sensitivity
For capacitance partitioning, the driving capacitance plate can be smaller than the sensing capacitance plate
for better sensing. However, reduced driving capacitance plate area means a higher test driving voltage is
required. Thus, there is a tradeoff between driving plate area and sensing plate area, and a suitable partitioning size must be decided for individual MEMS device. The overhead of the control circuit consists only
of a group of switches. Since the driving capacitance can also participate the normal operation, essentially
there is no area loss. If there is any, it is the gap among the capacitance plate partitions which is very small.
Fringe capacitance will be introduced due to the partitioning of fixed capacitance plates. Our ANSYS
simulations take fringe capacitances into account, and the results show that the BIST scheme works properly
for the MEMS devices. This indicates that the fringe capacitance effect is small and does not cause any
problem to the BIST scheme.
4.2.2 Calibration and Sensitivity BIST
The calibration process provides a specific stimulus with a known result to create a direct mapping between
sensor outputs and expected values. In this way, the graduations of the sensor are determined. Since a
special apparatus is needed to generate the stimulus of a real measurand, calibration is generally performed
by manufacturers before devices are released. Calibration is not convenient during in-field applications.
Sensitivity BIST uses some easily attained stimuli (such as electrostatic force, etc.)to mimic the action of
the measurand input, so it is suitable for in-field test. MEMS devices must be calibrated after fabrication,
and those with ”tolerable” defects will be calibrated and released as good devices. However, during infield usage, some unstable defects in a device may change the situation and cause the device to deviate
from its calibrated value. Such a change of unstable defects may not cause any left-right asymmetry, and
hence symmetry test cannot detect these defects. But, they may change the sensitivity of the device, so
that a sensitivity BIST can detect them. Furthermore, MEMS devices may still develop new defects during
in-field usage, e.g., broken beams, material fatigue, etc. Thus, an in-field sensitivity BIST is still very
necessary, even though MEMS devices are calibrated before releasing.
Take the surface-micromachined comb accelerometer as an example, if an impermanent elastic particle
blocks the movable mass, it may hinder the movement of the mass to a degree, but the mass is still somewhat
flexible. Since the particle only impact the central mass, it will not cause left-right asymmetry to the device
35
and symmetry test cannot detect this defect. The device may be calibrated and released as a good device.
However, during in-field usage, the particle may be released due to repeated movement of the mass. Thus,
the mass is free and the device sensitivity becomes larger than the calibrated value. Symmetry test cannot
detect this deviation, but an in-field sensitivity BIST can detect it.
For a bulk-micromachined accelerometer, the residual stress may cause the four beams to curl in the
same amount, so that the mass will be lifted above the central position of the capacitance gap. Symmetry test
cannot detect this defect since it does not cause any left-right asymmetry. Such a device may be calibrated
and released as a good device. During later usage, the stress is gradually released and the beams become
flat, thus the mass moves toward its central position. As a result, the zero position of the device will deviate
from its calibrated value. Symmetry test cannot detect it since it does not lead to left-right asymmetry.
However, sensitivity BIST can detect it, since the electrostatic force will change due to the change of the
driving capacitance gap.
For a microresonator, the residual stress may also cause the folded-beam to curl after fabrication, and the
movable fingers are lifted up uniformly for a certain amount. Since it does not cause left-right asymmetry,
symmetry test cannot detect this defect. It may be calibrated and released as good device. However, during
later usage, the residual stress is gradually released and the movable fingers move down toward the device
plane. The overlap area between fixed and movable fingers increases and for the same amount of driving
voltage, the resonator will have larger displacement output than the calibrated value. Symmetry test cannot
detect it since there is no left-right asymmetry. However, an in-field sensitivity BIST is able to detect it.
Thus, an in-field sensitivity BIST is still necessary.
4.2.3 Built-in Self Repair of MEMS Devices
A thorough BIST solution is the prerequisite for the built-in self-repair (BISR) of MEMS devices. MEMS
have found broad applications in many safety-critical fields, such as automobile, aerospace, etc. MEMS
reliability becomes a critical issue in these areas. Additionally, the low yield (compared to well-developed
VLSI fabrications) of MEMS has become a barrier for the commercialization of MEMS technologies. If
MEMS can have BISR features implemented, both reliability and yield can be greatly improved. Self-repair
techniques through redundancy for VLSI circuits have been well developed [28][29]. Similarly, the idea
of redundancy for repairing may also be a promising solution for MEMS devices. However, in reality, the
application of BISR implemented by redundancy to MEMS devices is much more difficult than that to VLSI
circuits. The reason is that, in a VLSI circuit, the replacement of a faulty circuitry by a redundant component
can be easily realized by using switches to prevent the defective one from contributing its signals. While
in a MEMS device, all the parts are mechanically connected as a whole, which makes it very difficult to
physically separate a faulty portion from the main device. For example, it is impossible to use a good
moving finger to replace a defective one by multiplexer switching.
A novel built-in self-repair solution for comb MEMS accelerometers through redundancy has been
36
developed in our work. In the BISR scheme, the comb accelerometer is divided into n number of identical
modules, and m number of redundant modules are available for self-repair. Before the BISR process can
be performed, the control circuit of a MEMS device needs to know whether each module is good or faulty.
Thus, each module must be implemented a BIST function and the BIST results will be reported to the
control circuit. If any of the working modules is faulty, it will be replaced with a good redundant module.
In this way, the MEMS device can be self-repaired into a good device given the number of faulty modules
is smaller than the number of redundancy. Hence, BIST is a prerequisite for the BISR of a MEMS device,
and this work has laid a cornerstone for this purpose.
37
38
Chapter 5
Built-in Self-repair of MEMS Comb
Accelerometers and Performance Analysis
5.1 BISR MEMS Comb Accelerometer Design
Due to the tiny size (in the range of microns) of MEMS accelerometers, its overall capacitance is generally
below 1pF, and the capacitance change in working mode is in range of fF. In order for the tiny capacitance
to be detected by the signal sensing circuit, it is desirable to enlarge the device capacitance which can
be achieved by increasing the capacitance area or decreasing the capacitance gap. However, the device
thickness and capacitance gap are limited by the capability of the fabrication process. Thus, the only way
to enlarge the device capacitance is to increase the number of capacitance groups. For simplification, only
32 finger groups are shown in Figure 2.1. But, a real MEMS comb accelerometer generally comprises of a
much larger number of repeated comb finger groups in a very compact manner. For example, an ADXL50
accelerometer contains 42 differential comb finger groups [8], while an ADXL150 accelerometer contains
54 differential comb finger groups [38]. A bulk-micromachined comb accelerometer based on siliconon-glass structure is shown in Figure 5.1 [39]. It contains 80 differential comb fingers groups. Such a
highly dense comb structure is necessary for the comb accelerometer, because it ensures the sum of all tiny
device capacitances between movable and fixed finger groups large enough to be detected by the sensing
circuit. Theoretically, the more the number of finger groups is, the easier the acceleration signal will be
detected. On the other hand, such a highly dense comb structure with many long and narrow capacitance
gaps is extremely vulnerable to various defects such as particle contamination, stiction [40]. Taking the
ADXL50 accelerometer as an example, the length of each movable finger is 120m, while the capacitance
gap between each pair of fixed and movable fingers is only 1.3m. If a conductive particle with diameter
larger than 1.3m falls into any of the 84 capacitance gaps, it will lead to a short-circuit of the device
capacitance and result in a failure of the entire device. In other words, all the 42 finger groups will be
useless because of the catastrophic defect occurred to one single finger pair. In the fabrication process of
39
a comb accelerometer, it is very likely to have particle contaminations occurred to the capacitance gaps.
Thus, a large number of finger groups unavoidably leads to the decrease in yield as well as reliability.
However, if the MEMS device has a modularized design with some redundant modules, it can repair itself
to a good device by replacing faulty modules with good (and redundant) ones, given the number of faulty
modules is smaller than the redundancy number. With this fault-tolerant feature, both in-field reliability and
manufacturing yield of the MEMS device integrated in a SoC design can be greatly enhanced.
Figure 5.1: SEM photo of a bulk-micromachined comb accelerometer device[39].
anchor
beam
mass
Ma
Mb
Mc
Md
Me
Mf
Figure 5.2: Modularized comb accelerometer structure.
The modularized comb accelerometer design with BISR feature is shown in Figure 5.2. Here, the device
consists of six identical modules, and each module has its own beams, mass and finger structures (fixed and
movable). By assumption, four modules are connected together as the main device, while the remaining
two modules serve as redundancy. The general case for the distribution of modules between the main
device and the redundant device can be extended easily. The movable parts of each module are physically
connected to those of adjacent modules through the common anchors. In fact, all movable parts of the
entire MEMS device are connected physically, and signals sensed by all movable fingers in the device are
40
connected to the sensing circuit directly. However, the fixed fingers of each module are connected to the
modulation signal circuit through switches made of analog muxes. By turning on or off these switches, we
can determine whether a module works as part of the main device or the redundant device. For example,
if the modulation signals of module M are turned off, then the movable fingers of M cannot sense any
signal. Thus, M is electronically disconnected from the MEMS device (though it is still connected to the
entire MEMS structure physically), and is not involved in the MEMS function. We will show later that the
analog multiplexers do not degrade the signal sensing, and an electronically isolated module will not result
in any capacitive load effect.
During the BISR mode, the device will first perform the BIST process for each individual module. The
dual-mode BIST technology presented in [30] will be used here to ensure a thorough test of each module by
combining sensitivity test and symmetry test together. The BIST result of each module is fed into the BISR
control circuit. If a module is tested as faulty, the control circuit will permanently exclude the module from
the main device, and it will not be used any more in the future. If the number of good modules is smaller
than four, which is the minimal number of good modules required in the main device for normal operation,
then the device is faulty and non-repairable. However, if the number of good modules is larger than four, the
entire device is repairable though some modules are faulty. Thus, after repairing, the main device can still
be ensured to work properly. In this case, four of the good modules are structured as the main device, while
the remaining good module(s) (if any) will serve as redundancy. If any of the working modules in the main
device is found faulty in the future, the control circuit will separate the faulty module from the main device,
and replace it with a good redundant module. Thus, by realizing the BISR technique through redundancy
and repairing, both the yield of the manufacturing process and the reliability of in-field applications of the
MEMS device in a SoC can be greatly enhanced. We emphasize that all moving modules are physically
connected, and this guarantees an enough number of moving and fixed fingers working together to generate
strong signals. However, isolated modules are still physically connected (but electronically isolated) to the
main device without affecting the function. The BISR technique is virtually applicable to any kind of local
defects within one or two modules. However, it cannot be applied to global defects where all the modules
are faulty and no functional module is available.
In order to analyze the benefits as well as its overhead for implementing the BISR feature into a comb
accelerometer, we need to make a comparison between the accelerometers with and without the BISR
feature. For the convenience of discussion, we call the MEMS comb accelerometer with (without) the
BISR feature as a BISR (non-BISR) accelerometer separately. The total device capacitance of a comb
accelerometer plays an important role in the design consideration, since it must be large enough to be
detected by the sensing circuit. Generally, such a capacitance is in the order of pF for MEMS comb
accelerometers. In order for a fair comparison, we assume that the capacitance of a non-BISR accelerometer
is equal to that of the main device of a BISR accelerometer. That is, the total number of finger groups of
the non-BISR accelerometer should equal to that of the main device in the BISR accelerometer. In this
way, when compared with the non-BISR design, the area overhead of the modularized BISR accelerometer
41
includes the area for two redundant modules, plus the area for extra beams in the BISR accelerometer.
5.2 Performance Analysis
5.2.1 Sensitivity Analysis
Assume the main device of the BISR MEMS accelerometer contains n modules. The sense mass of each
module falls to 1=n that of the corresponding non-BISR design. If the geometry parameters of all beams
are kept the same for both BISR and non-BISR designs, then the sensitivity of the BISR accelerometer will
fall to approximately 1=n of the non-BISR design. It should not be hard to understand that the sensed signal
out of the movable parts of the BISR MEMS device drops to 1=n of the original strength. However, in the
following analysis, we will find that this sensitivity loss due to modularized design can be easily compensated by adjusting the beam width and other parameters. Furthermore, the sensitivity of an accelerometer in
the open-loop mode can be effectively enhanced with an appropriate DC biasing voltage till infinity. [41].
Thus, the sensitivity loss due to device modularization is not a problem for the BISR accelerometer.
In order to find out the sensitivity of a comb accelerometer, dynamic analysis must be performed. A
MEMS comb accelerometer actually can be simplified by a spring-mass model. The width and length of
each tether (seismic mass) are represented by Wb (Wm ) and Lb (Lm ) separately, while the width and length
of each movable finger are denoted by Wf and Lf respectively. There are totally v number of movable
fingers, and the thickness of the device is h. Assume the density and the Young’s modulus of poly-Si as and E respectively. The sensing mass Ms of the accelerometer, which includes the seismic mass and all the
movable fingers attached to it, can be expressed as follows
Ms = h(Wm Lm + nf Wf Lf ):
In the above analysis, all beams act as a spring, while the central mass and all movable fingers act as the
mass. For the non-BISR accelerometer shown in Figure 2.1, the four beams can be treated as four springs
connected together in parallel. For each beam, the spring constant ks can given by [34]
ks =
12EIb EhWb3
=
:
L3b
L3b
Thus, the total spring constant ktot of all four beams is
ktot = 4 ks =
4EhWb3
:
L3b
Hence, the displacement of the movable fingers can be expressed as
Ms a
ktot
a(Wm Lm + nf Wf Lf )L3b
:
=
4EWb3
ye =
42
The deflective sensitivity Sd [34] of the device, which is defined as the displacement of movable fingers
per unit gravity acceleration (g) along the sensitive direction, can be expressed as
Sd =
Ms g g(Wm Lm + nf Wf Lf )L3b
=
:
ktot
4EWb3
The capacitive sensitivity S [34] is defined as the capacitance change of the device under unit gravity
acceleration (g) along the sensitive direction. Given a displacement x of the movable mass and fingers
where x is much smaller than the static capacitance gap d0 , the capacitance sensitivity S can be expressed
as
nf 0 h(Lf )Sd
d20
n h(L ) g(Wm Lm + vWf Lf )L3b
:
= f 0 f
4EWb3 d20
From the above equations, we can observe that the device sensitivities Sd and S are tightly related to
S =
the device geometry parameters. We can tune up the device sensitivity by adjusting its geometry parameters.
Among them, there are two important geometry parameters which can be used to adjust the device sensitivity very effectively, without causing great change to the entire device area. They are the beam width Wb and
the central mass width Wm . The above equations show that the device sensitivity is inversely proportional
to the cube of beam width Wb , but linearly proportional to the width of central mass Wm . Assume all other
parameters are fixed, the relationships between the device displacement sensitivity Sd and the beam width
Wb , central mass width Wm are shown in Figure 5.3. From the curves, we can see that the displacement
sensitivity increases sharply as the beam width decreases. It also increases with the width of central mass
Wm , but the change is not as significant as that caused by the beam width change. Thus, the beam width
Wb can be used as the most effective geometry parameter to adjust the device sensitivity.
Figure 5.3: The relationship between displacement sensitivity of accelerometer and beam/mass width.
As stated before, in order for a fair comparison between the BISR and non-BISR accelerometers, we assume the total number of finger groups of the non-BISR accelerometer is equal to that of the main device in
43
the BISR accelerometer. Thus, for each module in the main device of the modularized BISR accelerometer,
the sense mass is approximately 1=n of the non-BISR design in size. We denote the deflective sensitivity of
the BISR (non-BISR) accelerometer by Sdb (Sdnb ). If we keep the same geometry parameters (i.e., width
Wb , length Lb and thickness h) for the beams of both BISR and non-BISR designs, we have
Sdb =
Ms g
n
ktot
1
= Sdnb :
n
where Ms and ktot are the sense mass and spring constant of the non-BISR accelerometer.
From the above equation, the deflective sensitivity of the BISR accelerometer is approximately 1=n that
of the non-BISR design. Denoting the capacitive sensitivity of the BISR (non-BISR) accelerometer as Sb
(Snb ), we have
Sb =
nf 0 h(Lf
) (1=n)Sdnb 1
= Snb :
n
d20
Thus, the capacitive sensitivity of the BISR accelerometer also falls to approximately 1=n that of the nonBISR design.
Fortunately, many choices are available to compensate this sensitivity loss due to the modularized design. For the BISR accelerometer design, we can recover its sensitivity by reducing the beam width of
each module, by using folded beams instead of straight beams to increase the effective length (Lb ) of each
beam, or by enlarging the width (Wm ) of the central mass, etc. According to the above equations, all
these changes will help increase the sensitivity of the modularized accelerometer. However, as discussed,
tuning down the beam width Wb is the most efficient method to recover the sensitivity value back to normal. Take the beam width of the non-BISR accelerometer as Wb0 . If we reduce the beam width of the
BISR accelerometer to 0.63Wb0 and all other parameters are kept the same, the sensitivity of the BISR
accelerometer will become approximately equal to that of the non-BISR design. Thus, there is plenty of
room in adjusting the beam width to increase the sensitivity of the BISR accelerometer, and the sensitivity
loss due to device modularization will not be a concern for the BISR accelerometer. However, the beam
width cannot be reduced unlimitedly. If the beam is too narrow, it may be too fragile for fabrication. Thus,
the fabrication technology generally defines the minimum width for all beams. If the beam width of the
non-BISR accelerometer already approaches this minimum size (such as for high-sensitivity accelerometers), then the room for reducing the beam width is small and the other two choices may be considered. The
designer must compensate the sensitivity loss of a high-sensitivity BISR accelerometer by the combination
of the above three alternatives, according to the individual device requirements.
Even if all the device parameters (e.g., beam width and mass width) are fixed, the sensitivity of an
accelerometer in the open-loop mode can still be conveniently enhanced by electrostatic force with an
appropriate DC biasing voltage [41]. The electrostatic force acts as a spring with a negative spring constant.
This will help reduce the effective spring constant of the accelerometer and increase its sensitivity. Since
this electrostatic force actually deflects the mass, it is equivalent to the effect of an inertial force acting on
44
the mass. Such an enhancement of sensitivity has no side-effect at all. It differs from simply amplifying the
voltage signal in the circuit, which also amplifies the noise. Thus, the electrostatic force is a powerful tool
to enhance the sensitivity. A small DC biasing voltage (several volts) can increase the sensitivity to infinity,
and this offers a great flexibility in the sensitivity recovery.
5.2.2 Sensitivity Compensation through Electrostatic Force
Most of the capacitive MEMS accelerometers operate in either open-loop or closed-loop (force-balanced)
mode [42]. In open-loop mode, the suspended proof mass actually moves from the middle position, and
the displacement of the mass is measured by a sensing circuit to evaluate the experienced acceleration.
In closed-loop mode, however, an electrostatic force is used as a feedback on the mass to counteract the
displacement caused by the inertial force. Thus, the proof mass remains in the middle position. The required
feedback voltage to counteract the inertial force indicates the value of the acceleration. In open-loop mode,
the sensitivity of a modularized design can be easily compensated by applying a DC biasing voltage. But
in closed-loop mode, the sensitivity of the modularized design remains the same as the non-BISR design.
Thus, no sensitivity compensation is needed at all.
If the accelerometer works in open-loop mode, in the existence of DC biasing voltage VB , the electrostatic force will impose a positive effect on the displacement of the movable mass. In this way, the
displacement of the mass will further increase, which leads to the enhancement of the device sensitivity. A
small DC biasing voltage can easily increase the device sensitivity in open-loop mode till infinity, which has
been theoretically analyzed and experimentally observed in [41]. This offers a great opportunity to compensate the sensitivity loss in the MEMS redundancy repair technique. Since the electrostatic force physically
drives the movable mass, it is actually equivalent to the inertial force introduced by the acceleration. Thus, it
effectively enhances the device sensitivity. This is different from the case of simply amplifying the electrical
signal in the measurement circuit where the noise is also amplified. With the powerful tool of electrostatic
force, sensitivity loss will no longer be a problem for redundancy repair of MEMS accelerometers. In the
following analysis, the value of the required biasing voltage VB for the sensitivity compensation of a BISR
accelerometer will be analyzed.
As shown in Figure 5.4, the movable fingers experience a displacement of x along the horizontal
direction. For simplicity, only one finger group is shown in the figure. Voltages with opposite phases
([V (t) + VB ℄) are applied on the left/right fixed fingers separately. Here, VB is the DC biasing voltage,
V (t) is the high-frequency ( 1MHz ) square wave modulation voltage
V (t) = V0 sqr(!t):
Under this voltage biasing scheme, the differential capacitances 1 and 2 act as a voltage divider. The
voltage level in the movable finger VM can be expressed as
VM =
x
[V (t) + VB ℄:
d0 45
Vc (t)+V B
−Vc (t)−V B
Vs (t)
fixed
finger
F1
movable finger
fixed finger F2
f e2
f e1
d0+x
d −x
0
X
0x
Figure 5.4: The sensitivity increase of accelerometer due to DC voltage biasing.
