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Theses and Dissertations
2006-09-01
Design of Piezoresistive MEMS Force and Displacement Sensors Design of Piezoresistive MEMS Force and Displacement Sensors
Tyler Lane Waterfall Brigham Young University - Provo
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DESIGN OF PIEZORESISTIVE MEMS FORCE
AND DISPLACEMENT SENSORS
by
Tyler Lane Waterfall
A thesis submitted to the faculty of
Brigham Young University
in partial fulfillment of the requirements for the degree of
Master of Science
Department of Mechanical Engineering
Brigham Young University
Thesis CompletedDecember 2006
Copyright © 2006 Tyler Lane Waterfall
All Rights Reserved
BRIGHAM YOUNG UNIVERSITY
GRADUATE COMMITTEE APPROVAL
of a thesis submitted by
Tyler Lane Waterfall
This thesis has been read by each member of the following graduate committee and bymajority vote has been found to be satisfactory.
Date Brian D. Jensen, Chair
Date Larry L. Howell
Date Timothy W. McLain
BRIGHAM YOUNG UNIVERSITY
As chair of the candidate’s graduate committee, I have read the thesis of Tyler Lane Water-fall in its final form and have found that (1) its format, citations, and bibliographical styleare consistent and acceptable and fulfill university and department style requirements; (2)its illustrative materials including figures, tables, and charts are in place; and (3) the finalmanuscript is satisfactory to the graduate committee and is ready for submission to theuniversity library.
Date Brian D. JensenChair, Graduate Committee
Accepted for the Department
Matthew R. JonesGraduate Coordinator
Accepted for the College
Alan R. ParkinsonDean, Ira A. Fulton College ofEngineering and Technology
ABSTRACT
DESIGN OF PIEZORESISTIVE MEMS FORCE
AND DISPLACEMENT SENSORS
Tyler Lane Waterfall
Department of Mechanical Engineering
Master of Science
MEMS (MicroElectroMechanical Systems) sensors are used in acceleration, flow,
pressure and force sensing applications on the micro and macro levels. Much research
has focused on improving sensor precision, range, reliability, and ease of manufacture and
operation. One exciting possibility for improving the capability of micro sensors lies in
exploiting the piezoresistive properties of silicon, the material of choice in many MEMS
fabrication processes. Piezoresistivity—the change of electrical resistance due to an ap-
plied strain—is a valuable material property of silicon due to its potential for high sig-
nal output and on-chip and feedback-control possibilities. However, successful design of
piezoresistive micro sensors requires a more accurate model of the piezoresistive behavior
of polycrystalline silicon.
This study sought to improve the existing piezoresistive model by investigating the
piezoresistive behavior of compliant polysilicon structures subjected to tensile, bending and
combined loads. Experimental characterization data showed that piezoresistive sensitivity
is greatest and mostly linear for silicon members subject to tensile stresses and much lower
and nonlinear for beams in bending and combined stress states. The data also illustrated the
failure of existing piezoresistance models to accurately account for bending and combined
loads.
Two MEMS force and displacement sensors, the integral piezoresistive micro-Force
And Displacement Sensor (FADS) and Closed-LOop sensor (CLOO-FADS), were de-
signed and fabricated. Although limited in its piezoresistive sensitivity and out-of-plane
stability, the FADS design showed promise of future application in microactuator charac-
terization. Similarly, the CLOO-FADS exhibited possible feedback control capability, but
was limited by control circuit complexity and implementation challenges.
The piezoresistive behavior exhibited by the Thermomechanical In-plane Microac-
tuator (TIM) led to a focused effort to characterize the TIM’s behavior in terms of force,
displacement, actuation current and mechanism resistance. The gathered data facilitated
the creation of an empirical, temperature-dependent model for the specific TIM. Based on
the assumption of a nearly constant temperature for each current level, the model predicted
the force and displacement for a given fractional change in resistance. Despite the success
of the empirical model for the test TIM device, further investigation revealed the necessity
of a calibration method to enable the model’s application to other TIM devices.
ACKNOWLEDGMENTS
Many people and organizations have been influential and supportive during my
work on this thesis. I am grateful for the grant by the National Science Foundation which
funded this research. I greatly appreciate the encouragement, enthusiasm and kindness of
my advisor, Brian Jensen. From the beginning he has been a good friend to ‘study’ with.
Dr. Howell and Dr. McLain of my committee also provided consistent support and advice
relating to the design, testing and analysis of the mechanisms presented herein. I also feel
fortunate to have been a part of the MEMS component of the Compliant Mechanisms Re-
search group. For example, many test structures presented in Chapter 3 were designed by
Gary Johns, and all of data for SUMMiT-fabricated devices was acquired by Robert Mes-
senger. My daily interaction with intelligent and humorous peers in the basement of the
Clyde Building contributed to my enjoyable graduate experience at BYU.
I am indebted to my good family, for their generosity, encouragement, and belief in
me. I am happy to say that I have done my best to merit some of Grandma Lane’s pride
in her “little engineer”. Most importantly, I appreciate my kind, consistent and interested
wife, Amy, who always believed in me, prayed for me, and listened to my whiteboard
explanations of MEMS and piezoresistivity.
Table of Contents
Acknowledgements xiii
List of Tables xix
List of Figures xxi
1 Introduction 11.1 Importance of the Research . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 The Piezoresistive Effect of Silicon 52.1 Physical Phenomenon of Piezoresistance . . . . . . . . . . . . . . . . . . . 5
2.1.1 Crystalline Structure . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Energy Band Structure . . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 Carrier Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.4 Carrier Trapping at Grain Boundaries . . . . . . . . . . . . . . . . 9
2.2 Modeling the Piezoresistance Effect . . . . . . . . . . . . . . . . . . . . . 102.3 Silicon as a Piezoresistive Material . . . . . . . . . . . . . . . . . . . . . . 112.4 Gauge Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Piezoresistance Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5.1 Uniaxial Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5.2 Plane Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Gauge Factor Measurement Method . . . . . . . . . . . . . . . . . . . . . 182.7 Factors Influencing the Gauge Factor . . . . . . . . . . . . . . . . . . . . . 21
2.7.1 Fabrication Method . . . . . . . . . . . . . . . . . . . . . . . . . . 212.7.2 Crystalline Structure . . . . . . . . . . . . . . . . . . . . . . . . . 212.7.3 Dopant Concentration Level . . . . . . . . . . . . . . . . . . . . . 242.7.4 p-type vs. n-type Silicon . . . . . . . . . . . . . . . . . . . . . . . 252.7.5 Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.7.6 Operating Temperature . . . . . . . . . . . . . . . . . . . . . . . . 282.7.7 Orientation of Applied Stress . . . . . . . . . . . . . . . . . . . . . 302.7.8 Additional Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 312.7.9 Summary of Piezoresistivity . . . . . . . . . . . . . . . . . . . . . 31
2.8 Example of Piezoresistance Analysis: Uniaxial Tension . . . . . . . . . . . 312.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
xv
3 Investigation of Piezoresistive Property of Polysilicon in Bending 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3 Test Devices and Experimental Setup . . . . . . . . . . . . . . . . . . . . 393.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.1 Tensile Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.4.2 Bending Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4.3 Combined Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4.4 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 MEMS Force and Displacement Sensors 514.1 Future Trends: Sensor Integration . . . . . . . . . . . . . . . . . . . . . . 524.2 Design of an Integral Piezoresistive Force and Displacement Sensor . . . . 54
4.2.1 Design Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2.2 Fabrication Process Limitations: MUMPs and SUMMiT . . . . . . 554.2.3 Force and Displacement Measurement . . . . . . . . . . . . . . . . 574.2.4 Mechanical, Electrical and Thermal Interactions . . . . . . . . . . 584.2.5 Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Piezoresistive Force and Displacement Sensor, FADS . . . . . . . . . . . . 584.3.1 Preliminary Force Sensitivity of FADS . . . . . . . . . . . . . . . 614.3.2 Out-of-Plane Stability Analysis of FADS . . . . . . . . . . . . . . 63
4.4 Closed-loop Force and Displacement Sensor, CLOO-FADS . . . . . . . . . 644.5 An Alternative Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Characterization of the Piezoresistive Properties of the Thermomechanical In-plane Microactuator 695.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.4.1 Repeatability and Drift . . . . . . . . . . . . . . . . . . . . . . . . 775.4.2 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.4.3 Empirical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.6 Need for Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6 Conclusions and Recommendations 896.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.2.1 Piezoresistance of Monocrystalline Silicon . . . . . . . . . . . . . 90
xvi
6.2.2 Optimization of Piezoresistive Sensors . . . . . . . . . . . . . . . . 916.2.3 Calibration Method for TIM . . . . . . . . . . . . . . . . . . . . . 916.2.4 Multi-Physics Model of TIM . . . . . . . . . . . . . . . . . . . . . 92
A TIM Characterization Data 93
Bibliography 106
xvii
xviii
List of Tables
2.1 Piezoresistive coefficients of silicon. . . . . . . . . . . . . . . . . . . . . . 192.2 Comparison of longitudinal gauge factor for three types of silicon. . . . . . 222.3 Properties of beam in uniaxial tension. . . . . . . . . . . . . . . . . . . . . 332.4 Results of analysis for uniaxial tension example. . . . . . . . . . . . . . . . 34
3.1 Piezoresistance test structures. . . . . . . . . . . . . . . . . . . . . . . . . 403.2 Nominal dimensions of piezoresistive tensile and bending structures. . . . . 403.3 Dimensions of piezoresistance combined-load structures. . . . . . . . . . . 403.4 Published and measured piezoresistance gauge factors. . . . . . . . . . . . 48
4.1 Comparison of MUMPs and SUMMiT Fabrication Processes. . . . . . . . 564.2 Dimensions of FADS hat. . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3 Preliminary force resolution per unit resistance for FADS sensor. . . . . . . 63
5.1 Dimensions of TIM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Summary of user repeatability. . . . . . . . . . . . . . . . . . . . . . . . . 78
xix
xx
List of Figures
2.1 Diamond cubic crystal structure of silicon [1]. . . . . . . . . . . . . . . . . 62.2 Electron energy band structure for semiconductors. . . . . . . . . . . . . . 82.3 Hole transport of boron-doped (p-type) silicon due to external electric field. 82.4 Effect of tensile stress on constant energy surfaces in multiple crystal di-
rections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Stress element showing the principal stresses. . . . . . . . . . . . . . . . . 162.6 Original experimental design for measurement of piezoresistance in silicon
and germanium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.7 Wheatstone bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.8 Resistivity of boron-doped LPCVD polysilicon as a function of dopant con-
centration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.9 Longitudinal gauge factor as a function of doping concentration for boron-
and phosphorous-doped material. . . . . . . . . . . . . . . . . . . . . . . . 232.10 Longitudinal and transverse gauge factors as a function of dopant concen-
tration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.11 Longitudinal and transverse gauge factors as a function of anneal temperature. 272.12 Gauge factor as a function of boron-to-silicon ratio and annealing temper-
atures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.13 Theoretical curves for longitudinal gauge factor against grain size for p-
type material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.14 Π11 coefficients for various dopant levels as a function of temperature. . . . 292.15 Longitudinal and transverse gauge factors (K) as a function of temperature. 292.16 Gauge factor vs strain for epoxied 6H-SiC strain gauges of two doping levels. 302.17 Beam in uniaxial tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.18 Fractional change in longitudinal resistance for n-type silicon in uniaxial
tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.19 Fractional change in longitudinal resistance of p-type silicon. . . . . . . . . 35
3.1 A traditional cantilever-beam piezoresistive force sensor and integral micro-force sensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Test structures fabricated for the characterization of the piezoresistive effectof silicon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Linear force gauge and probe guide. . . . . . . . . . . . . . . . . . . . . . 423.4 Comparison of piezoresistance in tensile devices. . . . . . . . . . . . . . . 443.5 Piezoresistivity (as a function of force) of bent-beam and folded-beam de-
vices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.6 Piezoresistance behavior of S-Curl device. . . . . . . . . . . . . . . . . . . 46
xxi
3.7 Piezoresistive behavior for combined-load orientation of the Snake mecha-nism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.8 Comparison of piezoresistance. . . . . . . . . . . . . . . . . . . . . . . . . 493.9 Comparison of measured piezoresistance to the existing model. . . . . . . . 50
4.1 Piezoresistive (a) pressure sensor and (b) accelerometer. . . . . . . . . . . . 524.2 Determining biological cell penetration forces. . . . . . . . . . . . . . . . . 534.3 A traditional cantilever-beam piezoresistive force sensor and integral micro-
force sensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.4 Schematic of Thermomechanical In-plane Microactuator (TIM). . . . . . . 544.5 Theoretical ideal force-deflection curve for Thermomechanical In-plane
Microactuator (TIM). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.6 Integral piezoresistive force and displacement sensor designs. . . . . . . . . 594.7 Spring connection configuration for FADS sensor. . . . . . . . . . . . . . . 604.8 Schematic of the Piezoresistive Microdisplacement Transducer (PMT). . . . 614.9 Preliminary sensitivity plot for FADS force sensor. . . . . . . . . . . . . . 624.10 Theoretical force-displacement curves for the TIM at multiple current levels. 654.11 Preliminary piezoresistive force results for TIM. . . . . . . . . . . . . . . . 674.12 Preliminary piezoresistive displacement results for TIM. . . . . . . . . . . 67
5.1 Schematic of Thermomechanical In-plane Microactuator (TIM). . . . . . . 705.2 Theoretical constant-current curves for SUMMiT-fabricated TIM. . . . . . 715.3 Schematic of TIM characterization apparatus, including probe guide, force
gauge and optical vernier. . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.4 Optical vernier which employed for displacement measurement. . . . . . . 745.5 Comparison of force-voltage relationship for neighboring currents. . . . . . 765.6 Comparison of displacement-voltage relationship for neighboring currents. . 775.7 Resistance drift in ‘unloaded’ TIM at 20 mA. . . . . . . . . . . . . . . . . 785.8 Force sensitivity as a function of current level. . . . . . . . . . . . . . . . . 795.9 Displacement sensitivity as a function of current level. . . . . . . . . . . . 805.10 Force as a function of fractional change in resistance. . . . . . . . . . . . . 825.11 Schematic of the SRFBM. . . . . . . . . . . . . . . . . . . . . . . . . . . 845.12 Modeled force-displacement curve for SRFBM. . . . . . . . . . . . . . . . 845.13 Schematic of the Piezoresistive Microdisplacement Transducer (PMT). . . . 865.14 Calibration curve for TIM-PMT structure. . . . . . . . . . . . . . . . . . . 86
A.1 Complete force and displacement data for TIM. . . . . . . . . . . . . . . . 94A.2 Complete data of force as a function of voltage. . . . . . . . . . . . . . . . 95A.3 Complete data of displacement as a function of voltage. . . . . . . . . . . . 96A.4 Power as a function of force and displacement. . . . . . . . . . . . . . . . 97A.5 User repeatability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97A.6 Sourcemeter drift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
xxii
Chapter 1
Introduction
The purpose of this research was to review current theories and models of the
piezoresistance effect of silicon, provide preliminary piezoresistance data for the charac-
terization of polysilicon devices in combined loads, investigate the design of an integral
piezoresistive force and displacement sensor, and characterize the piezoresistive properties
of the Thermomechanical In-plane Microactuator (TIM).
1.1 Importance of the Research
The world of technology continues to become smaller and smaller. This miniatur-
ization of technology has been possible due to advances in MEMS (MicroElectroMechan-
ical Systems) devices, specifically in micro sensors and actuators. Methods of improv-
ing sensing accuracy and reliability while decreasing sensor size and power consumption,
therefore, are constantly being investigated. Many engineering applications, such as mi-
crosurgery and microrobotics, are limited primarily by the deficiencies of existing sensors
and actuators.
One source of improvement in MEMS sensor and actuator design lies in the ex-
ploitation of polysilicon’s piezoresistive behavior. Many studies on piezoresistance have
been conducted and numerous piezoresistive devices have been fabricated and tested to
expand current understanding of the piezoresistive effect and how it can be utilized more
efficiently and in more applications.
Most of the literature today, however, is limited to the piezoresistive effect in ten-
sile and compressive stresses and for thin films experiencing plane stress. In light of the
advantages of compliant MEMS devices, the simple tension/compression model is insuf-
1
ficient. A more general piezoresistance model which accurately accounts for bending and
combined loads typical to compliant mechanisms is needed.
Another exciting aspect of MEMS research relates to the implementation of feed-
back control systems in sensing and actuation applications. With the current level of un-
derstanding of feedback control theory, many feedback systems have been successfully
employed in a variety of fields. Greater devices are still to come, however, since no actu-
ator has been shown to provide reliable actuation and on-chip sensing. Such an all-in-one
sensing actuator would expand the possibilities of feedback sensors and actuators.
