EXPERIMENTAL CHARACTERIZATION OF THE TEMPERATURE
DEPENDENCE OF THE PIEZORESISTIVE
COEFFICIENTS OF SILICON
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory
committee. This dissertation does not include proprietary or classified information.
________________________________
Chun Hyung Cho
Certificate of Approval:
________________________________
Richard C. Jaeger, Co-Chair
Distinguished University Professor
Electrical and Computer Engineering
________________________________
Bogdan Wilamowski
Professor
Electrical and Computer Engineering
________________________________
Jeffrey C. Suhling, Co-Chair
Quina Distinguished Professor
Mechanical Engineering
________________________________
Greg Harris
Associate Professor
Mathematics and Statistics
________________________________
Joe F. Pittman
Interim Dean
Graduate School
EXPERIMENTAL CHARACTERIZATION OF THE TEMPERATURE
DEPENDENCE OF THE PIEZORESISTIVE
COEFFICIENTS OF SILICON
Chun Hyung Cho
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirement for the
Degree of
Doctor of Philosophy
Auburn, Alabama
August 04, 2007
iii
EXPERIMENTAL CHARACTERIZATION OF THE TEMPERATURE
DEPENDENCE OF THE PIEZORESISTIVE
COEFFICIENTS OF SILICON
Chun Hyung Cho
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon request of individuals or institutions and their expense.
The author reserves all publication rights.
Signature of Author
Date of Graduation
iv
VITA
Chun Hyung Cho, son of Nam Jin Cho and Sung Dong Hong, was born on
October 27, 1970 in Daejeon, Korea. He graduated from North Daejeon High School,
Daejeon, Korea in 1989, and from Seoul National University, Seoul, Korea in 1997 with
the degree of Bachelor of Science in Electrical Engineering. He started his M.S. Program
in the Electrical Engineering Department, Auburn University, Alabama, in 1998. He
received his M.S. in Electrical Engineering in 2001. He continued his Ph.D. program in
Auburn University from 2001. He married Eun Ok Jeong on December 29, 2001. On
September 30, 2002, their first son, Sung Am Cho, was born and on September 12, 2004,
their second son, Sung Mok Cho was born.
v
DISSERTATION ABSTRACT
EXPERIMENTAL CHARACTERIZATION OF THE TEMPERATURE
DEPENDENCE OF THE PIEZORESISTIVE
COEFFICIENTS OF SILICON
Chun Hyung Cho
Doctor of philosophy, August 04, 2007
(Master of Science, Auburn University, 2001)
(B.S. Electrical Engineering, Seoul National University, South Korea, 1997)
321 Typed Pages
Directed by Richard C. Jaeger and Jeffrey C. Suhling
In this work, the dependence of the silicon piezoresistive coefficients 11? , 12? ,
and 44? on temperature is investigated. Experimental calibration results for the
piezoresistive coefficients of silicon as a function of temperature are presented and
compared and contrasted with existing values from the literature.
Stress-sensing test chips are widely used to investigate die stresses occurring to
assembly and packaging operations. They incorporate resistor- or transistor-sensing
elements that are able to measure stresses through the observation of the changes in their
vi
resistivity/mobility. The piezoresistive behavior of such sensors can be completely
characterized through the use of three piezoresistive (pi) coefficients, which are electro-
mechanical material constants. In most prior investigations, calibration of the
piezoresistive coefficients has been performed at room temperature. Such restriction
limits the accuracy of test chip stress measurements made at other temperatures. In this
work, we have performed an extensive experimental study on temperature dependence of
the piezoresistive behavior of silicon. Calibration has been performed using four-point
bending of chip-on-beam specimens. A special four-point bending apparatus has been
constructed and integrated into an environmental chamber capable of temperatures from
-185 to 300oC. Finite element analysis has been used to calculate the stress states
applied to the calibration samples. Our test results show that the piezoresistive
coefficients for p- and n-type silicon decrease monotonically when temperature is
increased from -150 to 125oC.
Our goals in this work are to enable packaging stress measurements over a wide
range of temperature, to obtain a comprehensive set of piezoresistive coefficients over a
broad range of temperature, to resolve sign issues and demonstrate proper methods for
relating coefficients at different temperatures, and to obtain consistent formulation of
T) ,(R ? valid over wide temperature range.
vii
ACKNOWLEDGMENTS
I would like to express my respects and thanks to my co-advisor Dr. Richard C.
Jaeger and Dr. Jeffrey C. Suhling for their directions, patience, and encouragement.
Without their invaluable advices, this dissertation would not be finished. Thanks are also
due to Dr. Bogdan Wilamowski, and Dr. Greg Harris for their help and advice. They gave
me a challenging subject to solve.
In addition, I would like to thank Charles D. Ellis for his help in preparing the test
chips, and John Marcel for his efforts in designing the expanded hydrostatic test setup
required to complete this study. I would like to thank my colleague Md. S. Rahim for
extensive help on research. I also wish to acknowledge the financial support of the
Center for Advanced Vehicle Electronics (CAVE) and the Alabama Microelectronics
Science and Technology Center (AMSTC).
I would like to express special thanks to my parents, Nam Jin Cho and Sung Dong
Hong, for their belief in me without question. The memory that my father studied still
lives in my mind and gives me strength to finish this long journey. Also, I would like to
thank my wife, En Ok Jung. She takes care of the children while I stay in America. I
would like to express my love to my sons, Sung Am Cho and Sung Mok Cho. They have
grown strong even though I have not spent much time with them. Many thanks go to my
sisters for their love and support as a family.
viii
Style manual or journal used Guide to Preparation and Submission of Theses and
Dissertations
Computer software used Microsoft Word 2002 and Microsoft Excel 2002
ix
TABLE OF CONTENTS
LIST OF FIGURES ........................................................................................................xii
LIST OF TABLES......................................................................................................... xix
1 INTRODUCTION ................................................................................................ 1
2 LITERATURE REVIEW ..................................................................................... 7
3 REVIEW OF PIEZORESISTIVE THEORY ..................................................... 20
3.1 Resistance Change Equations for the (001) Silicon Wafer Planes
3.2 Resistance Change Equations for the (111) Silicon Wafer Planes
4 CALIBRATION OF SENSORS ON THE (111) SURFACE ............................. 27
4.1 Experimental Setup
4.1.1 Four-Point Bending Apparatus
4.1.2 The (111) Silicon Test Chips
4.2 Sensor Calibration for the (111) Silicon Test Chips
4.3 Simulation Results for the (111) Silicon Test Chips
4.4 Extraction of Piezoresistive Coefficients, B1 and B2
4.5 Relationship between Piezoresistive Coefficients with Different
Temperatures
4.5.1 General Resistance Change Equations at a fixed Temperature
Reference
4.5.2 General Resistance Change Equations with Varying Temperatures
4.6 Summary
5 HYDROSTATIC TESTS AND TCR MEASUREMENTS ............................... 59
5.1 Hydrostatic Tests
5.2 TCR Measurements and f(?T)
5.3 Analysis of Hydrostatic Tests and TCR Measurements
5.4 Summary
6 SILICON STRESS-STRAIN RELATIONS AND MEASUREMENT OF
YOUNG?S MODULUD OF SILICON .............................................................. 92
6.1 Silicon Stress-Strain Relations
6.2 Elastic Constants of Silicon by Equations
6.3 Measurement of the Elastic Constants by Deflection of Beams
x
6.4 Summary
7 VAN DER PAUW STRUCTURE.................................................................... 114
7.1 Van der Pauw?s Theorem
7.2 Experimental Results for the (111) Silicon
7.3 Sensitivity Magnification Factor and ?
7.4 Effects of Dimensional Changes of VDP and Resistor during Loading
7.4.1 Strain-effects of VDP Structures
7.4.2 Strain-effects of Resistor Sensors
7.5 Summary
8 TRANSVERSE STRESS ANALYSIS AND ERRORS ASSOCIATED
WITH MISALIGNMENT ................................................................................ 145
8.1 Transverse Stress Analysis
8.1.1 Resistor Sensors on the (111) Silicon Surface
8.1.2 Resistor Sensors on the (001) Silicon Surface
8.2 Off-Axis Error on the (001) Silicon Plane
8.3 Off-Axis Error on the (111) Silicon Plane
8.4 Summary
9 (001) TEST CHIP DESIGN AND CALIBRATION........................................ 181
9.1 Mask Alignment Using Wet Anisotropic Etching
9.2 The (001) Silicon Test Chips
9.3 Resistance Equations for the (001) Silicon
9.4 Strip-on-beam Test Samples
9.5 Simulation Results for the (001) Silicon Test Chips
9.6 Sensor Calibration for the (001) Silicon Test Chips
9.7 Summary
10 SUMMARY AND CONCLUSIONS ............................................................... 218
BIBLIOGRAPHY......................................................................................................... 222
APPENDICES .............................................................................................................. 232
A TYPICAL RESULTS OF S0 and S90 FOR THE (111) SILICON
AT DIFFERENT TEMPERATURES .............................................................. 233
B DETERMINATION OF PIEZORESISTIVE COEFFICIENTS....................... 242
C TYPICAL RESULTS FOR THE PRESSURE COEFFICIENT OF (111)
SILICON AT DIFFERENT TEMPERATURES.............................................. 246
D BEAM ROTATIONAL ERROR...................................................................... 250
E TYPICAL RESULTS FOR S0, S90, S45, AND S-45 FOR THE
xi
(001) SILICON VERSUS TEMPERATURE................................................... 257
F THE EFFECTS OF ERRORS ASSOCIATED WITH INITIAL RESISTANCE
ON THE DETERMINATION OF PI-COEFFICIENTS .................................. 269
G THE COMPARISONS OF PI-COEFFICIENTS BETWEEN STRIPS AND
DOUBLE-SIDED SILICON STRIP-ON-BEAM SAMPLES.......................... 281
H DETERMINATION OF THE STIFFNESS COEFFICIENT MATRIX
FOR THE UNPRIMED/PRIMED COORDINATE SYSTEM ........................ 292
I THE PROFILES OF CARRIER CONCENTRATION VERSUS DEPTH
IN SILICON...................................................................................................... 294
xii
LIST OF FIGURES
1.1 Basic concept of piezoresistive sensor............................................................... 2
2.1 Composite pi44 data collected from the literature for p-type silicon
as a function of impurity concentration at room temperature........................... 13
2.2 Literature data for pi11 versus concentration for p-type silicon at room
temperature ....................................................................................................... 13
2.3 Literature data for pi12 versus concentration for p-type silicon at room
temperature ....................................................................................................... 14
2.4 Literature data for pi11 versus concentration for n-type silicon at room
temperature ....................................................................................................... 16
2.5 Literature data for pi12 and pi44 versus concentration for n-type silicon
at room temperature .......................................................................................... 16
2.6 Calculated literature data for pi44 versus temperature with different
doping concentration for p-type silicon ............................................................ 18
2.7 Experimental literature data for pi44 versus temperature with different
doping concentration for p-type silicon ............................................................ 18
2.8 Experimental literature data for pi11 versus temperature with different
doping concentration for n-type silicon ............................................................ 19
3.1 An arbitrary oriented, filamentary conductor.........................................................20
xiii
3.2 General (001) silicon wafer ............................................................................... 22
3.3 General (111) Silicon Wafer.............................................................................. 24
4.1 Four-point bending loading fixture.................................................................... 30
4.2 The details about the parts of four-point bending fixture .................................. 31
4.3 A four-point bending fixture mounted inside the oven...................................... 32
4.4 A four-point bending fixture: Exterior of the oven............................................ 32
4.5(a) The (111) silicon test chip (JSE-WB100C)....................................................... 36
4.5(b) Microphotograph of Eight-Element Sensor Rosette ......................................... 36
4.6 A specially designed printed wiring board ....................................................... 36
4.7 A wire-bonded chip-on-beam structure (central part) ...................................... 37
4.8 A wire-bonded chip-on-beam structure ............................................................ 37
4.9 Mesh plot of chip-on-beam structure................................................................ 44
4.10 Mesh plot of silicon chip................................................................................... 44
4.11 The relative size and location of the sensor on the mesh plot .......................... 45
4.12 Contour plot of '11? at 25oC............................................................................... 45
4.13 Contour plot of '22? at 25oC.............................................................................. 46
4.14 Extracted B1 and B2 with temperature .............................................................. 50
4.15 The plot of resistance with temperature and stress ........................................... 54
5.1 Plot of p-type resistance with varying temperatures and forces ....................... 61
5.2 Plot of n-type resistance with varying temperatures and forces ....................... 62
5.3 Plot of ?R of a resistor p-type with varying temperatures and forces.............. 64
5.4 Plot of ?R of an n-type resistor with varying temperatures and forces ............ 65
xiv
5.5 Resistance of p-type sensors with varying temperatures for ? = 0 and ? = 90o
........................................................................................................................... 67
5.6 Resistance of n-type sensors with varying temperatures for ? = 0 and ? = 90o
.......................................................................................................................... 67
5.7 P-type resistance change with varying temperatures........................................ 68
5.8 N-type resistance change with varying temperatures ....................................... 68
5.9 f(?T) of p-type sensors with varying temperatures .......................................... 70
5.10 f(?T) of n-type sensors with varying temperatures .......................................... 70
5.11 Quarter model of JSE-WB100C for TCR and hydrostatic tests ........................ 75
5.12 Specially designed PCB for TCR and hydrostatic tests..................................... 75
5.13 Wire-bonded chip on the board for TCR and hydrostatic tests ......................... 76
5.14 Hydrostatic test chamber.................................................................................... 77
5.15 Hydrostatic test setup......................................................................................... 78
5.16 Expanded hydrostatic test setup for high and low temperatures........................ 78
5.17 An example of measured and temperature induced ?R/R for p-type resistors.. 79
5.18 Fluid temperature change with pressure for p-type resistors............................. 80
5.19 Adjusted hydrostatic calibration for p-type sensors........................................... 81
5.20 Pressure coefficient versus temperature for p-type sensors............................... 82
5.21 Pressure coefficient versus temperature for n-type sensors............................... 83
5.22 Combined pi-coefficients of p-type silicon versus temperature with neglect of ?
........................................................................................................................... 85