The interface circuit senses the high-frequency AC signal whose amplitude is related to the acceleration
a, but ignores the DC signal. Due to the voltage biasing, the movable finger experiences electrostatic forces
fe1 and fe2 introduced by left/right fixed fingers (Fig. 5.4). Assume there are nf number of finger groups.
The electrostatic forces experienced by the movable finger toward the right (Fe1 ) and left (Fe2 ) directions
are
nf [V (t) + VB VM ℄2 s
;
2(d0 x)2
Fe1 = nf fe1 =
Fe1 = nf fe1 =
nf [ V (t) VB VM ℄2 s
:
2(d0 + x)2
where s is the overlap area between the movable and fixed fingers in each finger pair, and is the dielectricity
constant. In working mode, the displacement of movable fingers is much smaller than the capacitance gap
(x d0 ) to ensure the device linearity. Thus, the voltage level VM in movable fingers is also much smaller
than the biasing voltage. For the calculation of electrostatic force, VM can be treated as approximately zero.
Further, since dx0 1, we have
x 2
d0 )
d12 (1 + dx )2 ;
1
1
=
(d0 + x)2 d20 (1 + dx0 )2
d12 (1 dx )2 :
(d0
1
1
=
x)2 d20 (1
The static capacitance C0 of the accelerometer is
C0 =
S
:
d0
0
0
0
0
where S (= nf s) is the total overlap area between each pair of fixed and movable fingers. Thus, the
electrostatic forces experienced by the movable fingers can be rewritten as
46
Fe1 = nf fe1 =
[V (t) + VB ℄2 C0
x
(1 + )2 ; and
2d0
d0
Fe2 = nf fe2 =
[V (t) + VB ℄2 C0
(1
2d0
x 2
):
d0
The total electrostatic force Fet is
Fet = Fe1 Fe2
2[V2 (t) + VB2 + 2V (t)VB ℄C0 x
:
=
d20
Since the working frequency of the biasing voltage is much higher than the resonant frequency of the
microstructure, the average effect of the electrostatic force is the time average of the above equation. The
time average values of V2 (t) and V (t) are V02 and 0 separately. Thus, we have
Fet =
2(V02 + VB2 )C0 x
:
d20
We can see that the net electrostatic force Fet is directly proportional to the displacement x, but it is
along the same direction of the displacement of the movable fingers. It can be treated as a spring with
”negative” spring constant kel
kel =
2(V02 + VB2 )C0
:
d20
As we derived before, the total restoring force of the four beams of the accelerometer is
Fm = ktot x =
4EhWb3
x:
L3b
where ktot is the spring constant of the accelerometer given by
ktot = 4 ks =
4EhWb3
:
L3b
The inertial force Fa is
Fa = Ms a:
The dynamic equilibrium equation of the accelerometer can be written as
Fa + Fel + Fm = 0:
47
The above equation can be rewritten as
Ms a = [ktot + kel ℄ x:
Finally, the displacement x after considering the electrostatic forces is
x=
Ms a
:
[ktot + kel ℄
The negative sign of x indicates that the displacement of movable fingers are in the opposite direction of
the acceleration. Since kel is negative, the electrostatic force acts as a negative spring. It helps lower the
effective spring constant of the device, and hence increases the device sensitivity.
Assume the interface circuit has the voltage gain of G. The interface circuit detects the amplitude of
the AC component of VM , which is V0 dx0 . The output voltage U0 when the movable mass moves with
displacement of x can be expressed as
x
U0 = GV0 :
d0
The displacement per unit gravity acceleration (1g
= 9:8m=s2 ) is therefore
x(a = 1g) =
Ms g
:
[ktot + kel ℄
Note that we only consider the absolute value of the displacement for sensitivity purpose. Thus, the negative
sign of x is omitted. The voltage sensitivity of the non-BISR accelerometer expressed as the voltage output
per unit gravity acceleration (1g ) is
Sa =
GMs gV0
(V=g):
(ktot + kel )d0
Figure 5.5 shows an example accelerometer design with DC biasing voltage for sensitivity enhancement.
As shown in the figure, when no DC biasing voltage is applied (VB = 0V ), the sensitivity Sa is 11:2V=g .
When we have VB = 4V , the sensitivity is increased to Sa = 111:8V=g . If we have VB = VBP = 4:21V ,
the sensitivity will become infinity.
Assume the spring constant of the non-BISR accelerometer is k , the static capacitance of the non-BISR
accelerometer as C0 , the voltage amplitude of the high-frequency carrier as V0 , and the DC biasing voltage
of the non-BISR accelerometer as VB 0 . For the BISR MEMS accelerometer, assume the main device is
divided into n number of modules. For each module in the main device of BISR design, the mass of
movable mass M reduces to n1 of that of the non-BISR device
Ms(module) = Ms =n:
48
Figure 5.5: The sensitivity increase of accelerometer due to DC voltage biasing.
However, the beam dimensions remain the same as that of the non-BISR device. That is, the spring constant
ktot of each module in BISR device remains the same as that of non-BISR device. Hence the sensitivity of
the BISR module is reduced to approximately 1=n of the non-BISR design
1
Sa(BISR) = Sa :
n
The capacitance of each module of the BISR device is also reduced to 1=n of that of the non-BISR device
as follows.
C0(BISR) = C0 =n:
Assume the original non-BISR device does not use DC biasing voltage, and the biasing voltage VB is
required for the BISR module to recover the sensitivity. Thus, we have
[ktot
G(Ms =n)V0
2(C0 =n)(V02 +VB2 )
d20
℄d0
=
[ktot
GMs V0
2C0 V02 ℄d :
0
d20
Solving this equation, the required biasing voltage VB for the BISR device to make full sensitivity compensation can be calculated by
VB =
s
(n 1)ktot d20
:
2C0
By selecting the above biasing voltage, the sensitivity of the BISR device can be fully compensated
back to that of the original non-BISR device. That is, the BISR MEMS accelerometer will not have any
performance degradation in its sensitivity compared to the non-BISR design. The application of DC biasing voltage will slightly degrade device stability and non-linearity. It also requires some additional offset
calibration. However, these negative effects only slightly degrade the device performance and will not be a
serious threat to the device function. As shown in [41], the degradation is not significant especially if we
only need to increase the BISR device sensitivity by four times. As we mentioned before, the BISR design
is for high-reliability and safety-critical usage, and it is assumed that for these applications, the overhead
can be tolerated in order to trade for high reliability.
49
5.2.3 Frequency Analysis
Resonant frequency is also an important feature which determines the dynamic performance of a MEMS
device. A higher resonant frequency means a wider working bandwidth but the device sensitivity will
be reduced. Thus, there is a tradeoff between the device sensitivity and the resonant frequency, and an
optimization must be made. For the modularized BISR design, the resonant frequency may be different
from that of the non-BISR design due to the change in device structure. Thus, frequency analysis must also
be considered.
Using the simplified spring-mass model discussed above for the comb accelerometer, the resonant frequency fnsr of the non-BISR design can be given by the following equation [8]
s
1 ktot
:
fnsr =
2 Ms
where Ms and ktot are defined in the last section. For the modularized BISR design, the resonant frequency
of the main device is actually that of each individual module. Assume the dimension of each beam in the
BISR design remains the same as that of the non-BISR design (i.e., ktot remains the same), but the comb
finger groups are divided into n identical modules (i.e., Ms is changed into n1 Ms ). The resonant frequency
fsr of the BISR accelerometer can thus be given by
1
fsr =
2
sk
p
tot
1 Ms = nfnsr :
n
Taking the case where n = 4, we have
fsr = 2fnsr :
That is, if the dimension of each beam remains unchanged, the resonant frequency of the BISR accelerometer will be doubled when compared to the non-BISR design. However, if the deflective sensitivity Sd is
compensated back to the value of the non-BISR device by adjusting the device parameters, the resonant frequency of the BISR design will also be adjusted back to the non-BISR value correspondingly. The reason
can be explained as follows. The deflective sensitivity Sd is given by
Sd =
where
have
Ms g
:
ktot
g is the unity gravity acceleration experienced by the accelerometer in its sensitive direction.
fnsr =
We
pg
p :
2 S
d
Consequently, if the deflective sensitivity of the BISR accelerometer is compensated to the same as that
of the non-BISR design by adjusting the design parameters (such as shrinking the beam width), the resonant frequency will also remain the same as that of the non-BISR device. However, if the sensitivity is
50
compensated by electrostatic force, the beam width of the BISR accelerometer remains the same as that
of non-BISR device. Thus, the resonant frequency of the BISR device will be two times as that of the
non-BISR device. This actually extends the device working frequency range.
51
52
Chapter 6
Yield Analysis of MEMS Redundancy
Repair
6.1 Yield analysis of the BISR comb accelerometer
6.1.1 Yield Model for MEMS Redundancy Repair
For the yield analysis of VLSI circuits through redundancy repair, many models have been available
[43][44]. However, MEMS devices have their own particular characteristics which are very different from
VLSI circuits. For example, embedded memory arrays may have higher similarity to the MEMS devices
than other VLSI circuits. However, generally, BISR for an embedded memory array is achieved by adding
two redundant memory blocks, one for row redundancy while the other for column redundancy [23]. It
is also possible that only one redundant memory block is added for either row redundancy or column redundancy. Any defect in the main circuit of the embedded array can be replaced by the redundant circuits
without any limitation to the topological location of the defect. That is, a defective memory module can still
be used by switching off the defective row or column signal (and a good row or column from the redundant
block will replace the bad one) by multiplexers. However, a defective MEMS module must be abandoned
due to the involvement of mechanical operations. For example, it is impossible to switch off a finger with
a stiction defect and replace it by another finger that might be far away at a redundant module. Thus, the
entire module containing the defective finger must be abandoned. From the other hand, MEMS module
has an advantage that a defective module can contain virtually infinite number of defects, since it will be
switched out of the normal operation eventually. But, this is not the case for memory array BISR. Thus, existent yield analysis models for redundancy repairing of VLSI circuits cannot be directly applied to MEMS
devices, and a new yield model must be developed though the defect distribution models of VLSI circuits
can be used. Currently, the yield model for MEMS devices has not been well developed yet, especially a
yield model considering redundancy repair is not existent.
Assume a set of defects can occur to N number of locations in a MEMS device, i.e., there are N
53
possible defects in the MEMS device. Further, assume every defect occurs independently to each other, and
the probabilities for each defect to occur is equal and defined as q . Thus, based on the defect distribution
discussed in [43], the probability P (X = x) that x number of indistinguishable randomly distributed
defects occurring to the MEMS device can be expressed as a binomial distribution
P (X = x) = (Nx )qx (1 q)N x :
If we assume that the number of possible defects N is large enough so that N
converges to , we can achieve a Poisson distribution of the defects
P (X = x) =
q (i.e., the average defect)
x e :
x!
The simple Poisson distribution is too pessimistic for yield estimation because the defect clustering effect
is not considered. Hence, the compound Poisson distribution is more popular by considering the normalized distribution of chip defect density clustering factor. The next problem is how the average defect distributes. In this work, we assume that the defect distribution function F () for is a gamma function
given by the following equation [43]
F () =
1
()k 1 e
(k)(Ab)k
=(Ab) :
where A is the device area, b is a defect density coefficient, is the average defect, and k is the clustering
parameter. Larger k number means less clustering effect. For k = 1, it is the random distribution of
defects. Further, the average defect density D0 is given by
D0 = b k:
The probability that x defects occur in a MEMS device with area A can be represented by the following
equation
Z 1 x
e F ()d
x!
Z 1 x+k 1 (1+1=Ab)
1
=
()
e
d
x! (k)(Ab)k 0
(x + k)(Ab)x
=
x! (k)(1 + Ab)x+k
1 k Ab x
= (xx+k 1 )(
)(
):
1 + Ab 1 + Ab
For a non-BISR MEMS accelerometer, the yield Y0 is the probability that no defect occurs (i.e., X = 0),
P (x; A) =
0
which can be expressed as
54
Y0 = P (0; A0 ) = (
1
)k :
1 + A0 b
where A0 is the area of the non-BISR accelerometer. From this equation, we see that the yield drops as the
device area increase. This is reasonable because the larger the device area is, the more likely it may suffer
from some defects.
For a modularized BISR accelerometer, since redundant modules are added, the total device area is
increased. This leads to a decrease in the yield. However, for a device with the number of faulty modules
smaller than the device redundancy, it can be self-repaired to a good device. In this way, the yield can be
greatly increased due to such redundancy repair. The net yield change of BISR MEMS accelerometers,
when compared with non-BISR design, is the combination of the above two conflicting factors. Assume
there are totally i possible defects in the BISR MEMS accelerometer, n number of modules in the main
device, and m number of redundant modules as shown in Figure 6.1. The yield of the BISR MEMS
accelerometer after redundancy repairing, denoted as Yr , equals the possibility that none of the i defects
occurs, plus the possibility that some among the i defects do occur but they all fall into no more than m
number of modules. The former is the case where all the (n + m) number of modules are healthy, while
the latter is the case in which some modules are faulty, but the device can still be self-repaired into a good
one through redundancy. Each of both cases will be investigated in the following discussions.
i faults
m redundant modules
n main modules
Figure 6.1: Fault distribution among the modules of BISR accelerometer.
Assume the main device of the BISR accelerometer has the same number of finger groups as the corresponding non-BISR design, the area of the non-BISR accelerometer is A0 , and the area of beams in each
module of the BISR accelerometer (and also the none-BISR accelerometer) is Ab , The area Ar of the BISR
accelerometer is given by
(n + m)(A0 Ab )
+ (n + m)Ab
n
(n + m)[A0 + (n 1)Ab ℄
=
:
n
Ar =
The possibility that none of the i number of defects occurs can be expressed as
P (0; Ar ) = (
1
)k :
1 + Ar b
55
For the second case in which some defects do occur but the device can still be self-repaired through
redundancy, it can be further divided into the following two sub-cases:
The total number (i) of defects is smaller than or equal to the number (m) of redundant modules,
i.e., i m. For this sub-case, regardless of whatever distribution for the defects, the BISR comb
accelerometer can always be repaired into a good device.
The total number (i) of defects is larger than the number of the redundant modules m; however, all
the defects fall into m number of modules or less. For this sub-case, the BISR accelerometer can still
be repaired into a good device.
Both the above two sub-cases must be counted into the yield of the BISR accelerometer. The first sub-case
can be easily solved, while the second sub-case requires an extensive analysis. For the first sub-case, the
possibility P1 that i number (i m) of defects occur in the BISR MEMS accelerometer can be expressed
as
x=m
P1 =
P (x; Ar ):
x=1
X
For the second sub-case (i m), the faulty device can be repaired into a good device only if all the i
number of defects fall into m number of modules or less. First, we examine the possibility that i defects
are distributed into j (j m) MEMS modules and each of the j modules contains at least one defect. So,
we distribute one defect to each of the j modules to ensure that each of the j modules contains at least one
defect. Thus, there are i j defects remaining to be distributed to the j modules with any number of defects
(maybe 0) for this distribution. As we know, there are (nr 0 +r 1 ) ways to distribute r identical balls into n0
distinct cells with any number of balls per cell. Here, we have r = i j and n0 = j and the total number
of ways of distribution is (ii 1j ). Finally, the probability that the BISR accelerometer (with n modules in the
main device, and m modules in the redundancy device)) can be repaired when a certain i number (i m,
sub-case 2) of defects occur is
R(m; n; i) =
P
Pmj=1(m+n) (i
1
i j)
j
:
min(i;m+n) m+n
( j ) (ii 1j )
j =1
In this equation, the numerator stands for all possible cases where all the i number (i m) of defects fall
into m (or less) number of modules so that the device can be self-repaired. The denominator stands for all
possible cases in which i number (i m) of defects are randomly distributed among the (m + n) modules.
Thus, the possibility P2 that more than m number of defects occur to a MEMS device, but it still can be
self-repaired into a good device can be expressed as follows.
P2 =
1
X
x=m+1
P (x; Ar ) R(m; n; x)
56
=
1
X
x=m+1
P (x; Ar ) P
Pmj=1(m+n) (x
1
x j)
j
:
min(x;m+n) m+n
( j ) (xx 1j )
j =1
That is, P2 equals to the sum over all probabilities that more than m number of defects occur and they can
be self-repaired.
According to the above discussion, the probability that defect-free or some defects do occur but the
BISR accelerometer can be self-repaired into a good device through redundancy is given by the sum of
above two sub-cases. Hence, the yield for the BISR accelerometer after redundancy repair is
Yr (m; n) = P (0; Ar ) + P1 + P2
=
1
X P (x; A ) + X
x=m
r
x=0
x=m+1
Pmj=1(m+n) (x
1
x j)
j
:
min(x;m+n) m+n
( j ) (xx 1j )
j =1
P (x; Ar ) P
The yield increase IY (m; n) of the BISR accelerometer, when compared with the corresponding non-BISR
accelerometer, can be given by
IY (m; n) = Yr (m; n) Y0
=
1
X P (x; A ) + X
x=m
x=0
r
x=m+1
Pm
m+n
x 1
j =1 ( j ) (x j )
min(x;m+n) m+n
( j ) (xx 1j )
j =1
P (x; Ar ) P
P (0; A0 ):
We can see that under this theory, if we set m = 0 and n = 1, then Yr (0; 1) comes back to P (0; A0 ) which
is exactly the initial yield of the non-BISR device. Thus, the general case of Yr (m; n) includes the initial
yield of the non-BISR device as a special case.
6.1.2 Yield Increase Analysis
Theoretically, with a larger number of redundancy in a BISR device, more faulty modules in the main device
can be repaired, and hence a higher repair rate (i.e., R(m; n; i)) can be achieved. Thus, for the benefit of
yield increase, we tend to have a large number of redundancy. On the other hand, this will lead to significant
area overhead accordingly. In a SoC design, the area available for MEMS devices may be limited. Further,
this increase in area can cause a higher probability of defect occurrence (i.e., P (x; A)), because it is possible
that more defects may occur within the enlarged area. The net effect depends on which factor (i.e., yield
increase due to redundancy vs. yield drop due to device area increase) is stronger and indirectly depends
on different parameters in the above equations. Thus, in a BISR accelerometer design, yield increase and
area overhead are both important factors we have to consider. A decision must be made to give the number
of modules in the main device and that in the redundant device, respectively. That is, we have to find a pair
57
of optimized m and n values to achieve the maximum yield increase due to BISR, while keeping the area
overhead within an acceptable range.
Based upon the MEMS yield model for redundancy repair, we can derive the relationship between the
yield increase and the non-BISR device yield for different m and n numbers. The defect clustering factor
k is decided by the individual fabrication process. For the following analysis in this section, we randomly
assume k = 1 for demonstration purpose. That is, we assume a certain amount of clustering effect for the
defects distribution. Figure 6.2 shows the simulation result for n = 4 and m = 2; 4; 6; 8 separately. From
this figure, we can observe that the BISR device always gives a positive yield increase regardless of the
initial yield. This demonstrates the effectiveness of the redundancy repair technique for MEMS devices.
If the initial yield is too low (approaches 0) or too high (approaches 1), the yield increase by redundancy
repair is not significant. This is a reasonable result. For a very low initial yield (approaches 0), the defect
density is extremely high and there are too many faulty modules in the main device. Compared with so
many defective modules in the main device, the redundancy is relatively deficient to repair all of them.
Thus, the yield increase by redundancy BISR is not significant. For a very high initial yield (approaches
1), the main device itself is highly likely to be fault-free, and hence repair is not necessary. Thus, the effect
of redundancy repair is not relatively significant. However, for a moderate initial yield (e.g., around 0.4),
the redundant modules can be fully utilized to repair as many faulty modules as possible, and a significant
yield increase due to redundancy BISR can be observed. Fortunately, the current yield of MEMS devices
does fall into this moderate range, and a significant yield increase can be achieved. The yield for bulkmicromachined accelerometer in Applied MEMS Inc. is about 50% [45]. According to [46], the yield
improvement of ADXL series accelerometers in Analog Device Inc. has also been a very difficult task and
tremendous effort was made on the yield of the accelerometers in order to make it profitable. Initially, the
MEMS manufacturing process had only a 10% yield. That is, ninety percent of the fabricated accelerometers
were defective. After many years’ effort, the yield of the ADXL accelerometers has been greatly improved.