1.2 Contributions of the Thesis
In addition to presenting an extensive literature review of piezoresistance, this thesis
reports and discusses data gathered to characterize the piezoresistive effect of polysilicon
for tensile, bending, and combined loads. The thesis extends the discussion of piezoresis-
tance to include the challenges involved in the design of an integral piezoresistive micro-
force and microdisplacement sensor. Finally, the thesis concludes with a description of the
characterization and modeling of the Thermomechanical In-plane Microactuator (TIM).
With the gathered data, an empirical model predicting force and displacement as a function
of current and voltage was created and the possibility of a future temperature-dependent
piezoresistive model is presented.
1.3 Outline of the Thesis
This thesis begins with an extensive review of literature about the piezoresistive
effect of silicon. A discussion of the strengths and inadequacies of piezoresistive models
found in the literature is provided and illustrated with a simple example of pure tension.
This thesis includes data and a brief discussion of the piezoresistive effect of polysilicon in
tension and combined loads.
Chapter 3 outlines considerations for the design of an integral piezoresistive force
and displacement sensor and the need for an improved understanding of the piezoresistance
effect in bending and combined loads. Several MUMPs-fabricated test structures used
2
for the characterization of piezoresistance in bending and combined loads are described.
Results from testing are presented and explained.
Applying the lessons learned from investigating the piezoresistance effect in bend-
ing, two possible force and displacement sensors, the FADS and CLOO-FADS, are de-
veloped and analyzed in Chapter 4. The advantages and weaknesses of these sensors are
discussed, as well as what could be done to make such sensors feasible.
After introducing the Thermomechanical In-plane Microactuator (TIM), Chapter 5
delineates the characterization of this thermal microactuator’s piezoresistive properties. In
addition to elucidating the experimental setup and procedures, this chapter demonstrates
the success of the characterization process in creating an empirical model which predicts
force and displacement as a function of temperature (current) and resistance (measured
voltage). It also points out the need for a calibration method to transform the empirical
model into a general, non-device-specific tool for TIM actuation and sensing.
3
4
Chapter 2
The Piezoresistive Effect of Silicon
Since its ‘discovery’ in 1954 by Charles Smith [2], the piezoresistive property of
silicon has been scrutinized in an attempt to explain the physical phenomenon, to create a
valid model of the behavior and to exploit this characteristic for use in MEMS sensing and
actuation. To understand the phenomenon of piezoresistivity, it is helpful to review pub-
lished research and explanations of the material property, existing theoretical and experi-
mental models and relationships, and factors which influence the magnitude and sensitivity
of the piezoresistance effect. The piezoresistance gauge factor of silicon and the effect of
crystal composition and structure, fabrication method, geometry, and environmental condi-
tions are also considered.
This section describes the aforementioned aspects of the piezoresistance effect in
silicon. Several considerations for the design of piezoresistive mechanisms are addressed.
Finally, a discussion of experimental data used to more fully characterize piezoresistance
in tensile and combined loading conditions is provided.
2.1 Physical Phenomenon of Piezoresistance
Piezoresistivity is a material property defined as the change in bulk resistivity due
to an applied stress or strain. Piezoresistive behavior has been observed in many materials,
but much of today’s research focuses on the piezoresistive behavior of silicon, as silicon
is the prime material used in MEMS fabrication processes. To comprehend the piezoresis-
tive effect, it is helpful to have a basic understanding of a few aspects of crystal physics,
including: atomic and crystalline structure, energy band theory, carrier transport, and car-
rier trapping at grain boundaries. Each of these aspects are described as they relate to the
piezoresistive behavior of silicon.
5
Figure 2.1: Diamond cubic crystal structure of silicon [1].
2.1.1 Crystalline Structure
Silicon is an ‘intermediate metal’. Since it typically forms an sp hybrid shell, sil-
icon has four valence electrons, which are available for covalent or ionic bonding with
other silicon or impurity atoms. As shown in Figure 2.1, pure silicon forms a highly sym-
metric diamond cubic crystal structure, so named because diamond also forms this crystal
structure.
In this diamond cubic crystal structure, each silicon atom is covalently bonded with
four adjacent atoms. When silicon is doped with impurity atoms, these impurities add
free electrons or holes to the crystalline structure, or lattice. For example, silicon doped
with boron (B) receives holes in the crystal lattice, and the resultant crystal is called p-
type silicon. Conversely, when doped with phosphorous (P), silicon receives extra free
electrons, resulting in n-type silicon. The crystal lattice that results from doping silicon has
an important effect on the material’s piezoresistive behavior.
2.1.2 Energy Band Structure
When pure silicon forms into the diamond cubic crystal structure, all four valence
electrons are covalently bonded with neighboring silicon atoms. This crystal structure has a
full valence band and an empty conduction band. These energy bands represent the energy
6
levels in which electrons can reside. The energy band is, in a sense, a region of available
energy states or orbitals for electrons in the crystalline solid. How easily electrons or holes
can reach the conduction band is key to determining the conductivity or resistivity of the
material.
For semiconductors in general, a small energy gap, called the energy band gap,
separates the conduction and valence energy bands. This band gap represents the energy
required by the highest electron in the valence band to enter the empty conduction band.
For insulators, the band gap is large, meaning that more external energy is required for
electrons in the valence band to ‘reach’ the conduction band and move throughout the
solid. When the band gap is reduced or the energy level of the highest valence electron is
increased, the conductivity of the crystal increases, since it becomes ‘easier’ for electrons
to conduct throughout the solid.
Once it has received the necessary energy (Eg) to overcome the band gap, an elec-
tron is able to move throughout the crystal structure (Figure 2.2). The movement of elec-
trons throughout the crystal is what constitutes current flow through a material. Electrons
moving through a crystal are called charge carriers, as they are the means of transporting
charge throughout a solid. Holes are also charge carriers since they have a charge equal in
magnitude but opposite in sign as the charge of an electron and can be transported through-
out the crystalline lattice.
2.1.3 Carrier Transport
For silicon doped with boron (Figure 2.3), the outer energy level of the valence band
is occupied by a hole. In Figure 2.3a, the impurity boron atom, which has three valence
electrons, bonds to the silicon crystal and contributes a hole to the lattice. When subject
to an external electric field, the hole is transferred to another crystal as it ‘switches’ with
the electron in that bond (Figure 2.3b). The hole in motion becomes the charge carrier. In
the presence of an electric field the holes located in p-type silicon will flow in a direction
opposite that of the electric field. The actual velocity of the hole—termed the carrier drift
velocity—does not follow a straight line but is scattered, since the carrier’s motion is im-
peded and redirected as it collides with other carriers, grain boundaries and dislocations.
7
Filled valence
band
Empty conduction
band
Band gap, Eg Hole in valence
band
(a) (b)
Ene
rgy
Figure 2.2: (a) Electron energy band structure for semiconductors. (b) Electron and holelocation within the energy band structure for p-type silicon (after [1]).
Si (4+)
Si (4+)
Si (4+)
Si (4+)
Si (4+)
Si (4+)
Si (4+)
Si (4+)
Si (4+)
Si (4+)
Si (4+)
Si (4+)
B (3+)
Si (4+)
Si (4+)
Si (4+)
hole
Si (4+)
Si (4+)
Si (4+)
Si (4+)
Si (4+)
Si (4+)
Si (4+)
Si (4+)
Si (4+)
Si (4+)
Si (4+)
Si (4+)
B (3+)
Si (4+)
Si (4+)
Si (4+)
(a) (b)
Figure 2.3: Hole transport of boron-doped (p-type) silicon due to external electric field(after [1]).
The complex band structure of p-type silicon complicates the analysis of its piezoresis-
tive behavior. Some physicists ascribe the piezoresistive behavior of p-type silicon to the
separation of heavy and light hole valence bands [3].
The piezoresistive behavior of phosphorous doped (n-type) silicon, on the other
hand, is attributed to the creation of an extra electron energy level near the top of the
energy band gap. Electrons in this energy level are more easily excited into the conduction
band under an applied stress or strain.
8
The ability of a material to transport carriers (electrons or holes) in this manner is
one way of describing the conductivity of the material. For a material to be highly con-
ductive, therefore, it requires a high number of mobile charge carriers. The concentration
of charge carriers in a material depends on the concentration of impurity atoms (dopant
level) as well as the temperature. Likewise, the carrier drift velocity—a function of the car-
rier mobility—is also affected by the dopant level and the magnitude of thermal vibrations
(temperature).
The conductivity, σ , of a material is a function of the number of charge carriers and
their respective mobility, as expressed by
σ = 1/ρ = n |e|µe + p |e|µh (2.1)
where n and p are the number of electrons and holes, respectively, and µe and µh are the
electron and hole mobility, respectively, and |e| is the absolute charge of an electron or
hole. Resistivity, ρ , is the inverse of conductivity. From this equation it is obvious that in-
creasing the concentration of holes or electrons and/or increasing their respective mobility
will increase the conductivity—and decrease the resistivity—of the material. However, it
should be noted that carrier mobility is a function of carrier concentration and an increase in
concentration does not directly result in an increase in conductivity of the same magnitude.
2.1.4 Carrier Trapping at Grain Boundaries
It is important to note that for a polycrystalline material, such as polysilicon, the
carrier mobility is affected by the presence of grains and grain boundaries. When single
crystal grains or ‘crystallites’ join together to form a polycrystalline solid, they form grain
boundaries. It has been shown that charge carriers can become ‘trapped’ in these grain
boundaries. This carrier trapping creates a charge buildup at the grain boundary, which sub-
sequently alters the material’s resistivity. Although the exact interaction of grain boundaries
with piezoresistance is not well understood, the interaction is important in piezoresistance
analysis. This phenomenon will be briefly presented in the discussion of the piezoresistive
gauge factor section.
9
In summary, the piezoresistance phenomenon has been described in terms of crystal
structure and electron band structure of silicon. The conductivity and resistivity of silicon
are highly dependent on the concentration and mobility of charge carriers. An externally
applied stress or strain changes the concentration and mobility of charge carriers and, sub-
sequently, induces the piezoresistive behavior of the material.
2.2 Modeling the Piezoresistance Effect
Many models and empirical graphs have been proposed which attempt to predict
the piezoresistive behavior of silicon under varied loading and environmental conditions.
Nevertheless, opinions vary as to which crystalline, environmental and loading properties
and conditions play a significant role in the piezoresistive behavior of a material.
After documenting the piezoresistance effect in germanium and silicon, Smith re-
mained uncertain of the actual mechanism behind the piezoresistive behavior he observed.
He noted that the change in volume and band gap energy of the material caused by uni-
axial stress resulted in a shear piezoresistance coefficient which was much larger than that
predicted by his band warping model. This unexpected finding left him in search of ‘an
essentially new mechanism’ to describe piezoresistance [2] (see also [4]).
In light of this, additional theories and models have been proposed in an attempt to
account for all influential factors of piezoresistance [5–7]. These models include:
• Dopant segregation model [9, 10]
• Many valley model (Figure 2.4) [8, 11]
• Carrier trapping model [8]
• Thermionic emission-diffusion model [12]
• Hole effective mass model
Each of these models seeks to provide greater accuracy by accounting for such
things as a grain size and grain boundary resistance. For example, according to the many-
valley model [8,13], or deformation potential theory, tensile stress imposed on an anisotropic
crystalline lattice alters the energy band structure in the longitudinal and transverse direc-
tion, as noted in Figure 2.4. As shown in this figure, tensile stress applied in the direction
10
[100]
[010]
[001]
•
•
J
longitudinal
transverse
Figure 2.4: Effect of tensile stress on constant energy surfaces (ellipses) in multiple crystaldirections; dotted lines denote the stressed condition [2, 3, 8].
noted causes the constant energy surface in the longitudinal direction to ‘shrink,’ while
the constant energy surfaces in the transverse direction ‘expand.’ This theory qualitatively
describes the anticipated anisotropic piezoresistive behavior of monocrystalline solids.
Another theoretical model was investigated by French and Evans, who concluded
that, especially for low doping levels, grain boundaries are sensitive to strain and, therefore,
affect the piezoresistance behavior of a polycrystalline material. Accordingly, this influence
was accounted for in their theoretical model, which was shown to produce greater accuracy
of gauge factor measurements than earlier models [14,15]. Other research, however, affirms
that there exists a sufficiently high doping level at which the effects of grain boundaries on
the piezoresistance are negligible [16]. Numerous tests have been conducted to validate
these piezoresistance models and theories [2, 5, 11, 17].
2.3 Silicon as a Piezoresistive Material
Silicon is widely used in semiconductor and microelectromechanical systems (MEMS).
The piezoresistive behavior of silicon was first documented by Smith in 1954 [2]. Sil-
icon is well suited for applications involving piezoresistivity for many reasons, includ-
ing: [11, 18, 19]
11
1. The gauge factor of semiconductors is more than an order of magnitude higher than
that of metals.
2. Silicon is a very robust material.
3. The possible integration of gauge and moving member eliminates the need to bond
the two components together, which eliminates hysteresis and creep and transmits
the strain perfectly from the moving member to the gauge.
4. The resistors can be limited to the surface of the element in bending or torsion where
the stresses are maximal.
5. Good matching of the resistors can be achieved, which is particularly useful if a
Wheatstone bridge configuration is employed.
6. The technique is very suitable for miniaturization of the sensors.
7. Mass fabrication can profit from the available technology of integrated circuits.
8. It is possible to integrate electronic circuitry directly on the sensor chip, for signal
amplification and temperature compensation.
The mechanical properties of silicon allow it to be used in mechanisms which re-
quire large deflections or motion. These compliant MEMS are important for a piezoresis-
tive system since the piezoresistive effect only becomes visible when a mechanism deflects
under an applied stress or strain. Finally, equations describing the large deflections inher-
ent in compliant mechanism motion are well documented and allow for accurate analysis
of these devices [20].
With its excellent material properties, the piezoresistive behavior of silicon is cur-
rently employed in many applications, including: [21–23]
1. Acceleration detection [24, 25]
2. Pressure sensing [12, 15, 26–29]
3. Flow sensing [30]
4. Displacement sensing and nanopositioning [31, 32]
5. Force and torque detection, as in atomic force microscopy [33–36], biological re-
search [37], and gauge calibration [38, 39]
6. Acoustic wave detection in microphones [40–43]
12
Many of these devices utilize the Wheatstone bridge, with one, two or variable
piezoresistors which are diffused on the surface of the beam, diaphragm or deflecting mem-
ber. To be successfully implemented in any of these applications, the piezoresistive behav-
ior of each resistor must be accurately modeled. The following section provides a discus-
sion of how the piezoresistance effect has been modeled as well as how certain aspects of
mechanism design and fabrication influence the piezoresistive sensitivity of silicon.
2.4 Gauge Factor
A key component of piezoresistance models is the gauge factor. The gauge fac-
tor, G, can be defined as the fractional change in resistance, ∆R/R, per unit strain, ε , as
expressed in:
G =∆RRε
(2.2)
where G is unitless.
The actual derivation of the gauge factor can be seen through mathematical manip-
ulations of the basic equations of Ohm’s Law and Hooke’s Law [3, 12, 44]. This derivation
begins by relating the resistance of a material to its resistivity and geometry by
R =ρLA
(2.3)
where ρ , L and A denote the material resistivity, the length in the direction of current flow
and the cross sectional area, respectively. Implicit differentiation of Equation (2.3) results
in∆RR
=∆ρ
ρ+
∆LL− ∆A
A(2.4)
The fractional change in area, ∆A/A, can be expressed in terms of the transverse strain by
∆AA
=∆ww
+∆hh
=−εt − εz (2.5)
13
where εt = εz. Including longitudinal strain (εl = ∆L/L), Equation (2.4) can be expressed
as (∆RR
)l=
(∆ρ
ρ
)l+ εl − εt − εz (2.6)
which can be simplified using Poisson’s ratio (ν =−εt/εl):
∆RR
=∆ρ
ρ+(1+2ν)εl (2.7)
Dividing both sides by ε , Equation (2.7) can be related to the gauge factor by
G =∆RRε
=∆ρ
ρ
1ε
+1+2ν (2.8)
Thus the gauge factor for silicon is dependent on Poisson’s ratio and the fractional change in
resistivity under a known strain. This fractional change in resistivity, ∆ρ/ρ , is the principle
source of piezoresistance behavior in semiconductors.
For metal strain gauges, on the other hand, the fractional change in resistivity is
nearly insignificant compared to the fractional change in resistance due to the specimen’s
change in length, ∆L/L. Therefore, the change in resistance induced by strain for a metal
is mainly due to the volumetric effects noted in Equation (2.8) by the 1+2ν term.
2.5 Piezoresistance Coefficients
The piezoresistance properties of silicon were first quantified using piezoresistance
coefficients, π , which relate the change in resistivity to stress and are expressed in Pa−1.