5.23 Pi-coefficients of p-type silicon versus temperature with neglect of ?............. 85
xv
5.24 Combined pi-coefficients of n-type silicon versus temperature with neglect of ?
........................................................................................................................... 86
5.25 Pi-coefficients of n-type silicon versus temperature with neglect of ?............... 87
5.26 Combined pi-coefficients of p-type silicon versus temperature
with consideration of ?........................................................................................ 88
5.27 Pi-coefficients of p-type silicon versus temperature with consideration of ?..... 89
5.28 Combined pi-coefficients of n-type silicon versus temperature
with consideration of ?........................................................................................ 90
5.29 Pi-coefficients of n-type silicon versus temperature with consideration of ?..... 90
6.1 Silicon wafer geometry....................................................................................... 96
6.2 E on the (001) silicon.......................................................................................... 97
6.3 E on the (001) silicon.......................................................................................... 98
6.4 ? on the (001) silicon .......................................................................................... 98
6.5 ? on the (001) silicon .......................................................................................... 99
6.6 E on the (111) silicon........................................................................................ 100
6.7 E on the (111) silicon........................................................................................ 101
6.8 ? on the (111) silicon ........................................................................................ 101
6.9 ? on the (111) silicon ........................................................................................ 102
6.10 E on the 0)1(1 silicon......................................................................................... 104
6.11 ? on the 0)1(1 silicon ........................................................................................ 104
6.12 Deflection of a beam in a four-point bending fixture ....................................... 106
6.13 Plot of 'F with respect to ? ............................................................................... 107
xvi
6.14 Plot of E for the ]211[ direction on the (111) silicon versus temperature ......... 109
6.15 Plot of E for FR-406 versus temperature .......................................................... 110
6.16 Plot of E for ME525 versus temperature .......................................................... 111
7.1 A flat sample of conducting material with uniform thickness............................. 114
7.2 A simple van der Pauw test structure................................................................. 115
7.3 Isotropic rectangular VDP structure .................................................................. 117
7.4 The (111) Silicon test chip, BMW-2.1.............................................................. 120
7.5 Typical stress sensitivity of p-type resistor sensors (R0) .................................... 122
7.6 Typical stress sensitivity of p-type resistor sensors (R90)................................... 122
7.7 Typical stress sensitivity of n-type resistor sensors (R0) .................................... 123
7.8 Typical stress sensitivity of n-type resistor sensors (R90)................................... 123
7.9 Typical stress sensitivity of p-type VDP sensors (R0).......................................... 126
7.10 Typical stress sensitivity of p-type VDP sensors (R90) ........................................ 127
7.11 Typical stress sensitivity of n-type VDP sensors (R0).......................................... 127
7.12 Typical stress sensitivity of n-type VDP sensors (R90) ........................................ 128
7.13 Isotropic rectangular VDP structure under uniaxial stress................................... 132
7.14 The plot of ?0 and ?90 at various stress levels .................................................... 139
8.1 Normalized % error in pi11? versus ?................................................................. 159
8.2 Normalized % error in pi12? versus ?................................................................. 160
8.3 Normalized % error in ?R45/R45 versus ?......................................................... 161
8.4 Normalized % error in ?R-45/R-45 versus ? ....................................................... 161
8.5 Normalized % error in pi44? versus ?................................................................. 166
xvii
8.6 Normalized % error in ?R0/R0 versus ?............................................................ 170
8.7 Normalized % error in ?R90/R90 versus ?......................................................... 170
8.8 Normalized % error in ?R45/R45 versus ?......................................................... 171
8.9 Normalized % error in ?R-45/R-45 versus ? ....................................................... 172
9.1 The alignment forks of both sides on silicon surface ....................................... 182
9.2 Alignment marks for subsequent masks ........................................................... 182
9.3 An example of an etched structure of alignment forks in one wafer ................ 185
9.4(a) The test chip on the (001) silicon surfaces ........................................................... 186
9.4(b) Microphotograph of the test chip on the (001) silicon surfaces.......................... 186
9.5 I-V characteristics of a p-type resistor after annealing..................................... 189
9.6 I-V characteristics of an n-type resistor after annealing................................... 189
9.7 The [100] and [110] strip-on-beam specimens ................................................... 190
9.8 Two directions cut from the (001) silicon wafer.............................................. 191
9.9 The obvious warp of a single-sided silicon strip-on-beam sample after
cooling from 150oC to room temperature ....................................................... 196
9.10 The almost warp-free double-sided silicon strip-on-beam sample after
cooling from 150oC to room temperature ....................................................... 196
9.11 Mesh plots of the [100] silicon strip-on-beam sample (quarter model)........... 197
9.12 Mesh plots of the [100] silicon strip (central part) .......................................... 198
9.13 Mesh plots of the [110] silicon strip-on-beam sample (quarter model)........... 198
9.14 Mesh plots of the [110] silicon strip (central part) .......................................... 199
9.15 Contour plot of ?11 on [100] silicon strip-on-beam at 25oC ............................ 200
xviii
9.16 Contour plot of ?22 on [100] silicon strip-on-beam at 25oC ............................ 200
9.17 Contour plot of ?'11 on [110] silicon strip-on-beam at 25oC............................ 201
9.18 Contour plot of ?'22 on [110] silicon strip-on-beam at 25oC............................ 201
9.19 pi44 for the (001) p-type silicon with temperature ............................................ 208
9.20 pi11 and pi12 for the (001) p-type silicon with temperature................................ 208
9.21 pi11 and pi12 for the (001) n-type silicon with temperature................................ 209
9.22 pi44 for the (001) n-type silicon with temperature ............................................ 209
9.23 pi11/pi12 for the (001) n-type silicon with temperature....................................... 210
9.24(a) piS for the (001) p-type silicon with temperature ............................................. 212
9.24(b) piS for the (001) p-type silicon with temperature. Fit to the average values
from Fig. 9.24(a)............................................................................................. 212
9.25(a) piS for the (001) n-type silicon with temperature ............................................ 213
9.25(b) piS for the (001) n-type silicon with temperature. Fit to the average values
from Fig. 9.25(a)............................................................................................. 213
9.26 Experimental data for pi44 versus temperature with different doping
concentration for p-type silicon .................................................................... 215
9.27 Experimental data for pi11 versus temperature with different doping
concentration for n-type silicon .................................................................... 215
xix
LIST OF TABLES
2.1 Composite data for pi11, pi12, and pi44 collected from the literature for p-type
silicon at room temperature (TPa)-1 .................................................................... 12
2.2 Literature data for pi11versus concentration for n-type silicon at room
temperature (TPa)-1 ............................................................................................. 15
2.3 Composite data for pi11, pi12, and pi44 collected from the literature for n-type
silicon as a function of impurity concentration at room temperature (TPa)-1..... 15
2.4 Composite data for pi44 collected from the literature for p-type silicon versus
temperature with different doping concentration (TPa)-1 ................................... 17
2.5 Composite data for pi11 collected from the literature for n-type silicon versus
temperature with different doping concentration (TPa)-1 ................................... 19
4.1 Dimensions of composite materials (Unit: mil).................................................. 38
4.2 S0 and S90 versus temperatures (Unit: N-1).............................................................. 40
4.3 Comparison between S0-S90 and A (= slope of R90/R0 versus F)
by measurements ......................................................................................................42
4.4 Elastic modulus of composite materials versus temperature (Unit: GPa) .......... 43
4.5 Simulation results of stresses around the sensor location at 25oC (Unit: GPa) . 47
4.6 ?'11 and ?'22 at the sensor location versus temperature (Unit: MPa) ................. 55
xx
4.7 ?'11 and ?'22 at the sensor location at 25oC (Unit: MPa)...................................... 55
4.8 Extracted B1 and B2 with temperature (Unit: TPa-1)........................................... 57
5.1 P-type resistance with varying temperatures and forces (Unit: kohm)............... 61
5.2 N-type resistance with varying temperatures and forces (Unit: kohm) .............. 62
5.3 ?R for a p-type with varying temperatures and forces (Unit: kohm) ................ 63
5.4 ?R for an n-type with varying temperatures and forces (Unit: kohm) ............... 64
5.5 Resistance with varying temperatures (Unit: kohm) .......................................... 66
5.6 Temperature coefficients of resistance with varying temperatures
(p-type resistors) ................................................................................................. 71
5.7 Temperature coefficients of resistance with varying temperatures
(n-type resistors) ................................................................................................. 71
5.8 Average of 1? of 32 specimens for p- and n-type sensors measured at a given
reference temperature (Unit: 10-3/oC) ................................................................. 72
5.9 Average of 1? (Unit: 10-3/oC) and 2? (Unit: 10-3/oC2) of 32 specimens for
p-type sensors measured at a given reference temperature................................. 72
5.10 Piezoresistive coefficients with room-temperature reference............................. 86
5.11 Piezoresistive coefficients with individual-temperature reference ..................... 86
5.12 Pressure coefficient data of p- and n-type versus temperature ........................... 82
5.13 Pi-coefficients of p-type silicon versus temperature with neglect of ?............... 84
5.14 Pi-coefficients of n-type silicon versus temperature with neglect of ?............... 86
5.15 Pi-coefficients of p-type silicon versus temperature with consideration of ?..... 88
5.16 Pi-coefficients of n-type silicon versus temperature with consideration of ?..... 89
xxi
6.1 Literature values for the stiffness coefficients of silicon [90] ............................ 94
6.2 E values for different directions and different authors (Unit: GPa).................. 105
6.3 Example: Measurement of E using deflection of beams .................................. 107
6.4 Measurement of E for several directions of silicon (Unit: GPa) ...................... 108
6.5 E for the ]211[ direction on the (111) silicon versus temperature (Unit: GPa) .. 109
6.6 E for FR-406 versus temperature (Unit: GPa) .................................................. 110
6.7 E for ME525 versus temperature (Unit: GPa) .................................................. 111
6.8 Summary: Measurement of E versus temperatures (Unit: GPa)....................... 112
6.9 The expressions of E and ? for each direction of silicon.................................. 113
7.1 Stress sensitivities of the (111) p-type resistor sensors (Unit: MPa-1) ................ 124
7.2 Stress sensitivities of the (111) n-type resistor sensors (Unit: MPa-1)................. 124
7.3 Stress sensitivities of the (111) p-type silicon VDP structures (Unit: MPa-1)..... 125
7.4 Stress sensitivities of the (111) n-type silicon VDP structures (Unit: MPa-1)..... 125
7.5 Analytically calculated magnification factor, M .................................................. 137
7.6 Experimental values of M ..................................................................................... 137
7.7 Comparison between Analytic and Experimental M ........................................... 137
7.8 ?0 and ?90 at various stress levels....................................................................... 139
7.9 The effective B1 and B2 (Unit: TPa-1)............................................................... 142
7.10 Modified B1 and B2 (Unit: TPa-1) ..................................................................... 142
7.11 The effective B1 and B2 (Unit: TPa-1)............................................................... 143
7.12 Modified B1 and B2 (Unit: TPa-1) ..................................................................... 143
7.13 Analytically calculated magnification factor, M, through the use of modified
xxii
B1 and B2................................................................................................................ 143
8.1 B1_(eff) and B2_(eff) versus temperature (Unit: TPa-1)............................................ 147
8.2 Modified B1 and B2 versus temperature with consideration of ? (Unit: TPa-1)
....................................................................................................................................... 147
8.3 Addition and subtraction of B1_(eff) and B2_(eff) versus temperature (Unit: TPa-1)
....................................................................................................................................... 148
8.4 Addition and subtraction of B1 and B2 versus temperature (Unit: TPa-1) ......... 148
8.5 ? with lateral diffusion (BMW-2.1) .................................................................. 149
8.6 ? with lateral diffusion (JSE-WB100C)............................................................ 150
8.7 Normalized % error in ?pi11 and ?pi12 versus ?...................................................... 159
8.8 Normalized % error in ?R45/R45 and ?R-45/R-45 versus ? ................................. 160
8.9 Normalized % error in ?pi44 versus ? .................................................................. 166
8.10 Normalized % error in ?R/R versus ? for ? = 0 and ? = 90o ........................... 169
8.11 Normalized % error in ?R45/R45 and ?R-45/R-45 versus ? ................................. 171
9.1 Sheet resistance measured by Van der Pauw?s method
(Unit: ohms per square) .................................................................................... 188
9.2 Expected resistance (Unit: ohm)....................................................................... 188
9.3 ?11 and ?22 (?'11 and ?'22 ) at the sensor location with temperature (Unit: MPa)
...................................................................................................................................... 202
9.4 Measurements of E with temperature (Unit: GPa) ........................................... 203
xxiii
9.5 Dimensions of composite materials of [100] silicon strip-on-beam (Unit: mil)
..................................................................................................................................... 203
9.6 Dimensions of composite materials of [110] silicon strip-on-beam (Unit: mil)
....................................................................................................................................... 203
9.7 S0, S90, S45, and S-45 for [100] p-type silicon with temperature (Unit: 10-6 N-1)
....................................................................................................................................... 205
9.8 S0, S90, S45, and S-45 for [110] p-type silicon with temperature (Unit: 10-6 N-1)
....................................................................................................................................... 205
9.9 S0, S90, S45, and S-45 for [100] n-type silicon with temperature (Unit: 10-6 N-1)
..................................................................................................................................... 206
9.10 S0, S90, S45, and S-45 for [110] n-type silicon with temperature (Unit: 10-6 N-1)
....................................................................................................................................... 206
9.11 pi11, pi12, and pi44 for (001) p- and n-type silicon with temperature (Unit: TPa-1)
....................................................................................................................................... 207
9.12 piS for the (001) p-type silicon with temperature (Unit: TPa-1) ......................... 211
9.13 piS for the (001) n-type silicon with temperature (Unit: TPa-1) ......................... 211
1
CHAPTER 1
INTRODUCTION
As VLSI chips have become highly integrated with advances in semiconductors
and microelectronics, their feature sizes have become smaller and smaller. Too, VLSI die
sizes have become larger and larger. Hence the effects of mechanical stresses on the
structural reliability of electronic packages have become an important issue. These
stresses are caused during both the fabrication and operation of an electronic package
from mechanical loads and from uneven expansions and contractions of the various
package materials.
Because of thermal and mechanical loadings, stresses in electronic packages
may cause not only premature mechanical failures but also alteration of the function of
the semiconductor devices. Thus stress related problems such as fracture of the die, die
bond failure, solder fatigue, severing of connections, and encapsulant cracking are
prevalent in semiconductor manufacturing. Especially, thermally induced stresses are
created during packaging procedures such as encapsulation and die attachment, as well as
during the application of the package in a thermally changing environment. Typical IC
packages are comprised of a variety of materials which expand and contract at different
rates and have different elastic moduli. Under heating and cooling of such assemblies of
materials, the coefficient of thermal expansion mismatches lead to mechanical stresses. In
2
addition, heat dissipated by high power density devices during operation produces
thermally induced stresses.
As the electronic industry continues to develop, the ratio of package size to chip size
becomes an issue for higher I/Os. Thus the stress distribution may change rapidly over
small scales.
Piezoresistive stress sensors are a powerful tool for experimental structural
analysis of electronic packages. Figure 1.1 illustrates the basic application concepts. The
structures of interest are semiconductor chips which are incorporated in electronic
packages. The sensors are an integral part of the structure to be analyzed. The stresses in
the chip induce resistance change in the sensors because of the piezoresistive effect and
may be easily measured. Thus the sensors are capable of providing non-intrusive
measurements of surface stress states on a chip even within encapsulated packages.
Silicon Wafer Chip
Electronic P ackage
Sensor RosetteLocal Die Stress State
Fig. 1.1 - Basic concept of piezoresistive sensor
3
If the piezoresistive sensors are calibrated over a wide temperature range,
thermally induced stresses may be measured. A full-field mapping of the stress
distribution over the surface of a die can be obtained using specially designed test chips
which incorporate an array of sensor rosettes (or resistors). In addition to being applied to
packaging stress measurements, piezoresistive sensors have widespread applications as
sensing elements in various transducers.
Accurate values of the piezoresistive coefficients of the sensing resistors, as well
as recognition of the many potential sources of error that may be present during
calibration and measurement, are required for the successful application of piezoresistive
stress sensors. Therefore, it is very important to search for the optimal wafer orientation
in order to minimize the associated calibration errors for (001) silicon wafer.
Misalignment with respect to the true crystallographic axes of the semiconductor crystals,
such as the tilt of wafer plane, affects the calibration values of piezoresistive coefficient.
Errors in misalignment with the given crystallographic axes are described and analyzed
because precise determination of the crystallographic orientation in (001) silicon wafers
is found to be essential for accurate determination of piezoresistive coefficients of silicon.
On the other hand, for (111) silicon wafers, errors associated with misalignment have no
effect on the calculation of piezoresistive coefficients of silicon due to the isotropic
characteristics of (111) silicon.
Enhanced calibration techniques are needed for accurate determination of
piezoresistive coefficients prior to application of piezoresistive sensors within packages.
For experimental techniques and methods, a four-point bending fixture system and a
hydrostatic pressure vessel system have been constructed. In order to extract a complete
4
set of pi-coefficients ( 441211 ?and , ?,? ) for both p- and n-type sensors, hydrostatic tests
are needed for (111) silicon. On the other hand, the hydrostatic tests are not required for
stress sensors on (001) silicon. In order to determine a complete set of pi-coefficients,
(001) silicon wafers are cut along two directions (e.g., the [100] axis and the [110] axis).
Associated beam rotational errors in piezoresistive coefficients induced by the rotational
misalignment of the strip on the supports with respect to the ideal longitudinal axis of the
strip are explained.
In the chip-on-board (or strip-on-beam) method, a die (or strip) is adhered to a
board which is subjected to pure bending. The general equation for an off-axis (0, 90,
+45, and -45) resistor oriented at some arbitrary angle is discussed. The fabrication and
calibration of the (001) or (111) silicon test chips for p- and n-type materials are
explained. Finite element analysis has been presented to calculate the stress states applied
to the calibration samples. In addition, several issues related to the regular four-point
bending calibration procedure have been described.
The hydrostatic pressure calibration technique has been reviewed for sensors
fabricated on the (111) silicon. In the case of hydrostatic calibration, a high capacity
pressure vessel is used to subject a single die to triaxial compression. The temperature
effects must be removed from hydrostatic calibration data before evaluating the pressure
coefficients, and accurate determination of the TCR (temperature coefficients of
resistance) of a sensor must be done prior to pressure coefficient measurement. Finally, a
combination of four-point bending and hydrostatic calibration tests has been shown to be
suitable for obtaining a comprehensive set of piezoresistive coefficients.
5
Stress analyses of electronic packages and their components have been performed
using experimental, numerical, and analytical methods. Experimental approaches have
included the use of test chips incorporating piezoresistive stress sensors, whereas
numerical studies have typically considered finite element solutions for sophisticated
package geometries. Analytical investigations have been concerned primarily with
finding closed-form elasticity solutions for layered structures.
In the current microelectronics industry, it is most common for silicon devices to
be fabricated using the (001) silicon wafers. The other customarily utilized wafer
orientation is (111). In this work, the general piezoresistivity theories at fixed and
variable temperatures are presented, and the general equations for sensors fabricated on
the (001) and the (111) silicon wafers along with the basic sensor rosette configurations
are reviewed and expanded. Further, the general expressions for the resistance changes
experienced by in-plane resistors fabricated on these two types of silicon wafers have
been reviewed. In particular, for each wafer type the normalized resistance change for an
in-plane resistor has been expressed as a function of the resistor orientation and a set of
linearly independent combined piezoresistive coefficients.
The directionally dependent nature of silicon crystals will be explained to help
with the understanding of physical properties of semiconductor materials and the linear
elastic silicon stress-strain relations have been presented in tensor notation. The
appropriate selection of stress-strain relations has been discussed.
Over the years, the VDP (Van der Pauw) structures have been used to measure
resistivity or sheet resistance of materials. Sheet resistivity of a flat conductive structure
can be calculated using the resistivity value measured using VDP structure. It is
6
noteworthy that dimensional changes of VDP structures during loading have not been
considered in prior investigations.
In this work, however, the dimensional changes of VDP structures and resistor
during loading will be discussed. Furthermore, strain-effects of VDP structures and
resistor sensors on piezoresistive coefficients and sensitivity magnifications will be
discussed and compared to the cases in which strain-effects are not considered.
Simultaneously, the VDP stress sensitivities will be compared with analogous resistor
sensors on the same wafer and with the same doping concentration.
7
CHAPTER 2
LITERATURE REVIEW
In 1932, Bridgman [1-3] observed that applied transverse and longitudinal stresses
in certain crystals changed their electrical resistance. Bridgman, credited with making the
first piezoresistance measurements, initially observed piezoresistive behavior in metals.
He subjected metals to tension and hydrostatic pressure. Experimental observations of the
piezoresistive effect in semiconductors (silicon and germanium) were first made by
Taylor [4], Bridgman [5], Smith [6], and Paul and Pearson [7].
Smith [6] described the piezoresistance effect that is a major sensing principle in
micro-mechanical sensors. In 1961, Pfann and Thurston [8] derived longitudinal and
transverse piezoresistance coefficients for various directions in cubic crystals of silicon.
These formulations were later clarified and formulized using tensor analysis techniques
by Mason and Thurston [9], Thurston [10], and Smith [11]. Since then, many
researchers have studied the piezoresistance coefficients of silicon both analytically and
experimentally, as a function of doping concentration mostly at room temperature. An
extensive derivation of the piezoresistive theory was given by Bittle, et al. [12-13]. A
detailed theory for silicon piezoresistive sensors was derived by Bittle et al. [14], and
Kang [15] explored piezoresistive theory for silicon on various wafer planes.