However, according to [46], a yield in the neighborhood of 75% is typical of a mature product [46]. This
indicates that with the BISR technique, the yield can potentially be further increased to 95%. That is, a
yield increase of 20% can still be achieved. Figure 6.2 clearly shows that, when the value of n is fixed at
n = 4, the value of IY(m,n) increases with an increasing value of m. This demonstrates that the gain in
yield by redundancy repairing overcomes the loss in yield due to area increase in this case (n = 4). We
have observed the same trend when n is fixed to different values. Figure 6.3 shows the yield after repair
(Yr ) v.s. initial yield (Y0 ) for different m numbers (m = 2; 4; 6; 8) when n = 4. This figure demonstrates
that the proposed BISR technique can significantly increase the yield (much higher than the initial yield).
Figure 6.4 shows the relationship between the yield increase and the initial non-BISR yield for different
n numbers when m is fixed as m = 4. From the figure, we can see that the BISR technique also demonstrates a positive yield increase for all initial yield values. The yield increase for too-low and too-high initial
yield is not significant. But, a significant yield increase can be observed for any moderate initial yield. If
the m number is fixed as m = 4, the IY value achieved by redundancy repair decreases with an increasing
58
Figure 6.2: The yield increase vs initial yield for different m numbers.
Figure 6.3: The yield after repair vs initial yield for different m numbers.
59
n value.
This is reasonable because if m is fixed, the repair ratio m=n decreases with the increase of n.
This can be further explained as follows. Assume the main device of the BISR MEMS contains only one
module (i.e., the main device is not modularized) and has area A. The redundant device area is m A with
the repair ratio equal m (i.e., mA=A). However, if the main device of the BISR MEMS is divided into n
(n > 1) modules, the redundant device area is (A=n)m (with repair rate m=n) which is much smaller than
mA. Consequently, a smaller repair rate gives a smaller IY value since the redundant device area is smaller.
Figure 6.4: The yield increase vs initial yield for different n numbers.
Based on the above results, to achieve the maximum yield increase, we should set m as infinity and n
as 1. However, with the increase of m and decrease of n, the area overhead increases drastically. Thus,
there is a trade-off and we must find a balance between the yield increase and the area overhead by a
pair of optimized m and n values. In order to verify the effectiveness of redundancy repair for moderate
initial yield, we randomly select A0 = 0:24mm2 , b = 1:8=mm2 , and k = 1, the yield of the non-BISR
accelerometer is 0.698. Simulation results on the yield increase for different m and n numbers are shown
in Figure 6.5 for the moderate initial yield of 0.698. We can see that, for a given n value, the yield increases
steadily with the increase of the m value till it finally approaches 1. But, for a given m value, the yield drops
slightly with the increase of n. For the BISR accelerometer with m = 2 and n = 4, the yield becomes
0.947 which is an increase of 35.7% (when compared with the initial yield 0.698). This demonstrates an
effective improvement on the yield for a comb accelerometer through redundancy repair.
6.1.3 Yield Model Revision Considering Defect Fatal Rate
In the above yield model, every random defect is assumed to be fatal. That is, any random defect on a
MEMS device will definitely lead to the device failure regardless of the defect location. Actually, this
assumption is too pessimistic. In reality, the device area consists of different portions and the defect sensitivity level of each portion may be different. Taking the comb accelerometer as an example, the device
area can be divided into four portions: beam, mass, finger groups and the empty field. Each finger group
60
Figure 6.5: The yield increase for different m and n numbers when Y0=0.698.
portion contains intensive narrow finger gaps. If a particle falls into the finger group portion, it is highly
likely that it will block the capacitance gap and lead to the device failure. However, in the empty field, no
functional part exists and any particle falling into this portion is non-fatal. Hence, defect fatal rate needs to
be considered for a more accurate MEMS yield model.
Defect fatal rate is defined as the possibility that a random defect will lead to the device failure, if
the defect falls into a certain area of the MEMS device. Assume the MEMS device area A consists of n
different portions (such as beam, mass, etc.) with area Ai at component i, we have
A=
Xn A :
i=1
i
The defect fatal rate for each component i is Fi (0 Fi 1) correspondingly. The average defect fatal
rate for a random defect falling into any portion of the MEMS device can be represented by
F=
Xn Ai F :
i=1
A
i
The critical area A of a MEMS device with area of A is defined as
A = F A:
The consideration of defect fatal rate makes the yield model of the MEMS device more complex. For
example, the initial yield without self-repair of the MEMS device includes the probability that no defect
occur, plus the probability that defects do occur, but all of these defects are non-fatal. The initial yield Y0
of MEMS device can be calculated as
Y0 =
1
X
(x+k
x=0
x
1 )(
Ab x
1 k
) (
) (1 F )x :
1 + Ab
1 + Ab
61
Computing the above infinity series is complex; however, the following lemma can be introduced for
yield analysis.
Lemma: If the average fatal rate for a random defect is F, the yield of a MEMS device with area of A is
equal to that of the case where every defect is fatal, but the device area is reduced to the critical area F A
[47].
The above lemma is equivalent to the following equation
1
X
P (x; A)(1
x=0
That is
1
X
(x+k
x=0
x
1 )(
F )x = P (0; F A):
1 k
Ab x
1
) (
) (1 F )x = (
)k :
1 + Ab
1 + Ab
1+F Ab
The lemma indicates that for the purpose of yield calculation, the case where defect fatal rate is considered is equivalent to the case where every defect is fatal, but the device area A is shrunk to critical area
F A. This equivalency can greatly simplify the yield calculation when defect fatal rate is considered.
In summary, in order to consider the defect fatal rate of each device component, we only need to replace
the MEMS device area A with critical area F A in the former yield analysis, and all the former results
remain valid after this replacement. Assume the average defect fatal rates of non-BISR and BISR devices
as Fnsr and Fsr separately. The initial yield without redundancy repair is
Y0 = P (0; Fnsr Ar ) = (
1
1 + Fnsr A b
)k :
The yield after redundancy repair Yr (m; n) can be expressed as
Yr (m; n) = P (0; Fsr Ar ) + P1 + P2
=
1
X P (x; F A ) + X
sr
r
x=m
x=0
x=m+1
Pmj=1(m+n) (x
1
x j)
j
:
min(x;m+n) m+n
( j ) (xx 1j )
j =1
P (x; Fsr Ar ) P
The yield increase after redundancy repair, IY (m; n), can be expressed as
IY (m; n) = Yr (m; n) Y0
=
1
X P (x; F A ) + X
sr
r
x=m
x=0
x=m+1
Pmj=1(m+n) (x
1
x j)
j
min(x;m+n) m+n
( j ) (xx 1j )
j =1
P (x; Fsr Ar ) P
P (0; Fnsr A0 ):
Using the same non-BISR/BISR device design as that in Section 6.1.2, we assign defect fatal rates for
the beam, mass, finger areas and empty field as 0.9, 0.1, 0.4, and 0 relatively. After calculation, we find that
62
the average fatal rates of non-BISR and BISR devices are Fnsr = 0:14and Fsr = 0:11 separately. Assume
the defect clustering factor k = 1, the yield increase IY versus initial yield Y0 after considering defect fatal
rate is shown in Figure 6.6. The yield increase versus initial yield with/without considering defect fatal rate
revision when m = 2, n = 4 and k = 1 is shown in Figure 6.7. From Figure 6.7, we can see that the yield
increase after considering defect fatal rate is even better than the case where each defect is considered as
fatal. For example, with m = 2 and n = 4, the yield increase when defect fatal rate is not considered takes
its peak value of IY = 0:323 at initial yield Y 0 = 0:46. However, when defect fatal rate is considered,
the yield increase takes its peak value of IY = 0:373 at initial yield Y 0 = 0:43. That is, the yield increase
peak value is increased by 0:05 when defect fatal rate is considered.
Figure 6.6: The yield increase vs initial yield for different m numbers when defect fatal rate is considered.
Figure 6.7: The yield increase vs initial yield with/without considering defect fatal rate (m
k = 1).
63
= 2, n = 4,
6.2 Monte Carlo Simulation of Point-stiction Defects
During the fabrication process or in-field usage, MEMS devices are vulnerable to various defect sources.
The occurrence and the location of these defects are random and cannot be precisely predicted. Such
stochastic behavior can be better predicted with statistical simulation methods, such as Monte Carlo simulation. Monte Carlo simulation is a stochastic technique used to approximate the probability of certain
outcomes by running multiple trial simulations using random number and probability statistics. Monte
Carlo methods were originally developed during World War II. Now, they are applied to wide range of
stochastic problems such as nuclear reactor design, econometrics, stellar evolution, stock market forecasting, etc. Monte Carlo methods randomly select values to create scenarios of a problem, and the values are
selected from a fixed range to fit a probability distribution. In a Monte Carlo simulation, the random selection process is repeated many times to create multiple scenarios. Each time, a value is randomly selected to
form one possible solution to the problem. Together, these scenarios give a range of possible solutions with
different possibilities. When the simulation is repeated for a large amount of times, the average solution
will give an approximate answer to the problem. The more scenarios are simulated, the more accurate the
Monte Carlo simulation result will be. However, the total simulation time will also increase with the number of scenarios. ANSYS FEM software supports the feature of Monte Carlo simulation in its probabilistic
design module.
Considering the feasibility of ANSYS Monte Carlo simulation, we simulate a specific defect denoted
as point-stiction. During the fabrication or the in-field usage of MEMS devices, the movable microstructure
may be stuck to substrate in one or multiple points. This is different from the stiction problem due to
surface forces in surface micromachining techniques, and we denote it as ”point-stiction”. The size and
the location of the stiction points vary from device to device. These local point-stictions can limit or
totally block the movement of the movable microstructure, and hence lead to device failure. An example
of point-stiction is illustrated in Figure 6.8. The point-stiction defects can be developed due to various
reasons. For example, a pinhole in the sacrificial layer may lead to such point-stiction. A particle on the
photolithography mask during the patterning of anchor area may also lead to a point-stiction. Furthermore,
a particle may randomly fall into the gap between a movable microstructure and the substrate, and it may
block the movable microstructure at that particular point. Even after the device is sealed, particle-resulted
point-stiction may still be developed during in-field usage. This is because some particles or debris may
be sealed in the package. These particles or debris may be released and free to move around later due to a
sudden shock impulse. The package itself may also generate some debris during shock. Thus, point-stiction
can be a common defect source for MEMS devices.
In our research, we use Monte Carlo simulation to simulate the device behavior with point-stiction
defects. We made the following assumptions and criteria in our simulation. First, according to Federal
Standard 209E, the typical particle size in a clean room is 0:1 5m in diameter. Thus, we set the size of
a point-stiction defect in the range of 0:1 6m. Point-stiction defects (square in shape) with random size
64
in this range will be generated and randomly distributed in the device area (including the surrounding empty
area). Second, we assume the point-stiction distribution is totally random without clustering effect. However, if clustering effect is considered, the MEMS device yield will be even higher. Third, we use a similar
sensitivity selection criterion as [48] for the simulated devices: devices with sensitivity deviation within
5% is treated as acceptable ”good” devices; sensitivity deviation from 5% 30% is treated as parametric defects; deviation larger than 30% will be treated as catastrophic defects. Devices with parametric or
catastrophic defects will be discarded in our yield analysis.
We simulated six cases of the non-BISR accelerometer design with the number of point-stiction defects
ranging from 1 to 6 separately. Since the BISR device has about 1.5 times of area when compared with the
non-BISR device, in order for fair comparison, we assume a constant defect density. That is, we simulated
six corresponding cases for the BWC (Beam Width Compensation) BISR device with the numbers of pointstiction defects equal 1.5, 3, 4.5, 6, 7.5, and 9 separately. There is a trick in simulating the ”half” defect.
If we double the device area that the point defect can fall into, the occurring possibility for this defect
in the device is reduced to half. Thus, it can be treated as a ”half” point-stiction defect. We simulated
1000 device samples and derived the device displacement sensitivities with such defects. The Monte Carlo
simulation mimics the behavior of the real MEMS device, hence, it offers a very trustable result. Based on
the simulations on both non-BSIR and BISR devices, we can make a comparison and see how the MEMS
yield can be improved due to the proposed BISR design. Although we simulated the case for the BWC
BISR device, the result for the EFC (Electrostatic Force Compensation) device should be almost the same
as that of the BWC BISR device for point-stiction defect. This is because the critical areas for both devices
are almost the same.
stiction in empty field
stiction in beam
stiction in finger
stiction in mass
stiction at the
end of beam
stiction in anchor
Figure 6.8: Point-stiction and its formation.
As we have observed in our simulation, the location and the size of point-stiction defects do matter in
65
determining the device behavior. For example, if a point-stiction defect is in the empty field or within the
anchor area, it will not cause any problem to the device. However, if it falls into the mass area, it will totally
block the mass movement and leads to device failure. The point defect on a beam will cause sensitivity
change of a certain amount. If it is near the end of a beam connecting to the anchor, it may cause very
small tolerable change in sensitivity, and the device still can be treated as a ”good” device. However, if
the point-stiction is at the beam end close to the central mass, it will be a catastrophic defect. Similarly,
the size of a point defect is also important. A very small point-stiction defect with size 0:2m falling into
the capacitance gap will not affect the device behavior. However, a large point-stiction defect with size
6m will totally block the capacitance gap and leads to the stiction of the movable finger at that point. An
example of random scattering of 1000 samples (two defects in each device) of point-stictions generated
during the Monte Carlo simulation is shown in Figure 6.9. The corresponding sensitivity results plot for
these 1000 samples in the Monte Carlo simulation are shown in Figure 6.10. From the figure, we can see
that depending on the location and size of each point-stiction defect, the device sensitivity of each defect
may remain unchanged as the defect-free value, changed to some extent, or even becomes zero (the movable
microstructure is totally stuck and cannot move at all).
Figure 6.9: The random scatter of a point-stiction defect for 1000 samples (two defects) in Monte Carlo
simulation.
The simulation results for sensitivity distribution among the 1000 non-BISR device samples (two defects in each sample) for the non-BISR device are shown in Figure 6.11. As shown in the figure, 757 out
of 1000 samples demonstrate sensitivity of 0:26m=100g of the good device response. The point-stiction
defects in these samples are either in the empty field or the anchor areas. Thus, they will not change the
device behavior. Hence, these samples will be accepted as good devices. There are 126 samples with zero
sensitivity. This means their movable components are totally stuck and cannot have any displacement. For
66
Figure 6.10: The sensitivity history for 1000 samples (two defects) in Monte Carlo simulation.
example, if the point-stiction defect is on the mass, it will totally fixed the movable components and lead
to a catastrophic failure. Other samples exhibit sensitivities between 0 and 0:26mu=100g , depending on
the locations of the point-stiction defects. Based on this statistical result, we can conclude that the yield for
the non-BISR device with two random point-stictions is 75.7%. The simulation results for good-modulenumber distribution among the 1000 BISR samples (3 defects in each device) for the BISR device is shown
in Figure 6.12. ANSYS extracts the sensitivity of each module in every BISR device sample. If the sensitivity of a module is within 5% deviation range of that of the good module response, it will be accepted
as ”good” module. The total number of good modules in each BISR device sample is summed up. As we
can see that the number of samples with four, five, and six good modules are 19, 231, and 750 separately.
These samples are guaranteed to work properly and they are accepted as ”good” devices. The devices with
good-module-number below 4 will be discarded in the yield analysis process. In this way, the yield for the
BISR device with three point-stiction defects is calculated as 100%.
The Monte Carlo simulation results for the non-BISR device are shown in Table 6.1. We simulated
the cases for the non-BISR device with 1-6 point-stiction defect(s). A device with sensitivity within 5%
deviation from that of the defect-free device value is treated as a ”good” device. With this criterion, the
corresponding yields are listed in the table depending on the number of defects. We can see that if there are
6 point-stiction defects, the yield will drop to 45.0%.
The Monte Carlo simulation results for the BISR device are shown in Table 6.2. We maintain the same
defect density to ensure a fair comparison. The device area of the BISR accelerometer is about 1.5 times
that of the non-BISR device. As a result, we simulated the cases for the BISR device with 1.5, 3, 4.5, 6, 7.5,
and 9 point-stiction defects. A module with sensitivity within 5% deviation from that of the defect-free
module value are accepted as a ”good” module. A device with four or more good modules is treated as a
”good” device. With this criterion, the corresponding yields are listed in the table. We can see that even if
67
Figure 6.11: The sensitivity distribution of 1000 non-BISR device samples (two defects).
Figure 6.12: The good-module-number distribution of 1000 BISR device samples (three defects).
68
Table 6.1: Monte Carlo simulation results for point-stiction defects in non-BISR device.
Case
#1
#2
#3
#4
#5
#6
No. of point-stiction defects
Good (5% devication)
parametric (5-30%)
catastrophic (30-100%)
Yield
1
878
0
122
87.8%
2
757
5
238
75.7%
3
652
10
338
65.2%
4
566
11
423
56.6%
5
494
14
492
49.4%
6
450
10
540
45.0%
there are 9 point-stiction defects, the yield is still as high as 97.7%.
Table 6.2: Monte Carlo simulation results for point-stiction defects in BISR device.
Case
#1
#2
#3
#4
#5
#6
No. of point-stiction defects
No. of devices with 6 good modules
No. of devices with 5 good modules
No. of devices with 4 good modules
No. of devices with less than 4 good modules
No. of good devices
Yield
1.5
862
135
3
0
1000
100%
3
750
231
19
0
1000
100%
4.5
72
287
39
2
998
99.8%
6
573
360
63
4
996
99.6%
7.5
505
383
102
10
990
99.0%
9
451
426
100
23
977
97.7%
Yield comparisons between the non-BISR and BISR devices are shown in Table 6.3. Since the area of
the BISR device is 1.5 times that of the non-BISR device, the defect number in the BISR device is also 1.5
times that of the non-BISR device in each case. As we can see, the yield of the BISR device in the presence
of point-stiction defects is apparently much higher than that of the non-BISR device. For the case of 6
defects in the non-BISR device (and correspondingly 9 defects in the BISR device), the yield of the nonBISR device is 45%, while the yield of the BISR device is 97:7%. A yield increase of 52:7% is observed, and
this indicates that a significant yield increase can be achieved for moderate initial yield (e.g., 45% incase 6).
This coincides with our theoretical prediction before. A visual comparison between the yields of non-BISR
and BISR devices for different number of point-stiction defects is shown in Figure 6.13. The yield increase
due to redundancy repair for six cases is shown in Figure 6.14. From the bar chart, it is clearly seen that
the BISR design leads to positive yield increase when compared with the non-BISR design for all the six
simulation cases. It can be observed that the yield decreases only slightly for the BISR design as the number
of defects keeps increasing.
Table 6.3: Comparison of Monte Carlo simulation results between non-BISR and BISR devices.
Case
#1
#2
#3
#4
#5
#6
No. of defects in non-BISR
No. of defects in BISR
Non-BISR device yield
BISR device yield
Net yield increase IY
1
1.5
87.8%
100%
12.2%
2
3
75.7%
100%
24.3%
3
4.5
65.2%
99.8%
34.6%
4
6
56.6%
99.6%
43.0%
5
7.5
49.4%
99.0%
49.6%
6
9
45.0%
97.7%
52.7%
69
Figure 6.13: The yield comparison between non-BISR and BISR devices.
Figure 6.14: The yield increase due to redundancy repair for six cases.
70
In the above Monte Carlo simulation, we simulated the cases for large ( 1) and moderate ( 0:5)
initial yields. In order to find out the yield increase for small ( 0) initial yield, we further increased the
number of defects in non-BISR/BISR devices in our Monte Carlo simulation. We performed Monte Carlo
simulations for the cases where there are 1, 10, 20, 30, 40 and 50 point-stiction defects in the non-BISR
device. Correspondingly, the number of defects in the BISR device is set as 1.5, 15, 30, 45, 60 and 75
separately. The simulation results for both non-BISR and BISR devices are shown in Table 6.4. The yield
comparison between the non-BISR and BISR devices for six simulation cases is shown in Figure 6.15. The
yield increase and the initial non-BISR device yield for different defect numbers is shown in Figure 6.16.
Monte Carlo simulation shows that when the defect number is too large (N =60 or above), eventually the
BISR device yield will also drop to zero, and the yield increase becomes zero. In the former analytical
(theoretical) results in Section 6.1.2 and 6.1.3, it has been shown that the yield increase due to redundancy
repair is most significant for moderate initial yield. If the initial yield is too large (approaching 1) or too
small (approaching 0), the yield increase due to redundancy repair is not significant. However, in the Monte
Carlo simulation result, it shows that the peak of yield increase shifts toward small initial yield. For example,
for a small initial yield of 6:8%, the yield increase still remains a large valule of 59:9%. This result actually
does not conflict with the former theoretical prediction. This is because in the former theoretical analysis
in Section 6.1.2 and 6.1.3, the clustering factor is selected as k = 1, which means a large defect clustering
factor. However, in Monte Carlo simulation, the defect distribution is totally random. Thus the clustering
factor in Monte Carlo simulation is k = 1. Since it is difficult to simulate the case for k = 1 in computer,
we simulate the theoretical analysis of the case when k = 2000 (a large number). The comparison between
the theoretical prediction for k = 2000 and the above Monte Carlo simulation results is shown in Figure
6.17. In Figure 6.17, the curve stands for the case of theoretical prediction with our MEMS yield model
when k = 2000, and the square dots stand for the Monte Carlo simulation results. From the figure, we can
see that the Monte Carlo simulation results coincide with the theoretical prediction very well. This proves
the correctness of our MEMS yield model for redundancy repair.