The derivation of the π coefficients begins by using Ohm’s Law to relate the electric field
vector, ~E, to the current vector, ~J, by a 3x3 resistivity tensor ρ [2, 3, 45]:
E1
E2
E3
=
ρ1 ρ6 ρ5
ρ6 ρ2 ρ4
ρ5 ρ4 ρ3
J1
J2
J3
(2.9)
As shown in equation (2.9), the resistivity tensor always reduces to 6 coefficients due to
crystal symmetry. For silicon—and other crystals in the cubic family—the first three resis-
14
tivity terms, ρ1, ρ2, ρ3, which represent resistivity along the <100> axes, are identical:
ρ1 = ρ2 = ρ3 = ρ (2.10)
and the last three terms, ρ4, ρ5, ρ6, which relate the electric field in one direction to a
perpendicular current, are zero:
ρ4 = ρ5 = ρ6 = 0 (2.11)
For a stressed crystal, these resistivity components can be expressed by
ρ1
ρ2
ρ3
ρ4
ρ5
ρ6
=
ρ
ρ
ρ
0
0
0
+
∆ρ1
∆ρ2
∆ρ3
∆ρ4
∆ρ5
∆ρ6
(2.12)
where ∆ρi is the resistivity change due to the stress. The change in resistivity terms is
related to the piezoresistance coefficients and the stress by a 6x6 tensor, which, for an
isotropic material with cubic crystalline structure, reduces to three non-zero terms: π11,
π12, and π44. These coefficients relate the fractional change in resistivity in the six crystal
directions (Figure 2.5) to the stresses by [45]
1ρ
∆ρ1
∆ρ2
∆ρ3
∆ρ4
∆ρ5
∆ρ6
=
π11 π12 π12 0 0 0
π12 π11 π12 0 0 0
π12 π12 π11 0 0 0
0 0 0 π44 0 0
0 0 0 0 π44 0
0 0 0 0 0 π44
σ1
σ2
σ3
τ1
τ2
τ3
(2.13)
This relationship can be applied to Equation (2.9) to relate the electric field with the applied
15
Figure 2.5: Stress element showing the principal stresses.
stress by
E1 = ρJ1 +ρπ11σ1J1 +ρπ12(σ2 +σ3)J1 +ρπ44(J2τ3 + J3τ2) (2.14)
E2 = ρJ2 +ρπ11σ2J2 +ρπ12(σ1 +σ3)J2 +ρπ44(J1τ3 + J3τ1) (2.15)
E3 = ρJ3 +ρπ11σ3J3 +ρπ12(σ1 +σ2)J3 +ρπ44(J1τ2 + J2τ1) (2.16)
To more easily analyze piezoresistance in various crystal directions and orientations, lon-
gitudinal, πl , and transverse, πt , piezoresistance coefficients have been calculated using a
statistical averaging technique [12, 15, 44]. These coefficients are related to the principle π
terms by
πl = π11 +2(π44 +π12−π11)(l21m2
1 + l21n2
1 +m21n2
1) (2.17)
πt = π12 +(π11−π12−π44)(l21 l2
2 +m21m2
2 +n21n2
2) (2.18)
where l, m, n are the direction cosines of the crystal lattice [3, 44]. The longitudinal and
transverse π coefficients allow the calculation of the fractional change in resistivity along
the direction of applied stress and transverse or perpendicular to applied stress, as expressed
by:∆ρl
ρl= πlσl (2.19)
∆ρt
ρt= πtσt (2.20)
16
The longitudinal and transverse piezoresistance coefficients have been tabulated for various
crystal directions [11, 13, 46] as well as for specific geometries and loading cases [44].
Although the πt and πl coefficients provide a more general application of the prin-
ciple piezoresistance coefficients, these should only be directly applied to single-crystal
silicon. In order to apply them to a polycrystalline material, a weighted average of the
piezoresistance effect in the various crystal directions must be employed. This is accom-
plished using the texture function, which expresses the probability of specific grain orien-
tations. Assuming completely random grain orientations, this texture function is unity and
the average longitudinal and transverse piezoresistance coefficients are calculated as
< πl >= π11−0.400(π11−π12−π44) (2.21)
< πt >= π11 +0.133(π11−π12−π44) (2.22)
For n-type polysilicon (4×1014 cm−3) with modulus of elasticity of 168 GPa and ν = 0.22,
the average longitudinal and transverse piezoresistance coefficients are−45.4×10−11 Pa−1
and 34.5× 10−11 Pa−1, respectively. The average longitudinal and transverse coefficients
for p-type polysilicon (1.5×1015 cm−3) are 58.8×10−11 Pa−1 and −18.4×10−11 Pa−1,
respectively. As will be discussed, the value of these coefficients varies greatly depending
on doping concentration, temperature and other factors.
2.5.1 Uniaxial Stress
For the case of uniaxial tension or compression, equations (2.8), (2.21), and (2.22)
can be used to derive the equations for the longitudinal, Gl , and transverse, Gt , gauge
factors, which are given as
Gl =1εl
(∆RR
)l= πlE +1+2ν (2.23)
Gt =1εl
(∆RR
)t= πtE−1 (2.24)
17
It is important to note that the transverse gauge factor is given in terms of the fractional
change in resistance in the transverse direction per longitudinal strain.
2.5.2 Plane Stress
Similar mathematical manipulations can be employed to derive an expression for
the parallel, Gpar, and perpendicular, Gper, gauge factors for the case of plane stress. These
are found to be
Gpar =E(πl +νπt)
(1−ν2)+
1(1−ν)
(2.25)
Gper =E(νπl +πt)
(1−ν2)+
(1−2ν)(1−ν)
(2.26)
With these equations, the fractional change in resistance for plane stress can be
related to the parallel and perpendicular gauge factors by
(∆RR
)l= Gparεl +Gperεt (2.27)
(∆RR
)t= Gperεl +Gparεt (2.28)
Detailed derivations of these equations can be found in the literature [3, 44].
2.6 Gauge Factor Measurement Method
To measure the piezoresistance coefficients, the change in resistivity must be mea-
sured given a known applied stress or strain. Smith’s original measurement of the piezore-
sistance of silicon involved a basic, uniaxial tension system. As shown in Figure 2.6,
low-doped silicon rods were placed in uniaxial tension. The change in longitudinal and
transverse voltage was measured for two crystal orientations. Note that, by varying the
location of the voltage and current and adjusting the crystal orientation, Smith was able to
obtain all three piezoresistance coefficients, as well as a combination of the three coeffi-
cients.
As shown in Table 2.1, the π11 coefficient has the greatest magnitude (−102.2) for
n-type silicon, meaning that n-type mono-crystalline silicon exhibits the greatest piezore-
18
Figure 2.6: Original experimental design for measurement of piezoresistance in silicon andgermanium by Smith in 1954 [2].
Table 2.1: Piezoresistive coefficients of silicon, given in 10−11 Pa−1 [2].Doping Resistivity (Ω·cm) π11 π12 π44n-type +11.7 -102.2 +53.4 -13.6p-type +7.8 +6.6 -1.12 +138.1
sistive sensitivity in the direction of principle stress. For this same material, the shear
piezoresistance factor (-13.6) is least sensitive to axial tension and would not have a sig-
nificant effect on the piezoresistive output. For p-type silicon, on the other hand, the shear
coefficient is greatest (+138.1).
An important tool for accurate measurement or application of the piezoresistive
behavior of silicon is the Wheatstone Bridge (Figure 2.7). With this configuration one, two
or four piezoresistive elements can be subjected to a strain, while the remaining elements
are unstrained.
The output voltage is related to the four resistors and the excitation voltage by
Vout
Vexc=
R3
(R2 +R3)− R4
(R1 +R4)(2.29)
19
Vout
R1
R3
R2
R4
Vexc
Figure 2.7: Wheatstone bridge
where resistors 1 and 3 as well as 2 and 4 are nominally equal. Under an applied strain, the
change in resistance of the elements results in a change in output voltage.
Using the Wheatstone bridge, the piezoresistive gauge factor can then be deter-
mined by
GFε =Vout
Vexc=
∆RnRnom
(2.30)
where n is equal to 1, 2 or 4 for a full-, half- or quarter-bridge configuration, respectively.
One advantage of using the Wheatstone bridge is its ability to compensate for tem-
perature variations. Further, the relatively high repeatability achievable by today’s MEMS
fabrication processes allow for nearly equivalent nominal resistors, which greatly improves
measurement accuracy.
The Wheatstone bridge has been used in many applications, such as the piezoresis-
tive pressure sensor and a microdisplacement sensor [32]. For the pressure sensor, a high
pressure above the diaphragm causes the diaphragm to deflect downward, placing all four
resistors in compression. Since all four resistors experience the compressive strain, the
pressure sensor is an example of a full-Wheatstone bridge.
20
2.7 Factors Influencing the Gauge Factor
Although the original piezoresistance coefficients reported by Smith [2] have been
widely used in research, experience has shown that the piezoresistive behavior of silicon
varies significantly, depending on crystal structure, dopant level and type, loading and en-
vironmental conditions, and geometry [12, 46–49].
Following is a discussion on how the fabrication method, material composition,
crystalline structure, geometry, loading conditions and electrical configuration affect the
piezoresistive gauge factor of silicon. While some research has been published for certain
factors described below, other factors are only discussed qualitatively.
2.7.1 Fabrication Method
Since nearly all of the known elements affecting the piezoresistance gauge factor
are dependent on the fabrication method used, it is important to understand the material
properties and geometries achievable by each method. Among the numerous MEMS fab-
rication processes in use today, only the MUMPS and SUMMiT processes are described
herein since these two processes were used to fabricate all test mechanisms examined in
this research.
Both SUMMiT and MUMPS processes create phosphorous-doped (n-type) poly-
crystalline silicon, although the dopant level and exact crystalline structure varies between
the two processes. In addition, each process results in mechanisms with different available
mechanical layers, surface finish and achievable geometry [50]. Each of these differences
result in mechanisms which exhibit slightly unique piezoresistive behavior.
2.7.2 Crystalline Structure
Sometimes called the crystal ‘texture’, the crystalline structure of silicon influences
its piezoresistive behavior. As described above, silicon typically forms a diamond cubic
crystal structure, which can be fabricated as single crystal (mono-crystalline) or as a col-
lection of randomly orientated molecular grains (polycrystalline). Understanding how the
21
Table 2.2: Comparison of longitudinal gauge factor for three types of silicon.Material Mono-crystalline Polysilicon Polysilicon(Model) (π11) (weighted-average: Eq. (2.21)) (empirical: Fig. 2.10)Gauge Factor -172 -75 -20
crystal structure impacts the piezoresistive gauge factor of the material is vital to correctly
applying a piezoresistance analysis.
In his original measurement of the piezoresistance coefficients, Smith used mono-
crystalline silicon, which can be thought of as one continuous crystal. Mono-crystalline
silicon can be created with a Czochralski crystal puller or by carefully controlling the diffu-
sion process. The name ‘mono-crystalline’ implies that the crystal contains no independent
grains or grain boundaries. The actual piezoresistive behavior of mono-crystalline silicon
is highly anisotropic, meaning that the gauge factor is dependent on the orientation of the
applied stress as well as the applied and measured current and voltage, respectively. As
evident in Table 2.2, the gauge factor of silicon depends greatly on its crystalline structure.
Single-crystal silicon exhibits greater sensitivity to piezoresistance, which is most likely
due to the effect of the grain boundaries of polycrystalline silicon in increasing its resistiv-
ity, as illustrated in Figure 2.8. The two gauge factors for polysilicon listed in Table 2.2
represent the discrepancy which exists among piezoresistance models.
Polycrystalline silicon only reaches 60-70% of the piezoresistive sensitivity of mono-
crystalline silicon [27]. However, many MEMs devices are composed of polycrystalline
silicon due to fabrication constraints or preferences. Due to its texture—the orientations
and sizes of individual grains throughout the material—polysilicon is typically assumed to
behave isotropically. Certain researchers have relied on this assumption to derive weighted-
average longitudinal and transverse piezoresistance coefficients (Equations (2.21) and (2.22))
which they then apply to polysilicon.
Many questions about the influence of crystal structure and grain boundaries re-
main unanswered. For example, some researchers defend that grain boundary effects must
be accounted for in piezoresistance analysis, especially for silicon with low dopant lev-
els [12, 14, 15, 47]. On the other hand, Gridchin, et al showed that the grain boundary
22
calculatedmeasured
polysilicon
monocystalline
Doping concentration [cm ]-31021 1022 102310-3
10-2
10-1
1
101
102
103
ρ [Ω
cm
]
Figure 2.8: Resistivity of boron-doped LPCVD polysilicon as a function of dopant concen-tration at room temperature [51].
Doping concentration [cm ]-3
1021 1022 1023
10
20
30
-10
-20
-30
Gau
ge fa
ctor
(p-ty
pe)
Gau
ge fa
ctor
(n-ty
pe)
Figure 2.9: Longitudinal gauge factor as a function of doping concentration for boron (—)and phosphorous (- - -) doped material, based on a grain size of 60 nm dominated by the< 110 > orientation. Curves b show the same calculation with the grain boundary assumedto be insensitive to strain [14].
23
interaction was negligible for high doping levels [6]. Therefore, a few of the pertinent
research questions concerning the influence of crystalline structure include:
• Under what conditions are grain sizes sufficiently small and grain orientations suffi-
ciently random to validate the assumption that polycrystalline silicon can be consid-
ered isotropic?
• Can polysilicon be doped to a certain level at which anisotropic effects are negligible?
• How accurate are the weighted averaged piezoresistance coefficients of Equations (2.21)
and (2.22) in the analysis of polysilicon?
• Is there a dimension or scale in mechanism geometry small enough to make the
effects of grains become significant?
• How will the crystalline structure and, therefore, the piezoresistive behavior of a
MEMS device vary among different fabrication methods, such as MUMPS or SUM-
MiT?
Although the models of French and Evans (Figure 2.9)—which include grain bound-
ary effects—are a step closer to including all crystalline effects on piezoresistance [12, 14,
15, 52], a more precise and all-encompassing theoretical and practical understanding of
the piezoresistance mechanism is needed. Finding answers to these questions will require
further research and experimentation involving mechanisms with fabrication-specific crys-
talline structure.
2.7.3 Dopant Concentration Level
As noted earlier, impurity atoms are added to silicon to optimize its electrical and
mechanical characteristics. These impurity atoms, called dopants, alter the conductivity
and piezoresistive sensitivity of silicon. In general, silicon is doped with either boron or
phosphorus, with a doping concentration ranging from 1016–1020 cm−3.
Experimental data show that the gauge factor typically decreases with increasing
doping concentration. As shown in Figure 2.10, the theoretical gauge factor increases
from 1022 cm−3 to a maximum at approximately 4×1022 cm−3. The actual measured
gauge factor for these doping concentrations only slightly increases toward the maximum
24
and then decreases as the doping concentration exceeds 1023 m−3. From Figure 2.10 it is
obvious that determining the actual gauge factor for a mechanism with a specific dopant
level brings with it some error, especially if a value is merely extrapolated from the graphs,
as will be illustrated.
2.7.4 p-type vs. n-type Silicon
Another influence on the piezoresistive gauge factor is the type of impurity atoms
diffused into the silicon. As evident in Figure 2.10, p-type silicon exhibits a larger em-
pirical gauge factor than that of n-type silicon. For the MUMPS and SUMMiT processes,
which create n-type silicon, the gauge factor will be somewhat lower than that of similar
mechanisms fabricated by other processes which create p-type silicon. Once again, cre-
ating a mechanism with maximum piezoresistive sensitivity (gauge factor) is somewhat
constrained to the available fabrication method.
2.7.5 Annealing
In addition to reducing residual stress in a mechanism, the process of annealing
increases the grain size of a polycrystalline material. This, in turn, improves the piezoresis-
tance sensitivity of a material, especially for low-doped polysilicon [49, 53]. As shown in
Figure 2.11, the longitudinal and transverse gauge factor for both n-type and p-type silicon
increases with increasing anneal temperature. This is also illustrated for different doping
concentrations of p-type silicon in Figure 2.12. From a theoretical approach, French and
Evans modeled the effect of grain size on piezoresistive gauge factor (Figure 2.13) and con-
cluded that, to achieve the maximum piezoresistive sensitivity with reasonable stability, a
mechanism should be composed of silicon fabricated by the LPCVD process at 560° C,
doped with boron at 1023 cm−3 and annealed at 1000–1100° C [14]. Mechanisms fabri-
cated in the SUMMiT process have similar material composition.
In industry, numerous annealing methods are employed, including: furnace anneal-
ing (FA), laser annealing (LA), and rapid thermal heating (RTA). Each method creates a
unique silicon structure with distinct piezoresistive properties. Consequently, the expected
25
Doping concentration [cm ]-3
Gau
ge fa
ctor
(n-ty
pe)
1021 1022 1023
10
20
Phosphorus-doped material
Longitudinal strain
Transverse strain
Doping concentration [cm ]-31021 1022 1023
Gau
ge fa
ctor
(p-ty
pe)
10
20
30
40
Transverse strain
Longitudinal strain
Boron-doped material
Figure 2.10: Longitudinal and transverse gauge factors for n-type and p-type polysiliconover a range of dopant levels (dots represent measured values) [12].