In the early work, stress was often applied by hanging a weight on a string fixed to
8
the end of the silicon cantilever. Only the largest coefficients, pi44 for p-type silicon and
pi11 for n-type material, are easily measured. The effects of crystallographic misalignment
and temperature errors were generally ignored. A design tool for precise determination
of the crystallographic orientation in the (001) silicon wafer using anisotropic wet etching
was introduced by Vangbo [16]. Jaeger and Suhling [17] showed that temperature
variations and measurement errors play a pivotal role in determining accuracy of the
results obtained during both calibration and application of piezoresistive stress sensors.
They demonstrated the significance of thermally induced errors in the calibration and
application of silicon piezoresistive stress sensors in [17]. Furthermore, Cordes, et al. [18]
presented optimal temperature compensated piezoresistive stress sensor rosettes.
Matsuda [19] measured the nonlinear piezoresistance effect in silicon and
presented the theoretical and experimental values of piezoresistive coefficients. Kanda
[20] offered a graphical representation of the piezoresistance coefficients in silicon, based
on the literature values of piezoresistive coefficients by Smith. He plotted the
theoretical longitudinal and transverse piezoresistance coefficients at room temperature as
a function of the crystal directions for orientations in the (100), (110), and (211)
crystallographic planes. Richter et al. [21] presented experimentally obtained results
for the piezoresistive effect in p-type silicon. They measured the longitudinal (piL) and
transverse (piT) components for the [110] direction of the (001) silicon. Vladimir [22]
produced numerical simulations of the piezoresistance effect in silicon using a relaxation
time formation. The results in these publications were for fixed temperature
The temperature dependence of the piezoresistance of high-purity silicon and
9
germanium was described by Morin [23]. The temperature dependence of the large
coefficients (pi44 for p-type silicon and pi11 for n-type silicon) has been measured by Tufte
and Stelzer [24-25] as a function of impurity concentration. Suhling, et al. [26-27] used
piezoresistive sensors to measure and investigate thermally-induced stresses. Lenkkeri
[28] presented experimental values of the piezoresistance coefficients at 77K and 300K.
Jaeger et al. [29] presented experimental results for the piezoresistive coefficients of
silicon, pi44 and piD, as a function of temperature (25oC~140oC), and Lund [30] measured
the piezoresistance coefficients in p-type silicon, using the 22.5? off-axis direction of
silicon, over the temperature range 5oC to 140?C. Gniazdowski [31] measured the
longitudinal (piL) and transverse (piT) components of the piezoresistance coefficient in p-
type [110] silicon over the temperature range 25oC to 105?C. Toriyama [32] derived an
approximate piezoresistance equation for p-type silicon as a function of impurity
concentration and temperature (-100oC~100?C) taking into account spin-orbit interaction.
Kozlovskiy [33] calculated the piezoresistance coefficients, pi44 in p-type silicon and pi11
in n-type silicon as a function of temperature for different impurity concentrations.
Yamada, et al. [34] described the nonlinearity of the piezoresistive effect.
The results of all these efforts indicate relatively good agreement in magnitude for
the large (and hence easily measured) coefficients as well as a small relative temperature
dependency. However, 11pi and 12pi in p-type material are much less well defined
with large discrepancies in magnitude and even sign among researchers. The overall
goal of this work is to try to resolve these discrepancies and produce a set of coefficients
for use in packaging measurements over 77K - 450K.
10
For experimental structural analysis of electronic packages, piezoresistive sensors
are a highly useful tool. The piezoresistive sensors are usually resistors that are
conveniently fabricated into the surface of the die using current microelectronic
technology, and are capable of providing non-intrusive measurements of surface stress
state on a chip even within encapsulated packages [35-40].
Stress sensors based on piezoresistive field effect transistors (PIFET's) were
proposed and designed [41-48] using the relations between the MOS drain current change
and applied mechanical stress. In addition, Mian, et al. studied the sensitivity of the
resistance of Van der Pauw structures to applied stress [49-50].
A four-point bending calibration procedure was discussed and utilized by Bittle, et al.
[12-13], Suhling, et al. [26-27], Beaty, et al. [51], Jaeger, et al. [17, 29, 52], and Van
Gestal [53]. A wafer-level procedure and calibration for piezoresistive stress sensors was
developed and utilized by Cordes [54], and Suhling, et al. [55-57]. This technique for
wafer-level calibration of stress sensing test chips was successfully applied to different
test chip designs by Cordes [54]. The errors associated with the design and calibration of
piezoresistive stress sensors in (100) silicon was analyzed by Jaeger, et al. [58-59]. Many
researchers have performed experimental studies using test chips with piezoresistive
stress sensors in the literature [60-66]. A hydrostatic calibration method for (111) silicon
test chips was developed and applied by Kang [15], and Suhling, et al. [67-68]. Finite
element simulations provide useful insight into the stress distributions produced in plastic
packages during die attachment, encapsulation, and reliability tests. Various package
processes and reliability tests can be investigated by means of finite element methods
11
[69-72].
Test chips with resistor sensors have been used to measure die stresses in various
packages [13, 73-77]. However, resistor sensors are typically made using serpentine
conduction paths to increase the unstressed resistance values (they have large numbers of
conductive squares and thus have a relatively large resistance values) in order to reduce
the measurement errors. In addition, resistor sensors have several drawbacks such as the
large sensor sizes and less sensitivities.
Van der Pauw (VDP) structures used as stress sensors have the potential to solve
the deficiencies of resistor based sensor. The VDP structure requires only one square of
material, and its characteristics are also size independent. Thus such sensors can be made
small enough to capture stress variation in a small area without any loss of sensitivity.
VDP structures have been used as piezoresistive stress sensors [50]. The van der Pauw
method is a widely used technique for measuring resistivity of arbitrary shaped samples
of constant thickness In addition, VDP structures have been used for Hall mobility
measurements. The techniques are based on the theoretical developments of van der
Pauw [78-79]. Since then, many researchers have extended the originally proposed ideas
to develop a variety of approaches for evaluating the resistivities of both isotropic and
anisotropic materials using VDP-type structures [80-87].
An extensive review of results from the literature are presented and compared in
this Chapter as discussed below:
12
circle6 Piezoresistive values at room temperature
The composite data for pi11, pi12, and pi44 collected from the literature for p-type silicon
at room temperature are displayed in Table 2.1, where the letter ?C? represents calculated
(theoretical) values from the literature and the letter ?E? represents experimental results.
Table 2.1 - Composite data for pi11, pi12, and pi44 collected from the literature for p-type
silicon at room temperature (TPa)-1
Literature data for pi44, pi11, and pi12 for p-type material as a function of impurity
concentration at room temperature appear in Figs. 2.1 through 2.3, respectively. The
results indicate that pi44, a relatively large coefficient, versus impurity concentration is
13
relatively well defined in magnitude and sign as shown in Fig. 2.1. On the other hand, pi11
and pi12 in p-type material are much less defined with large discrepancies in magnitude
and sign among researchers, as shown in Figs. 2.2 and 2.3.
pi44p with Impurity Concenration
at Room Temperature
pi44p = -96.231Ln(C) + 5047.2
R2 = 0.6972
0
200
400
600
800
1000
1200
1400
1.00E+16 1.00E+17 1.00E+18 1.00E+19 1.00E+20 1.00E+21 1.00E+22
Impurity Conc.
(1/
TP
a)
Fig. 2.1 - Composite pi44 data collected from the literature for p-type silicon as a function
of impurity concentration at room temperature
pi11p with Impurity Concentration
at Room Temperature
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
1.00E+16 1.00E+17 1.00E+18 1.00E+19
Impurity Conc.
(1
/T
Pa
)
Fig. 2.2 - Literature data for pi11 versus concentration for p-type silicon at room
temperature
14
pi12p with Impurity Concentration
at Room Temperature
-250
-200
-150
-100
-50
0
50
100
150
1.00E+16 1.00E+17 1.00E+18 1.00E+19
Impurity Conc.
(1/
TP
a)
Fig. 2.3 - Literature data for pi12 versus concentration for p-type silicon at room
temperature
Similarly, for n-type material, pi11, pi12, and pi44 from the literature are presented
as a function of impurity concentration at room temperature in Tables 2.2 and 2.3. Note
that the underlined values in Table 2.3 are generated by assuming the
approximation 1211 ?2? ?? [23]. In those cases, Spi is originally given instead of pi11 and
pi12. The largest coefficient pi11 for n-type material is relatively well defined versus
impurity concentration as displayed in Fig. 2.4. Unlike p-type material, the smaller
coefficients (pi12 and pi44 in n-type material) are also well defined with impurity
concentration despite the limited amount of data, as shown in Fig. 2.5.
15
Table 2.2 - Literature data for pi11versus
concentration for n-type silicon at room
temperature (TPa)-1
Table 2.3 - Composite data for pi11, pi12, and pi44 collected from the literature for n-type
silicon as a function of impurity concentration at room temperature (TPa)-1
16
pi11n with Impurity Concentration
at Room Temperature
pi11n = 71.695Ln(C) - 3739.6
R2 = 0.8788
-1400
-1200
-1000
-800
-600
-400
-200
0
1.00E+16 1.00E+17 1.00E+18 1.00E+19 1.00E+20 1.00E+21
Impurity Conc.
(1
/T
Pa
)
Fig. 2.4 - Literature data for pi11 versus concentration for n-type silicon at room
temperature
pi12n and pi44n with Impurity Concentration
at Room Temperature
pi44n = 15.677Ln(C) - 779.59
R2 = 0.6346
pi12n = -23.479Ln(x) + 1319.6
R2 = 0.8454
-300
-200
-100
0
100
200
300
400
500
1.00E+16 1.00E+17 1.00E+18 1.00E+19 1.00E+20 1.00E+21
Impurity Conc.
(1/
TP
a)
Fig. 2.5 - Literature data for pi12 and pi44 versus concentration for n-type silicon at room
temperature
17
circle6 Piezoresistive values with varying temperatures
Values of pi44 collected from the literature for p-type silicon versus temperature are
displayed in Table 2.4, whose plots are shown in Figs. 2.6 and 2.7. Also, pi11 in n-type
material versus temperature is shown in Table 2.5, whose plot is displayed in Fig. 2.8.
These literature data show that the piezoresistive coefficients pi44 in p-type material and
pi11 in n-type material decrease monotonically with rising temperature. This is in
agreement with the predictions [23] that the piezoresistance is linear in T-1 over most of
the temperature range. Note that few values of the small pi-coefficients, pi11 and pi12 for p-
type silicon and pi12 and pi44 for n-type silicon, versus temperature can be found in the
literature data.
Table 2.4 - Composite data for pi44 collected from the literature for p-type silicon versus
temperature with different doping concentration (TPa)-1
18
pi44p Vs. Temperature (Calculated)
600
700
800
900
1000
1100
1200
1300
1400
-150 -100 -50 0 50 100 150
T (Celsius)
(1/
TP
a)
C = 1.0E17 C = 1.0E18 C = 1.0E19 C = 1.0E20
Fig. 2.6 - Calculated literature data for pi44 versus temperature with different doping
concentration for p-type silicon
pi44p Vs. Temperature (Experimental)
500
700
900
1100
1300
1500
1700
-150 -100 -50 0 50 100 150
T (Celsius)
(1
/T
Pa
)
C = 8.0E17 C = 3.0E18 C = 8.2E18 C = 9.0E18 C = 5.0E19 [Gniazdowski]
Fig. 2.7 - Experimental literature data for pi44 versus temperature with different doping
concentration for p-type silicon
19
Table 2.5 - Composite data for pi11 collected from the literature for n-type silicon versus
temperature with different doping concentration (TPa)-1
pi11n Vs. Temperature (Experimental)
-2500
-2000
-1500
-1000
-500
0
-300 -200 -100 0 100 200
T (Celsius)
(1/
TP
a)
C = 1.3E16 C = 1.8E18 C = 8.8E18 C = 5.0E19 C = 5.2E19 C = 9.0E19 C = 2.1E20
Fig. 2.8 - Experimental literature data for pi11 versus temperature with different doping
concentration for n-type silicon
20
CHAPTER 3
REVIEW OF PIEZORESISTIVITY THEORY
A filamentary silicon conductor arbitrarily oriented in a crystallographic coordinate
system is shown in Fig. 3.1.
?x1
?x2
x1
x 2
r?e
1
r?e
2
x 3
?x 3
r?e
3
rn
Fig. 3.1 - An arbitrary oriented, filamentary conductor
The unprimed axes x1 = [100], x2 = [010], and x3 = [001] are the principal crystallographic
directions of the cubic silicon crystal. The primed coordinate system is arbitrarily rotated
21
with respect to this unprimed crystallographic system. For this conductor, the general
expression for the resistance change of a filamentary piezoresistive sensor in the plane of
the wafer may be obtained as follows [12, 15, 50, 99, 105]:
...]TT[
ml)(2nm)(2nl)(2
)n()m + ()l = (R?R
2
21
''''
6
''''
5
''''
4
2'''
3
2'''
2
2'''
1
+??+??+
?pi+?pi+?pi+
?pi+?pi?pi
??????
??????
Eq. (3.1)
where 6) 2,..., 1, = ,( ' ??pi?? are the off-axis temperature dependent piezoresistive
coefficients. 1? , 2? , ? are the temperature coefficients of resistance, and ?T = Tm - Tref
is the difference between the measurement temperature and reference temperature, and 'l ,
'm , and 'n are the direction cosines of the conductor orientation with respect to the '
1x ,
'
2x , and
'
3x axes, respectively. Equation (3.1) assumes that geometric changes are neglected.
When the primed axes are aligned with the unprimed (crystallographic) axes, Eq.
(3.1) reduces to
...]TT[2sin ??
sin)]?(?+ ??+ [?cos)]?(?+ ?? = [?R?R
2
211244
2
3311122211
2
3322121111
+??+??+?+
?+?+
Eq. (3.2)
where ? is the angle between the 1x -axis and resistor orientation.
For an arbitrarily oriented in-plane resistor, the resistance change equation can be
obtained by using Eq. (3.1).
22
3.1 Resistance Change Equations for the (001) Silicon Wafer Planes
A general (001) silicon wafer is shown in Fig. 3.2.
Fig. 3.2 - General (001) silicon wafer
A convenient wafer coordinate system may be used where the primed axes '1x , '2x
are chosen to be parallel and perpendicular to the primary wafer flat.
The '3x -axis is then be perpendicular to the wafer plane--that is, ]110[x '1 = , ]101[x '2 = ,
and ]001[x '3 = . For the unprimed and primed coordinate systems shown in Fig. 3.2, the
appropriate direction cosines for the primed axes are shown as follows:
23
[ ]
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?=
100
02121
02121
a ij
Eq. (3.1.1)
Substitution of the off-axis piezoresistive coefficients into Eq. (3.1) yields the
general expression for the resistance of a resistor which is oriented at an angle ? with
respect to the ]110[x '1 = axis on the (001) surface of a silicon wafer as follows [12-13, 15,
17, 27, 29, 35, 39, 49-50, 72, 92, 99, 105]:
+ ...] ?T?T + [2sin )?(?
sin ? 2 ??? + ?2 ??? +
cos ?2 ??? + ?2 ??? = R?R
2
21
'
1212113312
2'
22
441211'
11
441211
2'
22
441211'
11
441211
??++?pi?pi+pi+
??
?
?
??
? ?
?
??
?
? ++?
?
??
?
? ?+
??
?
?
??
? ?
?
??
?
? ?+?
?
??
?
? ++
Eq. (3.1.2)
where
0 = , and nsin , mcos = l ''' ?=? Eq. (3.1.3)
has been introduced. Equation (3.1.2) indicates that the out-of-plane shear stresses '13? and
'
23? do not influence the resistances of stress sensors fabricated on (001) wafers. This
means that a sensor rosette on (001) silicon may at best measure four of the six unique
components of the stress tensor. All three of the unique piezoresistive coefficients for
silicon ) and ,,( 441211 pipipi appear in Eq. (3.1.2).
24
3.2 Resistance Change Equations for the (111) Silicon Wafer Planes
The other common silicon crystal orientation used in semiconductor fabrication is
the (111) surface. A general (111) silicon wafer is shown in Fig. 3.3.
Fig. 3.3 - General (111) silicon wafer
The surface of the wafer is a (111) plane--that is, the [111] direction is normal to the
wafer plane. Since the principal crystallographic axes x1 = [100], x2 = [010], and x3 = [001]
do not lie in the wafer plane, they have not been indicated. As mentioned previously, it is
convenient to work in an off-axis primed wafer coordinate system where the axes '1x and
'
2x are parallel and perpendicular to the primary wafer flat. If Eq. (3.1) is used, the
resistance change of an arbitrarily oriented in-plane sensor may be expressed in terms of
the stress components derived in this natural wafer coordinate system. For the primed
25
coordinate system indicated in Fig. 3.3, the appropriate direction cosines for the primed
axes are as follows:
[ ]
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
=
3
1
3
1
3
1
6
2
6
1
6
1
02121
a ij
Eq. (3.2.1)
The general expression for the normalized change in resistance of a resistor that is
oriented at an angle of ? with respect to the ]011[x '1 = axis on the (111) surface of a
silicon wafer is given by [15, 35, 49-50, 72, 92, 99, 105]:
] . . . TT[2sin])BB()BB(2[2
sin])BB(22BBB[
cos])BB(22BBB[RR
2
21
'
1221
'
1323
2'
2332
'
333
'
221
'
112
2'
2332
'
333
'
222
'
111
+??+??+???+??+
???+?+?+?+
?????+?+?=?
Eq. (3.2.2)
where ? is again the angle between the '1x -axis and the resistor orientation. The
coefficients
3
??2?, B
6
??5?, B
2
???B 441211
3
441211
2
441211
1
?+=?+=++=
are a set of linearly independent temperature-dependent combined piezoresistive
parameters. These parameters must be calibrated before stress component values may be
extracted from resistance change measurements. It is noteworthy that the general resistance
change expression in Eq. (3.2.2) is dependent on all six of the unique stress components.