Table 6.4: Monte Carlo simulation results for more point-stiction defects.
Case
#1
#2
#3
#4
#5
#6
No. of defects in non-BISR
No. of defects in BISR
Non-BISR device yield
BISR device yield
Net yield increase IY
1
1.5
87.8%
100%
12.2%
10
15
26.2%
92.0%
65.6%
20
30
6.8%
66.7%
59.9%
30
45
1.2%
38.3%
37.1%
40
60
0.5%
18.8%
18.3%
50
75
0.1%
10.3%
10.2%
71
Figure 6.15: The yield comparison between non-BISR and BISR devices with more defects.
Figure 6.16: The yield increase due to redundancy repair with more defects.
Figure 6.17: The comparison between theoretical prediction and Monte Carlo simulation result.
72
Chapter 7
Circuit Support and Simulation Results of
BISR Accelerometer
7.1 Circuit Support for MEMS BISR
7.1.1 Differential Capacitance Sensing Circuit
Various circuit schemes [49]-[53] for signal detection of MEMS differential capacitance sensors have been
reported. Among them, capacitance measurement by sensing the current flow through a transducer [53]
is very attractive. Due to the high-frequency probe signal utilized, very high sensitivity and speed can be
achieved. Further, this circuit scheme is especially suitable for BISR MEMS device, since it ensures better
separation of the faulty modules in the BISR device. By turning off the corresponding analog switch, the
faulty module will have no contribution to the total current sensed by the OPAMP. Hence, the load effect of
the faulty module to the main device can be eliminated. The circuit diagram [53] is shown in Fig. 7.1.
C−V modulation
A
C1
B
S1
Vs
CC
S2 2
rectifier,amplifier
R1
Rf
−
A1
low−pass filter
C3
R3
R2
R4
−
VAC
S/H
A2
R5
−
A/D converter
R7
A3
R6
A/D converter
Digital Output
C4
C5
VDC
S/H
S/H
sensor
Vsig
Vref
Start
Controller
Figure 7.1: The differential capacitance sensing circuit.
As shown in Figure 7.1, the sensing circuit [53] consists of five stages: capacitance-to-voltage (C-V)
modulation, rectifier and amplifier, low-pass filter, S/H, and A/D converter. A high frequency (1MHz)
carrier Vs is used for signal modulation. The differential capacitance of the accelerometer is denoted as
C1 and C2 separately, as defined in section 2. The switches S1 and S2 stay in opposite states (on or off)
73
alternately, so that the signal detection circuit senses capacitances (C1 + C2 ) and (C1 ) in a time-sharing
scheme. Assume the OPAMP and switches are ideal, when S1 is on and S2 is off (phase 1), capacitance
C1 and C2 are connected in parallel to the voltage source Vs. The output voltage of OPAMP A1 at phase 1
(denoted as VAC 1 ) is:
VAC 1 = j!(C1 + C2 )Rf Vs :
This voltage will be converted into a dc signal and applied to the A/D converter as the reference voltage
Vref . When S1 is off and S2 is on (phase 2), the capacitance C1 is connected to Vs and C2 is shorted to
ground. The output voltage of OPAMP A1 at phase 2 (denoted as VAC 2 ) becomes
VAC 2 = j!C1 Rf Vs :
This voltage will also be converted into dc signal and applied to the A/D converter as signal voltage Vsig .
Thus, the ratio r of VAC 2 and VAC 1 is
r=
C1
VAC 2
=
:
VAC 1 C1 + C2
The ratiometric change x of the differential capacitance of the accelerometer can be given by [53]
2C1
C C2
=
1 = 2r 1:
x= 1
C1 + C2 C1 + C2
By measuring the ratio r of VAC 2 and VAC 1 , we can know the ratiometric differential capacitance change
x. This will in turn be converted into a digital representation by the A/D converter as the final output.
7.1.2 BISR Control Circuit Design
To support the BISR MEMS structure, the BISR control circuit must be designed to electronically isolate
a faulty module and replace it with a good redundant module. This replacement is implemented by a
group of analog switches made of transmission gates (TGs) as shown in Figure 7.2. Here, all six identical
modules of the BISR comb accelerometer are labeled with a f . For each module, the pair of differential
capacitances are labeled with 1 and 2 respectively. For example, Ce1 and Ce2 stand for the differential
capacitance pair of module e. The selection signals for all six transmission gates are Ta to Tf respectively.
The movable portions of all modules are connected together through common anchors, and are directly
connected to the OPAMP A1 for signal sensing. The left fixed fingers of each module are connected to the
voltage source Vs through an analog switch. The right fixed fingers of each module are connected to the
common node of two NMOS transistors. The on and off states of each transmission gate decide whether
the corresponding module is connected or separated from the main device. If the TG is on, the voltage
source Vs is applied to the working capacitances of the module. The resulted current flowing through the
capacitances of the module contributes to the total current sensed by the OPAMP A1. Hence, the module
is electronically connected into the main device. If the TG is off, the voltage Vs cannot be applied to the
74
working capacitances of the module. Thus, the current flowing through the module is zero, so the module
does not contribute any current to the OPAMP input. In this way, the module is electronically separated from
the main device. The BISR control circuit sets the selection signals to their appropriate states according to
the BIST result of each corresponding module.
Sensor
Modules (a~f)
TGa
Ta
Ta
TGb
TGc
Vs
TGd
Tb
Sb1
Tc
Sb2
Tc
Sc1
Td
Te
Te
TGf
Sa2
Tb
Td
TGe
Ca1
Sa1
Tf
Tf
Sc2
Ca2
Cb1
Cb2
Cc1
Cc2
Cd1
Sd1
Sd2
Rf
−
A1
+
Cd2
VAC
detector
...
Ce1
Se1
Se2
Ce2
Cf1
Sf1
Sf2
Cf2
SW
Figure 7.2: Switching circuit for redundancy repair.
When the device enters the BISR mode, it will first perform the BIST process [30] for each individual
module to determine whether it is good or faulty. According to the BIST result, the BISR controller will
set the selection signals Ta Tf to their appropriate values. If any of the four working modules (initially
assigned by default) becomes faulty, the BISR controller will set the corresponding TG selection signal of
the faulty module to 0 to separate it from the main device. Simultaneously, the BISR controller changes the
selection signal of a good redundant module from 0 to 1 to activate it as a working module, and connect it
to the main device electronically. After the BISR process is completed, the MEMS device enters the normal
mode. During the normal mode, four of the selection signals Ta Tf are set to ”high” and the other two
are set to ”low”. That is, four out of the six modules (restructured by the BISR controller) will be connected
as the main device, while the other two modules are separated as either redundant or defective ones.
7.1.3 Parasitic Capacitance Analysis
Due to the tiny size of MEMS devices, their working capacitances are generally very small (in the range of
1pF or below). The parasitic capacitances of a MEMS device can easily go beyond this value. Hence, it is
necessary to find out the influence of the parasitic capacitances, and a parasitic capacitance model needs to
be developed for the analysis.
Taking module a in Fig. 7.2 as an example, a model taking into account the parasitic capacitances is
75
shown in Figure 7.3 where all major parasitic capacitances are extracted by ANSYS. Note that Ca1 and
Ca2 are the differential working capacitances of module a, while Ca10 , Ca20 and Ca30 are the parasitic
capacitances between the left fixed fingers/right fixed fingers/movable fingers and the ground, respectively.
Further, Ca12 is the parasitic capacitance between the left fixed fingers and right fixed fingers. If the transmission gate TG and both NMOS transistors are treated as ideal switches, in phase 1, S1 is on and S2 is off,
and the circuit can be simplified as Figure 7.4 (a). In phase 2, S1 is off and S2 is on, and the circuit can be
simplified as shown in Figure 7.4 (b).
Ta
node 1
TGa
Ca1
Rf
Ca10
Ta
S1
Ca12
node 2
C a20
Vs
C a2
node 3
C a30
opamp
−
A1
VAC
+
S2
Vctr
Figure 7.3: Parasitic capacitance analysis for one BISR module.
Rf
Ca1 +Ca2
Vs
opamp
−
A1
VAC
+
Ca30
Ca10 +Ca20
(a) Parasitic capacitance equivalent circuit in phase 1
Rf
Ca1
Vs
Ca12 + Ca10
opamp
−
A1
VAC
+
Ca2 + Ca30
(b) Parasitic capacitance equivalent circuit in phase 2
Figure 7.4: Simplified parasitic analysis circuit in phase 1 and 2.
Assume the open-loop gain of the OPAMP A1 as A. The output voltage VAC in phase 1 can be expressed
as
Vin Rf !(Ca1 + Ca2 )
:
VAC 1 =
1 + 1+Rf !(Ca1A+Ca2 +Ca30 )
The output voltage VAC in phase 2 can be expressed as
VAC 2 =
Vin Rf !Ca1
1+
Rf !(Ca1 +Ca2 +Ca30 ) :
1+
A
From the above analysis, we can see that only parasitic capacitance Ca30 has influence on the output voltage, while other parasitic capacitances (e.g., Ca10 ; Ca20 ; Ca12 ) will not affect the output voltage.
76
Since the open-loop gain A of an OPAMP is generally very large (A 105 ), the denominators of the
above two equations are approximately 1. Hence, the influence of parasitic capacitances will not be
a problem for signal detection. In reality, the transmission gate and both NMOS transistors cannot be
ideal. Although their off-resistances can be treated as infinity, their on-resistance are about 16k and
32k
in our model. In this case, the equivalent circuits of both phases for parasitic effects can be derived. Here, mathematical expressions of the output voltage for both phases cannot be directly accessible,
and HSPICE simulations have been used for analysis. The transmission gates and NMOS switches are
designed with Magic and extracted for HSPICE simulation. The device parasitic capacitances are extracted through ANSYS. Simulation results showed that the BISR device maintained the same sensitivity
(VAC 2 = 6:83mV=g) with and without parasitic capacitances being considered. For example, even when
we have Ca1 = 0:15pF; Ca2 = 0:05pF; Ca10 = Ca20 = Ca30 = 2pF , the circuit can still work properly
for signal detection.
7.1.4 Analog Multiplexers and Signal Strength Analysis
In the BISR design, transmission gates are used to determine whether a module is connected to or separated
from the main device. The output signal of a MEMS device is generally weak due to its tiny size. It is
necessary to analyze the possibility of signal degradation caused by the added transmission gates. In order
to ensure that the signal strength is not weakened, it is important to avoid passing the MEMS signals through
transmission gates. In a MEMS accelerometer, generally, the signal is sensed from the movable portion.
Thus, in our BISR design, the movable portions of all modules are intentionally connected together through
the common anchors between neighboring modules. They are in turn directly connected to the OPAMP
without bypassing through transmission gates. This ensures that the small current signal sensed from the
movable fingers will not be degraded. The transmission gates are inserted between the signal source and
the left fixed fingers of each module to determine the selection of the modules, as shown in Fig. 7.2.
The internal resistance of the voltage source is very small when compared with the on-resistance of the
transmission gate Ron , which is around 16k . Assume the total capacitance of the MEMS device is about
1pF , and the frequency f of voltage source Vs is 1MHz . The impedance of the MEMS device capacitance
can be estimated as
Z =
1
= 159:2k
Ron :
2fC
Thus, the on-resistance of each transmission gate can be ignored when compared with the impedance of
the MEMS device capacitance. The signal source Vs can pass the transmission gate without significant
degradation.
In summary, all movable plates are jointed together and connected to the signal sensing circuit directly,
while a transmission gate is inserted between the left fixed fingers of each module and the voltage source.
In this way, the insertion of transmission gates will not degrade the signal strength sensed by the moving
fingers. This has been verified using HSPICE simulation. All transmission gates are designed using Magic
77
and the layout is extracted to HSPICE for simulation. This ensures that the parasitic effects from transmission gates are all considered. Simulation results show that the BISR device has nearly the same signal
strength as the non-BISR device.
7.1.5 Defective Module Isolation and Load Effect Analysis
The separation of a faulty module from the main device is accomplished through the transmission gates
inserted between the left fixed fingers and the voltage sources. As discussed, the movable portions of all
modules are physically connected together. When a faulty module is separated out of the main device by
turning off the corresponding transmission gate, its movable plate is still connected to the main device.
Whether this will cause any load effect to the sensing circuit must be analyzed. According to the working
principle of the BISR MEMS structure, the current flowing through the working capacitances in each module will be summed at the input of the OPAMP. If the transmission gate of the faulty module is turned off,
there will be no current flowing through the working capacitances of that module. Hence, the contribution
of current flow from the faulty module is zero. In this way, the capacitive load of a defective module will
not affect the signal strength of the accelerometer. This has been verified with HSPICE simulation. For
example, we ever changed the capacitance of a faulty module to different values (very large or very small),
and the output signal remains the same as before. This proves that the capacitance of a faulty module will
not cause any load effect to the sensing circuit. Even if the capacitance of the faulty module is changed to
extremely large or small value due to defects, the output signal of the sensing circuit is still not affected.
Further, we introduced a bridging defect by connecting a pair of movable and fixed fingers with a small
resistor (0.001 ), and the output signal is not affected either. This is because the fixed fingers of each
faulty module are connected to the insulation layer instead of the silicon substrate. In case of bridging defect between movable and fixed fingers in a faulty module, the fixed fingers of the faulty module just share
the same voltage level as the movable plates, and the movable fingers will not be shorted to ground. This
demonstrates that the BISR technique is effective to bridging defect between the movable/fixed fingers. Actually the BISR design is effective to almost any kind of local defect which falls into an individual module.
However, for global defects which occur to every module, the redundancy repair cannot be applied since
there is no working module.
7.1.6 BIST Circuit for BISR
In order to implement the built-in self-repair technique of a MEMS device, the device must first perform
built-in self-test for each individual module to decide whether it is faulty. The BIST result of each module
will be fed into the BISR control circuit. Based upon this information, the BISR control circuit will virtually
disconnect the faulty module and replace it with a good redundant module. In this way, MEMS BIST is
the prerequisite for MEMS BISR. Without an efficient and robust BIST solution, MEMS BISR cannot be
realized because the control circuit cannot know which module is good and which module is faulty. A
78
dual-mode BIST scheme has been proposed in [30]. Circuit support for the BIST method, especially the
voltage biasing scheme, has also been presented in [30]. The proposed dual-mode BIST can serve as an
effective BIST solution for the BISR of capacitive MEMS devices.
7.2 Design and Simulation of BISR Accelerometer
The geometry parameters of the BISR comb accelerometer with m = 2 and n = 4 are listed in Table
7.1, using a set of design rules comparable to ADXL accelerometers [8]. For comparison, a none-BISR
accelerometer with the same number of capacitance groups (as that at the main device of the BISR accelerometer) is also designed. The geometry parameters of the non-BISR accelerometer are also listed in
the same table. The simulation results for the performance of both BISR and none-BISR accelerometers are
shown in Table 7.2. From Table 7.2, we can see that, by narrowing the beam width, the sensitivity loss of
the BISR accelerometer due to device modularization can be fully compensated. The BISR accelerometer
shows approximately the same displacement sensitivity as that of the none-BISR accelerometer.
Table 7.1: Design of BISR/non-BISR accelerometers.
Design
Parameters
BISR
device
non-BISR
device
device area(m2 )
1500900
980900
6
6
thickness t (m)
no. of capacitance groups
capacitance gap d0 (m)
beam width Wb (m)
beam length Lb (m)
mass width Wm (m)
mass length Lm (m)
finger width Wf (m)
finger length Lf (m)
206
80
2
2
2
3.2
300
300
200
200
2206
880
4
4
200
200
Given a non-BISR accelerometer design, its sensitivity and resonant frequency can be determined.
As discussed before, in order to maintain the sensitivity, several alternatives are available. Assume the
sensitivity loss is compensated by shrinking the beam width and enlarging the mass width simultaneously.
How much should these two parameters be adjusted? The curves of displacement sensitivity Sd v.s. different
beam/mass width values are drawn with MathCAD as shown in Fig. 7.5. The sensitivity of the non-BISR
design is also shown in the same figure as a horizontal line. The cross points between the sensitivity curves
and the horizontal line suggest the possible solutions for the BISR design, with the same sensitivity as that
of the non-BISR design. The frequency compensation can also be performed in a similar way (Fig. 7.6).
As shown in Figure 7.5, the displacement sensitivity of the non-BISR accelerometer is demonstrated as
79
Figure 7.5: Sensitivity simulation results for the BISR accelerometer design.
Figure 7.6: Frequency simulation results for the BISR accelerometer design.
80
Table 7.2: Simulation results of BISR/non-BISR accelerometers.
Performance
BISR
non-BISR
main device
device
sensing mass Ms (g )
capacitance C0 (pF )
sensitivity Sd (nm=g )
sensitivity S (fF=g )
spring constant km (N=m)
frequency f0 (kHz )
0.844=3.36
0.1034=0.41
6.8
0.74=2.8
1.214=4.84
6.05
3.36
0.41
6.64
2.74
4.95
6.12
the horizontal line Sd = 8:1nm=g . The beam/mass width of the non-BISR design are 3.2m and 200m
separately, as given in Table 7.1. If the beam width of the BISR design is shrunk to Wb = 2m, then
displacement sensitivity will be approximately the same as that of the non-BISR device.
In order to demonstrate the effectiveness of the sensitivity compensation by adjusting the beam width,
ANSYS simulation has been performed for both non-BISR and BISR accelerometers. Due to the limitation in node number imposed by the available ANSYS version, simplified models for both non-BISR and
BISR accelerometers are used for simulation. ANSYS simulation results for the displacement sensitivity in
response to acceleration from 0 to 50g are shown in Figure 7.7. For the non-BISR accelerometer, the beam
width Wb is 3.2m. For the BISR MEMS accelerometer, the beam width is shrunk to 2m. According to
Figure 7.7, the sensitivity of the non-BISR accelerometer (expressed as the voltage output of OPAMP A1
in phase 2, i.e., VAC 2 ) is 6.68mV=g , while the sensitivity of the BISR accelerometer after compensation is
6.83mV=g . The sensitivity of the BISR device is about the same as that of the non-BISR device. Further,
the electrostatic force can act as a powerful tool in compensating the sensitivity loss of the BISR design
as illustrated in [41]. The sensitivity of the accelerometer can be increased till infinity by applying an appropriate DC biasing voltage given the accelerometer works in the open-loop mode. Thus, as suggested
by the simulation results, the sensitivity loss is not a concern for the BISR design. If the sensitivity loss
of the BISR design is compensated by adjusting the beam/mass width, this will introduce a yield decrease.
A more precise yield model taking into account of the device design parameters has been developed. The
simulation results showed that the yield revision introduced by the beam/mass adjustment for sensitivity
compensation is small. Due to page size limitation, the results will be published later.
81
Figure 7.7: ANSYS simulation results for sensitivity of BISR/non-BISR accelerometers.
82
Chapter 8
Reliability Analysis of MEMS Comb
Accelerometer
8.1 Introduction
In order for MEMS technologies to be used for real applications, yield as well as reliability are two very
important issues which need to be immediately addressed. Reliability analysis is required for almost every
commercial product. With the commercialization of MEMS devices, their reliabilities need to be thoroughly
studied. Second, MEMS will be integrated into System-on-Chip (SoC) design very soon. The reliability
of an entire SoC cannot be guaranteed if the reliability of MEMS is low. Further, MEMS is finding more
and more applications in safety-critical areas, such as aerospace, medical instruments, etc. For these applications, extremely high reliability is required. For example, during the launching process of a rocket,
the failure of a tiny MEMS device may easily lead to unpredictable disaster. An unreliable bioMEMS chip
embedded inside human body can be a serious threat to the health and may lead to the loss of life. Thus,
the reliability research for MEMS in safety-critical applications is an especially urgent need. However,
since MEMS is a newly developed discipline, MEMS reliability research still remains in its infant stage.
A well-developed MEMS reliability model is not available. Many MEMS failure mechanisms are still unclear. Researchers have begun to realize the importance and many efforts have been made [54][55][56][57].