26
Figure 2.11: Longitudinal and transverse gauge factors as a function of anneal temperaturefor n-type (- - -) and p-type (—) polysilicon [14].
Figure 2.12: Gauge factor as a function of boron-to-silicon ratio and annealing tempera-tures [49].
27
Gau
ge fa
ctor
10
15
20
25
30
Grain Size [nm]0 20 40 60 80 100
N = 1 x 10 cm23 -3
N = 3 x 10 cm22 -3
Trap density = 3.0 x 10 m16 -2
N = 1 x 10 cm22 -3
Figure 2.13: Theoretical curves for longitudinal gauge factor against grain size for p-typematerial [12].
gauge factor produced by the MUMPS and SUMMiT processes—which involve unique
annealing procedures—may differ significantly.
2.7.6 Operating Temperature
For MEMS devices which are subjected to relatively high and/or varying temper-
atures, the effect of temperature on the piezoresistive gauge factor becomes significant.
Experiments have shown that increasing the temperature decreases the gauge factor of a
material, as illustrated in Figure 2.14 [4, 48, 54]. The change in piezoresistive sensitiv-
ity due to temperature has been attributed to a change in the temperature coefficient of
resistance (TCR) [11, p 153–204] [14].
In the low temperature range (0 – 100°C), the longitudinal and transverse piezore-
sistive gauge factors decrease nearly linearly with increasing temperature (Figure 2.15). For
highly doped silicon, however, the effect of temperature on the piezoresistance coefficients
becomes less significant, as illustrated in Figure 2.14. Accordingly, for the MUMPS and
28
Figure 2.14: Π11 coefficients for various dopant levels (in cm−3) as a function of tempera-ture [48, 54].
Figure 2.15: Longitudinal and transverse gauge factors for p-type polysilicon(N=1019 cm−3) as a function of temperature [16].
29
Strain [ppm]0 200 400 600 800 1000
0.975
0.980
0.990
0.985
0.995
1.005
1.015
1.025
1.000
1.010
1.020
tension
compression
Nor
mal
ized
resi
stan
ce
Figure 2.16: Gauge factor vs strain for epoxied 6H-SiC strain gauges of two doping lev-els [55].
SUMMiT processes, the effect of temperature on piezoresistive sensitivity can be assumed
negligible for small temperature changes around 300K. This assumption may not be valid
for such devices as a thermal actuator since temperatures range from 20 – 700°C. The exact
correlation of operating temperature and piezoresistive gauge factor remains unknown.
2.7.7 Orientation of Applied Stress
As introduced in the discussion of longitudinal and transverse gauge factors, the ori-
entation of the applied stress on a mechanism greatly affects the piezoresistive behavior. It
could be expected that, for a simple beam, each loading configuration—tension, compres-
sion, bending, or a combination of all three—will produce different piezoresistance output.
Research has shown that the piezoresistive behavior of a beam in tension or compression
is nearly equal in magnitude, though opposite in sign, as illustrated in Figure 2.16. The
linear change in resistance per unit strain evident in the figure implies that the material’s
piezoresistance gauge factor is constant at all strain levels (G = ∆RRε
).
Although specific effects of loading configurations on piezoresistivity for various
types of materials has been presented qualitatively in the literature [3, 8, 16], several ques-
tions remain. For example, will a beam in ‘pure bending’ experience a net change in
30
resistance if the piezoresistance effect in the upper and lower surfaces—in tension and in
compression, respectively—are equal and opposite and cancel each other? Also, will the
gauge factor be constant for a specific material regardless of the loading orientation, or will
a unique gauge factor exist for tension, compression, bending, etc? These questions will be
further addressed in the Design Considerations and the Characterization sections.
2.7.8 Additional Factors
Similar to the effect of the orientation of the applied stress, the orientation of the
applied current and the measured voltage alters the expected piezoresistance output. This
should be obvious, given the previous discussion of longitudinal and transverse piezoresis-
tance effects.
Another possible influence on piezoresistance is mechanism geometry. Very little
has been published describing the extent to which a piezoresistive mechanism’s dimensions
and surface finish affect its gauge factor [44]. Intuitively, the geometry of a MEMS device
should affect its piezoresistive behavior, given that the dimensions of devices are on a
scale for which stiction and other atomic or molecular interactions become important. In
light of this, the effects of geometry on piezoresistivity should be investigated in future
experimentation.
2.7.9 Summary of Piezoresistivity
In discussing the piezoresistance gauge factor, many questions concerning the va-
lidity of current models and the physical explanation of piezoresistance have been raised.
Although several influential components of piezoresistance sensitivity have been addressed,
questions remain. What other factors influence piezoresistive behavior and how can these
influences be discovered and modeled accurately? In answer to these questions, further
investigation and experimental research have been performed and are presented below.
2.8 Example of Piezoresistance Analysis: Uniaxial Tension
To review the models and equations discussed, the simple case of uniaxial loading
will be analyzed. The beam in Figure 2.17 is assumed to be n-type polysilicon with the
31
b
F
h
R R
polycrystalline
Figure 2.17: Beam in uniaxial tension.
properties and dimensions listed in Table 2.3. It is desired to measure the fractional change
in resistance across the length of the beam due to an externally applied tensile load.
Before beginning the analysis, a few assumptions must be made. First, the polysili-
con is assumed to be isotropic. This implies that each material property, including Young’s
modulus, Poisson’s ratio, resistivity, and the piezoresistance coefficients, is the same in all
crystal orientations, and the weighted average values of these properties will be used.
Another assumption made is that the specimen is sufficiently long such that the
boundary conditions—the gripping at the fixed and free end—create a uniform stress through-
out the length of the beam. Finally, it is assumed that the piezoresistance behavior will
follow a similar linear pattern as the elastic stress and strain relationship, signifying that
the piezoresistive gauge factor is constant for all strain levels (independent of the applied
stress).
From stress-strain theory and Hooke’s Law, the resultant strain is related to the
applied force, F , by
σ =FAc
= Eεl ⇒ εl =F
EAc(2.31)
where εl is the longitudinal strain and Ac is the cross-sectional area normal to the stress
(Ac = bh). From Equations (2.2) - (2.8), the fractional change in resistance for uniaxial
32
Table 2.3: Properties of beam in uniaxial tension.Property Name Variable Value UnitsModulus of Elasticity E 164 GPaPoisson’s Ratio ν 0.22Average Resistivity ρ 117e−6 Ω·cmAxial Length L 100 µmBase Width b 3 µmOut-of-plane Thickness h 3.5 µmDopant Concentration N 1020 cm−3
Applied Force F 10 mN
stress is (∆RR
)l= Glεl (2.32)
where
Gl =1εl
(∆RR
)l= πlE +1+2ν (2.33)
At this point in the analysis, a decision must be made as to what longitudinal gauge factor,
Gl , or piezoresistance coefficient, πl , will be used. Using the averaged piezoresistance
coefficient will result in a higher gauge factor, since it does not account for the doping
level, operating temperature or annealing temperature. For this analysis, therefore, the
gauge factor will be taken from empirical data.
From Figure 2.10a, the longitudinal gauge factor of this n-type (1020 cm−3) material
can be estimated to be 20. Though this gauge factor accounts for the doping concentration
of n-type polysilicon, it does not include the effects of annealing and other fabrication-
specific effects. Thus, it should not be assumed that the gauge factor of a similar mechanism
fabricated by the MUMPS or SUMMiT process would be the same.
The anticipated fractional change in longitudinal resistance is now related to the
applied force by Equations (2.31) and (2.33) as
∆RR
= Gl
(F
EAc
)(2.34)
As noted in Table 2.4 and Figure 2.18, the anticipated change in resistance varies greatly
according to the gauge factor or piezoresistance coefficients used in the analysis. By us-
33
Table 2.4: Results for the uniaxial tension example showing the variation due to the methodof gauge factor calculation (σ = 952 MPa).
Calculation Method Gauge Factor ∆R/REmpirical -20 0.116Averaged π coefficients -75 0.424Monocrystalline π11 -172 0.973
0 200 400 600 800 1000−120
−100
−80
−60
−40
−20
0
Applied Axial Load, MPa
Fra
ctio
nal C
hang
e in
Res
ista
nce
(%)
Monocrystalline (G=−172)Averaged Coefficients(G=−75)Empirical Data (G=−20)
Figure 2.18: Fractional change in longitudinal resistance for n-type silicon in uniaxial ten-sion.
ing the empirical gauge factor (-20), the anticipated fractional change in resistance for
this load case was -0.116, meaning that, for a nominal resistance of 11.14 Ω, an applied
tensile stress of 952 MPa would cause a resistance drop of approximately 1.29 Ω (11%).
This estimated percent change in resistance exceeds the change measured (∼7.4%) for the
SUMMiT-fabricated tensile beam described in Chapter 3. The piezoresistance effect is sig-
nificantly greater when the averaged longitudinal πl coefficient is used (42%) or the single
crystal silicon piezoresistance π11 coefficient is used (97%).
34
0 200 400 600 800 10000
10
20
30
40
50
60
Applied Axial Load, MPa
Frac
tiona
l Cha
nge
in R
esis
tanc
e (%
)
Monocrystalline (G=11)Averaged Coefficients (G=100)Empirical Data (G=30)
Figure 2.19: Fractional change in longitudinal resistance of p-type silicon.
For comparison, Figure 2.19 shows the piezoresistance effect for p-type polysilicon
under the same uniaxial tensile load. Note how the resistance change is predicted to be pos-
itive, meaning that an applied tensile load increases the mechanism resistance. Further, the
model shows that, for p-type polysilicon, the expected piezoresistive sensitivity is slightly
higher than for n-type polysilicon.
2.9 Conclusion
Piezoresistance—the change in electrical resistance due to applied stress or strain—
has been shown to have numerous applications in sensing and actuation on the micro scale.
Several theories and mathematical and empirical models from the literature have been pre-
sented which illustrate past and present efforts to correctly characterize and model all the
influential components of this material property. To facilitate improved implementation
of the piezoresistive effect in MEMS sensing, future work will focus on characterizing
piezoresistance in bending and complex loads, as described in Chapter 3.
35
36
Chapter 3
Investigation of Piezoresistive Property of Polysilicon in Bending
3.1 Introduction
Since Smith documented the piezoresistive effect in silicon in 1954 [2], piezoresis-
tive MEMS devices have been implemented in a variety of sensing applications, including
pressure, acceleration, force and displacement sensing [11, 19, 21–23, 26, 56]. Piezoresist-
ivity—the change in electrical resistance due to applied stress or strain—facilitates the
measurement of stress or strain of a silicon member using an electrical signal, enabling the
use of piezoresistive devices in on-chip and feedback-control applications [31, 57].
The literature today contains extensive data, models, and theories describing the
piezoresistive effect in tension and compression [3, 5–8, 13–15, 17, 55]. However, as this
chapter shows, existing models fail to predict the piezoresistive behavior of polysilicon in
bending loads. A complete understanding of the piezoresistive effect of silicon and of the
factors which influence piezoresistive sensitivity is vital to the design and implementation
of innovative piezoresistive sensors.
With their versatility, reliability, and ease of manufacture, compliant mechanisms
are candidates for novel, integral piezoresistive sensors [20]. Such compliant sensors could
experience—and measure—tensile, compressive, bending and combined loads. However,
little, if any, data is available which describes polysilicon’s piezoresistive behavior in bend-
ing and combined loads. The purpose of this research is to present the design, testing and
results of several test structures which explore the piezoresistive property of polysilicon in
bending and combined loads.
37
3.2 Background
Although the physical mechanism of piezoresistivity is not completely understood,
several important trends of piezoresistance in tension and compression have been docu-
mented. It has been shown that the resistance of an n-type (phosphorus-doped) polysilicon
member subjected to uniaxial-tensile stress decreases linearly. For the same material, an
applied compressive stress causes a linear increase in resistance. The opposite is true in
both cases for p-type (boron-doped) polysilicon, with the resistance increasing in tension
and decreasing in compression.
The original model of piezoresistance relates the fractional change in electrical re-
sistivity, ∆ρ/ρ , in each crystalline direction to the the applied stress, σ , with a matrix of
piezoresistance, π , coefficients:
1ρ
∆ρ1
∆ρ2
∆ρ3
∆ρ4
∆ρ5
∆ρ6
=
π11 π12 π12 0 0 0
π12 π11 π12 0 0 0
π12 π12 π11 0 0 0
0 0 0 π44 0 0
0 0 0 0 π44 0
0 0 0 0 0 π44
σ1
σ2
σ3
τ1
τ2
τ3
(3.1)
where the π coefficients were determined experimentally [2, 3, 44, 45].
Understanding the trend of piezoresistance in tension and compression has permit-
ted the successful design and implementation of the sensors mentioned previously. How-
ever, a more complete model of piezoresistance—which accounts for the effect of com-
bined loads on resistance—is needed, as illustrated in the design of an integral piezoresis-
tive sensor.
For example, a traditional cantilever-beam force sensor, depicted in Figure 3.1a,
is composed of two basic components. First, a cantilever beam structure experiences the
physical phenomenon, in this case, an applied force. The second component—a piezore-
sistive element diffused on top of the cantilever—experiences a nearly pure-tensile or pure-
38
Top
Side
Piezoresistor
(a) (b)
Anchor
Figure 3.1: (a) Traditional cantilever-beam piezoresistive force sensor and (b) integralpiezoresistive force sensor.
compressive stress as the cantilever beam deflects [58]. The structure and the sensing
element are two separate objects, each requiring individual fabrication.
In contrast, the compliant u-shaped spring or bent-beam device, shown in Fig-
ure 3.1b, could function as an integral piezoresistive force or displacement sensor. In
the presence of an applied force, the spring would compress (or stretch) and the resultant
change in resistance of the entire structure would be measured and used to calculate the ap-
plied force or displacement. Thus, the structure itself experiences and senses or measures
the physical phenomenon.
This chapter describes experimental results of the piezoresistance effect in bending
and combined loads. Several test structures are presented with their corresponding piezore-
sistive behavior. The data illustrate the inadequacy of existing piezoresistance models and
provide motivation for the development of an improved model which accounts for the ef-
fects of combined loads on the piezoresistive effect of polysilicon.
3.3 Test Devices and Experimental Setup
Several test devices—fabricated in both the MUMPs and SUMMiT processes, which
both produce n-type polysilicon structures [59, 60]—were designed and tested to explore
39
Table 3.1: Piezoresistance test structures.Device Fabrication Process Desired Testing InformationTension Bars MUMPs/SUMMiT Tensile Model Validation; Influence of
Fabrication Method, Polysilicon Layerand Beam Length
Bent-Beam SUMMiT Effects of Bending LoadsFolded-Beam MUMPs Bending Loads and Boundary ConditionsS-Curl MUMPs Combined LoadsSnake MUMPs Combined Loads
Table 3.2: Nominal dimensions of piezoresistive tensile and bending structures, given inµm.
Device Fabrication Member In-Plane Out-of-Plane PolysiliconName Method Length, Lx Width, w Thickness, t Layer(s)L100 SUMMiT 100 1∗ 2.25 4L150p1p2 MUMPs 150 3 3.5 1,2L50p1 MUMPs 50 3 2 1L50p2 MUMPs 50 3 1.5 2L50p1p2 MUMPs 50 3 3.5 1,2Folded-Beam MUMPs 150 3 3.5 1,2Bent-Beam SUMMiT 90 2.75∗ 4.5 3,4
∗Nominal dimension: SUMMiT-fabricated devices exhibit 0.1 µm etch bias
the piezoresistance effect in bending and combined loads, as summarized in Table 3.1. As
shown in Figure 3.2, the diverse structures were designed in an attempt to capture many
of the probable loading conditions an integral piezoresistive sensor would experience. Di-
mensions for each design are included in Tables 3.2 and 3.3.
The tension structures were tested to validate the current piezoresistance model and
to characterize any differences in piezoresistive behavior due to fabrication process, fabri-
cation layers, and tensile member length. Piezoresistivity in simple bending was explored
using the Bent-Beam and the Folded-Beam devices. Finally, piezoresistance in combined
Table 3.3: Dimensions of MUMPs-fabricated piezoresistance combined load structures,given in µm.
Device Centerline In-Plane Out-of-Plane PolysiliconName Radius, R Width, w Thickness, t LayerS-Curl 4.5 3 2 1Snake 4.5 3 2 1
40
w
Lx
(a) Tensile
R w
Tension
Compression
(b) S-Curl
Lxw
(c) Folded-Beam
Anchor
Lxw
(d) Bent-Beam
t = out-of-plane thickness
R w
(e) Snake
Figure 3.2: Test structures fabricated for the characterization of the piezoresistive effect ofsilicon.
loads (bending and tension or compression) was investigated with the S-Curl and Snake
mechanisms.