26
Therefore, the potential exists for developing a sensor rosette which may measure the
complete three-dimensional states of stress at points on the surface of a die.
Theoretical analysis has established that properly designed sensor rosettes on the (111)
silicon wafer plane have several advantages relative to sensors fabricated using the standard
(001) silicon. In particular, optimized sensors on the (111) silicon may be used to measure
the complete state of stress (six stress components) at a point on the top surface of the die,
while optimized rosettes on the (001) silicon measure, at most, four stress components [26].
The additional stress components that may be measured are the out-of-plane (interfacial)
shear stresses. Knowledge of these shear stresses may be important for determining the
integrity of die interfaces or for the detection of interface delamination [99]. Further,
optimized sensors on the (111) silicon offer the unique capability of measuring four
temperature compensated combined stress components, whereas those on the (001) silicon
may only be used to measure two temperature compensated quantities. In our discussion,
?temperature-compensated? refers to the ability to extract the stress components directly
from the resistance change measurements (without the need to know the temperature
change ?T). This is a particularly important attribute, given the large thermally induced
errors that often may be found in stress sensor data. The four stress components that may
be measured in a temperature compensated manner using the (111) silicon sensors are the
three shear stress components and the difference of the in-plane normal stress components.
27
CHAPTER 4
CALIBRATION OF SENSORS ON THE (111) SURFACE
In this work, the (001) and (111) silicon test chips containing an array of optimized
piezoresistive stress sensor rosettes have been successfully applied. Calibrated and
characterized stress test chips are assembled into chip-on-beam specimens. The resistances
of the sensors are then recorded. The stresses on the die surface are calculated using the
measured resistance changes and the appropriate theoretical equations. For comparison
purpose, two-dimensional linear finite element simulations are also performed.
Three sets of test chip experiments are performed in this work. In the first set of
experiments, 200 x 200 mil JSE WB100C test chips fabricated on the (111) silicon [99,
105] are utilized to characterize the tensile die stresses on chip-on-beam specimens.
These test chips will be described in more detail later. The JSE WB 100C chip contains
wire bonding pads on the perimeter of the die. With this design, the wire-bonding is used
to provide electrical connection between the chip and PC board. In addition, soldering is
used for electrical connection between the outer pads on the board and external
instrumentation through wires. In the second set of experiments, test strips fabricated on
the (001) silicon are used to measure die stresses.
28
4.1 Experimental Setup
The general procedures of resistance measurements with the JSE-WB100C test
chips are now discussed.
The equipment we utilized in the experimental procedure included:
? Semiconductor Parametric Analyzer 4155C
The 4155C is an electronic instrument for measuring and analyzing the
characteristics of semiconductor devices. This instrument has four highly accurate
source/monitor units (SMUs), two voltage source units (VSUs). It has high resolution
enough for voltage/current measurement. In this work, sweeping voltage and
measurement of voltage/current through the resistor sensors is performed by using this
instrument. When measuring the sensor resistances on the test chip, the 4155C is used to
provide bias in the circuit, and to provide voltage to the measured resistors. In this work,
the voltage across the resistor is swept from 0.6V to 1.0V in 20mV steps. For an n-
substrate and p-well, the voltage is set to be 1V and -1V, respectively. This is for
electrical isolation between the doped surface resistor and the bulk of the chip by using
proper reverse biasing of the resistor and substrate regions. Measurement results are
saved to a 3.5 inch diskette.
? Environmental Test Chamber (Delta Design 2850)
This environmental chamber is capable of temperatures from -155 to +300oC with
a 0.1oC resolution. The temperature in the experiments reported here is typically swept
over a large range from a low temperature (as low as -150 oC) to a high temperature (as
high as 100 oC). For low temperatures, a liquid-nitrogen (LN2) tank is hooked up with
this chamber. A special four-point bending apparatus was constructed and integrated into
29
this environmental chamber. The increment of the temperature between sensor readings is
usually set to be 25 or 50 oC, and the temperature at each step is maintained for at least 10
minutes before measurements are taken to ensure a uniform temperature distribution on
the silicon chip.
? Digital Panel Mount Meter (DPM-3)
The DPM-3 panel instrument is a versatile, cost effective solution to a wide
variety of monitoring and control application. This instrument is easily set to produce an
accurate display of weight or load. Digital calibration of all ranges eliminates drift
associated with potentiometers found in non-microcomputer-based meters. In addition,
the fast read rate provides an accurate display of input and quick response in applications.
During loading in a four-point bending fixture, the precise reading of applied load is
essential. This instrument is used to determine the exact load on the chip-on-beam
specimens.
? Four-Point Bending Apparatus
A special four-point bending (4PB) apparatus has been constructed and integrated
into the Delta Design environmental chamber. Details will be discussed in a later section.
? Accessories
For TCR measurements and hydrostatic tests, a pair of edge connectors are used to
contact to the outer pads on the board. Those are needed to provide electrical connections
between the chip and external instrumentation.
30
4.1.1 Four-Point Bending Apparatus
In the four-point bending method, a rectangular strip containing a row of chips is
cut from a wafer and is loaded in a four-point bending fixture to generate a uniaxial stress
state. A general four-point bending loading fixture is shown in Fig. 4.1. Using the free
diagram and its corresponding 3 equations ( 0M and 0,F ,0F yx =?=?=? ), the
normal stress 11? induced at points on the top surface of the strip that are between the
bottom supports is given by
2bh D)3F(L? ?= Eq. (4.1.1)
where
beam of thicknessh
beam ofwidth b
?
?
?L distance between two top supports
D ? distance between two bottom supports
Fig. 4.1 - Four-point bending loading fixture
The details about parts of four-point bending fixture are given in Fig. 4.2. The
silicon strip is loaded on bottom supports. By controlling the micrometer, uniaxial stress
31
can be generated. F can be calculated from the output of the load-cell. A special four-
point bending apparatus is constructed and integrated into the environmental chamber as
shown in Fig. 4.3. Through a ceramic rod penetrating the bottom side of the oven, force
generated by a vertical translation stage is applied to the strip in the four-point bending
fixture inside the chamber (see Fig. 4.4).
Load
Cell
Spacer Block
Rotational Joint
Bottom Supports
Silicon Strip
Top Support
Vertical Translation Stage
Micrometer
Fig. 4.2 - The details about the parts of the four-point bending fixture
32
Fig. 4.3 ? A four-point bending fixture mounted inside the oven
Fig. 4.4 - A four-point bending fixture: Exterior of the oven
33
Four-point bending calibration produces a well-defined uniaxial stress state. Like
any other calibration techniques, the four-point bending method has its limitations. For
example, it necessitates cutting the wafer containing the fabricated test chips into strips.
Each of these strips must then be individually loaded, making the calibration process tedious.
Also, only sensor rosettes in the middle of each strip can typically be calibrated. If a wafer
strip along the 1x? direction is subjected to four point bending and a known uniaxial stress
?=??11 is applied in the 1x? -direction, the 0o and 90o oriented sensors yield the following
resistance changes:
circle6 (001) plane: with respect to the unprimed axes
+ ...] ?T?T + [2sin ??
sin)]?(?+ ??+ [?cos)]?(?+ ?? = [?R?R
2
211244
2
3311122211
2
3322121111
??+?+
?+?+
Eq. (4.1.2)
where ? is the angle between the 1x? axis and the resistor orientation. For the 0-90o oriented
sensors,
? ?= R?R 1111
0
0 and
1112
90
90 ? ?
R
?R = Eq. (4.1.3)
34
circle6 (001) plane: with respect to the primed axes
...]TT[ 2sin )?-?(???
sin ? 2 ??? + ?2 ???
cos ?2 ??? + ?2 ??? = R?R
2
21
'
121211
'
3312
2'
22
441211'
11
441211
2'
22
441211'
11
441211
+??+??+?++
??
?
?
??
? ?
?
??
?
? ++?
?
??
?
? ?++
??
?
?
??
? ?
?
??
?
? ?+?
?
??
?
? ++
Eq. (4.1.4)
For the 0o and 90o oriented sensors,
'11441211
0
0 ? )
2
???(
R
?R ++= Eq. (4.1.5)
? )2 ???(R?R '11441211
90
90 ?+= Eq. (4.1.6)
circle6 (111) plane: the primed axes
] . . . TT[2sin])BB()BB(2[2
sin])BB(22BBB[
cos])BB(22BBB[RR
2
21
'
1221
'
1323
2'
2332
'
333
'
221
'
112
2'
2332
'
333
'
222
'
111
+??+??+???+??+
???+?+?+?+
?????+?+?=?
Eq. (4.1.7)
For the 0o and 90o oriented sensors,
'
111
0
0 ? B
R
?R = and ? B
R
?R '
112
90
90 = Eq. (4.1.8)
35
4.1.2 The (111) Silicon Test Chips
When piezoresistive sensors are used in experimental stress analysis studies of
microelectronic packages, special test chips are typically designed and fabricated. The
test chip contains an array of the optimized eight-element dual polarity measurement
rosettes. Due to the piezoresistive effect, the stresses in the chip yield measurable
changes in the sensor resistance. The doped active region of a piezoresistive sensor is a
serpentine pattern in order to achieve acceptable resistance for measurement. Figure
4.5(a) shows the basic die image of JSE-WB100C resistor test chip on the (111) silicon
plane. In the chip-on-beam case, the sensors are located in the top surface of the die, a
free surface, and the sensor elements used here are the horizontal and vertical (0 and 90o)
sensor elements from the eight-element rosette. A microphotograph of an eight-element
sensor rosette is shown in Fig. 4.5(b) [92].
The die then is attached to the specially designed printed wiring board (PWB) with
die-attached material (ME525). For the test, an FR-406 PWB is designed using Lavenir
software as shown in Fig. 4.6. The (111) silicon test chip (JSE WB100C) is attached to
the center of the PWB.
36
Fig. 4.5(a) - The (111) silicon test chip (JSE-WB100C)
Fig. 4.5(b) - Microphotograph of eight-element sensor rosette
Fig. 4.6 - A specially designed printed wiring board
37
During the experiments, a chip-on-beam is placed on the bottom supports of the four-
point bending fixture. For electrical connection between chip and board, wire-bonding is
made between the inner pads on the board and the pads on the silicon chip as shown in
Fig. 4.7, where only the central part of a chip-on-beam structure is shown and enlarged.
The inner and outer pads on the boards are electrically connected and protected by a
solder mask. In addition, the outer pads on the board are electrically connected to external
instruments through wires by soldering. The picture of a wire-bonded chip-on-beam
structure is shown in Fig.4.8.
Fig. 4.7- A wire-bonded chip-on-beam structure (central part)
Fig. 4.8 - A wire-bonded chip-on-beam structure
38
The dimensions of the board material (FR-406), (111) silicon chip, and adhesive
material (ME525) are measured with a resolution of 0.05mil. In Table 4.1 the
dimensions of composite materials of chip-on-beam samples are presented. Following are
the average values of 10 specimens:
Table 4.1 - Dimensions of composite materials (Unit: mil)
board material (111) silicon adhesive material
length (l) 3400 200 200
width(b) 650 200 200
thickness(h) 22.67 25 1.60
4.2 Sensor Calibration for the (111) Silicon Test Chips
The eight sensors are configured as a parallel connection of four two-element
half-bridges in order to simplify the measurements. The n-substrate is maintained as 1 V,
and a bias of -1 V is applied to the p-well for electrical isolation between the bulk of the
chip and the doped surface resistor whose voltage is swept from 0.6 V to 1.0 V during
measurements.
The Four-point bending apparatus is used to generate the required stress. Since the
cross section of the chip-on-beam structure is three-dimensional at the die site, two-
dimensional states of stress are induced. Thus Eq. (4.1.1) for the calculation of uniaxial
stress is not applicable to this case. Hence, all the calibrations are performed with respect
to applied F. The following expression is for the stress-induced resistance changes for the
(111) silicon:
39
...] ?T?T[
2sin])?-B(B)?-B(B22[
sin])?-B(B22?B?B?[B
cos])?-B(B 22?B?B? [BR?R
2
21
'
1221
'
1323
2'
2332
'
333
'
221
'
112
2'
2332
'
333
'
222
'
111
+?+?+
?++
?++++
??++=
Eq. (4.2.1)
In the process, we consider two-dimensional states of stress ( '11? and '22? ) and defined
temperature terms [?1 ?T+?2 ?T2+?] as f (?T) in Eq. (4.2.1). The result is that Eq.
(4.2.1) simplifies to
T)f(?B?BR?R '222'111
0
0 ?++= Eq. (4.2.2)
T)f(?B?BR?R '221'112
90
90 ?++= Eq. (4.2.3)
We also adopt the following notations:
F?F, ??? ' F22'22' F11'11 ??
)R?R(dFdS
?
?
? ? Eq. (4.2.4)
Assuming 0)T(f =? in Eqs. (4.2.2) and Eq. (4.2.3) and using the notations above yield
the following results:
'
22F2
'
11F10 ?B?BS += Eq. (4.2.5)
?B?BS '22F1'11F290 += Eq. (4.2.6)
Solving for B1 and B2 in Eqs. (4.2.5) and (4.2.6) yield the results below:
40
2'
F11
2'
F22
0
'
F1190
'
F22
1 )(?)(?
S?S?B
?
?= Eq. (4.2.7)
2'
F11
2'
F22
90
'
F110
'
F22
2 )(?)(?
S?S?B
?
?= Eq. (4.2.8)
Measurements of S0 and S90 of the (111) silicon are performed over varying temperatures.
Typical results of S0 and S90 of the (111) silicon for each temperature are displayed in
Appendix A. The average values for each temperature are presented in Table 4.2 which
reflects the average values of 10 specimens.
Table 4.2 - S0 and S90 versus temperature (Unit: N-1)
T(Celsius) S0p S90p S0n S90n
-133.4 -2.46E-03 5.00E-03 1.34E-03 -1.52E-03
-93.2 -2.23E-03 4.55E-03 1.29E-03 -1.46E-03
-48.2 -2.05E-03 3.83E-03 1.17E-03 -1.42E-03
-23.6 -1.96E-03 3.63E-03 1.11E-03 -1.36E-03
0.6 -1.87E-03 3.43E-03 1.06E-03 -1.31E-03
25.1 -1.77E-03 3.20E-03 1.04E-03 -1.26E-03
49.9 -1.58E-03 2.83E-03 9.50E-04 -1.12E-03
75.1 -1.40E-03 2.51E-03 8.79E-04 -1.03E-03
100.6 -1.32E-03 2.30E-03 8.13E-04 -9.66E-04
Subtraction of Eq. (4.2.6) from Eq. (4.2.5) leads to
) ?)(?B(BSS ' F11' F2212900 ??=? Eq. (4.2.9)
41
We analyze
0
90
R
R in terms of F. In the process, we define A as the slope of
0
90
R
R
with respect to F,
0
90
R
R can be expressed as
)0,0(R )0,0(RAF)T,(R )T,(R
0
90
0
90 +=
??
?? Eq. (4.2.10)
Assuming f (?T) = 0 leads to
)0,0(R )0,0(RAF]?B?B1 ?B?B1[)0,0(R )0,0(R
0
90
'
222
'
111
'
221
'
112
0
90 +=
++
++ Eq. (4.2.11)
Then we let C)0,0(R )0,0(R
0
90 ? with the following result:
CAF]?B?B1 ?B?B1C[ '
222
'
111
'
221
'
112 +=
++
++ Eq. (4.2.12)
1?B and 1?B Assuming 21 <<<< , the result becomes:
C
AF)?)(?B(B
CAF)?B?B?B?B1C(
'
11
'
2221
'
221
'
112
'
222
'
111
=??
+=++??
Eq. (4.2.13)
1C ? for both p- and n-type yields
AF)?)(?B(B '11'2221 ??? Eq. (4.2.14)
Re-using the notation, ,F?and ?F?? ' F22'22' F11'11 ?? we arrive at the result below:
) ?)(?B(BA ' F11' F2221 ??? Eq. (4.2.15)
Comparing Eq. (4.2.9) and Eq. (4.2.15) yields
A(F)S(F)S 900 ??? Eq. (4.2.16)
42
By performing extensive measurements over temperatures, we determined the validity of
Eq. (4.2.16) as shown in Table 4.3.
Table 4.3 - Comparison between S0-S90 and
A (= slope of R90/R0 versus F) by measurements
T(Celsius) S0 p -S90p A(slope_p) S0 n -S90n A(slope_n)
-133.4 -7.45E-03 7.54E-03 2.85E-03 -2.57E-03
-93.2 -6.78E-03 6.75E-03 2.74E-03 -2.53E-03
-48.2 -5.88E-03 6.07E-03 2.59E-03 -2.45E-03
-23.6 -5.59E-03 5.71E-03 2.47E-03 -2.41E-03
0.6 -5.31E-03 5.23E-03 2.37E-03 -2.35E-03
25.1 -4.97E-03 4.76E-03 2.30E-03 -2.26E-03
49.9 -4.42E-03 4.32E-03 2.07E-03 -2.11E-04
75.1 -3.91E-03 3.93E-03 1.91E-03 -2.01E-03
100.6 -3.61E-03 3.63E-03 1.78E-03 -1.98E-03
By Table 4.3, it can be summarized that Eq. (4.2.16) is valid over the temperature range
of -150 to 100 oC.
4.3 Simulation Results for the (111) Silicon Test Chips
Finite element simulations are used to determine the actual states of stress in the
silicon chip. The finite element model predictions are used to approximate trends of the
various stress component distributions, so that the experimental data could be better
understood. In our simulations, ?'11 and ?'22 at the site of sensor are obtained when F =
1N is applied to both sides. Mesh plots of chip-on-beam are shown in Fig. 4.9.
43
In the finite element models, the materials are modeled as linear elastic.