Multiple energy domains are generally involved in the working principle of MEMS devices. Moreover,
most MEMS devices contain movable components. Hence, MEMS devices are vulnerable to much more
defect sources during its fabrication process and in-field usage compared to VLSI chips. This makes the
MEMS reliability research a challenging work. The understanding of various MEMS failure mechanisms
is also non-trivial. In this work, we tried to develop a MEMS reliability model, which can be used to quantitatively evaluate the reliability of MEMS devices. Based on this model, we evaluate the reliability of both
non-BISR and BISR MEMS accelerometers in different failure mechanisms: material fatigue, shock, and
stiction. The comparison between non-BISR and BISR MEMS devices demonstrates effective reliability
83
enhancement in response to different defect sources. The reliability model and the research strategies may
also be applied to other MEMS devices in a similar way.
8.2 Basic Concepts of Reliability
Reliability is the probability that a component, equipment, or system will perform the required function
under different operating conditions encountered for a stated period of time [58]. The reliability function
is denoted by R(t) (0 R(t) 1) where t is time. The larger the reliability function value is, the more
reliable the component, equipment or system will be. For a specific time t, if R(t) = 1, it means the
reliability is perfect and the system will never fail, if R(t) = 0, it means the system definitely fails and
will never be able to work. Meanwhile, the unreliability Q(t), or the probability of failure, is defined as
the probability that a component, equipment, or system will not perform the required function under the
operating conditions encountered for a stated period of time t. From the definitions, we can easily conclude
the relationship between reliability R(t) and unreliability Q(t) by
R(t) + Q(t) = 1:
The failure rate is expressed as the ratio of the total number of failures to the total operating time
=
K
:
T
Where K is the number of failures and T is the total operating time. For most products, is generally a
very small number. Its unit can be number of failures per 1 106 hours. As we can see, the larger the value is, the more unreliable the component, equipment or system will be.
The Mean Time To Failures (MTTF) is also used to quantitatively measure the reliability. The MTTF is the
reciprocal of the failure rate as defined below
1
MT T F = :
In reality, the failure rate is generally the function of time. That is, it changes during the service time
of the product. Thus, the failure rate can also be expressed as (t). The failure rate (t) of mechanical
components and VLSI chips follows the behavior of a bathtub curve as shown in Figure 8.1 [59]. It is
believed the failure rate of MEMS devices also follows the bathtub curve [60]. The bathtub function
consists of three regions. In the initial stage, the failure rate is high due to the latent defects in the device,
and falls off till time tinfant . After that, the device enters a stable stage with a constant failure rate. For high
reliability applications, this constant failure rate should be extremely small. Finally, after time toperation ,
the failure rate will increase sharply due to wear-out and the device comes to the end of its lifetime. The
useful time of the device with low constant failure rate is defined as
84
tuseful = toperation tinfant :
Generally, tinfant and toperation of a certain product cannot be given as definite values. The manufacturers
will give average values for their products based on statistical testing.
Figure 8.1: The bathtub curve of failure rate [60].
8.3 Reliability Models
There are several standard probability models available for describing the reliability of a system: the exponential reliability distribution, the binomial reliability distribution, the Poisson reliability distribution, and
Weibull reliability distribution [61].
1) The exponential reliability distribution
The exponential reliability distribution is the most common probability model used to predict the lifetime of a system. Thus, in our MEMS reliability analysis, the exponential reliability model is used. The
reliability function R(t) is expressed as
R(t) = e
t :
where is the defect failure rate and t is the time period. The unreliability Q(t) is therefore
Q(t) = 1 R(t) = 1 e
t :
The exponential reliability distribution is shown in Figure 8.2.
2) The binomial reliability distribution
The binomial reliability distribution is used for describing the reliability of a discrete distribution. It
can be expressed as
85
Figure 8.2: The exponential reliability distribution function.
(R + Q)n = 1:
where n is the total number of trials conducted. A typical example of binominal reliability distribution is
the case of flipping coins for heads and tails. If we define the probability of obtaining a head and a tail as R
and Q separately, wee have R = Q = 0:5. If two trials are made (n = 2), then
(R + Q)2 = R2 + 2RQ + Q2 = 1:
3) The Poisson reliability distribution
The Poisson reliability distribution is a discrete distribution which provides a useful tool in the case of
the binomial distribution. The reliability can be expressed as
R = 1 (Q1 + Q2 + Q3 + :::):
where Qi is the probability of exactly i failures occurring during time period t, and can be expressed as
Qi (t) = (t)i e
t =i!:
The probability of zero failure in the Poisson reliability model comes to the result of the exponential reliability distribution
R0 (t) = e
t :
That is, the exponential reliability distribution can be treated as a special case of Poisson reliability model
for i = 0.
4) The Weibull reliability distribution
86
The Weibull reliability function is expressed as
R(T1 ) = e [(T1 )=℄ :
For general reliability measurement, we consider = 0 and = 1= , hence:
R(t) = e (t) :
is the shape parameter which indicates whether the failure rate is increasing or decreasing. If
< 1:0, the failure rate is decreasing. If = 1:0, the failure rate is constant. If > 1:0, the failure rate is
increasing. If = 1:0, it comes to the result of exponential distribution.
where
R(t) = e (t) :
Thus, the exponential distribution can also be treated as a special case of the Weibull reliability function for
= 1:0.
The above reliability models describe the reliability of an individual component. However, sometimes
we also need to evaluate the reliability of a system consisting of multiple components. Such system reliability models enable us to calculate the reliability characteristics of a design, evaluate design alternatives,
and perform sensitivity analysis. Depending on the configuration of the system, different system reliability
models (such as series, parallel, and k-out-of-n, etc.) are available [59].
1) Series Reliability Model
If the functional operation of a system depends on the successful operation of all system components,
the reliability of the system can be calculated with a series reliability model. The reliability block diagram
of the series model is shown in Figure 8.3. Assume the system consists of n number of serial components,
and the failure rate of each component i is i , the reliability of the entire series system is
Rtot = ni=1 Ri (t)
= e 1 t e 2 t ::: e
i=n
= e (i=1 i )t :
n t
From the above equation, we have
tot = ii==1n i :
That is, the failure rate of a series system is the sum of the failure rates of all the series components. The
mean-time-to-failure (MTTF) of the system can be expressed as
MT T Ftot = ii==1n
87
1
:
MT T Fi
C1
C2
C3
......
Cn
Figure 8.3: The block diagram of series reliability model.
where MT T Ftot is the MTTF of the entire series system, MT T Fi is the MTTF of the ith component.
2) Parallel Reliability Model
The block diagram of a parallel reliability model is shown in Figure 8.4. The system functions properly
until all of the components (C 1 to Cn) fail. Hence, the reliability Rtot of the parallel system is given as one
minus the probability of failure for each component as shown below
Rtot = 1 ni=1 [1 Ri (t)℄
= 1 ni=1 (1 e i t ):
C1
C2
C3
......
Cn
Figure 8.4: The block diagram of parallel reliability model.
3) k-out-of-n Redundancy Reliability Model
The block diagram of a k-out-of-n redundancy reliability model is shown in Figure 8.5. Among the
n number of modulus, at least k number of modules need to be fault-free in order for the whole system
to work properly. In other words, a maximum of n k number of faulty modules are allowed without
losing the function of the whole system. Our BISR comb accelerometer is exactly an example of k-out-of-n
reliability model. If the BISR accelerometer contains n number of modules in main device and m number
of redundant modules, it is n-out-of-(n+m) model. Assume the reliability of each component as R . The
reliability Rtot of the k-out-of-n redundancy system is given as below
Rtot = Rn + nRn 1(Q ) + ::: + (nk )(R )k (Q)n k :
Assume the failure rate of each component as , the MTTF of the system can be calculated as [61]
1
1
1
1 1
+
+ ::: + ):
MT T F = ( +
n n 1 n 2
k
88
C1
C2
C3
k−out−of−n
redundancy
......
Cn
Figure 8.5: The block diagram of k-out-of-n redundancy reliability model.
8.4 MEMS Failure Mechanisms
The unreliability of a device, component or system is caused by failures. Thus, in the MEMS reliability
research, it is very important to understand various failure modes and mechanisms for MEMS. Based on
these, one can develop reliability models to quantitatively evaluate the MEMS reliability of each individual
failure category, and find out corresponding ways to improve the MEMS reliability. Due to the diversity
and multiple energy domains involved in MEMS working principles, MEMS devices are vulnerable to
many more failure mechanisms during their fabrication as well as in-field usage. Some possible failure
mechanisms for MEMS have been identified as shown in the following items [54][62][56]. Most of them
are MEMS-specific which are very different from those of VLSI chips.
material fatigue
mechanical fracture
stiction
wear
delamination
residual stress
Further, there are also some environmentally induced failure mechanisms listed below.
shock
vibration
humidity
particle contamination
electrostatic discharge
In the following, we will have a brief introduction to each of the above failure categories.
89
8.4.1 Material Fatigue and Aging
Most MEMS devices, including comb accelerometers, contain movable components. A MEMS device relies on the movable components to perform its specific function. However, the long-term cycling movement
will lead to the material fatigue and material aging. Fatigue of a material is the process of damage and failure due to cycling loading, so that the structure will crack even at stress well below the material’s ultimate
strength. The fatigue behavior of a ductile material such as metal has been well researched, and is generally
modeled with a stress versus life (number of cycles to failure) curve (also known as a S-N curve) as shown
in Figure 8.6.
Figure 8.6: A typical S-N curve of a ductile material.
From Figure 8.6, we can see that the cycling load leads to the decrease of the material fracture stress.
Eventually, the material will crack at a stress lower than its original material strength. If the material is
loaded at a higher stress level, the number of cycles to failure will be smaller. Once the amplitude of the
cycling load is known, one can predict the fatigue life (in number of cycles to failure) from the S-N curve.
Fatigue has been estimated to account for up to 80% to 90% of mechanical failures in engineering structures
[63]. Thus, fatigue is directly related to the mechanical long-term reliability. With the commercialization of
MEMS devices, researchers have also begun looking into the fatigue behavior in a MEMS material. Most
MEMS devices involve movable components, and this makes the material fatigue a major reliability concern. Whether the repeated movement of MEMS components will induce fatigue in its structural material
and how this will affect the reliability of MEMS remain very interesting research topics.
Besides the material fatigue under cyclic loading, it also has been observed that MEMS material properties, such as Young’s modulus, etc, will gradually shift after long-term repeated cycling load, which is
often in the order of billions of cycles. This material properties shift due to long-term cyclic loading is
sometimes referred to as material aging. The shift in Young’s modulus leads to the change of the device
resonant frequency as well as the sensitivity of an accelerometer (for example). It turns out this will degrade
the sensor output signal. Other properties, such as the dampening coefficient and resistance of a material,
90
may also change due to long-term cyclic loading. The mechanical and electrical parameters degradation of
a p+ silicon cantilever beam due to fatigue is shown in Table 8.1 [55]. As we can see from the table, after
about 1:9 109 cycles of loading, the Young’s modulus of p+ silicon shifts from 129MPa to 136MPa, an
increase of about 5.4%. This leads to a corresponding sensitivity decrease of 5.4%. For some applications,
this may be totally unacceptable and it will be treated as a device failure. Thus material fatigue and aging
are really important concerns for MEMS.
Table 8.1: Mechanical and electrical parameters degradation of p+ silicon cantilever beams due to aging.
Aging
0
1
2
3
4
excitation cycles
resonant frequency f0 (kHz)
Young’s modulus (GPa)
Dampping coeff. (Hz)
Resistance ( )
0
5.975
129
600.84
9.9
9.022E+08
6.000
130
644.69
11.3
1.464E+09
6.025
131
730.12
14.3
1.908E+09
6.150
136
749.56
14.5
2.05E+09
6.150
136
769.74
15.3
8.4.2 Mechanical Fracture
Mechanical fracture is the breaking of a uniform material into two separate sections, if the applied stress
exceeds its fracture strength. In a MEMS device, the fracture of mechanical structure generally leads to a
catastrophic failure. A MEMS device is designed so that it can endure the corresponding stress induced in
its normal operation. However, in some certain occasions such as shock, the over-range input stimulus leads
to a large displacement and stress in the movable microstructure. For an actuator with large displacement
(such as a micro-resonator), the mechanical fracture may also be a serious concern.
The fracture stress of a brittle material (such as silicon) is actually a stochastic parameter which requires
a statistical treatment [64][65]. When a large number of seemly identical structures are tested, it has been
found that the fracture strength actually scatters around a certain mean value. The probability of fracture
Pf (0 Pf 1) actually follows the Weibull distribution function [66][67]
Pf = 1 exp[
Z
A
[
a (x; y; z ) m
℄ dA℄:
0
where Pf is the fraction of the total number of devices that will fracture under the applied stress distribution
a (x; y; z ). It can also be termed as the fracture probability of the microstructure. Here, is the lowest
stress at which fracture will occur and 0 is a normalizing factor. The exponent m is known as the Weibull
modulus. It is a material parameter standing for the statistical scattering of fracture events: a high Weibull
modulus leads to a sharper switch around the fracture strength which means a low scattering of fracture
events. For brittle materials such as poly-silicon, is generally set to zero [66]. For an applied stress
close to the mean fracture strength, the fracture probability will gradually increase from 0 to 1, instead of
an abrupt change. A typical Weibull fracture probability curve of silicon material is shown in Figure 8.7.
The design of MEMS devices should put the induced stress in its normal operation well below the mean
91
fracture strength in order to ensure reliability.
Figure 8.7: Weibull distribution of the probability of fracture versus stress.
8.4.3 Shock
Shock [68] is different from vibration in the sense that it is a single accidental large acceleration impact
instead of cyclic vibration. Shock can lead to the lift-off of a microstructure from the bonded surface. It
can also lead to the adhesion and stiction of a movable microstructure to the substrate or fixed components.
Intensive shock can also lead to the fracture of MEMS structures. The fracture behavior in turn can be
described with Weibull fracture probability distribution, as discussed before. Shock may lead to different
results depending on its direction and strength. For example, a Z-axis shock toward +Z direction may lead
to the fracture of a beam, while a shock along -Z direction may lead to the contact of movable components
to the substrate. A shock toward X direction may lead to the contact and stiction of movable fingers to
fixed fingers. However, a shock in Y direction can be less destructive except it is very strong. The MEMS
reliability in a shock environment will be studied in detail for both non-BISR and BISR accelerometers in
our research.
8.4.4 Stiction
Stiction is one of the most dominant and bothersome failure mechanism in MEMS, especially in surface
micromachining. Due to its importance, significant efforts have been put into the research of stiction to
declare its failure mechanism, model development and stiction prevention [69][71][72]. When two MEMS
surfaces come into contact (either accidentally or intentionally), if the restoring force of the microstructure
cannot counteract the surface force, the two surface will be stuck to each other. This phenomenon is called
a stiction failure in MEMS. The surface force include capillary force, electrostatic force, Van der Waals
92
force, solid bridging and hydrogen bonding, etc [69] [70]. Various techniques [71] have been suggested
to prevent the stiction problem. Some techniques are roughening and skewing the surfaces, using antistiction coatings, using getters, and using leaky dielectrics, etc. Stiction can occur either during the final
releasing step of the micromachining process, or during the in-field usage due to over-range input stimulus
or electromechanical instability.
8.4.5 Wear
When one surface is moving over another contacted surface, some material on the contacted surface will
be removed due to the mechanical action, and this is called wear [73]. There are four main processes that
cause wear: adhesion, abrasion, corrosion, and surface fatigue [54]. They follow different rules and can
be described with different models. Wear is an important failure mechanism in some actuators such as
micro-engines and contact switches. Wear and friction can lead to significant reduction in service lifetime
of a MEMS device, and they must be carefully considered [74][75][76]. Increasing the smoothness of
contact surfaces help alleviate wear, but cannot totally avoid it. Since there is no contact surface during the
microstructure movement for comb accelerometers, wear is not a immediate concern for them.
8.4.6 Delamination
Delamination is a failure that occurs when the material interface loses its adhesive bond, so that two initiallybonded surfaces will separate from each other [77][78]. Delamination can be the result of various reasons,
such as particles on the wafer during processing, fatigue by long-term cyclic loading for structures with
mismatched thermal expansion coefficients, etc. For most of the cases, delamination can be a catastrophic
failure. The lifted-off structure will lose its mechanical support, and the device structure stability can be
destroyed. If the lifted-off material is free to move, it may block the displacement of other components and
lead to electrical shock.
8.4.7 Residual Stress
The residual stress in a thin film can lead to the bending or warping of MEMS devices [79][80]. Residual
stress mainly comes from the thermal expansion coefficient mismatch of materials during the deposition
process. The difference between the deposition temperature and room temperature leads to thermal mismatch. The defects during deposition of the material also contribute to residual stress. This failure mechanism is especially important in surface micromachined MEMS devices. The residual stress also affect the
device performance. The gradual releasing of residual stress during in-field usage also lead to the performance shift of a device from its calibrated value. The residual stress can be alleviated by optimizing the
fabrication process (such as thin-film deposition process, etc.). Using single-crystal silicon (instead of polysilicon) as the structure material in bulk-micromachining can partially avoid the stress problem. However,
93
surface-micromachining is still widely used due to its compatibility with the already mature CMOS VLSI
techniques.
8.4.8 Vibration
MEMS devices are often exposed to vibrating environment during their in-field usages. The vibration surroundings can be a large reliability concern for MEMS. The repeated cyclic vibration may lead to material
fatigue. Even worse, if the vibration frequency is near the resonant frequency of the MEMS device, it
can induce a large vibration amplitude to the device. This in turn can lead to the contact and stiction of
movable microstructures to the substrate or fixed components. It may also lead to the fracture of movable
microstructures such as beams. For aerospace applications, a MEMS device might experience very strong
vibrations during the launch process. Thus, the failure induced by vibration is especially a serious concern
for aerospace applications. Some research work has been reported on the MEMS reliability under vibration
environments [81].
8.4.9 Humidity
MEMS devices (especially surface-micromachined devices) are extremely hydrophilic in their surfaces
[82]. If there is high humidity in the environment, water will condense into small cracks and pores on the
surfaces of a MEMS microstructure. The condensation of water on surface micromachined devices can lead
to an increase in the residual stress of the MEMS material [54]. It will also result in strong capillary forces
which will lead to the stiction of a movable microstructure to the substrate or fixed components [69].
8.4.10 Particle Contamination
During the fabrication process of MEMS devices, particle contamination is an unavoidable defect source
[83]. Even after a device has been packaged, particle contamination is still possible. For example, assume
some particles were originally adhered to the substrate or package. During the in-field usage, if there is a
mechanical shock, such particles may be released and free to move. A package may also contribute some
particles and debris. These can all be potential threats to device operation. Particle contamination can
block the movement of a movable microstructure so that it cannot move freely. A conductive particle can
also lead to electrical short between two components. Thus, particle contamination is a vital threat to the
MEMS device reliability. Depending on the size and location of the particles, their influence to a device
varies. Particle contamination on the mask or photoresist during the photolithography step may also lead to
another defect, which is called ”point-stiction” in surface micromachining. This is because the sacrificial
layer in the particle location is unintentionally removed after photoetching. Thus, a movable microstructure
may be stuck to the substrate at this point. This is different from the general ”stiction” problem due to
surface force in surface-micromachining. Thus, it is called ”point-stiction”. The influence of point-stiction
94
to yield has been discussed before.
8.4.11 Electrostatic Discharge
Electrostatic discharge, or ESD, is a common problem for electronic devices. A human body can easily
develop an electrical potential of more than 1000V in a dry environment. If the electronic device is touched,
this voltage difference will be transmitted to the device and cause the damage of the device. The ESD
effect has been catastrophic in circuits. It can also have similar effect on MEMS [84][85]. Especially,
electrostatistically actuated MEMS devices may be vulnerable to the ESD effect. ESD can also destroy the
MEMS signal detection circuit. For a MEMS comb accelerometer device, ESD may lead to an electrostatic
force between movable and fixed comb fingers. It can easily result in the contact between the movable
and fixed finger surfaces. If the restoring force is not strong enough to counteract the surface force, the
contacted surfaces will be permanently stuck. Thus, ESD is also a serious reliability threat for MEMS.
In the above, we listed some common possible MEMS failure mechanisms. However, due to the variety
of MEMS devices and their complexities, MEMS failure mechanisms are far more than the above listed
categories. For the MEMS reliability research, a clear understanding of MEMS failure mechanisms and
their behaviors are required. We will try to develop MEMS reliability models using several different failure
mechanisms: material fatigue, shock, and stiction. The developed MEMS reliability models and their
strategies can be potentially applied to other MEMS devices in a similar way.
8.5 Reliability Analysis of MEMS Comb Accelerometer
In order to evaluate the effectiveness of the redundancy repair technique on MEMS reliability enhancement,
we have to develop the MEMS reliability models for both non-BISR and BISR devices. Based upon the
above knowledge, we can develop MEMS reliability models in the following discussions.