Each mechanism was fabricated with an attached micro-probe guide, force gauge,
and optical vernier, shown in Figure 3.3, which allowed for force application and mea-
surement. With its relatively low stiffness (k=9.2 µN/µm), the probe guide provided off-
axis stability to the microprobe used to apply the desired forces to the test structure. The
41
VernierForceGauge
ProbeGuide
Figure 3.3: Each test structure included a micro probe guide (top) and a linear force gauge(bottom) to reduce off-axis forces and deflections.
MUMPs-fabricated probe guide was designed with the same geometry and dimensions as
the folded-beam structure depicted in Figure 3.2 (refer to Table 3.2 for dimensions). A
comparable probe guide was attached to the SUMMiT-fabricated devices.
Force measurements were calculated using large deflection equations [20, 61] for
the relative compression and stretching of the folded-beam force gauge in Figure 3.3. For
the MUMPs-fabricated devices, the long flexures of the force gauge were 110 µm long
with an in-plane width of 3.5 µm and an out-of-plane thickness of 3.5 µm (POLY1 and
POLY2 fabrication layers). With eight flexures in each parallel set, the force gauge had a
linear spring constant of approximately k=147 µN/µm. A similar force gauge was attached
to the SUMMiT-fabricated devices.
The amount of compression or stretching of the force gauge was measured with the
optical vernier. Each row of vernier ‘teeth’ was offset from the facing row in such a way
that the alignment of each sequential set of teeth represented a displacement of 0.5 µm.
This allowed a displacement resolution of 0.5 µm, signifying a resolution of applied force
of approximately 73.5 µN.
Once fabricated and mechanically released, each test device was electrically con-
nected to a Hewlett Packard Model 4145A Semiconductor Parameter Analyzer using a stan-
42
dard wedge bonder. The change in resistance of each test structure was measured across
the deflecting or strained element. To measure the piezoresistive behavior of each device,
the nominal electrical resistance was measured. Then, a set of specific force steps were
applied, with the resistance being recorded at each step. Most devices were tested with a
progressive ‘sweep’ of applied forces. Finally, the undeflected, or nominal, resistance was
again measured in order to observe any thermal drift or plastic deformation.
3.4 Experimental Results
The following subsections present the experimental results for each test device and
provide a summary of the significance of each data set.
3.4.1 Tensile Loads
The goal of the tensile testing was, first, to observe the variation in piezoresis-
tive behavior due to fabrication process and, second, to validate the existing piezoresistive
model.
The results of both SUMMiT- and MUMPs-fabricated tensile elements of various
geometries illustrate how the electrical resistance decreased linearly with increasing ap-
plied tensile stress, as shown in Figure 3.4. This piezoresistive behavior in tension followed
the trend described in the literature for n-type polysilicon [2, 14].
For the SUMMiT tensile device, the piezoresistive sensitivity (as approximated by
the slope of the linear region) is 0.011% change in resistance per MPa. A similar linear
decrease in resistance was evident in the MUMPs-fabricated devices, with a sensitivity
ranging from 0.010 – 0.012% change in resistance per MPa. The variation of piezoresistive
behavior of the MUMPs tensile elements was due to differences in element length and
polysilicon layer, with a slightly higher sensitivity (steeper slope) corresponding to longer
tensile elements fabricated in POLY2. It should be noted, however, that the measured
difference in piezoresistive sensitivity between polysilicon layers was quite small.
43
0 50 100 150 200 250−3
−2.5
−2
−1.5
−1
−0.5
0
Applied Tensile Stress, MPa
Fra
ctio
nal C
hang
e in
Res
ista
nce
(%)
L=100 SUMMiTL=150 POLY1POLY2L=50 POLY1L=50 POLY2L=50 POLY1POLY2
Figure 3.4: Comparison of piezoresistance in tensile devices for MUMPs- and SUMMiT-fabricated devices of varying length and fabrication layer.
3.4.2 Bending Loads
The bent-beam and folded-beam devices were used to characterize the piezoresis-
tance effect for bending loads. The change in resistance was induced as the devices were
stretched or compressed with an applied force.
Figure 3.5 presents the data for the SUMMiT-fabricated bent-beam device and for
the MUMPs-fabricated folded-beam device. The results for both mechanisms reveal how
the electrical resistance increased nonlinearly with increasing applied force. This non-
linear increase in resistance was observed regardless of force direction. The fractional
change in resistance for a beam in bending is small at low bending stresses and becomes
more sensitive at higher stress levels.
44
0 100 200 300 400 500 600 7000
0.5
1
1.5
2
2.5
3
Applied Force, µ N
Fra
ctio
nal C
hang
e in
Res
ista
nce
(%)
Folded−BeamBent−Beam
Figure 3.5: Piezoresistance behavior of SUMMiT-fabricated bent-beam device and offolded-beam device fabricated with the POLY1 and POLY2 layers of the MUMPs process.
3.4.3 Combined Loads
To explore the effects of combining tension, compression and bending loads on
piezoresistive behavior, the S-Curl and Snake devices were tested. With each of the com-
bined loading devices, acquiring numerous data points at high applied forces was inhibited
by the out-of-plane stability of the designs when connected to the force gauge. Conse-
quently, only the data for relatively low applied forces are presented here.
The results of the S-Curl testing, shown in Figure 3.6, illustrate how the resistance
rose when the device was compressed. This increase in resistance became slightly steeper,
i.e., more sensitive, at higher applied forces, much like the simple bent-beam devices. This
45
−600 −400 −200 0 200 400 600 800−0.3
−0.2
−0.1
0
0.1
0.2
0.3
Applied Force, µ N
Fra
ctio
nal C
hang
e in
Res
ista
nce
(%)
CompressionTension
Figure 3.6: Piezoresistance behavior of MUMPs-fabricated S-Curl device.
behavior could be attributed to the combined or additive effect of the compressive and
bending stresses of the device, both of which induce an increase in resistance.
When the S-curl was put in tension and bending, its resistance quickly decreased
initially, followed by a leveling off at higher applied forces. Just as the compressive and
bending stresses combined to cause an increase in resistance, initially the tensile stress in
the S-curl dominated, resulting in a net, linear decrease in resistance. As the applied force
continued to rise, however, the effect of bending stresses began to dominate, counteracting
the decrease in resistance caused by the tensile stress.
The piezoresistance effect in combined loads was further shown with the Snake
mechanism. As Figure 3.7 shows, the resistance of the Snake device initially decreased
sharply, began to level off at approximately 800 µN, and then increased nonlinearly. Just
46
0 500 1000 1500 2000 2500−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Applied Force, µ N
Fra
ctio
nal C
hang
e in
Res
ista
nce
(%)
Figure 3.7: Piezoresistive behavior for the Snake mechanism.
as with the S-Curl device, the Snake device experienced both tension and bending loads.
Initially, when stretched, or placed in tension and bending, the dominant stress state was
tension, resulting in the decrease of resistance. Only at higher stress levels did the bend-
ing stresses and resultant rise in resistance become significant enough to overpower the
resistance drop due to tension.
3.4.4 Summary of Results
As summarized in Table 3.4, the experimental data show that tensile stress induces
a linear decrease in resistance with a piezoresistive sensitivity similar to that of published
empirical values for n-type polysilicon [12]. The piezoresistive sensitivity has been de-
scribed in terms of a unitless piezoresistive gauge factor, G, calculated as:
G =(
1ε
)∆RR
= πlE +1+2ν (3.2)
47
Table 3.4: Published and measured piezoresistance gauge factors for tensile stress, unitless.Published Values SUMMiT MUMPs
Monocrystalline Averaged π Empirical [12] 3,4∗ 1∗ 2∗ 1,2∗
-175 -75 -20 -14.8 -16.5 -18.1 -16.4∗Polysilicon fabrication layer(s)
for which ε is the applied axial strain, πl is the longitudinal1 piezoresistance coefficient, E
is the modulus of elasticity and ν is Poisson’s ratio. The gauge factor for monocrystalline
silicon relied on the calculation of πl using Smith’s original piezoresistance coefficients [2]
while the averaged-π gauge factor accounted for an averaged effect of grains on piezore-
sistance for polycrystalline silicon [3, 12].
Although much lower than the gauge factors for monocrystalline silicon (-175) and
the averaged-π estimate of polysilicon (-75), the experimental values calculated in this
work showed reasonable agreement with previous empirical data for polycrystalline silicon
(-20).
Figure 3.8 compares the resistance drop for tensile members with the nonlinear rise
in resistance due to simple bending loads and the additive effect of combined loads on
piezoresistive behavior. The authors believe that these findings represent the first published
data for the piezoresistance effect of polysilicon in bending and combined loads. Although
the piezoresistive sensitivity in bending is slightly inferior at lower force levels, the ef-
fect may be advantageous for a piezoresistive device which operates—and demands high
piezoresistive sensitivity—at higher bending stresses. Further, the bending and combined
loading devices would be better suited for applications requiring significant displacement
or deformation of the senors during operation. Such deformation is not feasible with sim-
ple tensile members or thin-film elements, such as the piezoresistive elements used in the
cantilever-beam sensor in Figure 3.1a.
The experimental results also show the inadequacy of the current linear piezoresis-
tance models which are based on Smith’s π coefficients. Figure 3.9 compares the measured
to the predicted change in resistance as a function of applied force for the bent-beam de-
1The longitudinal π coefficients and gauge factors describe the fractional change in resistance of thepiezoresistive element in the direction or along the path of the applied principal stress.
48
0 200 400 600 800 1000−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Applied Force, µ N
Fra
ctio
nal C
hang
e in
Res
ista
nce
(%)
Tension BarFolded−BeamS−Curl CompressionS−Curl TensionSnake
Figure 3.8: Comparison of piezoresistive behavior in tension, bending and combined loads.
vice. The two prediction curves were generated using finite-element and finite-difference
analyses involving the superposition of Smith’s piezoresistance model.
As evident from these experimental results, the current piezoresistance model failed
to capture the trend and the magnitude of the piezoresistive effect for this simple bending
device. In fact, regardless of the sign and magnitude of the piezoresistance coefficients
used in the analyses, the model predicted a decrease in resistance due to bending stresses.
Consequently, an improved model of piezoresistance is needed to predict the effects of
bending and combined loads on piezoresistance. Such a model would facilitate the design,
optimization and implementation of integral piezoresistive force and displacement sensors
and may provide greater insights into the physical phenomenon of piezoresistivity.
49
0 20 40 60 80 100 120 140 160 180 200−6
−5
−4
−3
−2
−1
0
1
2
Applied Force, µN
Fra
ctio
nal C
hang
e in
Res
ista
nce
(%)
FEAFinite DifferenceMeasured
Figure 3.9: Comparison of measured and predicted piezoresistive property in bending,showing inadequacy of existing piezoresistance model.
3.5 Conclusion
This chapter has described an experimental investigation of the piezoresistance ef-
fect of polysilicon in bending and combined loads. The design and experimental setup of
several n-type polysilicon test structures was provided, as well as the results from experi-
mentation. As shown in Figure 3.8, bending stresses induce a nonlinear rise in resistance
while a combination of tension, compression and bending stresses ‘add’ together, with
the tensile and compressive stresses dominating at lower applied force levels. The results
demonstrated the failure of existing piezoresistance models to predict the piezoresistive ef-
fect of polysilicon in bending and combined loads. Future work will focus on developing
an all-encompassing model of piezoresistance which accurately predicts piezoresistance in
bending. Such a model may shed greater light on the physical phenomenon of piezoresis-
tance and will facilitate the design and optimization of integral piezoresistive sensors.
50
Chapter 4
MEMS Force and Displacement Sensors
As technology drives sensors and actuators smaller and smaller, the challenges in-
volved with micro-scale force and displacement measurement have increased. Recently,
for example, researchers have proposed heart surgery on human fetuses that exhibit early
signs of congenital heart disease [62]. Of the numerous challenges involved in this pro-
cedure, many deal specifically with force and displacement measurement limitations of
existing MEMS sensors and actuators. To perpetuate the progression of technology in this
and other medical and engineering fields, it necessary to improve sensing and actuation
techniques and mechanisms.
Research in recent decades has focused on exploiting the excellent mechanical and
electrical properties of silicon in MEMS sensing and actuation applications. Specifically,
silicon is currently being employed in many piezoresistance-based applications, includ-
ing: [21–23, 53]
1. Acceleration detection (Figure 4.1a) [24, 25]
2. Pressure sensing (Figure 4.1b) [12, 15, 26–29]
3. Flow sensing [30]
4. Displacement sensing and nanopositioning [32]
5. Force and torque detection, as in atomic force microscopy [33–36], biological re-
search [37, 63], and gauge calibration [38, 39]
6. Acoustic wave detection in microphones [40–43]
51
R1
R2
R3
R4Longitudinal
Transverse
Proof Mass SideView
TopView
Flexure
TopView
a) b)
TopView
Flexible Diaphragm
Figure 4.1: Piezoresistive (a) pressure sensor and (b) accelerometer.
4.1 Future Trends: Sensor Integration
As the piezoresistance model of silicon and the piezoresistive sensor design method-
ology improve, piezoresistive sensors will replace many existing systems and will be em-
ployed in novel applications, such as: [57]
• Smaller scale nanopositioning and displacement sensing for lens or mirror position-
ing
• Bio-MEMS applications such as biological cell manipulation using force feedback
(Figure 4.2) [56, 62, 64–67]
• Autonomous acceleration or threshold sensing [24]
• Micro robotic manipulation and micro-assembly [68–72]
• Electrical and thermal control in satellite applications [73]
• Force and displacement measurement for material property calculations of nano-
materials, residual stress measurement, fingerprint sensing, and characterization of
micro electrical contacts and switches [74–79]
Successful implementation of these applications depends upon increased measurement sen-
sitivity, decreased sensor and actuator size, decreased power requirements, more straight-
forward and reliable calibration techniques, improved dynamic range and sensor band-
width, and reduced manufacturing costs and complexity. One solution to these design
52
FAPPLIED
Biological Cell
Holding Pipette
Injection Pipette
Figure 4.2: Determining biological cell penetration forces.
requirements centers on the idea of an integral piezoresistive sensor, as introduced in Chap-
ter 3.
Existing piezoresistive MEMS devices typically involve a thin piezoresistive ele-
ment attached to the surface of a deflecting element, as in the piezoresistive pressure sen-
sor [11]. This design is advantageous since the piezoresistive element only experiences
significant tensile or compressive loads, thus simplifying the analysis.
A comparison of the current cantilever-beam force sensor and a possible integral
micro-force sensor, as shown in Figure 4.3, illustrates one attempt to provide the necessary
sensing and actuation improvements [58]. Unlike the cantilever-beam sensor, which is
typically composed of one to four thin piezoresistive elements diffused on the top of the
cantilever beam at its base, the integral sensor would be fabricated as one mechanical part,
reducing both the complexity (and cost) involved in manufacture and potentially reducing
the overall sensor size. In addition, the integral sensor would be employed in a feedback
control circuit, expanding its dynamic range and giving rise to the possibility of feedback
control in micro-actuation devices.
Optimizing the integral piezoresistive sensor, such as the one shown in Figure 4.3b,
however, will require an improved model of piezoresistive behavior of polysilicon. During
the development and validation of such a model, integral sensor design has principally
relied on experimental data and engineering judgment to provide basic design guidelines.
53
Top
Side
Piezoresistor
(a) (b)
Figure 4.3: (a) Traditional cantilever-beam piezoresistive force sensor and (b) an integralpiezoresistive force sensor.
Lx
Ly
t = in-plane thicknessw
anchor
Figure 4.4: Schematic of Thermomechanical In-plane Microactuator (TIM).
4.2 Design of an Integral Piezoresistive Force and Displacement Sensor
Many of the preliminary guidelines for the design of an integral piezoresistive sen-
sor center around increasing sensitivity while maintaining sensor geometry within fabri-
cation limitations. Reducing the measurement noise and uncertainty is another key aspect
of design [80]. The design guidelines, therefore, focused on meeting these objectives for
one specific sensing application: characterizing the force-deflection behavior of a target
object—such as a biological cell—using the Thermomechanical In-plane Microactuator
(TIM), drawn schematically in Figure 4.4.
54
To improve positioning and actuation precision and accuracy on the micro scale,
a more accurate and straightforward method of characterizing micro-force actuators is
needed. For example, having a relatively quick and reliable way of verifying or calibrat-
ing microactuators, such as the TIM, with various geometries and of various fabrication
batches, could facilitate the use of such an actuator in numerous industrial and research
applications. With an accurate force/deflection model, or a ‘self-calibrating’ mechanism
built into the actuator itself, the TIM would provide reliable actuation for several of the
applications previously mentioned.