Temperature dependent elastic modulus E is displayed in Table 4.4. Poisson's ratio ? of
ME525, (111) silicon and FR-406 are assumed to be 0.3, 0.262, and 0.117, respectively
[97]. Solder is neglected to simplify the analysis in the finite element models.
Table 4.4 - Elastic modulus of composite materials
versus temperature (Unit: GPa)
T (Celsius) ME525 (111) silicon FR-406
-151.0 19.81 173.5 28.82
-133.4 18.46 172.9 27.41
-93.2 15.99 172.3 25.57
-48.2 13.70 170.9 25.12
-23.6 12.85 170.2 24.80
0.6 12.00 169.5 24.68
25.1 10.43 169.1 23.73
49.9 9.85 168.5 22.05
75.1 8.75 167.9 20.26
100.6 7.72 167.0 18.55
125.9 4.98 166.6 16.37
151.5 0.98 165.5 14.87
The silicon chip is meshed into 24 x 24 x 3 elements as shown in Fig. 4.10. The
relative size and location of the sensors are presented in Fig. 4.11. The location of the
sensor corresponds to the node (#49788). It also should be mentioned that the size of 4
combined resistor sensors is 1.2 times larger than that of one element on the mesh plot. In
order to load a uniform force on the line in chip-on-beam structure (see Fig. 4.9), the
width between two nodes is kept constant. From Table 4.1, the ratio of silicon chip to
board is 200:650 in width. Hence the board should be 78-element width.
44
Fig. 4.9 - Mesh plot of chip-on-beam structure
Fig. 4.10 - Mesh plot of silicon chip
45
Fig. 4.11 - The relative size and location of the sensor on the mesh plot
In Fig. 4.12 and Fig. 4.13, the contour plots of ?'11 and ?'22 in the silicon chip are
presented. The colors of contour represent the stress value at the rosette site.
Fig. 4.12 - Contour plot of '11? at 25oC
46
Fig. 4.13 - Contour plot of '22? at 25oC
It is to be emphasized that the direction of ?'22 is parallel to the direction of the beam
in our case. At room temperature, ?'11 and ?'22 at the sensor location appear in Table 4.5
where the other 8 nodes surrounding node #49788 are presented as well. The comparison
of the average of 9 nodes (node #49788 and 8 surrounding nodes) with the value at node
#49788 proved the two to be very close, as expected. For further measurements, the value
at node #49788 is chosen.
Similar tests are also performed on the same resistor sensors over the temperature
range of -150 oC to 100 oC. It is observed that ?'11 and ?'22 at sensor location increases in
magnitude with increasing temperatures even though contour plots for different
temperature looks similar. This phenomenon is due to the uneven change in mechanical
property such as E with temperature among silicon, die attachment adhesive (ME 525),
47
and PCB material (FR-406). In Table 4.6, ?'11 and ?'22 at sensor location (assuming node
#49788) appear.
In our simulations, Poisson?s ration ? of ME 525, (111) silicon and FR-406 are
assumed to be 0.3, 0.262 and 0.117, respectively [97].
Table 4.5 - Simulation results of stresses around the sensor location at 25 oC
(Unit: MPa)
node # ?'22 ?'11 ?'33 ?'12 ?'23 ?'13
49787 7.976 -1.866 -6.131E-03 -6.015E-02 7.003E-03 1.711E-02
49788 8.051 -1.883 -5.649E-03 -1.211E-01 1.474E-02 1.727E-02
49789 8.1792 -1.907 -4.433E-03 -1.832E-01 2.437E-02 1.758E-02
49798 7.843 -1.848 -8.494E-03 -8.816E-02 6.940E-03 2.963E-02
49799 7.916 -1.868 -8.080E-03 -1.777E-01 1.461E-02 2.991E-02
49800 8.041 -1.883 -6.997E-03 -2.694E-01 2.417E-02 3.045E-02
49776 8.052 -1.879 -5.195E-03 -3.048E-02 7.028E-03 7.942E-03
49777 8.129 -1.898 -4.670E-03 -6.129E-02 1.479E-02 8.015E-03
49778 8.259 -1.926 -3.373E-03 -9.261E-02 2.443E-02 8.150E-03
Average 8.050 -1.884 -5.891E-03 -1.205E-01 1.534E-02 1.845E-02
170:center 8.052 -1.877 -5.099E-03 1.111E-10 2.243E-11 5.142E-11
Table 4.6 - ?'11 and ?'22 at the sensor location
versus temperature (Unit: MPa)
T (Celsius) ?'22 ?'11 ?'22-?'11
-151.0 7.608 -1.749 9.357
-133.4 7.690 -1.772 9.462
-93.2 7.795 -1.809 9.604
-48.2 7.886 -1.824 9.709
-23.6 7.937 -1.846 9.783
0.6 7.993 -1.869 9.863
25.1 8.051 -1.883 9.934
49.9 8.262 -1.945 10.207
75.1 8.515 -2.020 10.535
100.6 8.735 -2.089 10.824
125.9 8.932 -2.128 11.060
48
Assuming that the chip-on-beam structure is made of one material, finite element
simulations have given ?'11 and ?'22 at sensor location at 25 oC (assuming node #49788)
as shown in Table 4.7.
Table 4.7 - ?'11 and ? '22 at the sensor location at 25oC (Unit: MPa)
Composite material ?'22 ?'11 ?'22-?'11
Silicon 4.538 -1.424 5.962
FR-406 4.442 -1.087 5.529
As the table shows, ?'11 and ?'22 at sensor location is about half compared with the real
case. If we consider a strip without a silicon chip and adhesive material, only uniaxial
stress is induced as reflected in the following:
2bh D)3F(L? ?= Eq. (4.3.1)
where F = 1N, (L-D) = 2 x 10-2 m, b = 650 mil (1.651 x 10-2 m), and h = 22.67 mil (5.758
x 10-4 m). Substitution of these dimensions into the equation above yields ?'22 = 10.961
MPa.
4.4 Extraction of Piezoresistive Coefficients, B1 and B2
For a given fixed temperature, the piezoresistive coefficients can thus be evaluated by
performing controlled experiments where the resistance changes of the resistor sensors are
monitored as a function of applied force. Large errors can be induced in the measured
resistance changes, and thus in the values of the extracted piezoresistive coefficients, if the
temperature varies between measurements during experiments. Hence, much attention has
49
been given to minimize the errors in the resistance change by keeping a given fixed
temperature for sufficient time duration. As seen in Section.4.2, using an individual-
temperature reference, B1 and B2 are given by
2'
11F
2'
22F
90
'
11F0
'
22F
22'
11F
2'
22F
0
'
11F90
'
22F
1 )(?)(?
S?S?B ,
)(?)(?
S?S?B
?
?=
?
?= Eq. (4.4.1)
where S0 and S90 of (111) silicon are determined by experiments performed over
temperature, and ?'11F and ?'22F at sensor location are obtained by computer simulations.
It is obvious that ?'11F and ?'22F for a 1-N force are ?'11 and ?'22, respectively. Hence the
piezoresistive coefficients B1 and B2 can be extracted. Using Table 4.2 and Table 4.6, B1
and B2 with temperatures are presented in Table 4.8 and their corresponding plots are
shown in Fig. 4.14.
Table 4.8 - Extracted B1 and B2 with temperature (Unit: TPa-1)
T( oC) B1p B2p B1n B2n
-133.4 608.2 -179.6 -166.2 135.3
-93.2 542.1 -157.3 -155.4 127.3
-48.2 447.3 -154.9 -152.7 112.7
-23.6 422.8 -148.6 -146.8 105.7
0.6 398.7 -142.6 -141.0 101.0
25.1 366.2 -133.9 -133.7 97.4
49.9 315.3 -117.3 -114.5 88.1
75.1 271.0 -100.1 -102.5 78.9
100.6 239.6 -92.4 -93.2 70.2
50
Piezoresistance Coefficients with Temperature
B1p = -1.556E+00T + 3.931E+02
R2 = 9.933E-01
B2n = -2.741E-01T + 1.004E+02
R2 = 9.912E-01
B1n = 3.113E-01T - 1.324E+02
R2 = 9.128E-01
B2p = 3.554E-01T - 1.344E+02
R2 = 9.391E-01
-300
-200
-100
0
100
200
300
400
500
600
700
-150 -100 -50 0 50 100 150
T (Celsius)
(T
Pa
)-1
Fig. 4.14 - Extracted B1 and B2 with temperature
Test results show that all coefficients decrease monotonically with increasing temperature
in magnitude.
4.5 Relationship between Piezoresistive Coefficients with Different
Temperatures
In this section, the relationship between piezoresistive coefficients with different
temperatures will be explained. Also, determination of the values of ??s will be described.
In order to extract a complete set of pi-coefficients ( 441211 ?and , ?,? ) for both p- and n-
type sensors, B3 (as well as B1 and B2) is needed. To be discussed later is the requirement
of hydrostatic tests for the extraction of B3.
51
4.5.1. General Resistance Change Equations at a fixed Temperature Reference
In most prior investigations, calibration of the piezoresistive coefficients has been
performed at room temperature. Such restriction limits the accuracy of test chip stress
measurements made at other temperatures. In this work, we have performed an extensive
experimental study on temperature dependence of the piezoresistive behavior of silicon.
From Chapter 3, the general resistance change equations are described. For convenience
of discussion, those equations are repeated here:
circle6 (001) plane: with respect to the unprimed axes
+ ...] ?T?T + [2sin ??
sin)]?(?+ ??+ [?cos)]?(?+ ?? = [?R?R
2
211244
2
3311122211
2
3322121111
??+?+
?+?+
Eq. (4.5.1)
circle6 (001) plane: with respect to the primed axes
...]TT[ 2sin )?-?(???
sin ? 2 ??? + ?2 ???
cos ?2 ??? + ?2 ??? = R?R
2
21
'
121211
'
3312
2'
22
441211'
11
441211
2'
22
441211'
11
441211
+??+??+?++
??
?
?
??
? ?
?
??
?
? ++?
?
??
?
? ?++
??
?
?
??
? ?
?
??
?
? ?+?
?
??
?
? ++
Eq. (4.5.2)
52
circle6 (111) plane: with respect to the primed axes
...] ?T?T[
2sin])?-B(B)?-B(B22[
sin])?-B(B22?B?B?[B
cos])?-B(B 22?B?B? [BR?R
2
21
'
1221
'
1323
2'
2332
'
333
'
221
'
112
2'
2332
'
333
'
222
'
111
+?+?+
?++
?++++
??++=
Eq.(4.5.3)
Simply, the general expression of resistance is expressed as follows:
?(?T)]?}[?)T(f1){0,0R(T)R( ++?+=?? , Eq. (4.5.4)
where T)]??([? ?+ is the summation of combined stress terms and
. . . T?2T?T)f( 221 +?+?=? Eq. (4.5.5)
: ?1, ?2 . . . temperature coefficients of resistance
. . . T?2T?T)?( 2(2)(1) +?+?=? Eq. (4.5.6)
: ?(1), ? (2). . . temperature coefficients of piezoresistance
in which ?T = Tm-Tref is the difference between the measurement temperature and the
reference temperature at which the reference resistance R(0,0) is measured. From Eq.
(4.5.4), the normalized change in resistance is given as follows:
T)]??([?T)f(
0) R(0,
)0 ,0(R)T ,(R
R
?R
?++?=
???=
Eq. (4.5.7)
in which we let T) ,(R
refT
?? be the resistance with the change in stress ? and the change
in temperature ?T with respect to the reference temperature.
53
Much attention has to be given in order to reduce the error induced by measurement
of temperature for the calculation of ? . To minimize the discrepancy from the set point,
maintaining temperature of the oven at the set point for a long period of time is necessary.
Mathematical calculations based on the many-valley model predict a decrease of the
piezoresistance effect with increasing temperature [23, 96]. Based on the quantum
physics, the doping concentration can be calculated and the piezoresistance coefficients
decrease with increasing impurity concentration. For p-type, when the temperature is into
the range where most of the carriers are freezing out onto donors and acceptors, a similar
tendency is observed.
In our case, two-dimensional states of stress are induced. For a 0-degree resistor
sensor on the (111) silicon surface, the term T)]??([? ?+ in Eq. (4.5.7) is given by
'22B2'11B1 )]T(B[)]T(B[
21
???++???+ Eq. (4.5.8)
Similarly, for a 90-degree resistor sensor on the (111) silicon surface, the term
T)]??([? ?+ in Eq. (4.5.7) is given by
'11B2'22B1 )]T(B[)]T(B[
21
???++???+ Eq. (4.5.9)
In Eqs. (4.5.8) and (4.5.9), )T(
1B
?? and )T(2B
1
?? are given by
])T)(5([61)T(
])T)(([21)T(
N)N(
44
)N(
12
n
1N
)N(
11B
N)N(
44
)N(
12
)N(
11
n
1N
B
2
1
????+?=??
??+?+?=??
?
?
=
= Eq. (4.5.10)
54
4.5.2 General Resistance Change Equations with Varying Temperatures
Now, consider two reference temperatures, A and B. Typical responses of
resistance subjected to the change in temperature and applied force are shown in Fig. 4.15
where resistance changes with varying temperatures and applied force (or stress). The
equation of relationship between piezoresistive coefficients with different temperatures is
derived below.
Fig. 4.15 - The plot of resistance with temperature and stress
From PA to PG in Fig. 4.15, the resistance equation becomes:
T)]?(?[? (0,0)RT)]f([1 (0,0)R } T)]?(?[?T)f((0,0){1R T) , (?R
AAAA
AAAA
?++?+=
?++?+=? Eq. (4.5.11)
Similarly, from PB to PG, the resistance equation may be expressed as
? ? (0,0)R(0,0)R ?}?(0,0){1R0) , (?R
BBB
BBB
+=
+= Eq. (4.5.12)
55
in which we let T) ,(R
refT
?? be the resistance with ? (change in stress) and ?T (change
in temperature) with respect to refT . By comparing Eq. (4.5.11) and Eq. (4.5.12), it is
apparent that
(0,0)RT)]f([1 (0,0)R 0), , (?RT), (?R BABA =?+=? Eq. (4.5.13)
in which ? ABTTT refmea ?=?= . Subtracting Eq. (4.5.12) from Eq. (4.5.11) leads to
?(0,0)?RT)]?(?(0,0)[?R BBAAA =?+ Eq. (4.5.14)
To get the same ? in the equation as in Eq. (4.5.14)
(0,0)R T)](?(0,0)[?R?
B
AAA
B
?+= Eq. (4.5.15)
Hence
BB
A
B
AA )]T(f1[?)0,0(R
)0,0(RT)](?[? ??+==?+ Eq. (4.5.16)
Substitution of ABT ?=? into Eq. (4.5.16) yields
TAAA ? )]T(f1[T)](?[? ?+?+=?+ Eq. (4.5.17)
Hence
ATAA ?? )]T(f1[T)(? ??+=? ?+ Eq. (4.5.18)
From Eq. (4.5.7),
AA )]T(fRR[1)T( ??????=?? Eq. (4.5.19)
Substitution of the equation above into Eq. (4.5.17) yields
)]T(fRR[1)T(f1 1TA ?????+=? ?+ Eq. (4.5.20)
In addition, substitution of Eq. (4.5.17) into Eq. (4.5.7) yields
56
???++?=
???+?+?=???=
?+ TA
AA
A
AA
)]T(f1[)T(fR?R
)]T([)T(f)0, 0(R )0, 0(R)T,(RR?R
Eq. (4.5.21)
In differentiating both terms of Eq. (4.5.21) with respect to ? (stress) we derive at the
following:
TA? )]T(f1[)R?R( ?+?+=??? Eq. (4.5.22)
Thus
)R?R()T(f1 1? TA ????+=?+ Eq. (4.5.23)
In Section 4.2, the reference temperature for each case is varied. On the other hand, in
order to have a constant reference temperature (e.g. room temperature) for all cases, the
general equations for any ?T are given by using Eq. (4.5.21):
)BB( T)]f([1 T)f( RR '222'111
0
0 ?+??++?=? Eq. (4.5.24)
)BB( T)]f([1 T)f( RR '221'112
90
90 ?+??++?=? Eq. (4.5.25)
For any small ?T, it is obvious that 1)T(f1 ??+ . Hence the equations become
)BB( T)f( RR '222'111
0
0 ?+?+?=? Eq. (4.5.26)
)BB( T)f( RR '221'112
90
90 ?+?+?=? Eq. (4.5.27)
57
Re-using the notations, ,F?and ?F?? ' F22'22' F11'11 ?? and )R?R(dFdS
?
?
? ? in Eqs.
(4.5.24) and (4.5.25) leads to
]?B?B)][T(f1[S '22F2'11F10 +?+= Eq. (4.5.28)
]?B?B)][T(f1[S '22F1'11F290 +?+= Eq. (4.5.29)
Solving for B1 and B2 in Eq. (4.5.28) and Eq. (4.5.29) yields the findings below:
2'
11F
2'
22F
90
90'
11F
0
0'
22F
2
2'
11F
2'
22F
0
0'
11F
90
90'
22F
1
)(?)(?
)]T(f1[
S?
)]T(f1[
S?
B
)(?)(?
)]T(f1[
S?
)]T(f1[
S?
B
?
?+??+=
?
?+??+=
Eq. (4.5.30)
Assuming 0)]T(f1[)T(f1 ?+=?+ and 90)]T(f1[)T(f1 ?+=?+ yields
2'
11F
2'
22F
90
'
11F0
'
22F
2
2'
11F
2'
22F
0
'
11F90
'
22F
1
)(?)(?
S?S?
)T(f1
1B
)(?)(?
S?S?
)T(f1
1B
?
?
?+=
?
?
?+=
Eq. (4.5.31)
in which 1B and 2B are combined piezoresistive parameters with the change in
temperature ?T. For instance, Eq. (4.5.31) reduces to Eq. (4.4.1) when ? 0T = . Accurate
calibration of f (?T) is essential for this purpose.