8.5.1 Reliability Model of Non-BISR MEMS Device
The structural diagram of the non-BISR MEMS accelerometer for this analysis is shown in Figure 8.8, and
its corresponding reliability model is shown in Figure 8.9. As shown in Figure 8.8, the non-BISR device
consists of 4 beams, one mass, N number of movable fingers, 2N number of left/right fixed fingers, and
four anchors. All these components must be fault-free in order for the entire device to function correctly.
If any of these components becomes faulty, the whole device will malfunction. For example, if one of the
beams is broken, a movable finger is stuck to the fixed finger, or the mass is stuck to the substrate, then the
device will not be able to work properly. Although the fixed components such as anchors and fixed fingers
are less likely to be faulty than the movable components, they also must be fault-free in order to ensure
the proper function of the device. For example, if one left fixed finger is shorted to its neighborhood right
fixed finger, this will lead to a short circuit to the signal detection circuit. Or, if an anchor is lifted off the
95
Ran
Rlf Rrf
Rb
Rm
Rf
Figure 8.8: The structural diagram of a non-BISR MEMS device.
4 beams
1 mass
N left
fixed fingers
N movable
fingers
N right
fixed fingers
4 anchors
Figure 8.9: The reliability model for the non-BISR MEMS device.
96
substrate, the device structure will be unstable. Thus, the reliability of a non-BISR device can be described
as a series model, as shown in Figure 8.9. Since the four beams have exactly the same dimension and equal
loading, we assume all four beams have the same reliability Rb . Similarly, we assume the reliabilities of
the mass and each movable finger as Rm and Rf separately. The reliabilities of one fixed finger and one
anchor are denoted as Rff and Ran individually. Finally, the reliability of the entire non-BISR device can
be expressed as
2N R :
Rnsr = Rb4 Rm RfN Rff
an
In reality, movable components of the MEMS device are vulnerable to more defect sources when compared with fixed components. Thus, the reliabilities of fixed components can be much higher than those
of movable components. The defects of fixed components mainly result from device fabrication. The devices with catastrophic defects in fixed components can be filtered out during manufacturing test. During
in-field usage, it is less likely for the fixed components to develop new defects compared to the movable
components. Thus, in the following discussion, we will mainly concentrate on the reliability of the movable
components, and the reliability of the fixed parts can be treated as constant 1. In this way, the reliability of
the non-BISR device can be expressed as
Rnsr = Rb4 Rm RfN :
From the reliability model of the non-BISR device, we can see that the major threat to the device
reliability comes from the large index (i.e., N ) of Rf . For example, if the reliability of a single movable
finger is 0.99 and N equals 42, this will reduce the entire device reliability to 0.656 even though the
beam and mass are assumed perfect. This is the major impetus for us to implement the redundancy repair
technique for MEMS comb accelerometers. By modularizing the device, each module contains a smaller
number of comb fingers. The reliability of each module will be higher than the original non-BISR device.
By implementing the redundancy repair technique, even higher reliability can be achieved.
Assume the failure rates for the beam, mass and finger as b , m and f separately. The reliability
functions of the beam, mass and finger can be expressed as
Rb = e
b t ;
Rm = e
m t ;
Rf = e
f t :
Hence, the reliability function of the non-BISR device is
Rnsr (t) = e nsr t
= e (4b +m +Nf )t :
97
where the failure rate nsr of the non-BISR device is
nsr = 4b + m + Nf :
Once we know the failure rates b , m and f , we can derive the reliability of the non-BISR device at
certain time t based on the equation for Rnsr (t).
8.5.2 Reliability model of BISR MEMS device
Rlf Rrf
Rf
Rm
Ran
module a
module b
module d
module c
module e
module f
Figure 8.10: The structural diagram of the BISR MEMS accelerometer.
module a
module b
module c
module d
module e
module f
4 beams
1 mass
N/4 movable
fingers
N/4 left
fixed fingers
N/4 right
fixed fingers
4 anchors
4 beams
1 mass
N/4 movable
fingers
N/4 left
fixed fingers
N/4 right
fixed fingers
4 anchors
4 beams
1 mass
N/4 movable
fingers
N/4 left
fixed fingers
N/4 right
fixed fingers
4 anchors
4 beams
1 mass
N/4 movable
fingers
N/4 left
fixed fingers
N/4 right
fixed fingers
4 anchors
4 beams
1 mass
N/4 movable
fingers
N/4 left
fixed fingers
N/4 right
fixed fingers
4 anchors
4 beams
1 mass
N/4 movable
fingers
N/4 left
fixed fingers
N/4 right
fixed fingers
4 anchors
4 of 6
redundancy
Figure 8.11: Reliability model for BISR device.
The structural diagram of the BISR comb accelerometer is shown in Figure 8.10 and its corresponding
reliability model is shown in Figure 8.11. The BISR comb accelerometer consists of six identical modules.
In each module, it consists of four beams, one smaller mass, (N=4) number of movable fingers, (N=2)
number of left and right fixed fingers, and four anchors. Note that N is the number of movable fingers for
the non-BISR device. Again, the reliability of each module can be described with a series model. Assume
the reliabilities of each beam, movable finger and mass as Rbr , Rfr and Rmr separately, and the reliabilities
of each fixed finger and anchor as Rff and Ran individually. The reliability Rmod of each BISR module
can be expressed as
98
4 R RN=4 RN=2 R :
Rmod = Rbr
an
mr
ff
f
Similarly, if we only concentrate on the reliability of movable components, the reliability of each module
can be expressed as
4 R RN=4 :
Rmod = Rbr
mr
f
Assume the failure rates for the beam, mass and finger of each BISR module as br ,
separately. The reliability functions of the beam, mass and finger can be expressed as
Rbr = e
br t ;
Rmr = e
mr t ;
Rf = e
mr
and f
f t :
Hence, the reliability function of the entire BISR module is
Rmod (t) = e (mod t)
= e [4br +mr +(N=4)f ℄t :
where the failure rate mod of each module is
mod = 4br + mr + (N=4)f :
The unreliabilty Qmod of each module is
Qmod = 1 Rmod :
There are totally six identical modules in the BISR comb accelerometer. Among them, only four modules are required to be fault-free to ensure the proper function of the device. The reliability of such a system
can be calculated using k of n redundancy reliability model. The reliability Rsr of the BISR accelerometer
can be expressed as
6 + 6R5 Q
4
2
Rsr = Rmod
mod mod + 15Rmod Qmod
= (e mod t )6 + 6(e mod t )5 (1 e mod t ) + 15(e
mod t )4 (1
e
mod t )2 :
This is the reliability function of a 4-of-6 redundancy BISR comb accelerometer. For the general case,
assume there are n number of modules in main device and m number of modules as redundancy. The
reliability of the BISR accelerometer can be expressed as
Rsr = (Rmod )n+m + (n + m)(Rmod )n+m 1 (Qmod ) + ::: + (nn+m )(Rmod )n (Qmod )m :
99
8.5.3 Reliability Enhancement and Reliability Analysis
After we have developed the reliability models for both non-BISR and BISR MEMS devices, we can derive
the reliability increase IR due to redundancy repair by
IR = Rsr
Rnsr :
In the BISR MEMS comb accelerometer, the large number of comb finger groups are divided into several
modules. Each module contains a smaller number of comb finger groups, and thus the risk for each module
to be faulty is lowered. The redundancy further improves the reliability of the BISR device. However,
compared to the original non-BISR device, the BISR device contains more beams. This will lead to the
decrease of the reliability. The net reliability increase or decrease depends on the interaction between these
counteracting factors. In fact, the reliability of the MEMS device is tightly linked to the corresponding
defect sources and failure mechanisms. Different failure mechanisms lead to different device reliability
behavior.
In order to evaluate the MEMS reliability, it is important to look into the possible MEMS failure mechanisms. In the following MEMS reliability analysis, we evaluate the reliabilities of example designs for
both non-BISR and BISR devices with different fault mechanisms. The non-BISR device has a beam width
of Wbnsr = 3:2m. For the BISR device, we consider two cases: one is the case with beam width compensation (which is called BWC device, Wbbw = 2:0m = 0:63Wbnsr ), and the other is the case with
electrostatic force compensation (which is called EFC device, Wbef = 3:2m = Wbnsr ). The design
parameters of both the non-BISR and BISR devices are shown in Table 8.2.
Table 8.2: Non-BISR/BISR accelerometers design parameters for reliability evaluation.
Design parameters
non-BISR device
BWC BISR module
EFC BISR module
beam width Wb
beam length Lb
mass width Wm
mass length Lm
movable finger width Wf
movable finger length Lf
number of finger groups N
device thickness t
3.2m
310m
100m
534m
4m
160m
48
2.0m
2.0m
310m
100m
132m
4m
160m
12
2.0m
3.2m
310m
100m
132m
4m
160m
12
2.0m
8.6 Reliability of MEMS accelerometer in Material Fatigue
8.6.1 Fatigue Analysis and Cycles to Failure
The fatigue behavior of metal materials in MEMS, such as aluminum used in digital micromirror device
(DMD), can use the already existing theories and models. Silicon has been widely used in MEMS for its
100
superior electrical and mechanical properties [86]. As a brittle material, silicon was believed not to be
susceptible to dynamic fatigue. However, in 1992, the fatigue of silicon was demonstrated in [87] using
a specimen with dimension in the range of microns. Ever since, many researchers have made tremendous efforts to explore the fatigue mechanism and behavior of both single crystal silicon [88][89][90] and
polysilicon [91][92].
Currently, the fatigue mechanism and model for MEMS materials such as silicon and poly-silicon are
still far from being mature. In [93], 3:5m thick and 50m wide poly-silicon tensile specimens under
cycling loading were used to investigate the long-term mechanical fatigue behavior of poly-silicon material
for MEMS applications. Based on the experimental data, the S-N curve of poly-silicon under cycling
loading is plotted. They observed that the tensile strength = 1:1GP a of virgin samples is reduced by
about 35% to a fatigue strength of f = 0:70GP a after 109 cycles. They varied the test frequency between
20 and 6000Hz, and no influence of the test frequency on the fatigue behavior was observed in this range.
That is, the number of cycles Nf to failure does not depend on the frequency f of the cyclic loading.
Since the mean-time-to-failure (MTTF) of a device can be calculated by
MT T F = Nf =f;
i.e., the MTTF does depend on the frequency of the cycling load. The samples which are experienced
cycling load with higher frequency will fail after a shorter time. Further, no endurance limit (stress below
which failure will never occur) was observed in their experiments. That is, even very small stress in cycling
load will also induce material fatigue of poly-silicon. Hence, in some MEMS devices, even if the displacement (and thus the induced stress) in normal operation is very small, fatigue still exists in the poly-silicon
material. The only difference is the fatigue life will be much longer compared to the case of large stress.
Based on the experiment data, the S-N curve for poly-silicon material can be plotted. The experimental data
can be fitted with a power law as shown in Figure 8.12. According to [93], the number of cycles (Nf ) to
failure of the poly-silicon sample in cycling load can be predicted using the following experimental formula
f 1=m
) :
where is the mean tensile strength of poly-silicon, = 1:10GP a in their research. Further, f is
the applied maximum stress during cycling (fatigue strength), and m is a constant. Using the least square
fit analysis on the experimental data, they found m = 0:02. This experimental equation is very useful
Nf = (
to estimate the lifetime (number of cycles to failure) of the material under cycling load, or the maximum
allowed peak stress for a required lifetime if the initial strength is known.
8.6.2 Reliability Analysis by Cycles to Failure
In reality, the mean tensile strength ( ) of brittle materials (such as poly-silicon) shows a large scattering.
The above results are also derived using sinusoidal wave loading, while other loading waveforms may or
101
Figure 8.12: The S-N curve of poly-silicon tensile specimens with cyclic loading [93].
may not change the behavior. However, the above equation is very helpful for us to have an order-ofmagnitude evaluation on the device reliability. In our analysis, we first use ANSYS to extract the stress
distribution of the comb accelerometer in response to a given cyclic input stimulus (acceleration) with
frequency f0 . With this result, we can find the maximum stress in the device. With this maximum stress
value, we can derive the number of cycles Nfbnsr to failure for one beam using the above equation by
considering the poly-silicon material fatigue. The lifetime or the MTTF of one beam under cyclic loading
can be expressed as
MT T Fbnsr =
Nfbnsr
:
f0
Since there are four beams in the non-BISR device and they can be described with series reliability
model, the total mean-time-to-failure MT T Fnsr of the non-BISR device is
N
1
MT T Fnsr = MT T Fbnsr = fbnsr :
4
4f0
Thus, the failure rate of the device can be calculated as
nsr =
1
4f0
=
:
MT T Fnsr Nfbnsr
Finally, the reliability function Rnsr (t) of the MEMS device can be expressed as
Rnsr (t) = e
nsr t :
This is our strategy in evaluating the reliability of the non-BISR MEMS device for material fatigue due to
cyclic loading.
Similarly, we use ANSYS to extract the maximum stress in each BISR module in response to the same
cyclic loading as that of the non-BISR device. With this maximum stress value, we can derive the number
of cycles Nfbmod to failure of one beam in a BISR module. The lifetime or the MTTF of one beam under
cyclic loading can be expressed as
102
MT T Fbmod =
Nfbmod
:
f0
Since there are four beams in each BISR module, the total mean-time-to-failure
BISR module is
MT T Fmod
of one
N
1
MT T Fmod = MT T Fbmod = fbmod :
4
4f0
Thus, the failure rate of one BISR moduel can be calculated as
mod =
4f0
1
=
:
MT T Fmod Nfbmod
Finally, the reliability function Rmod (t) of one BISR module can be expressed as
Rmod (t) = e
mod t :
The reliability function of the whole BISR device based on 4-of-6 redundancy is
6 + 6R5 Q
4
2
Rsr = Rmod
mod mod + 15Rmod Qmod
= (e mod t )6 + 6(e mod t )5 (1 e mod t ) + 15(e
mod t )4 (1
e
mod t )2 :
By comparing the reliability functions of both non-BISR and BISR devices, we can quantitatively evaluate
the reliability enhancement of the BISR design.
8.6.3 Maximum Stress and Beams
In our experiment, we applied sinusoidal wave acceleration along the X direction (the sensitive direction of
the accelerometer) as the cyclic load. The acceleration frequency is 1kHz , and the acceleration amplitude
is 500g . This mimics the cyclic loading of the comb accelerometer in accelerated testing. The maximum
vibration amplitude A(f ) of the accelerometer in sinusoidal actuation can be expressed as
A(f ) =
F0 =m
:
2
[(f02 f 2 ) + ( f Qf0 )℄1=2
where F0 and f are the amplitude and frequency of the cyclic loading separately, f0 and m are the resonant
frequency and mass of the accelerometer respectively, and Q is the quality factor of the accelerometer. Since
the frequency of cyclic loading (acceleration) is lower than the resonant frequencies of non-BISR/BISR
devices, and the maximum vibration amplitude of the accelerometer is limited by the capacitance gap, the
resonant vibration effect is ignored in our analysis.
103
An example ANSYS stress simulation of a BWC BISR module is shown in Figure 8.13. According
to the stress distribution contour plots of ANSYS simulation, it is clearly shown that the maximum stress
occurs at the end of each beam (either the anchor end or the end connected to the movable mass) in response
to the X axis acceleration input. This indicates that these locations on the beams are most vulnerable to the
fatigue failure due to cyclic loading. This also demonstrates that the beams have the lowest reliability
compared to other components (mass, fingers, etc) in material fatigue. Thus, their lifetimes determine the
lifetime of the entire comb accelerometer. The lifetimes of other components (mass, movable finger, anchor,
left/right fixed fingers, etc.) can be treated as infinity, and their failure rates can be treated as zero. Assume
an accelerated fatigue testing with cyclic loading of 500g along X direction (device sensitive direction) with
frequency of 1kHz , the ANSYS stress simulation results and estimated failure rates for one beam in both
non-BISR and BISR devices are shown in Table 8.3.
Figure 8.13: ANSYS stress analysis of one BWC BISR module.
The ANSYS stress simulation results are shown in Table 8.3.
Table 8.3: The ANSYS stress analysis result for one beam of both BISR and non-BISR devices.
Items
non-BISR
device
BWC BISR
module
powered EFC
module
unpowered EFC
module
Displacement
Maximum stress f
No. of cycles to failure Nf
Lifetime (MTTF)
Failure rate (se 1 )
1.3m
22.67 MPa
1.3m
13.30MPa
1.3 mum
22.7 MPa
0.3 m
5.53MPa
1:98 1084
7:53 1095
1:98 1084 8:58 10114
1:98 1081se 7:53 1092se 1:98 1081 8:58 10111se
5:05 10 82 1:33 10 93 5:05 10 82 1:17 10 112
8.6.4 Reliability Comparisons
Based on the expected lifetime data derived above, we can evaluate the reliability functions of both nonBISR and BISR devices using the MTTF of a beam n the non-BISR device. The failure rate bnsr of one
104
beam can be determined by
bnsr =
1
= 5:05 10 82 se 1 :
MT T Fbnsr
The reliability function Rnsr (t) of the non-BISR device is thus
81
Rnsr (t) = e 4bnsr t = e 2:0210 t :
Similarly, the failure rate bmod of one beam in BWC BISR module is
bmod =
1
= 1:33 10 93 se 1 :
MT T Fbmod
The failure rate mod of each BISR module is
mod = 4bmod = 5:32 10 93 se 1 :
The reliability function Rmod of one BISR module is therefore
Rmod = e
mod t
= e 5:3210
93
t :
Finally, the reliability function Rsr (t) of the BWC BISR device is
6 + 6R5 Q
4
2
Rsr (t) = Rmod
mod mod + 15Rmod Qmod
= (e mod t )6 + 6(e mod t )5 (1 e mod t ) + 15(e mod t )4 (1 e mod t )2
93
93
93
93
= (e 5:3210 t )5 (1 e 5:3210 t ) + 15(e 5:3210 t )4 (1 e 5:3210 t )2 :
The reliability function of the EFC BISR device can be calculated in a similar way. Following the above
analysis, the reliability function curves for both non-BISR and BWC/EFC BISR devices are shown in Figure
8.14. From the curves, we can see that the MTTF of the non-BISR device is about 1080 sec. The MTTF
of the BWC device is increased to about 1092 se. This is because the beam width of BWC BISR device
is shrunk, and hence for the same displacement the maximum stress in the beam of the BWC BISR device
is smaller than that of the non-BISR device. This leads to longer fatigue life compared to the non-BISR
device. For the EFC BISR device, we analyzed both the unpowered and powered cases. In the powered
case, the electrostatic force compensates the device sensitivity to the same value as the non-BISR device.
The beam width of the EFC device is the same as that of the non-BISR device. Thus, the maximum stress in
the EFC device beam is almost the same as that in non-BISR device. This leads to almost the same fatigue
life (MTTF) for the powered EFC device as that of the non-BISR device. However, for the unpowered EFC
device, the displacement for the same acceleration input is almost 14 as that of the non-BISR device. This
leads to reduced maximum stress in the beam. As a result, the fatigue life of the unpowered EFC BISR
device is increased to about 10111 sec.
105
Figure 8.14: The reliability curves for BISR and non-BISR device (t = 10x ).
On the other hand, we can also see that the lifetime of both non-BISR and BISR devices are long
enough for normal application. This is because in normal operation, the displacement of the movable mass
is very small. Hence, the resulted maximum stress on each beam is much lower than the fracture strength
( = 1:10GP a). Although no endurance limit is found in poly-silicon material, a very low maximum
stress level on the beams leads to extremely large Nf value. As a result, both non-BISR and BISR devices
exhibit extremely long lifetime in material fatigue failure mode. This indicates that for normal operation
of the MEMS comb accelerometer, the crack of a beam due to material fatigue in cycling loading is very
rare, and it is not a major concern for device reliability. The device reliability may therefore be determined
by other defect sources and failure mechanisms. However, this does not mean that the material fatigue is
not a threat for MEMS device reliability. For other MEMS devices such as aluminum micromirror devices
(DMD), the material fatigue of aluminum leads to much lower reliability in its working mode. Thus,
for these MEMS devices, the material fatigue due to cyclic loading is a serious concern and needs to be
addressed carefully. Meanwhile, MEMS material under cyclic loading develops aging at a life-time much
shorter than that of fatigue. The MEMS material aging has been observed after cyclic loading of as short
as 1:0 109 cycles. Assume this aging lead to the failure of the device, and the frequency of the cyclic
loading is 1kHz, the MTTF for aging can be calculated as
MT T F = Nf =f0 = 1:0 109 =(1 103 )
= 106 se = 11:5days:
This shows that the material aging can be a serious concern for MEMS comb accelerometers. However,
the aging mechanism of MEMS material has not been well understood yet. A quantitative MEMS aging
model is not available at this time. As a result, we cannot quantitatively evaluate the MEMS reliability
increase of BISR design in material aging. However, as we saw before, the reduce in mass in modularized
design help reduce the maximum stress level in the material. This will help alleviate the aging in the material
106
and lead to prolonged MTTF for aging failure. As a result, the device reliability of the BISR design in case
of material aging will be enhanced.