4.2.1 Design Constraints
The force and displacement sensor would be employed with a TIM having an ideal
force-deflection curve shown in Figure 4.5 (Refer to Chapter 5 for a more detailed discus-
sion of the TIM). Consequently, the piezoresistive sensor was designed to have a force and
displacement range of 0–1100 µN and 11.6 µm, respectively.
The sensor design was constrained to the fabrication limitations of the MUMPs
and SUMMiT processes. In addition, care was given to design a sensor and circuit with
minimal mechanical, electrical and thermal noise inherent in such a sensing application.
Further, the sensor was designed to maximize piezoresistive sensitivity and to allow for fu-
ture experimentation into the possibility of feedback control of the thermal actuator/sensor
system.
4.2.2 Fabrication Process Limitations: MUMPs and SUMMiT
One primary design constraint was the geometry limitations imposed by the avail-
able fabrication processes. With the SUMMiT process, developed at Sandia National Lab-
oratories, MEMS devices with three or four mechanical layers (MMPOLY1, MMPOLY2,
MMPOLY3, and MMPOLY4) with line widths as small as 1 µm are achievable. In con-
trast, the MUMPS process is capable of producing mechanisms with two mechanical layers
(MMPOLY1 and MMPOLY2) with a minimum line width of approximately 2–2.5µm [59].
Despite its line width limitations, the MUMPs process benefits from an optional 0.5 µm
55
0 2 4 6 8 10 120
200
400
600
800
1000
1200
TIM Displacement, µ m
Forc
e, µ
N
Figure 4.5: Theoretical ideal force-deflection curve for Thermomechanical In-plane Mi-croactuator (TIM).
Table 4.1: Comparison of MUMPs and SUMMiT Fabrication Processes.Fabrication Layer Thickness µm Min. LineProcess POLY0 POLY1 POLY2 POLY3 POLY4 Width µmMUMPs 0.5 2.0 1.5 n/a n/a 2.5SUMMiT 0.3 1.0 1.5 2.25p 2.25p 1.0
p Planarized layer
metal layer, currently unavailable in the SUMMiT process.The specific geometry limita-
tions and mechanical layer thicknesses are summarized in Table 4.1.
The in-plane line width proved to be a major constraint in the design of the force
and displacement sensor, especially for the MUMPs process. For example, in an applica-
tion for which maximum tensile stresses with low external forces are desired, a minimum
cross-section is also desired. With a minimum cross-section achievable of approximately
4.5 µm2, the MUMPs-fabricated device would require high externally applied forces to
induce a significantly large signal. Although the SUMMiT process permits a smaller cross-
56
section of approximately 1.5 µm2, the required actuation forces for a SUMMiT-fabricated
sensor are also relatively large.
Chapters 2 and 3 illustrated the inadequacies of existing linear piezoresistance mod-
els in predicting the piezoresistive behavior of a mechanism subject to a complex loading.
Consequently, efforts to find the optimal sensor design—the design with the highest sensi-
tivity, or fractional change in resistance per applied force or deflection—relied on experi-
mental results and engineering judgment. From the data, it was determined that elements
loaded in axial tension would avail the greatest sensitivity while providing negligible de-
flection. In contrast, a mechanism relying on bending or combined loads would allow for
greater actuator displacement while requiring lower actuation forces, but would produce
limited piezoresistive output. This design trade-off proved to be an important component
of integral sensor design.
4.2.3 Force and Displacement Measurement
Two sensor configurations for the characterization of the TIM were possible: a
‘back-end’ and a ‘front-end’ sensor. Attaching the sensor to the back of the TIM shuttle
necessitated long-travel capabilities, implying the use of a ‘bending’ mechanism in place of
a purely tensile or compressive element. However, the piezoresistive sensitivity for flexures
in bending is significantly lower than the sensitivity of elements in tension or compression.
Furthermore, a sensor trailing the actuator would not able to measure the force being ap-
plied by the actuator; the sensor only could only measure shuttle displacement.
The ‘front-end’ configuration, on the other hand, lent itself more useful in measur-
ing applied forces, since it lay in-line with the source of the applied force and the resul-
tant reaction force. With the sensor placed on the front of the TIM shuttle, nevertheless,
the sensor was required to travel with the shuttle without inducing a significant change in
its piezoresistive output. In addition, with the TIM’s specific force-displacement curve,
the sensor needed to be subjected to a displacement-specific force without compressing or
stretching significantly, since such a change in mechanism length would alter the maximum
applied force of the TIM at that point.
57
4.2.4 Mechanical, Electrical and Thermal Interactions
In the absence of an electrically insulating layer in the MUMPs and SUMMiT pro-
cesses, mechanisms connected mechanically are also connected electrically and thermally.
The effort required to overcome the challenges of eliminating or reducing the potential elec-
trical noise caused by this fabrication limitation were shown in Messenger’s work on the
Piezoresistive Micro-displacement Transducer (PMT) [32]. As with the PMT, the piezore-
sistive sensor attached to the TIM would require complex circuitry in order to provide a
high signal-to-noise ratio.
4.2.5 Feedback Control
The ultimate goal of an integral, piezoresistive force and displacement sensor is to
provide on-chip sensing, easily calibration and actuator operation without complicated op-
tical measurement processes or micro-manipulation [81]. Ideally, such a feedback sensor-
actuator would facilitate the simple application of a specific force and/or displacement and
the force and displacement characterization of a target device by a straightforward process
of recording the control signal (input current) to the thermal actuator.
Accomplishing this goal required a sensor design which simultaneously measured
the actuator displacement and the applied actuation force while relying solely on electrical
input/output. With these characteristics, a calibrated sensor/actuator system could form
part of a control circuit in which the change in resistance experienced by the force and
displacement sensor could be converted into an appropriate control current to actuate the
thermal actuator.
4.3 Piezoresistive Force and Displacement Sensor, FADS
One possible piezoresistive Force And Displacement Sensor, or FADS, was devel-
oped to meet the design requirements described above. The FADS mechanism was com-
posed of the force sensor, or TIM ‘hat’, shown in Figure 4.6, which extended off the front
of the TIM shuttle as to lie between the actuation shuttle and the target object. The TIM
actuation force induced tensile stress in the pair of thin vertical elements of the sensor. For
58
Figure 4.6: Integral piezoresistive force and displacement sensor design.
both the MUMPs and SUMMiT processes, the thin tensile flexures were designed with the
minimum in-plane and out-of-plane dimensions, as noted in Table 4.1.
The FADS hat relied on the change in resistance of the slender vertical elements (de-
picted in Figure 4.7) produced by the ‘applied’ axial tension during TIM actuation. Tensile
members were chosen for force sensing to provide linear piezoresistive sensitivity without
significant compression or deflection of the sensor, which would complicate the displace-
ment calculations. In both MUMPs and SUMMiT processes, the FADS ‘hat’ was designed
to give the smallest cross-sectional area so as to increase the piezoresistive sensitivity for a
given applied actuation force, as summarized in Table 4.2.
An alternative ‘sandwich’ design was feasible with the SUMMiT fabrication pro-
cess. In the sandwich design, the sensing elements were fabricated in POLY3 and were
‘encased’ by a solid POLY1/POLY2 and POLY4 frame in order to abate the possible out-
of-plane stability issues faced by the MUMPs-fabricated FADS hat design.
59
Table 4.2: Dimensions of FADS hat, given in µm.Tensile Element In-Plane Out-of-Plane
Design Name Length Width ThicknessMUMPs 10 10 2.5 1.5MUMPs 20 20 2.5 1.5SUMMiT 10 10 1.0 2.25SUMMiT Sandwich 10 1.0 2.25
anchor
Sensing Element
Upper Springs
Lower Springs
TIM
Shu
ttle
Figure 4.7: Spring connection configuration for FADS sensor (only half shown due tosymmetry).
Creating an electrical circuit for the FADS hat proved to be a challenge due to the
sensor’s motion with the TIM shuttle. Figure 4.7 depicts the set of symmetric, flexible
folded-beam connections designed to allow for four-probe testing of the sensor without
significant reduction of the actuation force range. The ‘spring connections’ were designed
to create the least mechanical resistance to the moving TIM while providing an electrical
path for the sensor. In the final design, the upper and lower spring connections had spring
constants of approximately 3.8 and 0.32 µN/µm, respectively.
The four electrical connections created at the fixed end of the spring connections
were used as probe pads. The four-probe measurement technique was chosen for this appli-
cation because it could accurately measure the change in resistance across the two sensing
elements while filtering out the change in resistance expected in the deflecting spring con-
60
Figure 4.8: Schematic of the Piezoresistive Microdisplacement Transducer (PMT) [31,82].
nections. It was assumed that the resistance change of the sensing elements would not be
significantly affected by the increased temperature of the TIM shuttle.
The actuator displacement could be measured in one of two ways. First, the mea-
surements could be made with the PMT sensing bridge rigidly attached to the back of the
TIM shuttle, as illustrated in Figure 4.8 [31, 32, 82]. The displacement of the TIM shuttle
would subject the PMT sensing beam in compression and bending, causing the sensing
beam’s electrical resistance to rise. Measuring the change in resistance with a Wheatstone
Bridge configuration allows for accurate temperature compensation of the sensor.
Alternatively, the change in resistance experienced across the long folded-beam
spring connections during TIM actuation could be used to calculate the TIM displacement.
With the force and displacement measurements, the FADS system could provide convenient
actuator calibration and possible feedback control.
4.3.1 Preliminary Force Sensitivity of FADS
To determine the feasibility of the FADS design, preliminary force-sensitivity and
out-of-plane stability analyses were performed. The force-sensitivity analysis relied on the
experimental data presented in Chapter 2 to calculate the expected piezoresistive output,
given in absolute fractional change in resistance per unit applied force, µN−1.
61
0 500 1000 1500 2000 25000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Applied Force, µ N
(abs
) Fr
actio
nal C
hang
e in
Res
ista
nce
(%) SUMMiT100
M50p1M50p2M50p1p2M150p1p2
Figure 4.9: Preliminary sensitivity plot for FADS force sensor.
One important assumption made in the preliminary force analysis was that the effect
of temperature on the sensor resistance was negligible. However, if the heat dissipated
by the TIM shuttle—which can exceed temperatures of 400°C—is taken into account, an
alternate FADS hat should be designed to incorporate a Wheatstone Bridge [83].
As shown in Figure 4.9, the sensitivity of the SUMMiT100 tensile elements was
greatest, at nearly 2.23e−5 µN−1, while the MUMPs-fabricated tensile element of length
50 µm and MMPOLY1 and MMPOLY2 layers (M50p1p2) was the lowest, at approxi-
mately 0.454e−5 µN−1. This meant that, for the SUMMiT-fabricated device, a difference
in force of 10 µN resulted in a resistance change of about 0.022% or 191 mΩ, given a
nominal resistance of 862 Ω. For the M50p1p2 elements, on the other hand, the equivalent
change in resistance was 150 mΩ, given a nominal resistance of 1.274 kΩ.
The results of the sensitivity analysis, summarized in Table 4.3, show that, assum-
ing a resistance measurement resolution of 10 mΩ, it would be possible to achieve a force
resolution of 0.67 µN (0.8%) for the M50p2 device and 0.52 µN (1.2%) for the SUM-
62
Table 4.3: Preliminary force resolution per unit resistance for FADS sensor, given in µN.Mechanism 1 mΩ % Signal 10 mΩ % Signal 100 mΩ % SignalMUMPs 50p1 0.41 0.3 4.1 3.5 41 35MUMPs 50p2 0.07 0.1 0.67 0.8 6.7 8MUMPs 50p1p2 0.40 0.2 4.0 1.8 40 18MUMPs 150p1p2 0.12 0.1 1.2 0.7 12.4 7SUMMiT 100 0.05 0.1 0.52 1.2 5.2 12
MiT100 device. However, if the relative percent of the signal to full-range output (%) is
taken into consideration, perhaps a more accurate measurement resolution for the M50p2
device is closer to 100 mΩ, or a force resolution of 6.7 µN (7.9%).
This analysis reaffirmed the notion that sensor sensitivity increased as element
cross-sectional area decreased and was only slightly influenced by element length and
polysilicon layer. Further, it illustrated that, with a force resolution of less than 1 µN
and approximately 6 µN for the SUMMiT- and MUMPs-fabricated devices, respectively,
the FADS design has notable potential as an integral force and displacement sensor.
4.3.2 Out-of-Plane Stability Analysis of FADS
In a first-order out-of-plane stability analysis of the FADS hat, the sensor was mod-
eled as a rigid beam connected to ground via a torsional hinge. The buckling force was
drawn collinear with the beam’s initial horizontal position. Instability was defined as the
point at which the applied force—which translated into an applied torque as the rigid beam
rotated—exceeded the resistance force exerted by the torsional spring. Equating and ma-
nipulating the equations relating the applied torque and the resistance torque resulted in:
Fmax =Ebt3
lal f(4.1)
where E represents the Modulus of Elasticity, b and t are the in-plane and out-of-plane
beam thicknesses, and la and l f denote the lengths of the thin axial elements and of the
rigid hat frame, respectively.
To maintain stability in the MUMPs-fabricated device, a conservative maximum
applied force was calculated as 110 µN for the sensor with a tensile member length of
63
20 µm and 271 µN for the sensor with a member length of 10 µm. Similarly, for the
SUMMiT devices, expected forces of 190 µN and 502 µN were calculated for sensors of
length 20 µm and 10 µm, respectively.
Although the preliminary out-of-plane stability analysis results are discouraging,
they represent a conservative estimate and do not account for stress stiffening or the stability
provided by the spring connections. Future analysis including these stabilizing effects, as
well as validation testing should show a significant improvement to the FADS stability,
which would eliminate the fear that the FADS design does not possess sufficient stability
for its intended application.
4.4 Closed-loop Force and Displacement Sensor, CLOO-FADS
Another possible force and displacement sensing device was the Closed-Loop Force
and Displacement Sensor (CLOO-FADS). The CLOO-FADS was comprised of the Piezore-
sistive Microdisplacement Transducer which measured the TIM displacement (via piezore-
sistivity) and calculated the TIM’s force based on the actuation current supplied by a control
circuit to the TIM. The accuracy of the sensor depended primarily on electrothermome-
chanical models and experimental data of the force-displacement-current characteristics of
the TIM, as illustrated in Figure 4.10 [84, 85].
Using this model, a nominal TIM displacement was chosen for the CLOO-FADS
which lay sufficiently beyond the location of maximum TIM force in order to prevent the
TIM from buckling backwards, beyond its initial position. The nominal displacement was
also selected to allow for the maximum range of applied current.
Calibrating the CLOO-FADS would rely on the measurement tools and techniques
introduced in the characterization of piezoresistance of polysilicon discussion of Chapter 2
and would proceed as follows:
1. The TIM is actuated to the nominal displacement as verified by the piezoresistive
output signal of the PMT.
2. The nominal TIM actuation current is recorded.
3. Using the nominal output voltage from the PMT as the reference input, a known
external force is applied against the deflected TIM shuttle.
64
0 2 4 6 8 10 12 14 160
50
100
150
200
250
300
350
400
450
TIM Displacement, µ m
Forc
e, µ
N
Nominal Displacement
Figure 4.10: Theoretical force-displacement curves for the TIM at multiple current levels.
4. As the TIM is ‘pushed’ back, the control circuit increases the actuation current to
the TIM until the TIM shuttle returns to the nominal displacement and the reference
voltage of the PMT is restored.
5. The control current required for that applied force is recorded.
6. The process is repeated for a range of known forces, with the control current being
recorded at each force step.
With the recorded force-current data, a model or look-up table for the CLOO-FADS could
be generated. A comparison of the measured data with the existing TIM model data could
be used for model or sensor validation. Once calibrated, the CLOO-FADS would be used
for force and displacement sensing by following the calibration steps listed above, except
that the unknown applied forces would be determined using the look-up table.
To determine the feasibility of the CLOO-FADS design, the closed-loop sensitivity,
SCL, was estimated. This sensitivity describes the smallest measurable change in force, δF ,
per change in control current, δ I:
SCL =δFδ I
(4.2)
65
A preliminary estimate of SCL depended on the force-displacement-current data of the TIM,
the TIM force-current sensitivity (estimated as 0.1 µN/mA), the displacement resolution of
the PMT (approximately 30 nm), and the precision of the feedback circuit. Actual testing
would be necessary to determine an accurate measure of sensitivity for the CLOO-FADS.
Despite the CLOO-FADS potentially high displacement sensitivity (30 nm), the
design presents several challenges in its design and implementation. For example, the
CLOO-FADS would demand a complex control circuit, due to the non-linear characteristic
of the actuation and measurement processes. Further, the current sensor design is limited
by the fact that, to measure force and displacement, a target object must be pushed against
the TIM shuttle rather than the TIM pushing against the target. This awkward measurement
process would not be feasible for most applications.