58
4.6 Summary
In this chapter, finite element analysis is used to calculate the stress states applied to
the calibration samples. Specially, stressing sensing test chips are used to measure the
mechanical stresses on the rosette site. Stress values (?'11 and ? '22) monotonically
increase with rising temperatures in magnitude. Relationship between piezoresistive
coefficients with different temperatures is derived. It is observed that all coefficients
decrease monotonically with increasing temperature in magnitude.
59
CHAPTER 5
HYDROSTATIC TESTS AND TCR MEASUREMENTS
5.1 Hydrostatic Tests
Calibration of (111) test chips may not be accomplished completely by using
four-point bending tests. For extracting a complete set of pi-coefficients
( 441211 ?and , ?,? ) for both p- and n-type sensors, hydrostatic tests are needed. If a
sensor is subjected to hydrostatic pressure ( p??? '33'22'11 ?=== ), the resistivity is
expressed as the following [50]:
)T(fp?)Tf()p?2(?)0,0?( )0,0?(?(?,?T)??? p1211 ?+=?++?=?= Eq. (5.1.1)
In addition, p? is called ?pressure coefficient? and is given by )?2(?? 1211p +?=
)BB(B 321 ++?= .
In Eq. (5.1.1), )T,(? ?? is the stressed resistivity component with temperature change ?T,
and ?(0,0) is the unstressed resistivity component. It is noteworthy that Eq. (5.1.1) is
independent of the sensor orientation on both (100) and (111) wafer planes, implying that
any silicon conductor remains isotropic under a hydrostatic pressure. If we neglect the
dimensional changes of resistor sensor during loading, the result is
60
)T(fp)T(f)p?2(????R?R p1211 ?+pi=?++?=? Eq. (5.1.2)
where p? may be evaluated if R?R , p, and T)f(? are known. For determining T),f(?
further TCR (temperature coefficient of resistance) measurements are required. The
expected values for the pressure coefficients are small in both p- and n-type silicon, so
direct measurement of the values is quite difficult. In fact, it has been shown theoretically
that 1211 2pi??pi in n-type silicon so that p? should be zero for n-type material [23].
5.2 TCR Measurements and f(?T)
During the application of pressure, a change in T is inevitable. For example, it has
been observed that the temperature of the hydraulic fluid changes by 0.6~0.7 oC because
of a change in p by 13 MPa at room temperature. Thus temperature-compensated
hydrostatic measurements are not possible. In order to evaluate p? accurately,
temperature effects must be removed from hydrostatic calibration data. For 0p = , Eq.
(5.1.2), T)f(? may be extracted by measuring the normalized resistance change with
respect to a temperature change of a resistor sensor in a temperature controllable chamber.
The resistance value of p- and n-type silicon with varying temperatures and forces
applied to the Chip-on-Board structure appear in Tables 5.1 and 5.2 for p-type and n-type
resistors, respectively. Also, their corresponding plots are shown in Figs. 5.1 and 5.2. In
this work, the case in which ? =0 is considered.
61
Table 5.1 - P-type resistance with varying temperatures and forces
(Unit: kohm)
T (Celsius) F = 0 F = 0.15 N F = 0.3 N F = 0.45 N
-133.4 13.888 13.883 13.877 13.871
-93.2 10.365 10.362 10.358 10.356
-48.2 9.927 9.923 9.921 9.918
0.6 10.163 10.160 10.158 10.155
25.1 10.368 10.366 10.363 10.360
49.9 10.812 10.810 10.807 10.805
75.1 11.283 11.280 11.278 11.275
100.6 11.840 11.838 11.836 11.833
P-type Resistance with Temperature
8
9
10
11
12
13
14
15
-150 -100 -50 0 50 100 150
T (Celsius)
R
(k
oh
m)
F = 0 F = 0.15 N F = 0.3 N F = 0.45 N
Fig. 5.1 - Plot of p-type resistance with varying temperatures and forces
62
Table 5.2 - N-type resistance with varying temperatures and forces
(Unit: kohm)
T (Celsius) F = 0 F = 0.15 N F = 0.3 N F = 0.45 N
-133.4 1.742 1.743 1.743 1.744
-93.2 1.900 1.900 1.901 1.901
-48.2 2.049 2.049 2.050 2.050
0.6 2.218 2.218 2.219 2.219
25.1 2.309 2.309 2.310 2.310
49.9 2.421 2.422 2.422 2.422
75.1 2.518 2.518 2.519 2.519
100.6 2.619 2.619 2.619 2.620
N-type Resistance with Temperature
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
-150 -100 -50 0 50 100 150
T (Celsius)
R(
ko
hm
)
F = 0 F = 0.15 N F = 0.3 N F = 0.45 N
Fig. 5.2 - Plot of n-type resistance with varying temperatures and forces
As seen in Figs. 5.1 and 5.2, the change in resistance due to varying force at fixed
temperature is relatively small. Resistance is measured over the temperature range of
63
-133 to 100 oC. Freeze-out is observed for the p-type sample during low temperature
operation. In the freeze-out temperature region, the concentration of electrons and holes
drops significantly, which leads to an increase in resistance. Contrary to the case for p-
type, the slope of resistance over the whole temperature range for the n-type samples is
observed to be linear due to heavy doping. That is, freeze-out is not observed for the n-
type resistor sensors.
In Tables 5.3 and 5.4, ?R with applied force at each temperature of measurement is
presented. The plots are shown in Figs. 5.3 and 5.4 for p-type and n-type resistors
respectively. The reference value at each temperature of measurement is the unstressed
(F = 0) resistance.
Table 5.3 - ?R for a p-type resistor with varying temperatures and forces
(Unit: kohm)
T (Celsius) F = 0 F = 0.15 N F = 0.3 N F = 0.45 N
-133.4 0 -5.660E-03 -1.139E-02 -1.709E-02
-93.2 0 -3.390E-03 -6.780E-03 -9.590E-03
-48.2 0 -3.300E-03 -5.620E-03 -8.590E-03
0.6 0 -2.780E-03 -5.350E-03 -8.170E-03
25.1 0 -2.360E-03 -5.260E-03 -7.970E-03
49.9 0 -2.450E-03 -5.210E-03 -7.850E-03
75.1 0 -2.510E-03 -5.140E-03 -7.570E-03
100.6 0 -2.090E-03 -4.250E-03 -6.445E-03
64
P-type Resistor: ?R with Temperature
(Reference: F = 0)
-1.8E-02
-1.6E-02
-1.4E-02
-1.2E-02
-1.0E-02
-8.0E-03
-6.0E-03
-4.0E-03
-2.0E-03
0.0E+00
-150 -100 -50 0 50 100 150
T (Celsius)
????R
(k
oh
m)
F = 0 F = 0.15 N F = 0.3 N F = 0.45 N
Fig. 5.3 - Plot of ?R for a p-type resistor with varying temperatures and forces
Table 5.4 - ?R for an n-type resistor with varying temperatures and forces
(Unit: kohm)
T (Celsius) F = 0N F = 0.15 N F = 0.3 N F = 0.45 N
-133.4 0 4.700E-04 9.400E-04 1.310E-03
-93.2 0 3.800E-04 7.400E-04 1.160E-03
-48.2 0 3.700E-04 7.500E-04 1.130E-03
0.6 0 3.500E-04 7.700E-04 1.070E-03
25.1 0 3.300E-04 6.800E-04 1.070E-03
49.9 0 3.200E-04 6.700E-04 1.003E-03
75.1 0 2.700E-04 6.300E-04 9.730E-04
100.6 0 2.500E-04 6.200E-04 9.630E-04
65
N-type Resistors: ?R with Temperature
(Reference: F = 0)
0.0E+00
2.0E-04
4.0E-04
6.0E-04
8.0E-04
1.0E-03
1.2E-03
1.4E-03
-150 -100 -50 0 50 100 150
T (Celsius)
????R
(k
oh
m)
F = 0 F = 0.15 N F = 0.3 N F = 0.45 N
Fig. 5.4 - Plot of ?R for an n-type resistor with varying temperatures and forces
The general expression of resistance is repeated as shown below:
T)]?)?([?T)f(R(0,0){1T) ?, R( ?++?+=? Eq. (5.2.1)
where T)]??([? ?+ is the summation of combined stress terms as presented in Eq.
(4.5.10). The equation of normalized change in resistance is given as follows:
T)]??([?T)f(R?R ?++?= Eq. (5.2.2)
in which the condition ? 0T = leads to
? ? R?R = Eq. (5.2.3)
66
For our heavily-doped n-type resistor, R is observed to increase monotonically with
rising temperature, whereas ? decreases monotonically with rising temperature.
Therefore, the two effects tend to cancel out for n-type, which results in a relatively small
change in ?R with varying temperatures as shown in Fig. 5.4. The same cancellation still
holds for our more lightly-doped p-type samples from -50 oC up to 100 oC. On the other
hand, below -50 oC, both R and ? increase with decreasing temperature resulting in a
drastic increase in ?R (see Fig. 5.3). Although the magnitude of ?'11 and ?'22 increases
monotonically with rising temperature (see Table 4.6), the change in R and ? with
temperature is relatively large in Eq. (5.2.3).
Typical resistance changes over the temperature range of -175 to 125 oC are presented
in Table 5.5. The plots are shown in Fig. 5.5 and Fig. 5.6 for p-type and n-type resistors
respectively, for the F = 0 case.
Table 5.5 - Resistance with varying temperatures (Unit: kohm)
T (Celsius) R0 p R90 p R0 n R90 n
-175 20.411 21.096 1.631 1.621
-150 14.359 15.101 1.693 1.685
-125 11.642 12.121 1.758 1.751
-100 10.394 10.737 1.833 1.828
-75 9.871 10.137 1.913 1.908
-50 9.730 9.936 1.997 1.994
-25 9.826 9.987 2.084 2.083
0 10.085 10.209 2.174 2.174
25 10.476 10.566 2.269 2.271
50 10.943 10.994 2.361 2.367
75 11.465 11.496 2.452 2.500
100 12.048 12.057 2.544 2.554
125 12.615 12.661 2.639 2.648
67
Furthermore, the resistance of 0 and 90 degree sensors for p- and n-type silicon with
varying temperatures are presented and compared in Table 5.5. Both orientations are very
close to each other for p- and n-type resistors (see Figs. 5.5 and 5.6) as should be
expected.
P-type Resistance Change with Temperature
0
5
10
15
20
25
-200 -100 0 100 200
T (Celsius)
R
(ko
hm
)
R0p
R90p
Fig. 5.5 - Resistance of p-type sensors with varying temperatures
for ? = 0 and ? = 90o
N-type Resistance Change with Temperature
1.5
1.7
1.9
2.1
2.3
2.5
2.7
2.9
-200 -100 0 100 200
T (Celsius)
R
(ko
hm
)
R0n
R90n
Fig. 5.6 - Resistance of n-type sensors with varying temperatures
for ? = 0 and ? = 90o
68
As described in Section 4.5, accurate calibration of f (?T) is essential to determine the
pressure coefficients with temperature. To this purpose, extensive calibrations of
resistance with varying temperatures are performed in a temperature controllable
chamber. Typical experimental resistance change over the temperature range of -180 to
130 oC with a step size 2.5oC is shown in Figs. 5.7 and 5.8.
P-type Resistance Change with Temperature
0
5000
10000
15000
20000
25000
30000
-200 -150 -100 -50 0 50 100 150
T (Celsius)
Re
sis
tan
ce
(oh
m)
Fig. 5.7 - P-type resistance change with varying temperatures
N-type Resistance Change with Temperature
0
500
1000
1500
2000
2500
3000
-200 -100 0 100 200T (Celsius)
Re
sis
tan
ce
(oh
m)
Fig. 5.8 - N-type resistance change with varying temperatures
69
As depicted in Fig. 5.7 and Fig. 5.8, the resistance of the p-type sensor elements
increases rapidly at low temperatures as a result of carrier freeze-out in lightly doped p-
type silicon material. Clearly, the temperature dependence of resistance cannot be
modeled by a linear term, but requires a more general formulation for f (?T). On the
other hand, the n-type sensor elements have a much higher doping level and are much
less affected by freeze-out over the measurement range.
From Fig. 5.7 and Fig. 5.8, the normalized change in resistance induced by ?T,
defined as ),T(f ? may be evaluated assuming ? = 0:
...TT?T? T)f(R(0,0)R(0,0)-T) 0, R(R?R 33221 +??+?+?=?=?= Eq. (5.2.4)
Through the use of Eq. (5.2.4), temperature coefficients of resistance for p- and n-type
sensors may be obtained, as shown in Figs. 5.9 and 5.10, respectively. For p-type sensors,
)T(f ? may clearly not be modeled by a linear term but requires higher order terms. The
n-type sensors also exhibit some curvature in )T(f ? .
70
f(?T) of P-type Sensors
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
-250 -200 -150 -100 -50 0 50 100 150
?T = T - 25oC
f( ????
T)
Fig. 5.9 - f (?T) of p-type sensors with varying temperatures
f(?T) of N-type Sensors
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
-300 -200 -100 0 100 200
?T = T- 25oC
f( ????
T)
Fig. 5.10 - f (?T) of n-type sensors with varying temperatures
71
Tables 5.6 and 5.7 present the results of least square fits to the )T(f ? data in Figs.
5.9 and 5.10, based upon 24 sensors of each type. For the p-type resistors, a 5th-order
equation is used. On the other hand, for n-type, a 3rd-order equation provides a good
curve fitting.
Table 5.6 - Temperature coefficients of resistance with varying temperatures
(p-type resistors)
?1 ?2 ?3 ?4 ?5
Average 1.416E-03 9.258E-06 1.738E-08 -2.710E-10 -4.572E-12
Std. Dev 7.398E-05 1.283E-06 1.401E-08 2.559E-10 1.260E-12
Table 5.7 - Temperature coefficients of resistance with
varying temperatures (n-type resistors)
?1 ?2 ?3
Average 1.663E-03 2.497E-07 -4.784E-09
Std. Dev 1.119E-05 7.945E-08 2.984E-10
On the other hand, at a given reference temperature, a quadratic equation is enough to fit
)T(f ? over any small range of temperature, especially for the hydrostatic tests, for both
p-type and n-type samples. In Table 5.8, only 1? of the average of 32 specimens is
presented at each reference temperature. From Table 5.8, it is observed that 1? increases
with rising temperature for p-type. However, for n-type sensors, 1? is less defined at low
temperatures, but decreases with rising temperature in higher temperature regions. In
Table 5.9, 2? as well as 1? measured at a given reference temperature for p-type sensors
is displayed.
72
Table 5.8 - Average of 1? of 32 specimens for p- and n-type sensors
measured at a given reference temperature (Unit: 10-3/oC)
T (Celsius) P- type sensors N-type sensors
-151.0 -24.5 1.81
-133.4 -12.6 1.72
-93.2 -3.42 1.74
-48.2 -0.191 1.73
0.6 1.18 1.74
25.1 1.56 1.70
49.9 1.76 1.61
75.1 1.82 1.58
100.6 1.98 1.44
125.9 2.11 1.38
Table 5.9 - Average of 1? (Unit: 10-3/oC) and 2? (Unit: 10-3/oC2)
of 32 specimens for p-type sensors measured at a given reference
temperature
T (Celsius) ?1 ?2
-151.0 -22.0 9.39E-01
-133.4 -10.7 4.04E-01
-113.4 -6.60 -7.43E-02
-93.2 -3.45 5.55E-02
-71.4 -1.16 -2.02E-02
-48.2 -0.191 1.50E-02
-23.6 0.573 -4.09E-03
0.6 1.14 -7.28E-03
25.1 1.49 9.06E-03
49.9 1.73 8.11E-04
75.1 1.76 4.90E-02
100.6 2.04 8.24E-03
125.9 2.06 1.91E-03
In Table 5.10 and Table 5.11, experimental calibration results for the piezoresistive
coefficients of silicon with room-temperature reference as a function of temperature are
presented and compared and contrasted with the values from individual reference
temperature by using Eq. (4.5.31):
73
2'
11F
2'
22F
90
'
11F0
'
22F
2
2'
11F
2'
22F
0
'
11F90
'
22F
1
)(?)(?
S?S?
)T(f1
1B
)(?)(?
S?S?
)T(f1
1B
?
?
?+=
?
?
?+=
Eq. (5.2.5)
Table 5.10- Piezoresistive coefficients with room-temperature reference (1/TPa)
T(Celsius) B1p B2p B1n B2n
-133.4 607.2 -164.7 -165.9 136.3
-93.2 548.7 -151.6 -155.2 127.8
-48.2 447.9 -151.5 -152.5 113.0
0.6 397.3 -141.1 -140.7 101.0
25.1 366.2 -133.9 -133.7 97.4
49.9 314.9 -118.0 -114.6 87.9
75.1 269.6 -100.7 -104.6 78.2
100.6 238.2 -93.3 -93.3 70.0
Table 5.11 - Piezoresistive coefficients with a given reference temperature (1/TPa)
T(Celsius) B1p B2p B1n B2n
-133.4 608.1 -179.6 -166.2 135.3
-93.2 542.1 -157.3 -155.4 127.3
-48.2 447.3 -154.9 -152.7 112.7
0.6 398.7 -142.6 -141.0 101.0
25.1 366.2 -133.9 -133.7 97.4
49.9 315.3 -117.3 -114.5 88.1
75.1 271.0 -100.1 -102.5 78.9
100.6 239.6 -92.4 -93.2 70.2
As seen from Table 5.10 and Table 5.11, the piezoresistive coefficients of silicon as a
function of temperature are in good agreement. Furthermore, by using )T(f ? for a fixed
reference temperature (e.g., at room temperature), we may obtain )T(f ? for arbitrary
reference temperature. From the general equation of resistance with 0=? ,
74
T)]f(R(0,0)[1T) 0, R( ?+=? Eq. (5.2.4)
Assuming "'refm TTTTT ?+?=?=? , where mT is the measurement temperature, leads
to
)]T(f1)][T(f1)[0,0(R )]T(f1)[T,0(R)]TT(f1)[0,0R()TT,0(R "
T
'
"
T
'"'"'
'
'
?+?+=
?+?=?+?+=?+?