8.6.5 Weak Device and Material Fatigue
ANSYS simulation result turns out that beam fracture due to material fatigue is not a serious threat for a
defect-free MEMS comb accelerometer in its normal operation. However, some MEMS actuators (such
as microresonator, vibratory gyroscope, etc.) have large vibration amplitude in their working modes. This
may cause the maximum stress in a device large enough to cause the material fatigue failure in relatively
short lifetime cycles. For these MEMS devices, material fatigue may become a major failure mechanism
and needs thorough consideration. ANSYS can be used to guide the design optimization in order to reduce
the maximum stress while maintaining the device performance requirement.
For an accelerometer device with certain defects, material fatigue can become a major concern. For
example, as shown in Figure 8.15, the accelerometer has a point stiction on the beam, and this can greatly
reduce the device sensitivity. But, if the beam section from the point stiction to the central mass is narrowed
to 0:1m, it will greatly decrease the spring constant of the defective beam. Assume the length of the
narrowed beam section as 6m. If both defects (beam stiction plus narrowing) occur simultaneously, they
mask each other and cause the device sensitivity to be within the tolerable range. This defective device may
still be accepted as a ”good” one and released to the market. However, during the in-field usage, such device
is very unreliable. The narrowed beam will have very large stress in it during in-field operation, and this
may cause the device lifetime due to fatigue failure becomes much shorter. For a non-BISR device, this can
be a serious reliability threat. However, for a BISR device, even if one or two modules have such combined
defects, other modules can still be connected to guarantee the device function. Thus the reliability of the
BISR device can still be very high.
We assume a cyclic loading of 360g acceleration is applied and simulated both the non-BISR, WBC
and EFC BISR devices for the above combined defects. The simulation results are shown in Table 8.4.
The lifetime of non-BISR device is estimated to be about 6:0 105 sec, which is about 6.9 days. Thus
material fatigue failure does become a serious threat to the reliability of the non-BISR device when such
combined defects exist. The fatigue lives of BWC/unpowered-EFC/powered-EFC BISR devices becomes
5:47 1033 sec, 2:04 1038 sec and 1:08 1010 sec separately. However, due to the redundancy repair, the
material fatigue lifetimes of BWC/unpowered-EFC/powered-EFC devices remain about 1092 sec, 1081 sec
and 10111 sec separately, which are far more than enough for normal operation. There may be still some
other combined defects cases in which the non-BISR device will have a serious reliability problem, while
the BISR device can still maintain high reliability due to redundancy repair. In reality, due to the imperfection in device fabrication process, even after careful manufacturing testing, there may be still some
hard-to-detect defects in the released ”good” devices. Even worse, some new defects may still be developed
during in-field usage. Some of these defects may be a potential threat to the device lifetime due to material
107
fatigue. Thus, it is necessary to have the redundancy repair to ensure the high device reliability.
Figure 8.15: The accelerometer with combined defects (stiction + narrowed-beam).
Table 8.4: The reliability analysis for accelerometers with combined defects.
Items
Displacement
Maximum stress f
No. of cycles to failure Nf
Beam lifetime (MTTF) (sec)
device lifetime (MTTF) (sec)
non-BISR
device
0:38m
BWC BISR
module
0:11m
unpowered EFC
module
powered EFC
module
713.9MPa
202.6MPa
0.09m
164.14MPa
0.38m
693.05MPa
2:4 109 5:47 1036 2:04 1041
2:4 106 5:47 1033 2:04 1038
6:0 105 1092
1081
1:08 1010
1:08 1010
10111
8.7 The Reliability of MEMS Accelerometer in Shock Survival
MEMS devices are designed to measure acceleration or other physical changes in a certain range. For
example, MEMS accelerometers for automobile airbag deployment are designed to measure acceleration
range between 50g . However, during the in-field usage, it cannot be guaranteed that the device will not
experience acceleration beyond this range. Actually, in the lifetime of an accelerometer, it will always experience several times (or even more) of accidental shocks with extremely high-level of acceleration [94].
In order to ensure high reliability, the device must be able to stand such accidental high-level shocks. Thus,
it is very important to measure the device reliability under such high-level shocks. MEMS reliability under shock environment has been researched recently [68][94][95]. In [94], the mechanical drop reliability
test for ADXL50 accelerometer used a minimum of 4000g shock from 1m dropping onto concrete within
250ms. They also measured the device reliability in withstanding high-level accelerations in z-axis (direction perpendicular to the device plane) on a 30,000g (294; 210m=s2 ) acceleration [94]. This value was
chosen based on the maximum acceleration applied during centrifuge testing of the accelerometer. In [68],
108
in order for the reliability test of a surface-micromachined micro-engine in shock environments, haversine
shock pulses with widths of 1 to 0.2ms in the range from 500g to 40,000g were used. Thus, in our experiment, we will also evaluate the shock survival reliability of both non-BISR and BISR devices in response
to shock levels from 500g to as high as 80,000g.
8.7.1 Assumptions
In order to simplify the problem, we made several assumptions in the following discussions. First, we only
consider the case of a negative Z-axis shock. That is, the shock acceleration is along the direction perpendicular to the device plane, and point to the negative Z direction. As a result, the movable microstructure
will deflect along positive Z-axis, as shown in Figure 8.16. For a shock along the positive Z direction, it
will lead to stiction between the movable structure and the substrate. A shock along the X direction will
lead to stiction between movable and fixed fingers, and will be discussed later in the stiction section of this
chapter. Second, during high-level acceleration shock, a packaged MEMS device may fail due to various
defect sources. For example, the wire-bonding may be lifted off the anchor, or the debris/particles sealed in
the package may fall out and move to the capacitance gap of the accelerometer, etc. These various defect
sources may cause the device to be faulty in lower acceleration shock levels. However, in order for simplification, in our research we only concentrate on the structure fracture due to a large stress caused by the large
displacement in high-g level shocks. Since both ends of each beam are the most stress-concentrated area
in MEMS comb acceleration, the structural fracture will first occur on the ends of the beams. Third, we
ignore the air damping effect in the shock testing. But, we need to be aware that the air-damping effect may
slightly alleviate the amplitude of the mass movement. Hence, it will help increase the durability of MEMS
devices to high-g shock. However, this effect is not significant and will not be considered in our reliability
comparison. Fourth, we assume both non-BISR and BISR devices are unpowered when the Z-axis shock
is experience. This means the devices will not have electrostatic force and the deflection of the movable
microstructure is purely due to the applied acceleration shock.
Figure 8.16: Deflection of accelerometer under Z-axis shock.
109
8.7.2 Reliablity Analysis for Shock
In [66], the Weibull fracture probability function Pf and the expected mean fracture stress f of a lateral
accelerometer structure were derived. In a similar way, we can calculate the Pf and f of our accelerometers. Since there is no frame tether in our accelerometer design, if we only consider the fracture behavior
of one beam, there are 2 instead of 16 identical beam flank surfaces, i.e., the right and left side-walls of the
beam. Thus, the Weibull fracture probability Pf of one beam can be expressed as
Pf = 1 exp[
Wb Lb max m
(
) ℄:
m + 1 0
where Wb and Lb are the width and the length of a beam, m is the Weibull modulus, max is the maximum
stress in the beam when it is deflected, and 0 is a normalizing factor. The Weibull modulus m and the
normalizing factor 0 are extracted by fitting the chi-squared method to the experimental measurement data
[66][96]. The maximum stress max in the beam under a given acceleration shock can be extracted by
ANSYS simulation.
For the non-BISR device, first we extract the maximum stress in the non-BISR device and one BISR
module in response to a given acceleration shock with ANSYS. Then, we use the Weibull distribution
function to derive the fracture probability Pfbnsr of one beam for the non-BISR device and the fracture
probability Pfbmod of one beam in a BISR module. The shock-survival probability Rfnsr , which is the
reliability of the non-BISR device under a given shock acceleration, can be calculated as
Rfnsr = (1 Pfbnsr )4 :
The reliability Rfmod of one BISR module can be derived by
Rfmod = (1 Pfbmod )4 :
The reliability of the entire BISR device can be calculated based on the reliability model of 4-of-6 redundancy repair derived before by
6
5 (1 R
4
2
Rsr = Rfmod
+ 6Rfmod
fmod ) + 15Rfmod (1 Rfmod ) :
The z-shock survival probability increase IR of the comb accelerometer can be calculated by
IR = Rsr
Rnsr :
This is our strategy to evaluate the reliability of both non-BISR and BISR devices for a shock failure
mechanism.
110
8.7.3 Mean Fracture Acceleration and Safety Factor Analysis
Two other concepts, mean fracture acceleration af and safety factor SF , can also be used to evaluate the
safety of a MEMS device. The expected mean fracture stress f of one beam in the accelerometer can be
calculated as [66]
1
m + 1 1=m
)
(1 + ):
f = 0 (
W b Lb
m
where is the Gamma function. The mean fracture acceleration af is defined as the negative Z-axis acceleration input resulting in the mean fracture stress f . When f is known, we can use ANSYS simulation
to find out the corresponding mean fracture acceleration af . The larger the mean fracture acceleration af
is, the safer the device is in its operation. Safety factor SF of one beam is defined as the ratio between the
mean fracture stress f and the maximum stress max in the beam for a given input shock acceleration, and
can be represented by [66]
SF =
f
:
max
The safety factor measures approximately how many times larger of the maximum stress the device can
withstand for a given input shock acceleration. The larger the SF factor is, the safer the device is in its
operation.
Our strategy for evaluating af and SF are described as follows. For a given input negative Z-axis
acceleration shock, we use ANSYS to extract the maximum stress max in one beam for both non-BISR
and BISR devices. The width Wb and length Lb are given in Table 8.2. With the experimental data for
normalizing factor 0 and Weibull modulus m, we can calculate the mean fracture stress f using above
equations. With this f value, we can use ANSYS to extract the corresponding mean fracture acceleration
af . Further, since we already know the f and max values, we can calculate the safety factor SF for a
given input acceleration shock.
8.7.4 Simulation Results
Displacement and Maximum Stress
In [66], the fracture behaviors of both thick (thickness=10m) and thin (thickness=2m) film beams were
tested. The measured Weibull fracture probabilities for both thick and thin film beams in [66] are shown
in Figure 8.17. Their results showed that the extracted parameters (Weibull modulus, mean fracture stress
etc.) are different for thick and thin film beams. In our experiment, the thickness of the comb accelerometer
is set as 2m. That is, we use the thin film parameters extracted in [66]. The values we used for Weibull
modulus m and normalizing factor 0 in our analysis are shown in Table 8.5.
Based on the above discussions, we simulated the stress and shock-survival probability for the nonBISR and BWC/EFC BISR devices under Z-axis shocks from 500g to 95000g. The maximum displacement
111
Figure 8.17: Weibull fracture probabilities for thick/thin film beams in [66].
Table 8.5: The measurement results of [66].
Parameters
Value
Weibull modulus m
Normailizing factor 0 (MPa)
11
102.2
and stress of non-BISR/BISR devices versus each input shock acceleration are shown in Figure 8.18 and
8.19 separately.
Figure 8.18: The maximum displacement versus input shock acclerations.
Reliability Analysis Result
Relationships for the Weibull fracture probability of one beam versus each input Z-axis shock acceleration
for non-BISR/BISR devices are shown in Figure 8.20. The plots of the Z-axis shock survival probability
versus each shock acceleration of non-BISR/BISR devices are shown in Figure 8.21. The Z-axis shock
survival probability increase (IR) versus each shock acceleration is shown in Figure 8.22. From Figure
8.20, it is clearly shown that the beam of the BISR modules begins to fracture at much higher acceleration
compared to the non-BISR device. That is, the maximum Z-axis shock acceleration the accelerometer can
112
Figure 8.19: The maximum stress versus input shock accelerations.
withstand is greatly extended due to the redundancy repair design. This result is intuitive because in each
BISR module of both BWC and EFC devices, the mass is reduced to 1=4 that of the non-BISR device. In
the BWC BISR device, the beam thickness remains the same, but the beam width is shrunk to 0.63 that of
the non-BISR device. That is, the Z-direction spring constant reduces to 0.63 that of the non-BISR value.
This translates a smaller maximum stress in the BISR device under the same Z-axis acceleration input. For
the EFC BISR device, the beam spring constant remains the same as that of the non-BISR device. Thus, the
maximum stress for the EFC BISR device is even smaller than that of the BWC BISR device. Hence, the
maximum input Z-axis acceleration shock the BISR devices can withstand is greatly increased. Figure 8.22
also shows the Z-shock survival probability increase (IR) versus input shock acceleration. We can see that
if the input Z-shock acceleration falls into the range between the mean fracture stress of non-BISR device
and BISR devices, a (maximum) reliability increase as large as 1 can be achieved due to the BISR design.
Figure 8.20: The Weibull fracture probabilities for non-BISR device and BISR modules.
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Figure 8.21: The z-axis shock survival probability for non-BISR/BISR devices.
Figure 8.22: The shock survival probability increase of BWC/EFC BISR devices.
114
Safety Analysis Result
The calculated mean fracture stress and its corresponding mean fracture acceleration for both non-BISR and
BISR devices are shown in Table 8.6. The SF factors for non-BISR/BISR devices in a shock acceleration
of 4000g are also shown in the table. We used this shock value (i.e., 4000g) because it is the minimum
shock acceleration for the mechanical drop reliability test by [94]. The bar charts of the mean fracture
accelerations and safety factors for both non-BISR and BISR devices are shown in Figure 8.23 and Figure
8.24 separately.
Table 8.6: The mean fracture stress for non-BISR/BISR devices.
Item
non-BISR device
BWC BISR device
EFC BISR device
mean fracture stress f
mean fracture acceleration af
Maximum stress max in device at 4000g
Safety factor SF at 4000g
805.56MPa
19369g
166.4MPa
5
840.72MPa
39850g
84.4MPa
10
805.56MPa
76764g
42.0MPa
19
Figure 8.23: The mean fracture accelerations of BWC/EFC BISR devices.
Figure 8.24: The safety factors of BWC/EFC BISR devices at 4000g.
115
According to Table 8.6, we can see that the mean shock fracture acceleration of one beam is increased
from 19369g (non-BISR device value) to 39850g (BWC device value) and 76764g (EFC device value) separately. This indicates that the maximum shock level one beam in the non-BISR device can withstand is
greatly extended after redundancy repair. From Table 8.6, we can see that for the minimum 4000g shock
in mechanical drop reliability test of accelerometers, the safety factor SF is increased from 5 (non-BISR
device) to 10 (BWC BISR device) and 19 (EFC BISR device) separately. This indicates a significant reliability enhancement for the comb accelerometer to Z-shock induced fracture failures. This is especially
helpful for some special applications such as aerospace, etc. During the launching and landing of a spaceship or rocket, extremely high level shocks can be expected. The MEMS device must be able to withstand
these intensive shocks to ensure the reliability of the whole system.
8.8 Reliability of MEMS Accelerometer in Stiction
There are two kinds of stiction in MEMS: one is the stiction between the movable microstructure and
fixed device components (such as the stiction between movable and fixed fingers in comb accelerometer),
while the other is the stiction between the movable microstructure and the substrate. These two stiction
mechanisms will be discussed separately in our analysis. Physical models describing the stiction mechanism
have also bee developed. In [97][98], a dimensionless number named peel number was proposed to predict
the stiction behavior in MEMS. The peel number Np is the ratio between the elastic strain energy stored in
the deformed microstructure and the work of adhesion between the contacted MEMS surfaces. If Np > 1,
the device will peel off the contact surface and stiction will not occur. If Np 1, stiction will occur between
the contact surfaces. For example, for a double clamped beam, the peel number can be calculated as [98]
Np =
128Et3 h2
4R L2 256 h 2
[1
+
+
( ) ℄:
5L4 Wa
21Et2 2205 t
where E is the Young’s modulus of the beam material, t is the thickness, h is the initial gap between two
surfaces, L is the length of the beam, Wa is the work of adhesion between two surfaces, and R is the
residual stress in the beam. Due to the complex mechanisms involved in stiction, the work of adhesion is
a stochastic concept and its value cannot be determined as an exact distinct value. The work of adhesion
varies within a certain range [99]. Other factors also affect the stiction behavior of the MEMS surfaces.
Examples of such factors include the surface roughness variation during fabrication process, humidity in
environment, residual stress, chemical properties change of surfaces, defects in the beam, etc. These factors
may be fluctuated with the fabrication variance or the environment.
Thus, for beams with exactly the same peel number near 1, some of the beams may have stiction while
others are stiction-free. Also, there will be some structures with peel number greater than 1 but they are
stuck; while some other structures with peel number less than 1 but they are not stuck. Thus, in reality
it is really difficult (if not impossible) to develop a model to precisely predict whether a specific beam
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will definitely stick or not [100]. That is, the stiction behavior can be better described as a probability
distribution, rather than a single value. When the peel number is close to 1, there will be a gradual change
in the stiction probability. This offers advantages for the MEMS redundancy repair technique to enhance
the device reliability in stiction failure mechanism. The stiction probability Ps is defined as the probability
that two surfaces will remain stuck after they come into contact with each other. We can also define stiction
survival probability Pv as the probability that two surface will remain unstuck even after they come into
contact. We have
Pv = 1 P s :
In order for a stiction failure to be developed during in-field usage, two steps are required. First, two
surfaces need to be brought into contact with each other by either over-range input stimulus or electromechanical instability. Second, the surface force should be large enough so that the restoring force of the
microstructure cannot separate the surfaces apart after contact has occurred. The first step is determined
by how sensitive the microstructure will respond to the input stimulus. The second step is determined by
the peel number or the stiction probability, after two surfaces come into contact. Our strategy in evaluating the device reliability of both non-BISR and BISR accelerometers are described as follows. First, we
evaluate the device reliability in resisting the input over-ranged stimulus from contacting with each other.
We will derive the device sensitivity in response to the input stimulus. Based upon this, we can find out
the over-range stimulus needed in order to bring two surfaces into contact. The larger the required input
stimulus is, the higher reliability the device will have in stiction failure mechanism. Second, if two surfaces
are accidentally contacted with each other, how large is the probability that two surface will remain stuck
with each other due to strong surface force. This can be evaluated as the stiction survival probability Pv .
The larger the Pv value is, the higher reliability the MEMS device has in stiction failure mechanism.
8.8.1 MEMS Reliability in Resisting Contact
The MEMS device reliability in the first step of stiction failure can be evaluated by measuring the capability
of the movable microstructure in resisting the over-range input stimulus. There are two possible stiction
modes in comb accelerometers. The first mechanism is the case where the movable fingers are brought
into contact with substrate due to over-range acceleration along positive Z direction (i.e., the direction
perpendicular to device plane pointing out of the substrate). The second mechanism is the case where the
movable fingers are brought into contact with fixed fingers by over-range acceleration along the X direction,
or by over-range electrostatic voltage between movable and fixed fingers.
Contact by Z-axis Shock Acceleration
We use ANSYS to find out the Z-axis shock acceleration needed in order to bring the movable fingers and
substrate into contact in unpowered condition for both non-BISR and BISR devices. We assume the gap
117
between the movable microstructure and the substrate as 1:6m. The simulation results are shown in Table
8.7. The bar chart of threshold acceleration for Z-axis contact is shown in Figure 8.25.
Table 8.7: The Z-axis acceleration required for contact of movable microstructure and substrate.
Device
Non-BISR device
BWC BISR device
EFC BISR device
Z-axis acceleration to trigger contact
1504g
3137g
4145g
Figure 8.25: The threshold acceleration for Z-axis contact.
From the simulation results, we see that the required Z-axis acceleration to bring the movable microstructure into contact with the substrate for non-BISR devices is 1504g. The required Z-axis acceleration for the BWC and EFC BISR devices are increased to 3137g and 4145g separately. Thus, the device
reliability in resisting the input Z-shock acceleration from contacting with the substrate is enhanced for the
BISR devices. This is an intuitive result and we can give a rough estimation. For non-BISR devices, the
minimum Z-axis input acceleration az which brings the movable microstructure into contact can be derived
by solving the following equation
M az = K dg =
4EWb t3 dg
:
L3b
where M is the mass of movable mass, dg is the gap between the movable microstructure and the substrate,
E is Young’s modulus, Wb and Lb are the width and length of a beam, and t is device thickness. From the
above equation, we have
az =
For BWC BISR devices,
we have
4EWb t3 dg
:
ML3b
Wb is changed to 0:63Wb , and M
is also changed to approximately
az (BW C ) = (0:63=0:25)az = 2:52az :
Similarly, for EFC BISR devices, we have
az (EF C ) = (1=0:25)az = 4az :
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(1=4)M , so
The ANSYS values of az are slightly smaller than the above rough estimation. This is because the
mass in a BISR device is larger than (1=4)M due to the extra portion of mass in both left and right ends of
each BISR module. Thus, the ANSYS simulation results should be more trustable.