4.5 An Alternative Approach
During the design of the FADS mechanisms, a promising alternative method of
force and displacement actuation and sensing was observed in the Thermomechanical
In-plane Microactuator (TIM). During actuation, the temperature-dependent piezoresis-
tive effect of the TIM became apparent, as illustrated in Figures 4.11 and 4.12 by the
change in voltage across the TIM at varying force and displacement points. At each force-
displacement point of the TIM, a constant actuation current was applied for which a unique
voltage across the TIM was measured. This behavior provided hope that the TIM could be
employed as a self-contained force-displacement actuator with possible feedback control.
Exploiting this piezoresistive behavior of the TIM would necessitate additional testing of
the device.
4.6 Conclusion
The piezoresistive Force And Displacement Sensor presented herein was designed
for use with the thermal actuator (TIM) in characterizing the force and displacement char-
acteristics of micro-scale objects, such as biological cells. In a preliminary analysis, the
FADS was shown to have a high preliminary force resolution (<1 – 6 µN) and low out-
of-plane stability (271 – 502 µN). Although the device was fabricated but not tested, it is
66
4.5 5 5.5 60
20
40
60
80
100
120
140
160
180
200
Measured Voltage, V
Forc
e, µ
N
Figure 4.11: Preliminary piezoresistive results of TIM, with respect to output force.
2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7−0.5
0
0.5
1
1.5
2
Measured Voltage, V
Dis
plac
emen
t, µm
Figure 4.12: Preliminary piezoresistive results of TIM, with respect to TIM displacement.
67
anticipated that future testing of the FADS devices will validate the predicted force resolu-
tion and out-of-plane stability and provide further insight into possible design iterations.
In addition to testing and improving the current design and implementation of the
FADS and CLOO-FADS devices, future research could focus on creating similar FADS
and CLOO-FADS sensors as single layer, monocrystalline structures. This could permit
the use of MEMS fabrication processes besides the MUMPs and SUMMiT processes. Fur-
ther, as was discussed in Chapter 2, the higher piezoresistive sensitivity of monocrystalline
silicon compared to polycrystalline silicon could provide the additional sensitivity the cur-
rent polysilicon designs lack. Using monocrystalline devices would necessitate additional
testing of its anisotropic mechanical and electrical properties.
Finally, an alternative method of force and displacement sensing and actuation was
observed in the TIM which may provide the necessary resolution and repeatability for many
MEMS applications, including those requiring feedback control. Work done to characterize
the TIM as a possible sensor and actuator is presented Chapter 5.
68
Chapter 5
Characterization of the Piezoresistive Properties of the Thermomechan-ical In-plane Microactuator
5.1 Introduction
As design and implementation of thermal actuators improve, they will be employed
in more complex applications such as: injection of RNA/DNA into biological cells, mirror
positioning and orientation, microrobotic manipulation and assembly, high density data
storage, and material property measurement of nano-materials and structures. Realization
of many of these applications, however, requires higher precision and reliability in force
and displacement sensing and actuation. Most systems will require some type of feedback
control to replace existing actuation and measurement methods.
For biological manipulation such as the extrasensory surgery, on-chip measurement
devices are needed. Lab on a chip, for example, is one effort to create such a self-contained
measurement device. Similar on-chip devices would reduce the size of measurement equip-
ment, facilitate small-scale operation, and improve measurement procedure. In addition,
such devices would require less power and space, making them desirable for military,
aerospace and computer applications.
This chapter describes promising results of an existing thermal actuator’s capability
to be used as both a force and displacement sensor in addition to a self-contained actuator.
By characterizing the force, displacement and temperature-dependent piezoresistive behav-
ior of the thermal actuator, a preliminary statistical model was created which may enable
feedback control of force and displacement measurement and micro-actuation.
69
Lx
Ly
t = in-plane thicknessw
anchor
Figure 5.1: Schematic of Thermomechanical In-plane Microactuator (TIM).
5.2 Background
Since its inception in 1999, the bent-beam thermal actuator—named Thermo me-
chanical In-plane Microactuator (TIM)(Figure 5.1)—has been employed in diverse actu-
ation applications [86–90]. Significant research has focused on understanding and opti-
mizing the TIM’s force and displacement behavior while reducing power consumption,
actuation noise and uncertainty in geometry due to current fabrication processes [32, 83–
85, 91–93].
Existing micro-actuation methods pose several limitations. Electrostatic actuators
like the comb drive have relatively high displacement (∼30 µm) yet lack actuation force
(5–25 µN). Piezoelectric actuators, on the other hand, possess high forces but very small
displacement. In addition, piezoelectric and magnetic actuators typically require compli-
cated fabrication and assembly. Although bimorph actuators have shown relatively high
displacement and forces, these actuators do not provide rectilinear motion and frequently
involve complicated fabrication sequences [94]. In contrast, the Thermomechanical In-
plane Microactuator (TIM) provides high actuation forces and displacements with rela-
tively low DC power requirements and straightforward circuitry.
To achieve its motion, the TIM relies on the expansion due to ohmic heating of its
long, thin flexures induced by an externally-applied electrical current. The slight angle of
the thin flexures (and symmetry of its design) effectively transfers the compressive forces
70
0 2 4 6 8 10 12 14 160
50
100
150
200
250
300
350
400
450
TIM Displacement, µ m
Forc
e, µ
N
Figure 5.2: Theoretical constant-current curves for SUMMiT-fabricated TIM.
generated by thermal expansion to linear deflection (and force) of the central shuttle. An
applied current–or voltage potential across the thin flexures–causes the temperature of the
thin flexures to reach as high as 400° C [83].
At a given current level, the TIM displaces to the point of force equilibrium. The-
oretically, therefore, an unloaded TIM displaces along the constant-current points lying on
a zero-force line (horizontal axis), as illustrated in Figure 5.2.
For applications involving the actuation and force and/or displacement characteri-
zation of a mechanism, such as with a bistable mechanism, the applied current is increased
incrementally [95]. Once again, at each current step the TIM displaces to the point at which
force-equilibrium occurs. With thermomechanical models of the TIM and finite-element
software, the actuation force required to displace the target mechanism can be calculated
for use in creating a force-displacement data for such a mechanism.
An important characteristic of MEMS sensors and actuators, including the TIM,
is their inherent piezoresistivity. First documented in 1951 by Smith [2], piezoresistance
of silicon describes the change in electrical resistance of a beam or element induced by an
71
Table 5.1: Dimensions of the MUMPs-fabricated TIM.Parameter Symbol Value UnitsBeam Length Lx 250 µmBeam Offset Ly 5.5 µmIn-Plane Width w 3 µmOut-of-Plane Thickness t 3.5∗ µm
∗ POLY1 and POLY2
applied stress or strain [18]. Since its discovery, the piezoresistive effect of silicon has been
exploited in many MEMS applications, including pressure and flow sensing, acceleration
detection and nanopositioning [14, 22, 25, 32, 34].
5.3 Experiment
Although the Thermomechanical In-plane Microactuator (TIM) has been studied
for several years, its piezoresistive behavior was never investigated. However, a recent re-
view of existing force, displacement, current and voltage data of two TIM devices revealed
that, during constant-current actuation, the voltage across the TIM rose as an externally
applied resistance force increased. This discovery led to the design and testing of the
MUMPs-fabricated TIM, shown schematically in Figure 5.1. The goal of the experimen-
tation was to accurately characterize the force, displacement, and resistance—voltage and
current—behavior of the TIM in order to create a thermal piezoresistance model which
could facilitate the TIM’s implementation in feedback-control systems.
5.3.1 Setup
The TIM used in this study was designed to provide high actuation forces and large
displacement. Many parameters of the TIM design summarized in Table 5.1, such as the
in-plane width, were determined by the constraints imposed by the MUMPs fabrication
process [59]. The resultant TIM was a phosphorus-doped (n ∼ 1× 1022 cm3) polysilicon
device with a footprint of approximately 500×250 µm.
Each test TIM structure was designed with the folded-beam force gauge, micro-
probe guide, and optical vernier, shown in Figure 5.3, which allowed for accurate dis-
placement measurement and force application. With its low stiffness (k=9.2 µN/µm), the
72
Force GaugeProbe Guide
Vernier
TIM
Figure 5.3: Schematic of TIM characterization apparatus, including probe guide, forcegauge and optical vernier.
linear folded-beam probe-guide provided off-axis stability without significantly inhibiting
the TIM motion. Force measurements were calculated using Hooke’s Law and equations
for large deflections of the relative compression and stretching of the folded beam force
gauge [20, 61]. A fairly flexible force gauge (k=36 µN/µm) permitted high resolution in
force measurements.
Similarly, the displacement of the TIM and the amount of compression or stretching
of the force gauge were measured with the optical vernier, depicted in Figure 5.4. Each row
of vernier ‘teeth’ was offset from the facing row in such a way that the alignment of each
sequential set of opposing teeth represented a displacement of 0.5 µm. In other words,
the resolution of displacement measurement was 0.5 µm which meant that the externally
applied force could be varied with a resolution of approximately 18 µN. The outer teeth of
the vernier were used to determine the absolute displacement of the microprobe, and the
73
AnchorProbe
TIM
Figure 5.4: Optical vernier which employed for displacement measurement.
inner set measured the relative compression or stretching of the force gauge. Both inner
and outer sets of teeth were necessary to calculate the absolute displacement of the TIM.
5.3.2 Method
Characterizing the TIM involved collecting a significant number of force, displace-
ment, current and voltage data points so as to represent the TIM actuation space modeled
in Figure 5.2. This was accomplished by first determining a set of discrete input currents,
based on the predicted TIM force-displacement behavior. After randomizing the current
level run order, each force, displacement, current and voltage data point was acquired by
the following process:
1. The initial TIM displacement (zero) and resistance (zero current) across the TIM
were recorded to note any visible residual stress, deformation, or anomalies in the
TIM device.
2. The specified current as determined by the randomized order was applied to the TIM,
and the initial displacement and voltage were measured.
3. While maintaining the actuation current, external forces were applied via the micro
probe guide. The applied force was varied by moving the micro probe the smallest
visible increment (0.5 µ m) which, in turn, stretched the folded-beam force gauge.
4. At the applied force level, the TIM displacement and voltage were measured and
recorded.
74
5. The externally applied force was removed by moving the micro probe back into its
initial, non-interfering position.
6. The subsequent (randomized) force was applied, and the displacement and voltage
were again measured and recorded.
7. The process of applying the randomized forces and recording the voltage and dis-
placement was repeated until all desired force levels were tested, at which point the
probe was removed, and the initial displacement and voltage were again recorded.
8. To account for the spring resistance of the microprobe guide, the probe was used to
compress the force gauge to the point of zero force (i.e., when the force gauge was
neither stretched nor compressed), at which point the displacement and voltage were
recorded.
9. The probe was removed and the initial TIM displacement and voltage were again
recorded, followed by a pause in testing.
10. The actuation and measurement process was repeated for all randomized current lev-
els.
The user repeatability for displacement measurement and force application was also
assessed. At actuation currents of 10, 16 and 18 mA, the user actuated the TIM and ap-
plied the median force for the given current level, measured the voltage and displacement,
released the TIM, and removed the current. This process was repeated 30 times. The stan-
dard deviations of voltage measurement and externally applied force were then used to
determine a user repeatability value.
As with any thermal and electrical system, it was desired to investigate thermal
drift in the TIM device. To do this, a relatively high current was supplied to the unloaded
TIM, and the voltage was measured at numerous time intervals. The process was repeated
during different times of the day and with multiple actuation currents. In addition, a test
was performed to assess the performance of the source meter by using a standard resistor
as the test device.
75
4.7 4.75 4.8 4.85 4.9 4.95 5 5.05−50
0
50
100
150
200
Voltage, V
For
ce, µ
N
15 mA15.5mA
Figure 5.5: Comparison of force-voltage relationship for neighboring currents.
5.4 Results
Force, displacement and voltage measurements were taken at 21 current levels from
10 to 21 mA at 0.5 mA increments. As was expected, at a constant current level, the
measured voltage across the TIM increased as the externally applied force increased. Fig-
ures 5.5 and 5.6 show the degree to which the voltage changed for two neighboring current
levels, with error bars denoting measured uncertainty in force (4.4 µN) and in displacement
(0.25 µm). Complete characterization data plots are presented in Appendix.
As the plots show, the voltage increased for increasing applied force but decreased
for increasing displacement. This behavior is best understood by considering the force-
displacement plot of the TIM shown in Figure 5.2. For a specific current level, the TIM
always displaces to the location of force equilibrium, which, for an unloaded TIM, is its
maximum displacement at that current level. Applying an external force while maintaining
the same current, therefore, increases the compressive stresses experienced by the thin
76
4.7 4.75 4.8 4.85 4.9 4.95 5 5.051
1.5
2
2.5
3
3.5
4
Voltage, V
Dis
plac
emen
t, µ
m
15 mA15.5mA
Figure 5.6: Comparison of displacement-voltage relationship for neighboring currents.
flexures. The increase in compressive load in the thin flexures induces the increase in
voltage, as has been shown in previous research [96, 97].
5.4.1 Repeatability and Drift
The user repeatability of voltage measurement and force application is provided
in Table 5.2. Since both the calculated voltage and force repeatability were close to the
physical uncertainty inherent in the source meter and the optical vernier, it was determined
that no significant error or uncertainty was generated by the user. It is important to note
that this repeatability represents error in user measurement rather than lack of resolution of
the thermal actuator itself.
Results from two drift assessments performed are plotted in Figure 5.7. As readily
apparent, the resistance of the TIM slowly increased with time in the presence of a high
current (20 mA). A drift test performed at a lower current level, or lower temperature,
exhibited a similar rise in resistance with time, though of a much smaller magnitude.
77
Table 5.2: User repeatability for force application and voltage measurement for three cur-rent levels.
Current [mA] Voltage [mV] Resistance [mΩ] Force [µN]10 1.0 102 2.116 2.6 164 4.418 5.3 292 3.7
0 5 10 15 20 25 30
339.2
339.4
339.6
339.8
340
340.2
340.4
340.6
340.8
Time, min
Res
ista
nce,
Ω
Trial 1Trial 2
Figure 5.7: Resistance drift in ‘unloaded’ TIM at 20 mA.
Although initially it seemed that the rising temperature of the TIM and the sur-
rounding substrate could combine to increase the resistance, it was determined that more
interactions are present in this thermal system. The observed drift in Figure 5.7 could,
for example, be affected by room temperature, humidity, or drift in the sourcemeter (see
Appendix).
5.4.2 Sensitivity
The sensitivity of the TIM as a function of current was calculated as the slope of a
line fit (using the least-squares method) for each constant-current data set, as reported in
Figures 5.8 and 5.9. The results show that, for a change of 1 µN at an actuation current of
16 mA, the change in voltage is 0.6 mV. In terms of fractional change in resistance, such
78
10 11 12 13 14 15 16 17 18 19
0.2
0.4
0.6
0.8
1
Current, mA
Forc
e Se
nsiti
vity
, m
V /
µN
MeasuredMoving Average
Figure 5.8: Force sensitivity as a function of current level.
a change in voltage corresponds to 0.00012, or an increase of 35 mΩ for the TIM whose
nominal resistance was 300 Ω. Similarly, a displacement of 0.1 µm at 16 mA corresponded
to a decrease of 7.5 mV, or a drop in resistance of 373 mΩ.
Figures 5.8 and 5.9 also illustrate how piezoresistive sensitivity increases somewhat
exponentially with increasing current level, or temperature. Although the maximum force
and displacement sensitivity occurs at higher currents, the higher temperature level associ-
ated with larger currents also introduced greater uncertainty in both force and displacement.
This rise in thermal and mechanical noise was observed during testing when the TIM began
to buckle out-of-plane at relatively low force levels (∼100 µN). It was assumed that this
non-ideal buckling behavior was due, in part, to the low out-of-plane aspect ratio and the
high slenderness ratio [85].
5.4.3 Empirical Model
With such small changes in resistance at each current level representing changes
in force and displacement, it was assumed that the change in mechanism temperature, or
79
10 11 12 13 14 15 16 17 18 19−200
−150
−100
−50
0
Current, mA
Dis
plac
emen
t Sen
sitiv
ity,
mV
/ µmMeasuredMoving Average
Figure 5.9: Displacement sensitivity as a function of current level.
power, given by P = I2R, was almost entirely dependent on current and not on resistance.
This meant that each actuation current level represented a relatively constant temperature
level. The non-constant (increasing) force and displacement sensitivity illustrated in Fig-
ures 5.8 and 5.9 confirmed the temperature dependence of the TIM’s piezoresistivity. In
other words, the piezoresistive sensitivity of the TIM, or piezoresistive gauge factor, in-
creased as the mechanism temperature increased.