?
?
Eq. (5.2.5)
In the equation, )T(f "T' ?? is defined as )T(f "? with the reference temperature 'T? .
Combining Eq. (5.2.4) and Eq. (5.2.5) yields
)T(f1 )T(f)TT(f)T(f1 )T(f)T(f)T(f '
'"'
'
'
"
T' ?+
???+?=
?+
???=?
? Eq. (5.2.6)
3
3
2
21 TTT)T(f ??+??+??=? in Eq. (5.2.6) yields
3'
3
2'
2
'
1
"2'
3
'
21
2"'
32
3"
3"
T )?T(?)?T(??T?1
T])?T(?3?T?2?[)?T()?T?3(?)?T(?)T(f
' +++
?+++++=?
?
Eq. (5.2.7)
Therefore
)T(f1)(
)T(f1
T3)(
)T(f1
)T(3T2)(
'
3
T3
'
'
32
T2
'
2'
3
'
21
T1
'
'
'
?+
?=?
?+
??+?=?
?+
??+??+?=?
?
?
?
Eq. (5.2.8)
where 'T1)( ?? , 'T2 )( ?? , and 'T3 )( ?? are the temperature coefficients of resistance at a
reference temperature 'ref TT ?+ . From Eq. (5.2.8), once )T(f ? is obtained for a fixed
reference temperature, f(?T) at any other reference temperature may be determined .
75
In our cases, extensive TCR measurements are performed especially around the
temperature of measurement.
5.3 Analysis of Hydrostatic Tests and TCR Measurements
Once the TCR measurements are completed, the die is then ready for hydrostatic tests.
In our work, hydrostatic tests are carried out on the resistor sensors of the JSE-WB100C
test chip (see Fig. 5.11). The (111) silicon test chips have dimensions of 100 x 100 mil.
Fig. 5.11 - Quarter model of JSE-WB100C for TCR and hydrostatic tests
One corner of the die is attached to a specially designed printed circuit board (PCB) using
a small amount of die attachment adhesive (ME 525). Lavenir software is used to design
the PCB and its picture is shown in Fig. 5.12. The resistor sensors on the die are wire-
bonded to the pads on the PCB to get the electrical access as shown in Fig. 5.13. During
the TCR measurements, the Chip-on-Board is inserted into the connector inside the test
fixture as shown in Fig. 5.14.
76
Fig. 5.12 - Specially designed PCB for TCR and hydrostatic tests
Fig. 5.13 - Wire-bonded chip on the board for TCR and hydrostatic tests
Only one corner of the die is attached to the board to satisfy the condition
p??? '33'22'11 ?=== . Then, the wire-bonded die is subjected to temperature change with
monitoring each resistance of the sensors through the use of computer-controlled GPIB
devices. An OMEGA CN3251 temperature controller is used for controlling the
temperature of calibration. However, the recording of the actual temperature is made by a
thermistor inside the vessel. It is noteworthy that the temperature reading by the
thermistor is very close to that of the temperature controller. The hydrostatic experiments
77
are performed on the test chips over the temperature range of -25 oC to100 oC. In order to
increase the temperature of fluid, a resistance heater is used inside the pressure vessel. To
lower the temperature of fluid, we use liquid nitrogen, which is injected into a specially
designed box surrounding the pressure vessel. In addition, another microprocessor-based
temperature controller CN76000 is used to monitor the temperature inside the box. The
input type is a ?K? thermocouple. In order to reach the equilibrium between the
temperature inside the box and the fluid temperature inside the vessel, a longer duration
of time is required.
Fig. 5.14 - Hydrostatic test chamber
78
Fig. 5.15 - Hydrostatic test setup
Fig. 5.16 - Expanded hydrostatic test setup for high and low temperatures
79
For TCR measurements, a program is used to control the oven temperature. Like
hydrostatic tests, the measurements of resistances are made by computer-controlled GPIB
devices. During the TCR measurements, no stress is applied. During hydrostatic tests, the
die is put into the pressure vessel, whose set up is shown in Fig. 5.14. A pump connected
to the vessel is used to generate pressure, as shown in Fig. 5.15. In addition, the expanded
hydrostatic test setup for high and low temperatures is shown in Fig. 5.16.
Typical change in resistance with varying temperatures is shown in Fig. 5.17 for a p-
type resistor in which the measured and temperature-induced normalized resistance
changes are plotted together. The shape of the curve is slightly parabolic. Similar
behavior is observed for n-type resistors.
Measured and Temperature Induced ?R/R
Measured
Temperature induced
0.0E+00
5.0E-04
1.0E-03
1.5E-03
2.0E-03
2.5E-03
3.0E-03
3.5E-03
4.0E-03
0 5 10 15
Pressure, p (MPa)
?R
/R
Fig. 5.17 - An example of measured and temperature induced ?R/R for p-type resistors
Subtraction of the effect of temperature from the resistance change determines the
pressure coefficient. A nonlinear change in the temperature of the hydraulic fluid for a p-
80
type resistor is observed during the application of pressure, as shown in Fig. 5.18. Similar
behavior is observed for n-type resistors.
Temperature Change with Pressure
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 5 10 15
Pressure, p (MPa)
Fig. 5.18 - Fluid temperature change with pressure for p-type resistors
Adjusted resistance change appears to be linear with pressure as expected in Eq.
(5.1.2), as shown in Fig. 5.19, whose slope of the curve corresponds to the piezoresistive
coefficient p? as given in the following equation:
p?
)pBB(B
)p2?(?)T(fR?R
p
321
1211
=
++?=
+?=??
Eq. (5.3.1)
Typical results of the pressure coefficient of (111) silicon for each temperature are
displayed in Appendix C.
81
?R/R- f(?T): Adjusted Hydrostatic Calibration
0.00E+00
5.00E-04
1.00E-03
1.50E-03
2.00E-03
2.50E-03
0 2 4 6 8 10 12 14 16
Pressure, p (MPa)
????R
/R
-f(
T)
Fig. 5.19 - Adjusted hydrostatic calibration for p-type sensors
The pressure coefficient data for p-type and n-type sensors over the temperature
range of -25oC to100oC are presented in Table 5.12, and graphs of the data appear in Figs.
5.20 and Fig. 5.21. It is observed that the pressure coefficient for both the p- and n-type
decreases with rising temperature. Compared with p-type silicon, the pressure coefficient
of n-type silicon has relatively small value. This phenomenon is in good agreement with
the experimental results of the hydrostatic-pressure coefficient of n-type silicon [25],
which showed that the hydrostatic-pressure coefficient is small at all concentration in n-
type silicon at 300K. It may be noted that the values in Table 5.12 reflect only the
counted specimens among 32. The values in Table 5.8 are used for the determination of
temperature-induced normalized resistance change. It is noteworthy that the unit is 1/TPa
for the pressure coefficient and all pi-coefficients throughout this dissertation.
82
Table 5.12 - Pressure coefficient data for p- and n-type sensors versus temperature
T (oC) # of sensors (p-type) Average (1/TPa) Std. Dev. (1/TPa) # of sensors (n-type)) Average (1/TPa) Std. Dev. (1/TPa)
-23.6 13 165.4 20.8 11 40.0 11.9
0.6 13 153.4 30.5 10 36.3 9.9
25.1 30 145.9 34.1 22 31.0 8.1
49.9 26 135.1 29.1 15 19.2 24.5
75.1 20 119.7 35.3 10 3.8 22.1
100.6 22 108.9 33.5 16 -3.1 14.9
Pressure Coefficient (p-type sensors)
y = -4.507E-01x + 1.550E+02
R2 = 9.919E-01
100
110
120
130
140
150
160
170
-50 -25 0 25 50 75 100 125
T (Celsius)
1/T
Pa
Fig. 5.20 - Pressure coefficient versus temperature for p-type sensors
As mentioned earlier, it has been argued theoretically that the pressure coefficient should
be zero for n-type silicon and these results are consistent with the theory.
83
Pressure Coefficient (n-type sensors)
y = -3.712E-01x + 3.512E+01
R2 = 9.570E-01
-10
0
10
20
30
40
50
60
70
80
-50 -25 0 25 50 75 100 125
Temperature (Celsius)
1/T
Pa
Fig. 5.21 - Pressure coefficient versus temperature for n-type sensors
As described in Chapter 4, four-point bending tests are used to measure B1 and B2,
whereas hydrostatic tests are used to measure B3. A complete set of pi-coefficients
( 441211 ?and , ?,? ) can be extracted by combining hydrostatic tests and four-point
bending tests:
3
?2??B
6
?5??B
2
???B
441211
3
441211
2
441211
1
?+=
?+=
++=
Eq. (5.3.2)
32144
3212
32111
B2BB
BB2
B3B3B
?+=pi
?=pi
+?=pi
Eq. (5.3.3)
84
circle6 Pi-coefficients with neglect of ? ( ? = 1)
In this work, ? is defined as the ratio of axial portion to the sum of axial and
transverse portion of a resistor. More details about ? will be discussed in Chapter 7. Pi-
coefficients of p-type silicon versus temperature with neglect of ? are presented in Table
5.13, whose graphs of the data appear in Figs. 5.22 and Fig. 5.23. As reflected in Table
5.13, all the coefficients in p-type material decrease monotonically with increasing
temperature over the temperature range of -25oC to 100oC.
Table 5.13 - Pi-coefficients of p-type silicon versus temperature with neglect of ?
T(Celsius) B1p B2p B3p pi11 p pi12 p pi44 p pip
-23.6 422.8 -148.6 -439.6 -450.2 142.4 1153.4 165.4
0.6 398.7 -142.6 -409.5 -402.0 124.3 1075.1 153.4
25.1 366.2 -133.9 -378.2 -366.7 110.4 988.7 145.9
49.9 315.3 -117.3 -333.1 -332.1 98.5 864.2 135.1
75.1 271.0 -100.1 -290.6 -300.5 90.4 752.1 119.7
100.6 239.6 -92.4 -256.1 -251.5 71.3 659.4 108.9
85
Piezoresistive Coefficients
B1p
B3p
B2p
-500
-400
-300
-200
-100
0
100
200
300
400
500
-50 -25 0 25 50 75 100 125T (Celsius)
1/T
Pa
Fig. 5.22 - Combined pi-coefficients for p-type silicon versus temperature
with neglect of ?
The smallest coefficients pi11 and pi12 exhibit discrepancies in sign with the data from
the literature [6].
Piezoresistive Coefficients
pi11p
pi12p
pi44p
-600
-400
-200
0
200
400
600
800
1000
1200
1400
-50 -25 0 25 50 75 100 125
T (Celsius)
1/T
Pa
Fig. 5.23 - Pi-coefficients for p-type silicon versus temperature with neglect of ?
86
Likewise, pi-coefficients of n-type silicon versus temperature with neglect of ? are
presented in Table 5.14, whose graphs of the data appear in Figs. 5.24 and 5.25. The data
in Table 5.14 indicate that B3 and pi44 in n-type material exhibit an opposite tendency in
magnitude. It may be noted that they are the smallest coefficients in n-type material. The
other coefficients are well defined.
Table 5.14 - Pi-coefficients for n-type silicon versus temperature with neglect of ?
T(Celsius) B1n B2n B3n pi11 n pi12 n pi44 n pip
-23.6 -146.8 105.7 1.1 -460.6 210.3 -43.3 40.0
0.6 -141.0 101.0 3.7 -432.9 198.3 -47.4 36.3
25.1 -133.7 97.4 5.3 -410.0 189.5 -46.9 31.0
49.9 -114.5 88.1 7.2 -357.2 169.0 -40.8 19.2
75.1 -102.5 78.9 19.8 -279.8 138.0 -63.2 3.8
100.6 -93.2 70.2 26.1 -225.5 114.3 -75.2 -3.1
Piezoresistive Coefficients
B1n
B3n
B2n
-200
-150
-100
-50
0
50
100
150
-50 -25 0 25 50 75 100 125
T (Celsius)
1/T
Pa
Fig. 5.24 - Combined pi-coefficients for n-type silicon versus temperature
with neglect of ?
87
Piezoresistive Coefficients
pi11n
pi44n
pi12n
-600
-500
-400
-300
-200
-100
0
100
200
300
-50 -25 0 25 50 75 100 125
T (Celsius)
1/T
Pa
Fig. 5.25 - Pi-coefficients for n-type silicon versus temperature with neglect of ?
circle6 Pi-coefficients with consideration of ?
For effective and realistic pi-coefficients, consideration of ? is necessary. As will
be discussed in Chapter 7, 21 BB + is constant, regardless of ?. Thus B3 in Eq. (5.3.2)
and pi44 in Eq. (5.3.3) are independent of ?. The pi-coefficients of p-type silicon versus
temperature with consideration of ? are presented in Table 5.15, whose graphs of the data
also appear in Figs. 5.26 and 5.27. As observed in the cases ? = 1, the smallest
coefficients pi11 and pi12 also exhibit discrepancies in sign with the data from the literature
[6]. However, the consideration of ? significantly reduces pi11 and pi12 in magnitude.
88
Table 5.15- Pi-coefficients for p-type silicon versus temperature
with consideration of ?
T(Celsius) B1p B2p B3p pi11 p pi12 p pi44 p pip
-23.6 491.2 -217.0 -439.6 -176.6 5.6 1153.4 165.4
0.6 463.5 -207.4 -409.5 -142.8 -5.3 1075.1 153.4
25.1 426.1 -193.8 -378.2 -127.1 -9.4 988.7 145.9
49.9 367.1 -169.1 -333.1 -124.9 -5.1 864.2 135.1
75.1 315.4 -144.5 -290.6 -122.9 1.6 752.1 119.7
100.6 279.4 -132.2 -256.1 -92.3 -8.3 659.4 108.9
Piezoresistive Coefficients
B1p
B3p
B2p
-600
-400
-200
0
200
400
600
-50 -25 0 25 50 75 100 125
T (Celsius)
1/T
Pa
Fig. 5.26 - Combined pi-coefficients for p-type silicon versus temperature
with consideration of ?
89
Piezoresistive Coefficients
pi44p
pi12p
pi11p-400-200
0
200
400
600
800
1000
1200
1400
-50 -25 0 25 50 75 100 125
T (Celsius)
1/T
Pa
Fig. 5.27 - Pi-coefficients for p-type silicon versus temperature with consideration of ?
Likewise, pi-coefficients of n-type silicon versus temperature with consideration of
? are presented in Table 5.16, whose graphs of the data are also shown in Figs. 5.28 and
5.29. The data in Table 5.16 indicate that B3 and pi44 in n-type material exhibit an
opposite tendency in magnitude. It may be noted that they are the smallest coefficients in
n-type material. The other coefficients are well defined.
Table 5.16 - Pi-coefficients for n-type silicon versus temperature
with consideration of ?
T(Celsius) B1n B2n B3n pi11 n pi12 n pi44 n pip
-23.6 -177.0 135.9 1.1 -581.4 270.7 -43.3 40.0
0.6 -170.0 130.0 3.7 -548.8 256.3 -47.4 36.3
25.1 -161.4 125.1 5.3 -520.8 244.9 -46.9 31.0
49.9 -138.8 112.4 7.2 -454.4 217.6 -40.8 19.2
75.1 -124.2 100.6 19.8 -366.6 181.4 -63.2 3.8
100.6 -112.8 89.8 26.1 -303.9 153.5 -75.2 -3.1
90
Piezoresistive Coefficients
B1n
B3n
B2n
-200
-150
-100
-50
0
50
100
150
-50 -25 0 25 50 75 100 125
T (Celsius)
1/T
Pa
Fig. 5.28 - Combined pi-coefficients for n-type silicon versus temperature
with consideration of ?
Piezoresistive Coefficients
pi11n
pi44n
pi12n
-600
-500
-400
-300
-200
-100
0
100
200
300
-50 -25 0 25 50 75 100 125
T (Celsius)
1/T
Pa
Fig. 5.29 - Pi-coefficients for n-type silicon versus temperature with consideration of ?
91
5.4 Summary
In this chapter, TCR measurements and hydrostatic tests have been described. The
pressure coefficient ppi may be determined by combining TCR measurements and
hydrostatic tests. It may be stressed that the pressure coefficient ppi , for the (111) silicon
test chips, is orientation independent because the conductor remains isotropic under
hydrostatic pressure. In order to determine ,ppi the data of adjusted resistance versus
pressure are essential. By subtraction of temperature-induced resistance change from the
total resistance change at each data point, adjusted resistance versus pressure data may be
obtained. For n-type, ppi is very small in each data point, as expected from the
approximation 1211 ?2? ?? [23]. On the other hand, for p-type, ppi obviously decreases in
magnitude with rising temperature. In this chapter, a complete set of pi-coefficients
( 441211 ?and , ?,? ) may be extracted by performing hydrostatic tests.
92
CHAPTER 6
SILICON STRESS-STRAIN RELATIONS AND MEASUREMENT OF YOUNG?S
MODULUS OF SILICON
6.1 Silicon Stress-Strain Relations
Silicon exhibits linear elastic material behavior and the generalized Hooke?s Law, the
most general formula of linear elastic stress-strain relations, is given by [15]:
klijklij ?C? = Eq. (6.1.1)
where ij? and kl? are the stress and strain components, and ijklC are the components of
the stiffness tensor. Inverting Eq. (6.1.1) gives
klijklij ?S? = Eq. (6.1.2)
where ijklS are the compliance components. Also, the transformation relations for the
reduced index stress and strain components can be expressed as indicated below [15]:
'?1??? ?T? ?= Eq. (6.1.3)
'?t??? ?T? = Eq. (6.1.4)
where the coefficients T?? are elements of a six by six transformation matrix related to
the direction cosines for the unprimed and primed coordinate systems. Also, note that ??