Contact by X-axis Shock Acceleration
Similarly, we can evaluate the minimum X-axis shock acceleration needed to bring the movable fingers into
contact with fixed fingers in unpowered condition. ANSYS simulation results are shown in Table 8.8 based
on the assumption that the capacitance gap d0 is 2m. The bar chart of threshold acceleration for X-axis
contact is shown in Figure 8.26.
Table 8.8: The simulation results of X-axis shock.
Device
Non-BISR
BWC BISR device
EFC BISR device
X-axis shock acceleration to trigger sticion ax
769g
769g
3030g
Figure 8.26: The threshold acceleration for X-axis contact.
From the result, we can see that the minimum X-axis acceleration ax required for BWC BISR device
to bring the movable fingers into contact with fixed fingers is the same as that of the non-BISR device.
However, the ax for EFC BISR device is increase to about 4 times that of the non-BISR device. Thus,
the device reliability in resisting the X-axis acceleration from contacting of movable and fixed fingers is
either the same or increased due to BISR design. We can also do a similar rough estimation. For non-BISR
device, we have
M ax = K d0 =
4EtWb3 d0
:
L3b
From above equation, we have
4EtWb3 d0
:
ML3b
For BWC BISR devices, Wb is changed to 0:63Wb , and M is also changed to approximately (1=4)M . So,
ax =
we have
119
ax(BW C ) = (0:633 =0:25)az = ax:
Similarly, for EFC BISR devices, we have
ax(EF C ) = (1=0:25)ax = 4ax :
Thus, the critical acceleration along the X direction to bring movable fingers into contact with fixed
fingers for the BWC BISR device is the same as that of the non-BISR device. However, for the EFC BISR
device, its ax (EF C ) is four times that of the non-BISR device. That is, it is more reliable in resisting the
contact of movable fingers to fixed fingers. The ANSYS simulation results coincide with this theoretical
analysis result well.
8.8.2 Stiction Survival Probability after Contact
Stiction Survival Probability after X-direction Contact
First, we will consider the case of stiction between the movable fingers and fixed fingers. This can be caused
by the over-range input acceleration along the X direction, the over-range input of driving voltages for its
BIST function, or simply an electrostatic discharge. These accidental causes are hard to predict, while they
are also unavoidable during in-field usage. After the movable fingers are brought into contact with fixed
fingers accidentally, there is a probability that the fingers will remain stuck with each other if the restoring
force cannot withstand the surface force. The stiction between movable and fixed fingers after contact can
be better characterized as stochastic behavior and described with a statistical model[100][101]. However,
a theoretical distribution to precisely predict the stochastic behavior of the stiction between movable and
fixed fingers is hard to get, due to the complex factors involved in stiction [102].
For simplification, we assume that the probability for each movable finger to remain stuck with a fixed
finger in a specific MEMS device is the same, and denote this finger stiction probability as pf (0 pf 1) .
The value of pf depends on many factors such as device design, finger surface roughness, humidity, surface
chemical properties, etc. For example, if the stiffness of an accelerometer along the X direction is very
small, then the restoring force is much less than the surface force, and all the fingers may be stuck with
fixed fingers after their contact. In this case, pf = 1. However, if a special surface chemical coating
is used, this may help to decrease pf to nearly zero. For other cases, pf may take any value within the
range of 0 to 1. Actually in many cases, there are one or several movable fingers stuck to the fixed fingers.
As demonstrated in Figure 8.27 [54], the SEM photo shows only one movable comb finger is stuck to the
substrate, while the other movable fingers are free-standing without stiction. For these cases, it is reasonable
to assume the pf value stays in the range of 0 to 0.2.
Assume there are totally N number of movable fingers in a non-BISR device, and the stiction probability
of each finger is pfnsr (0 pfnsr 1). For a stiction along the X direction, all the contact surfaces
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Figure 8.27: SEM photo showing the stiction of one comb finger to substrate [54].
are those between movable and fixed fingers. After they are accidentally brought into contact, all the
movable fingers should remain stiction-free in order for the whole device to work properly. Thus, this can
be described with a series reliability model. The survival probability P vnsr of the non-BISR device for the
X-direction finger stiction failure in a single contact event can be expressed as
P vnsr = (1 pfnsr )N :
In each BISR module, there are (N=4) number of movable fingers. Assume the stiction probability of
each movable finger in the BISR device for a single contact event is pfsr (0 pfsr 1). The survival
probability P vmod of each BISR module for the X-direction finger stiction failure in a single contact event
is
P vmod = (1 pfsr )N=4 :
The survival probability P vsr of the BISR device can be expressed as
6 + 6P v5 (1 P v ) + 15P v4 (1 P v )2 :
P vsr = P vmod
mod
mod
mod
mod
The survival probability increase IPv of the BISR device compared to non-BISR device is
IP v = P vsr P vnsr
= [(1 pfsr )N=4 ℄6 + 6[(1 pfsr )N=4 ℄5 [1 [(1 pfsr )N=4 ℄℄
+15[(1 pfsr )N=4 ℄4 [1 [(1 pfsr )N=4 ℄℄2 [(1 pfnsr )N ℄:
The surface force on each movable finger in contact with its fixed fingers is determined by the surface
property and the contact area. Thus, the surface force for each finger in both non-BISR and BISR devices
121
are the same. For the BWC BISR device, the spring constant is about (1=4) that of the non-BISR value
(Knsr ), and the number of movable fingers is (1=4)N . Assuming capacitance gap as d0 , we have the
restoring force frsr on each movable finger in the BISR device equal
1K d
frsr = 4 1nsr 0
4N
K d
= nsr 0 = frnsr :
N
It can be observed that the stiction probability pfsr of the BWC BISR device is approximately the same
as pfnsr of the non-BISR device. For the EFC BISR device, the spring constant is the same as that of the
non-BISR value (Knsr ), the number of movable fingers is (1=4)N , and the restoring force frsr on each
movable finger in the BISR device is
Knsr d0
1N
4
4Knsr d0
= 4frnsr :
=
N
frsr =
Thus, the stiction probability pfsr of the EFC BISR device should be smaller than pfnsr of the non-BISR
device, and it is more reliable in resisting the stiction after contact. We can find a lower bound for the
survival probability increase due to BISR design in the worst case, that is, in the case of the BWC BISR
device. We assume pfsr = pfnsr = pf (worst case analysis), and the stiction survival probability increase
due to BISR design is
IP v = P vsr P vnsr
= [(1 pf )N=4 ℄6 + 6[(1 pf )N=4 ℄5 [1 [(1 pf )N=4 ℄℄
+15[(1 pf )N=4 ℄4 [1 [(1 pf )N=4 ℄℄2 [(1 pf )N ℄:
The simulation results of stiction survival probabilities for both non-BISR and BISR devices when
N = 60 is shown in Figure 8.28. From the figure, we can see that the stiction survival probability of the
BISR device is apparently higher than that of the non-BISR device for N = 60. The simulation results of
stiction survival probability increase versus pf for different N numbers (N=100, 80, 60, and 40) are shown
in Figure 8.29. From Figure 8.29, we can see that if pf = 0 or pf = 1, the stiction survival probability
increase is zero. This is a reasonable result. If pf = 0, no movable finger will be stuck to fixed fingers.
Thus, the stiction survival probability of both non-BISR device and BWC BISR device is 1. If pf = 1, all
movable fingers will be stuck and none will survive. Thus, the redundancy repairing method is not helpful,
and the survival probability increase is also zero. However, for pf = 0 to 0:2, an effective stiction survival
122
Figure 8.28: Stiction survival probabilities vs pf for non-BISR/BISR devices (N
= 60).
Figure 8.29: Stiction survival probability increase versus pf for different N numbers.
123
probability increase due to BWC BISR design can be observed. There is a peak for certain pf values.
This value is different for different N numbers. With the decrease of N number, this peak value will shift
gradually toward the right direction. The location of each peak value can be derived by solving the equation
d[P vsr (pf ) P vnsr (pf )℄
= 0:
d(pf )
After rearranging the equation, we have
7 (1 pf )N
12 (1 pf ) 4 N + 5 (1 pf ) 2 N = 0:
5
3
Due to the complexity of the above equation, an analytical solution cannot be derived. However, if we
know the number N , we can numerically solve the equation for pfpeak . We find out that the maximum
stiction survival probability increase IP vpeak for different N number are almost the same, and the value is
IP vpeak = 0:514:
This indicates a significant maximum stiction survival probability increase one can achieve in BWC BISR
design. Taking N = 48 as an example, we find the numerical solutions as follows
pfpeak = 0:025;
IP vpeak = 0:514:
The above relationship can be helpful in guiding the design to find the optimized number N according
to pf value in order to achieve a good stiction survival probability increase while meeting the performance
requirement. The above result is the worst case scenario for BWC BISR device. However, for EFC BISR
device, the stiction survival probability is even better. Thus, the stiction survival probability can be improved
due to the BISR design, and this in turn can increase the device reliability in the stiction failure mechanism.
Stiction Survival Probability after Z-direction Contact
Fr
movable microstructure
Fs
substrate
Figure 8.30: Stiction of movable microstructure to substrate.
An over-range acceleration shock along positive Z-direction can bring the movable components (beams,
mass and movable fingers) of the accelerometer into contact with the substrate. The movable components
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will experience both surface force Fs toward the substrate and the restoring force Fr from the beams. If the
surface force is larger than the restoring force, the movable components will be stuck to the substrate and
a stiction failure is developed. If the surface force is smaller than the restoring force, a stiction will not be
developed after contact. The restoring force of the beams can be treated as a spring. A simplified model is
shown in Figure 8.30. The condition to prevent a stiction of the movable microstructure after Z-direction
contact with substrate is
Fr > F s :
We define an anti-stiction number as
= Fr =Fs :
If > 1, stiction will not occur. If 1, a stiction failure will be developed. The larger the antistiction number is, the less likely the movable microstructure will remain stuck to the substrate after contact.
Assume the restoring force and surface force of the non-BISR device as Frnsr and Fsnsr , the stiction number
nsr of the non-BISR device is
nsr = Frnsr =Fsnsr :
For each module in BWC BISR devices, the contact surface is reduced to about (1=4) that of the nonBISR device. Hence, the surfaces force is also reduced to (1=4)Fsnsr . However, the spring constant of each
BWC BISR module is reduced to 0.63 of the non-BISR device value. The restoring force of a BWC BISR
module becomes 0:63Frnsr . The anti-stiction number bw of each BWC BISR module becomes
bw =
0:63Frnsr
= 2:52nsr :
0:25Fsnsr
Similarly, the spring constant and the restoring force Fref of each EFC BISR module remain the same as
non-BISR value, but the surface force Fsef reduces to (1=4)Fnsr . Hence, the anti-stiction number ef of
each EFC BISR module becomes
ef =
Frnsr
= 4nsr :
0:25Fsnsr
The bar chart of anti-stiction numbers of non-BISR/BISR devivces is shown in Figure 8.31. From the
results, we see that the anti-stiction numbers of the BWC and EFC BISR module are increased to 2.52 and
4 times of the non-BISR value. This helps increase the device reliability in the stiction failure mechanism
when the movable microstructure is brought into contact with substrate. If the initial anti-stiction number
of the non-BISR device is close to but slightly below 1, the increase in anti-stiction number due to BISR
design may lead to significant improvement in stiction survival probability when the movable microstructure
is brought into contact with the substrate.
125
Figure 8.31: The anti-stiction numbers of non-BISR/BISR devices.
126
Chapter 9
Conclusions and Future Work
In this thesis, both built-in self-test and self-repair for capacitive MEMS devices are studied. BIST and
BISR techniques have been successfully applied to VLSI technologies, and they are also believed to be the
solutions to MEMS testing and yield/reliability enhancement. The major contributions of this thesis include:
(1) a powerful BIST method with two test modes is proposed; (2) an effective BISR method is proposed
using the idea of redundancy repairing. The implementation of the dual-mode BIST technique can enhance
the testability of the device, reduce the test time and test cost, increase the fault coverage, enable the user to
test the device in-field, and support the task of faulty module identification for the proposed BISR technique.
The implementation of redundancy-based BISR technique can enhance the yield as well as the reliability
of the MEMS device. However, it has the limitations of sensitivity loss and area overhead of 50%. The
sensitivity loss can be compensated by electrostatic force. However, the same technique can also be applied
to non-BISR device to increase its sensitivity. Thus, it is considered as performance overhead. The BISR
technique pays the above overheads for gains in yield and reliability. The proposed BISR technique can be
used for safety-critical applications where the reliability is the first priority and overhead can be tolerated. It
can also be used for the cases where low yield is a problem and cannot be improved by other ways. Further,
just like VLSI memories, it is reasonable to expect the complexity of MEMS device may also increase
dramatically in future. For example, with the development of nanofabrication, future MEMS device may
contains thousands of comb fingers or other components. As a result, it may become a must for MEMS
redundancy repair in the future.
9.1 Contributions of This Work
1. A powerful dual-mode BIST method
In our BIST research, a dual-mode built-in self-test technique for capacitive MEMS devices has been
proposed. The BIST technique partitions the fixed capacitance plate(s) at each side of the movable microstructure into three portions: one for electrostatic activation and the other two equal portions for capac127
itance sensing. Due to such a partitioning method, the BIST technique can be applied to surface-, bulkmicromachined MEMS devices and other technologies. Further, the sensitivity and symmetry dual BIST
modes based on this partitioning can also be developed. Each of the sensitivity and symmetry BIST has its
own fault coverage. The combination of both BIST modes covers a larger defect set, so a more robust testing result for the device can be expected. The BIST technique is verified by three typical capacitive MEMS
devices: surface-micromachined comb accelerometer, bulk-micromachined accelerometer and poly-silicon
comb resonator. The voltage biasing schemes for the dual mode BIST of these three MEMS devices are
proposed. The control circuit for the dual-mode BIST implementation consists of some analog Muxes.
ANSYS and HSPICE simulations have been performed on various defects such as stiction, etch variation
and finger height mismatch. The effectiveness of the dual-mode BIST in detecting the defects on fixed
capacitance plates is also discussed, such as broken-via defect in the fixed fingers of comb accelerometer,
gap variation defect of bulk-accelerometer, side-etch variation in microresonator, etc. Simulation results
show that the proposed technique is an effective BIST solution for various capacitive MEMS devices. The
dual-mode BIST technique can also be extended to other MEMS devices as well in a similar way.
2. An effective BISR technique
In our BISR research, a built-in self-repair technique for comb accelerometer devices is proposed. The
technique uses modularized design for the MEMS comb accelerometer. The device consists of six identical
modules. Among them, four modules work jointly as the main device to ensure the device operation, while
the other two modules act as redundancy. If any module in the main device is found faulty by the BIST, the
control circuit will separate out the faulty module and replace it with a good redundant module. In this way,
the device can be self-repaired into good device given the total number of faulty modules is less than the
number of redundancy. The above developed dual-mode BIST technique can be used here for the testing of
each module.
3. Performance compensation for the BISR technique
Performance analysis shows that the sensitivity is of the BISR design is reduced to (1=4) of the nonBISR value due to modularized design. However, this sensitivity loss can be compensated by adjusting the
device design parameters such as by shrinking the beam width to 0.63 that of the non-BISR value. If the
beam width is already the minimum line width allowed in the design rule, the sensitivity can still be recovered by electrostatic force compensation. Electrostatic force acts as a spring with negative spring constant.
Thus, it helps further deflect the mass and increase the device sensitivity in open-loop operation. A small
DC biasing voltage (several volts) can easily increase the device sensitivity to infinity. Thus electrostatic
force can be a powerful method for the sensitivity compensation of the BISR device.
4. Capacitive MEMS yield model and analysis
In order to evaluate the effectiveness of the BISR scheme on the yield increase, a yield model for
MEMS redundancy repair is developed. Based on the statistical analysis, the yield after repair and yield
increase for the BISR devices is obtained as a function of m and n values as well as the initial yield Y 0.
128
With the yield model, design optimization on the m and n values to achieve required yield increase with
tolerable overhead can be performed. Simulation results turn out that the yield increase of the BISR scheme
is most effective for moderate initial yield. The yield model further considers different defect fatal rates on
different portions of the MEMS device. The yield with defect fatal rate revision is even higher. Further,
Monte Carlo simulation is used to verify the developed yield model in the case of point-stiction defects.
The Monte-Carlo simulation mimics the random defect distribution in a real comb accelerometer device.
Point-stiction defects with random size from 0:1 to 6m are distributed randomly within the device area.
Monte Carlo simulation by ANSYS extracts the sensitivity of both non-BISR device and the BISR device
modules. The simulation result clearly indicates the effectiveness of the BISR scheme in yield increase for
different initial yields. It also proves the correctness of the above achieved yield model and its predictions
on the yield increase behavior.
5. BISR circuit support and parasitic analysis
A BISR control circuit is also designed. The control circuit takes the BIST result of each module as
input, and makes corresponding switches to replace the faulty module with a good one. The control circuit coordinates the four modes of the device: normal working mode, BIST mode, BISC mode and BISR
mode. A current sensing circuit is used for the differential capacitance sensing, because it ensures better
separation of the faulty modules. The loading effect of each separated faulty module is also simulated. The
ANSYS/HSPICE simulation results show that the control circuit works properly regardless various defects
in the faulty module. This indicates that the faulty module can be well separated out of the main device
without loading effects. The parasitic effects of the control circuit have also been analyzed. Simulation results demonstrate that the parasitic capacitance does not threaten the device operation due to the capacitance
sensing technique used.
6. Reliability analysis for the proposed BISR design
In order to quantitatively evaluate the effectiveness of the BISR scheme in increasing the device reliability, a reliability model for MEMS has been developed. A series model is used to describe the reliability
of the non-BISR device, while a combined series and k-out-of-n redundancy model is used for the reliability
evaluation of the BISR MEMS device. Based on the reliability model, the reliabilities of both non-BISR
and BISR devices for different failure mechanisms are discussed. The discussed failure mechanisms include
material fatigue, shock, and stiction. For the failure mechanism of material fatigue, ANSYS simulation is
used to extract the maximum stress in the beam under cyclic loading. The number of cycles to failure for the
accelerometer is calculated based on the reported stress-life (S-N) behavior of poly-silicon material. The
results turn out that both BWC and EFC BISR accelerometer have longer fatigue life than the non-BISR
device, and therefore the reliability of the BISR devices is increased in material fatigue failure mechanism
due to redundancy repair. Reliability analysis for intensive shock and stiction defect (developed during
normal operation) also demonstrates that the BISR device has a much higher reliability than the non-BISR
device. The reliability model and the strategies for reliability analysis in this work can be applied to other
MEMS devices as well.
129
9.2 Future Work
A few possible research directions that can be pursued in the future has been outlined in this section.
1. Built-in self-calibration of the BISR design
In our future work, built-in self-calibration (BISC) of the BISR design will be researched. Due to
process variations during the fabrication flow, the structure parameters and performance of all modules in
a BISR MEMS device cannot be perfectly identical. For example, due to process variation, one module
may have slightly narrower beam width than others. This leads to a slightly larger sensitivity of the module
than others. During the in-field usage, if any of the working module is found to faulty, it will be replaced
with a good redundancy. In order to maintain the device performance after redundancy repair, such a tiny
difference between modules cannot be ignored, and the new device configuration must be recalibrated.
Generally, special apparatus is used for the calibration of MEMS devices before they are released to the
market. But, during the in-field usage, such an expensive apparatus is not easily available. Hence, to
complete this research, we will develop a BISC technique for the BISR design of comb accelerometers.
2. Extension of BISR to other MEMS devices
In this work, MEMS comb accelerometer is used as an example to implement the BISR technique.
However, the proposed BISR technique can be extended to other MEMS devices as well. Further, we find
some MEMS devices where the implementation of BISR will not lead to performance loss. For example,
some MEMS devices such as MEMS moisture sensor [103] do not have movable microstructure. Hence,
each of them can be revised into a modularized design without performance loss. For piezoelectric-driven
[104] or thermal-driven [105] MEMS actuators, the driving amplitude is totally decided by the beams and
is not related to the mass. In this way, the reduction in mass due to modularized design will not lead to
any degradation in the driving performance. It is hopeful that we may also find some other extensions for
the BISR technique. However, due to the complexity of MEMS devices and their working principles, the
implementation of BISR remains a very challenging work. This will be left as one of our future research
works.
130
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