In light of this observation, various statistical models attempting to capture the
temperature-dependence of piezoresistive coefficients, as described in Chapter 2, were in-
vestigated [12, 48]. The models relied on the empirical data, excluding data at currents
above 19 mA, for which out-of-plane buckling and other thermal and mechanical interac-
tions introduced significant uncertainty in the data.
Several statistical models were explored in an attempt to accurately capture the
temperature-dependent piezoresistive effect of the TIM. These models were created by
fitting a polynomial to the measured data points or using matrix manipulation to solve
for a coefficient matrix. For example, a current-specific—in other words, temperature-
80
specific—statistical model for the TIM was created which to obtain the measured force, F ,
and displacement, D, from the measured voltage and applied current by:
[F D] = Aa (5.1)
where the data matrix, A, was composed of the fractional change in resistance measured at
each point, G:
A =[G G2 G3 G4] (5.2)
The fractional change in resistance, G, was calculated as:
G =R−Rnominal
Rnominal(5.3)
The coefficient matrix, a, was found by solving Equation ((5.1) with the pseudo-inverse for
the non-square A matrix:
a =(AT A
)−1 AT · [F D] (5.4)
This model contained temperature-dependence data in the fractional resistance terms, which
were calculated using Ohm’s law to relate resistance, R, to voltage, V , and current, I, by:
R =VI
First-, second-, third- and fourth-order piezoresistance models were investigated,
with each corresponding coefficient matrix being used to recreate the measured data. In
addition to graphically comparing the modeled and measured data, an R2 comparison was
examined to determine each model’s goodness of fit. Figure 5.10 shows the graphical com-
parison of force as a function of fractional change in resistance for neighboring current
levels. As readily apparent, the force varies quite linearly with fractional change in resis-
tance, suggesting that the piezoresistance model should be linear, or first-order. A similar
trend was observed for displacement as a function of fractional change in resistance.
Figure 5.10 also reveals an interesting phenomenon of the TIM at zero and negative
force. As the externally applied force decreases from positive, to zero, to negative (pulling
81
0.055 0.06 0.065 0.07 0.075 0.08 0.085−50
0
50
100
150
200
Fractional Change in Resistance
For
ce, µ
N
15 mA Model15 mA Measured15.5 mA Model15.5 mA Measured
Figure 5.10: Sample data of force as a function of fractional change in resistance for first-order, temperature-dependent piezoresistance model.
or stretching out) magnitude, the fractional change in resistance decreases more slowly. It
is suspected that this behavior, noted by the change in slope at the zero-force line, is caused
by the stress stiffening resulting from the TIM’s being pulled beyond its force-equilibrium
displacement point. A similar trend was observed in displacement, where, as the applied
force became increasingly negative, the slope of the displacement to fractional change in
resistance decreased abruptly.
Additional piezoresistance models were considered, including voltage- and frac-
tional change in voltage-based models, power models and combination models. One such
model incorporated force, power (I2R) and their interaction in solving for the fractional
change in resistance, G:
G = a1F +a2P+a3F ·P (5.5)
82
The generated model (coefficients) was then manipulated and used to predict the force:
F =G
a1 +a3P− a2P
a1 +a3P(5.6)
This process was followed to create a similar displacement model as well. Although this
model embodied a strong influence of temperature, it failed to accurately predict force and
displacement of the test TIM at multiple current levels. Refer to Appendix A for a plot of
power as a function of force and displacement.
5.5 Application
In order to validate the characterization process and thermal piezoresistance model,
the TIM was used to actuate and measure the force and displacement of the Self-Retracting
Fully-compliant Bistable Micromechanism (SRFBM, Figure 5.11) [98]. A TIM was placed
at the back (left) of the SRFBM shuttle to push it into its second stable-equilibrium position,
at which point another TIM, placed in front (right) of the SRBM, was used to measure the
force required to return the device to its initial stable-equilibrium position. After the TIM
nominal resistance was measured, the applied current was increased incrementally, with
the voltage across the TIM being recorded at each current step.
The finite element analysis presented in Masters’ work predicted the SRFBM’s
maximum return force to be 132 µN at approximately 3 – 4 µm from its second stable-
equilibrium position [99]. Masters reported a measured return force of 240 µN, which
was well above the predicted value, possibly due to friction. These predicted and measured
maximum return forces were used as a benchmark for validating the temperature-dependent
piezoresistance model prediction of the SRBM behavior.
Using the first-order piezoresistive model, the measured fractional change in resis-
tance of the TIM actuating the SRFBM was converted into force and displacement for the
SRFBM’s return to its initial stable-equilibrium position, as plotted in Figure 5.12.
From Figure 5.12, it appears that the predicted data points are shifted to higher
forces and and lower displacements, as compared to the predicted and measured data. Al-
though some of the discrepancy in the data could be attributed to the initial separation of
83
Figure 5.11: Schematic of Self-Retracting Fully-compliant Bistable Micromechanism(SRFBM).
−6 −5 −4 −3 −2 −1 0 1 2 3 4250
300
350
400
450
500
550
600
Displacement, µm
For
ce, µ
N
Figure 5.12: Force-displacement curve for SRFBM return to the initial stable-equilibriumposition, calculated by first-order piezoresistance model.
84
the TIM and SRFBM and the rough engagement of the two devices, a significant portion of
this shift, or error, could have resulted from the difference in the resistivity, grain structure
and geometry of the SRFBM TIM, which was rotated 90°with respect to the TIM used in
the characterization process.
While the test TIM possessed a nominal resistance of 299 Ω, the nominal SRFBM
TIM resistance was 327 Ω, a difference of nearly 10%. With a higher nominal resistance,
the measured voltage of the SRFBM TIM at a given current level was nearly 10% higher
than that of the characterized TIM, corresponding to a higher force and lower displacement
for a given current level, as illustrated previously in Figures 5.5 and 5.6. This effect can
also be thought of in terms of a 10% increase in TIM temperature at each current level,
which corresponds to a greater level of thermal expansion in the thin flexures.
5.6 Need for Calibration
As evidenced by the models’ inability to predict the force-displacement curve for
the SRFBM, a method of calibrating each TIM device is needed. One promising calibration
possibility utilizes the Piezoresistive Microdisplacement Transducer (PMT), illustrated in
Figure 5.13, which could provide a straightforward calibration procedure for each TIM
device.
With its excellent displacement resolution (∼30 nm), the well-documented PMT
would measure the TIM displacement, in the form of an electrical signal, as the TIM
‘stepped along’ the force-displacement curve of the PMT, depicted in Figure 5.14. At
each actuation current level, the voltage across the TIM would be measured and the TIM
force would be calculated using the TIM displacement and a simple finite element analy-
sis of the PMT. The force, displacement, current and voltage data points acquired in this
manner would serve as a calibration standard for the TIM and would obviate the need for
complicated and tedious optical, laser or other measurement methods.
One challenge of using the PMT for calibration pertains to the possible need of
calibrating the PMT itself. Calibrating a calibration device would be time consuming,
could introduce additional uncertainty in the calibration process, and would hinder the TIM
performance even more. Future work in characterizing and modeling the TIM, therefore,
85
Figure 5.13: Schematic of the Piezoresistive Microdisplacement Transducer (PMT) [31,82].
0 2 4 6 8 10 12 14 160
50
100
150
200
250
300
350
400
450
Displacement, µm
Forc
e, µ
N
PMT
Figure 5.14: Calibration curve for TIM-PMT structure.
86
should focus on validating the effectiveness of the PMT or exploring other calibration or
characterization methods.
5.7 Conclusions
This research has shown that the Thermomechanical In-plane Microactuator was
successfully characterized in terms of force, displacement, current and voltage. Based on
the characterization data, an empirical first-order, temperature-dependent piezoresistance
model was developed which predicted the TIM force-displacement pairs. Insights were
presented concerning the piezoresistive effect and the possibility of implementing a fully-
characterized (or generally modeled) thermal actuator as both a sensor and actuator in a
feedback circuit.
As shown, certain empirical models succeeded in predicting force and displace-
ment for the test TIM and yet were unable to accurately predict the behavior of other TIM
devices. From this, it was asserted that, though the TIM was sufficiently characterized to
be modeled, further work must be performed to calibrate this and other TIM devices of
interest.
In addition to examining calibration methods, great insights into the the depen-
dence of piezoresistance on temperature and stress orientation could be gained by further
research. Such efforts could focus on the development of a multi-physics model of the TIM
force, displacement, current (or temperature) and voltage (or resistance) which could then
be applied generally to other TIM devices or thermal actuator designs. With a calibrated
TIM and/or general piezoresistive thermal-actuator model, numerous feedback-control ap-
plications could be exploited.
87
88
Chapter 6
Conclusions and Recommendations
The purpose of this thesis research was to provide a review of current theories and
models of the piezoresistance effect of silicon, provide preliminary piezoresistance charac-
terization data for polysilicon devices in tension and combined loads, investigate the design
of an integral piezoresistive force and displacement sensor, and characterize the Thermo-
mechanical In-plane Microactuator (TIM). This chapter presents several key points drawn
from this research and proposes recommendations for future work in these areas.
6.1 Conclusions
Piezoresistivity, the change in electrical resistance due to applied stress or strain,
was demonstrated to have many applications in silicon-based MEMS sensors and actuators.
In addition, preliminary efforts to characterize piezoresistance for bending and combined
loads revealed that existing, linear piezoresistive models fail to predict the piezoresistive
effect for bending and combined loads. The piezoresistance data gathered in Chapter 3
suggest that bending loads cause a non-linear increase in resistance, while tensile stresses
cause a fairly linear drop in resistance. Characterization data also illustrated the superposi-
tion characteristic of piezoresistance for combined loads.
Many challenges and considerations related to the design of an integral piezore-
sistive MEMS sensor were discussed in Chapters 3 and 4. These design guidelines were
applied to the design of two piezoresistive MEMS force and displacement sensors: the
FADS and the CLOO-FADS. These force and displacement designs showed moderate pre-
liminary force sensitivity and out-of-plane stability and require future testing to validate the
predicted sensor performance and stability and provide insight into future design iterations.
89
In addition to testing and improving the current design and implementation of the
FADS and CLOO-FADS devices, future research could focus on creating similar FADS
and CLOO-FADS sensors as single layer, monocrystalline structures. This could permit
the use of MEMS fabrication processes besides the MUMPs and SUMMiT processes. Fur-
ther, as was discussed in Chapter 2, the higher piezoresistive sensitivity of monocrystalline
silicon compared to polycrystalline silicon could provide the additional sensitivity the cur-
rent polysilicon designs lack. Using monocrystalline devices would necessitate additional
testing of their anisotropic mechanical and electrical properties.
This research has shown that the Thermomechanical In-plane Microactuator was
successfully characterized in terms of force, displacement, current and voltage. Based on
the characterization data, an empirical first-order, temperature-dependent piezoresistance
model was developed which predicted the TIM force-displacement pairs. The empirical
model was unable to accurately predict the force-displacement curve of a different TIM
used to study the forward- and return-actuation forces and displacements of the SRFBM.
An accurate method of calibration for each TIM is needed.
6.2 Recommendations
With the comprehensive review of piezoresistance effect of silicon, the preliminary
characterization of the effect in polysilicon, and the investigation into integral piezoresistive
sensors, improved exploitation of the piezoresistive property of silicon into MEMS force
and displacement sensors is possible. In addition, the results of the characterization of
the Thermomechanical In-plane Microactuator provide the groundwork for future multi-
physics modeling of the TIM and other thermal actuators. Advancing the research in these
areas will be best achieved by addressing the items described below.
6.2.1 Piezoresistance of Monocrystalline Silicon
Chapter 3 reported characterization results for several SUMMiT- and MUMPs-
fabricated polycrystalline devices. To more fully characterize the piezoresistive effect in
silicon, piezoresistive testing on monocrystalline-silicon devices is necessary. Such char-
acterization data would broaden the application of the conclusions drawn on the piezore-
90
sistance effect in bending, tension and combined loads. Further, designing monocrystalline
piezoresistive test structures and integral piezoresistive sensors could benefit from faster
turnaround and additional design possibilities, since many commercial and in-house fabri-
cation processes are available. Use of monocrystalline silicon in these applications would
require additional testing to characterize anisotropic effects.
6.2.2 Optimization of Piezoresistive Sensors
With an improved model of piezoresistance, future integral piezoresistive force and
displacement sensors may benefit from design optimization. For example, the geometry
of the Piezoresistive Microdisplacement Transducer (PMT) sensor could be optimized to
improve sensitivity or linearity using any one of several optimization algorithms. A more
feasible FADS device may be conceived with the assistance of design optimization.
Also, the variation of material properties and exact geometry among devices of dif-
ferent fabrication runs and even among devices located across the same silicon wafer ne-
cessitates the use of robust design optimization and/or easy piezoresistive calibration [100].
The statistical techniques for robust design optimization investigated by Wittwer [61, 85]
could be used to create piezoresistive mechanisms with the lowest possible variation or
uncertainty in geometry for the MUMPS or SUMMiT process.
6.2.3 Calibration Method for TIM
As was demonstrated in Chapter 5, one TIM structure was successfully character-
ized, from which an empirical model was derived. The inability of that model to predict
the TIM used to actuate the SRFBM demonstrated the need for a method of calibrating
each TIM device before the temperature-based piezoresistance model can be applied to
all identical but separate TIM devices. The Piezoresistive Microdisplacement Transducer
(PMT) may provide a viable calibration method. Nevertheless, other displacement- or
force-sensing structures should be explored as possible calibration methods.
91
6.2.4 Multi-Physics Model of TIM
With additional experimentation and computer-aided modeling, a complete electro-
thermomechanical model of the TIM could be created. The multi-physics model could
integrate temperature distribution models of the TIM to investigate and characterize the
temperature-dependence of piezoresistivity in polysilicon. In addition, future experimenta-
tion and characterization of multiple TIM designs might lead to an expanded, or parametric,
TIM model capable of predicting temperature-based force and deflection behavior of any
TIM. Such a model could then be used to optimize TIM design to meet specific force, dis-
placement and power needs. Also, with a calibrated TIM or parametric temperature-based
piezoresistive model, numerous feedback-control applications could be exploited.
92
Appendix A
TIM Characterization Data
This Appendix contains additional plots of the data gathered to characterize the
piezoresistive properties of the TIM, as described in Chapter 5. Figures A.1, A.2 and A.3
present complete TIM characterization data of force and displacement as a function of
measured voltage. Several piezoresistance models involving power were investigated in
Chapter 5. TIM power, P = I2R, is plotted as a function of both force and displacement in
Figure A.4. Although a piezoresistance model based on power provided information about
the temperature dependence of piezoresistance, it was unable to completely predict TIM
force and displacement.
Finally, in Figure A.5, the user repeatability inherent in the characterization process
is shown, and, in Figure A.6, the measured sourcemeter drift is plotted.
93
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5−50
0
50
100
150
200
250
300
Displacement, µ m
For
ce, µ
N
10 mA10.5 mA11 mA11.5 mA12 mA12.5 mA13 mA13.5 mA14 mA14.5 mA15 mA15.5 mA16 mA16.5 mA17 mA17.5 mA18 mA18.5 mA19 mA19.5 mA20 mA20.5 mA21 mA
Figure A.1: Complete force and displacement data for TIM characterization.
94
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5−50
0
50
100
150
200
250
300
For
ce, µ
N
TIM Voltage, V
10 mA10.5 mA11 mA11.5 mA12 mA12.5 mA13 mA13.5 mA14 mA14.5 mA15 mA15.5 mA16 mA16.5 mA17 mA17.5 mA18 mA18.5 mA19 mA19.5 mA20 mA20.5 mA21 mA
Figure A.2: Complete data of force as a function of measured voltage for TIM.
95
3 3.5 4 4.5 5 5.5 6 6.5 7 7.50
1
2
3
4
5
6
Dis
plac
emen
t, µ
m
TIM Voltage, V
10 mA10.5 mA11 mA11.5 mA12 mA12.5 mA13 mA13.5 mA14 mA14.5 mA15 mA15.5 mA16 mA16.5 mA17 mA17.5 mA18 mA18.5 mA19 mA19.5 mA20 mA20.5 mA21 mA
Figure A.3: Complete data of displacement as a function of measured voltage for TIM.
96
Figure A.4: Surface plot of power as a function of force and displacement for TIM.
0 5 10 15 20 25 30−0.05
0
0.05
0.1
0.15
0.2
0.25
Measurement
Fra
ctio
nal C
hang
e in
Res
ista
nce
(%)
10 mA18 mA
Figure A.5: User repeatability.
97
0 5 10 15 20 25 306.494
6.496
6.498
6.5
6.502
6.504
6.506
6.508
6.51
6.512
Time, min
Vol
tage
, V
MeasuredQuadratic Fit
Figure A.6: Thermal drift for the Keithley Sourcemeter, measured using a standard 300 Ω
resistor.
98
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