93
and ?? are the stress and strain tensor components in the unprimed system, respectively,
whereas ?'? and ?'? are those components in a rotated primed coordinate system. Inverting
Eq. (6.1.4) leads to
?)T(? 1t' ?= Eq. (6.1.5)
If Eq. (6.1.2) is plugged into Eq. (6.1.5), the result is
S?)T(? 1t' ?= Eq. (6.1.6)
Finally, substitution Eq. (6.1.3) into Eq. (6.1.6) yields the relations between stress and
strain in a rotated primed coordinate system as follows:
'11t' ?TS)T(? ??= Eq. (6.1.7)
If an unprimed coordinate system is assumed, 1t )T( ? and 1T? in Eq. (6.1.7) simplify to
unit matrices. Thus Eq. (6.1.7) reduces down to Eq. (6.1.2).
6.2 Elastic Constants of Silicon by Equations
Very few tests have been performed on silicon. However, the three independent
elastic constants, stiffness coefficients, of silicon were measured by several researchers.
The relationships for the piezoresistive effect based on the strain components, and the
expressions relating the piezoresistive coefficients and the elastoresistive coefficients
using the elastic coefficients for cubic crystals were given in [9-10]. For instance,
McSkimin [88-89] obtained those constants by using ultrasonic measurement techniques
in which ultrasonic waves were transmitted into a specimen and measurement of the
reflections within specimen yielded values for the velocities of wave propagation and the
elastic constants. Using this technique, second-order elastic constants of single crystals
94
were experimentally measured. The first row in Table 6.1 presents the summary of
literature values for the stiffness coefficients of silicon by Wortman [90]. The three
compliance coefficients can be evaluated by using Eq. (6.1.6). The results are s11=7.68 x
10-12 Pa-1, s12=-2.14 x 10-12 Pa-1 and s44=1.26 x 10-11 Pa-1 as presented in the second row in
Table 6.1.
Table 6.1 - Literature values for the stiffness coefficients
of silicon [90]
c11 c12 c44
165.7 GPa 63.9 GPa 79.6 GPa
s11 s12 s44
7.68 x 10-12 Pa-1 -2.14 x 10-12 Pa-1 1.26 x 10-11 Pa-1
Another method is the use of the three compliance coefficients ( 11s , 22s , and 44s ). Based
on the values in Table 6.1, the elastic modulus and Poisson?s ratio can be evaluated. The
cubic nature of silicon lattice leads to orthotropic properties. Hence, Young?s Modulus E
and Poisson?s ratio ? are dependent on direction. Generally, simple isotropic values for
the elastic properties are commonly used. However, in some situations, greater accuracy
may be needed and achieved by employing the directional nature of these properties.
Equation (6.1.7) allows direct calculation of Young?s modulus. In Eq. (6.1.7),
95
?
?
?
?
?
?
2?
2?
2?
?
?
?
?
?
?
?
?
?
and
?
?
?
?
?
?
?
?
?
?
?
?
'
12
'
23
'
13
'
33
'
22
'
11
'
12
'
23
'
13
'
33
'
22
'
11
'
6
'
5
'
4
'
3
'
2
'
1
'
12
'
23
'
13
'
33
'
22
'
11
'
6
'
5
'
4
'
3
'
2
'
1
??
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
=
??
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
=
??
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
=
??
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
Eq. (6.2.1)
To obtain Young?s modulus in one direction (e.g., x1), setting all other stresses to
zero and solving for '
1
'
1
?
? allows direct calculation of E
1. Similarly, setting all other
stresses to zero and solving for '
1
'
2
?
?? gives Poisson?s ratio ?
1.
The direction cosines li, mi, and ni are required in order to determine 1-1-t and T) (T
in Eq. (6.1.7). If the conductor orientation is rotated counter-clockwise by ? from the
specified axis (e.g., x?1 axis [110]), li, mi, and ni are determined by solving 3 simultaneous
equations as follows. For instance, i =1
circle6 Plane Equation: The conductor is on (001) plane.
0)1(n)0(m)0(l 111 =++ barb2right 0n1 = Eq. (6.2.2)
circle6 Inner Product: Note that the angle between the x?1 axis and the resistor orientation
is ? .
?=++ cos(0)n)21(m)21(l 111 barb2right ?=+ cos2ml 11 Eq. (6.2.3)
circle6 Unit Vector: The magnitude of the unit vector is unity.
1)nm(l 212121 =++ Eq. (6.2.4)
96
There are two roots in these simultaneous equations. It may be noted that one of two
corresponds to the root for the clockwise rotation.
For the following discussion, the geometry for silicon wafers of interest here is
given in Fig. 6.1. Miller indices are introduced to describe directions and planes in crystal.
The rules for selecting the crystallographic axes are presented in [93].
Fig. 6.1 - Silicon wafer geometry
For the (001) plane, the direction cosines are
??
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
+?+??
+?+?
=
??
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
++?++?
+?+?
=
?
?
?
?
?
?
?
?
?
?
1 0 0
0 )4?cos( )4?sin(
0 )4?sin( )4?cos(
1 0 0
0 )2?4?sin( )2?4?cos(
0 )4?sin( )4?cos(
n m l
n m l
n m l
333
222
111
Eq. (6.2.5)
97
where ? is the angle of counter-clockwise rotation from the x?1 axis [110]. By Eq. (6.1.7),
E and ? are expressed as
44
2
12
2
11
2 )s2(cos)s2(cos2)s2(sin2[1
4E
?+?+?+= Eq. (6.2.6)
44
2
12
2
11
2
44
2
12
2
11
2
'
1
'
2
12 s)2(coss)2(cos2s)]2(sin1[2
s)2(coss)]2(sin1[2s)2(cos2
?
??
?+?+?+
???++??=?= Eq. (6.2.7)
In Eqs. (6.2.6) and (6.2.7), the periods of E and ? on the (001) silicon plane are 2?
respectively (see Figs. 6.2-6.5). The maximum and the minimum values of E are 169.1
GPa and 130.1 GPa respectively. For ?, the maximum and the minimum values are 0.278
and 0.062 respectively. E and ? show anisotropic characteristics on the (001) silicon
plane. Note that three compliance coefficients 11s , 22s , and 44s are based on the literature
value of stiffness coefficients 11c , 22c , and 44c from Wortman [90].
E of silicon: (001) plane
100
110
120
130
140
150
160
170
180
190
200
0 45 90 135 180 225 270 315 360
Degree from [110]
E
(G
Pa
)
Fig. 6.2 - E on the (001) silicon
98
E of silicon: (001) plane
-200
-150
-100
-50
0
50
100
150
200
-200 -150 -100 -50 0 50 100 150 200
Fig. 6.3 - E on the (001) silicon
Fig. 6.4 - ? on the (001) silicon
99
Fig. 6.5 - ? on the (001) silicon
E and ? of the (001) silicon at varying angular locations with respect to [110] axis are
plotted in Figs. 6.2 through 6.5. For example, the positive x-axis is the [110] direction
and 45o represents the [010] direction. Similarly, -45o is the [100] direction.
On the (111) silicon plane, the direction cosines are
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??+??+??
?????+?
=
?
?
?
?
?
?
?
?
?
?
1 1 1
6
2cos-
6
cos
2
sin
6
cos
2
sin
62sin- 2cos6sin 2cos6sin
n m l
n m l
n m l
333
222
111
Eq. (6.2.8)
where ? is the angle of counter-clockwise rotation from the x?1 axis 0]1[1 . By Eq. (6.1.7),
E and ? are given by
100
441211 s2s2s
4E
++= Eq. (6.2.9)
441211
441211
'
1
'
2
12 3s6s6s
s10s2s
?
??
++
?+?=?= Eq. (6.2.10)
In Eqs. (6.2.9) and (6.2.10), the periods of E and ? are infinity. That is, E and ? are
constant with ? as shown in Figs. 6.6 through 6.9, where E and ? exhibits an isotropic
characteristic on the (111) silicon plane where the elastic properties (E and ?) are
independent of direction (E = 169.1 GPa, ? = 0.262).
E of silicon: (111) plane
140
150
160
170
180
190
200
0 90 180 270 360
Counter-clockwise rotation from [1-10]
E
(G
Pa
)
Fig. 6.6 - E on the (111) silicon
101
E of silicon: (111) plane
-200
-150
-100
-50
0
50
100
150
200
-200 -100 0 100 200
Fig. 6.7 - E on the (111) silicon
? of silicon, (111) plane
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 90 180 270 360
Counter-clockwise rotation from [1-10]
Fig. 6.8 - ? on the (111) silicon
102
? of silicon: (111) plane
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
Fig. 6.9 - ? on the (111) silicon
For the 0)1(1 plane, if we assume that ? is the angle of clockwise rotation from the axis
[111], the direction cosines are
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?????+???+??
?+???+??+?
=
?
?
?
?
?
?
?
?
?
?
0 21 21
6
2cos
3
sin
6
cos
3
sin
6
cos
3
sin
3
cos
6
2sin
3
cos
6
sin
3
cos
6
sin
n m l
n m l
n m l
333
222
111
Eq. (6.2.11)
103
If ?= 0,
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
0 21 21
6
2
6
1
6
1
3
1
3
1
3
1
n m l
n m l
n m l
333
222
111
Eq. (6.2.12)
By Eq. (6.1.7), E and ? on the 0)1(1 silicon plane are given by
]12 ) sin225(sin sin2268) cos)[(1s2s36(2ss
1E
2
22
44121111
????++?+???=
Eq. (6.2.13)
]12 ) sin225(sin sin2268) cos)[(1s2s36(2ss
) sin4232sin221)(122ss(ss
?
??
2
22
44121111
244
121112
'
1
'
2
12 ???
?++?+???
??????+
?=?= Eq. (6.2.14)
Similarly, the periods of E and ? on the 0)1(1 silicon plane are ?, respectively, as shown
in Figs. 6.10 and 6.11. The maximum and minimum values of E are 187.9 GPa and 130.1
GPa respectively. For ?, the maximum and minimum values are 0.35 and 0.15
respectively. E and ? are dependent on the direction on the 0)1(1 silicon plane. For
example, E and ? of [111] direction are 187.9 GPa and 0.180 respectively.
104
E of silicon: (1-10) plane
120
130
140
150
160
170
180
190
200
0 45 90 135 180 225 270 315 360
Clock-wise rotation from [111]
E(
GP
a)
Fig. 6.10 - E on the 0)1(1 silicon
? of silicon: (1-10) plane
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0 45 90 135 180 225 270 315 360
Clock-wise rotation from [111]
Fig. 6.11 - ? on the 0)1(1 silicon
105
In summary:
circle6 For any crystallographic direction of silicon, E and ? may be expressed in terms
of compliance coefficients (s11, s12, and s44).
circle6 E and ? are dependent on the direction of the silicon.
It should be noted that E and ? are based on the literature values of stiffness coefficients,
c11, c12, and c44 from Wortman [90]. Further, using other literature values of stiffness
coefficients from different authors, E values are presented and compared in Table 6.2.
Table 6.2 - E values for different directions and different authors
(Unit: GPa)
Author\Direction [100] [010] [001] [110] [1-10] [11-2] [111]
Mcskimi 130.80 130.80 130.80 169.71 169.71 169.71 188.38
Wortman 130.13 130.13 130.13 169.10 169.10 169.10 187.85
Hall 130.02 130.02 130.02 168.96 168.96 168.96 187.68
6.3 Measurement of the Elastic Constants by Deflection of Beams
Kang [15] used a strain gauge technique to measure E of silicon using a four-point
bending fixture in which strain gauges are mounted on the surface of specimen strips.
Usually, a micro-tester has been used to measure E. However, due to some limitations of
a micro-tester for measuring E of stiff materials such as silicon, the ?Deflection of Beams?
theorem [91] with a four-point bending fixture is introduced, as presented in Fig. 6.12, in
our analysis.
106
Fig. 6.12 - Deflection of a beam in a four-point bending fixture
In Fig. 6.12, L is the distance between two supports and the deflection at distance ?a?
from the adjacent support is
6EI 4a)(3LFa?
2 ?
= Eq. (6.3.1)
In Eq. (6.3.1), the new notation 'F is defined as
6I 4a)(3LFaF
2
' ?? Eq. (6.3.2)
where the moment of inertia I is defined as 12bh
3
for the rectangular of beam in which b is
the width and h is the thickness of the beam. Deflection ? may be measured by reading a
micro-positioner. The applied force F is measured by a load-cell. In our case, the
sensitivity shows 8.58 x 10-3 mV/gram. Combining Eq. (6.3.1) and Eq. (6.3.2) yields
E ?F' = where E is easily obtained by evaluating the slope of ? with respect to 'F . An
Example is presented in Table 6.3.
107
Table 6.3 - Example: Measurement of E using deflection of beams
? (?m) load cell (mV) W (gram) F (N) F?
0 55.691 0.00 0.0000 0.00E+00
50 55.901 24.5 0.1199 6.35E+12
100 56.117 49.7 0.2433 1.29E+13
150 56.333 74.8 0.3666 1.94E+13
200 56.554 100.6 0.4929 2.61E+13
250 56.772 126.0 0.6174 3.27E+13
In these measurements, L = 6.05 cm, a = 1.20 cm, b = 1.016 cm (= 400 mil), and h = 6.35
x 10-2 cm (= 25 mil), respectively. In order to obtain E, 'F in Eq. (6.3.3) is plotted with
respect to ? as shown in Fig. 6.13.
E calculation by deflection: E[100]
y = 1.302E+11x
R2 = 9.999E-01
0.0E+00
5.0E+12
1.0E+13
1.5E+13
2.0E+13
2.5E+13
3.0E+13
3.5E+13
0 50 100 150 200 250 300
? (?m)
F'
Fig. 6.13 - Plot of 'F with respect to ?
Further, extensive measurements are performed for several directions of silicon. For
each direction, 10 specimens are measured. Comparing the measured values in Table 6.4
with the literature values in Table 6.2, one finds a good agreement. In Table 6.4, E for the
108
[100] direction is 130.0 GPa, but E for the ]2[11 and ],01[1 ],101[ ],110[ directions are
about 169 GPa. It may be noted that E[100] = 132.8 GPa and E[110] = 170.3 GPa from the
literature [15].
Table 6.4 - Measurement of E for several directions of silicon (Unit: GPa)
#\Direction E [100] E [110] E[-110] E [1-10] E [11-2]
1 130.2 168.7 168.4 168.4 167.0
2 130.6 166.9 171.1 169.6 168.9
3 131.6 167.8 166.6 167.1 170.2
4 125.9 165.5 169.4 169.0 169.6
5 130.3 167.6 169.1 169.4 169.2
6 130.9 167.8 165.3 170.1 169.1
7 130.2 167.7 166.3 170.4 170.7
8 129.6 166.9 166.7 170.5 171.0
9 130.6 169.6 171.2 169.2 171.3
10 129.8 167.1 170.6 169.5 170.6
Avg. 130.0 167.6 168.5 169.3 169.8
Std. 1.53 1.09 2.16 1.00 1.28
In Table 6.5, the temperature dependence of E for the ]2[11 direction is presented.
For this work, a special four-point bending apparatus was constructed and integrated into
an environmental chamber capable of temperatures from -185 oC to +300 oC. As
expected, it is observed that E for the ]2[11 direction decreases monotonically with
increasing temperature over the temperature range -150 oC to +150 oC as plotted in Fig.
6.14.
109
Table 6.5 - E for the ]211[ direction on the (111) silicon versus temperature (Unit: GPa)
#\T(?C) -133.4 -93.2 -48.2 0.6 25.1 49.9 75.1 100.6 125.9 151.5
1 173.9 172.7 171.4 168.7 168.0 167.0 166.5 165.5 164.3 164.2
2 174.1 172.2 170.6 169.8 168.2 168.0 166.7 166.0 165.6 165.5
3 173.8 172.1 172.0 169.6 168.8 169.4 169.0 167.9 166.4 165.9
4 173.6 172.2 169.2 168.5 168.9 168.8 168.0 168.2 167.7 164.2
5 172.5 172.8 170.4 169.6 169.6 169.3 168.1 166.2 167.4 164.2
6 173.4 173.3 171.4 170.0 168.2 167.5 166.9 167.8 166.0 165.0
7 171.6 172.5 171.3 169.7 170.7 169.7 168.5 166.4 168.1 167.1
8 170.5 170.1 171.4 170.3 169.6 168.7 168.6 167.0 167.0 165.6
9 170.7 171.9 171.9 170.8 170.5 168.6 168.6 166.7 167.1 166.2
10 174.5 173.4 169.6 168.2 168.5 168.0 167.7 168.0 166.6 166.7
Avg. 172.9 172.3 170.9 169.5 169.1 168.5 167.9 167.0 166.6 165.5
Std. 1.45 0.91 0.95 0.81 0.97 0.88 0.88 0.94 1.12 1.05
E[11-2] vs. temperature
165
166
167
168
169
170
171
172
173
174
175
-150 -100 -50 0 50 100 150 200
T (Celsius)
E
(G
Pa
)
Fig. 6.14 - Plot of E for the ]211[ direction on the (111) silicon versus temperature
Also, E for PCB material (FR-406) is measured over temperatures ranging from -133 oC
to +151 ?C as shown in Table 6.6. In this work, 10 specimens are used.
110
Table 6.6 - E for FR-406 versus temperature (Unit: GPa)
#\T(?C) -133.4 -93.2 -48.2 0.6 25.1 49.9 75.1 100.6 125.9 151.5
1 27.4 24.8 24.9 24.7 23.6 21.2 19.1 19.0 17.0 14.9
2 27.8 26.1 25.3 24.8 22.6 22.9 20.7 18.6 16.6 14.8
3 27.0 26.0 25.2 24.4 22.2 20.9 19.1 18.6 16.7 15.0
4 27.2 25.7 24.3 24.4 23.5 22.0 21.5 17.6 16.7 15.0
5 27.9 25.8 25.4 24.4 25.1 21.8 20.0 18.7 17.1 16.3
6 26.7 25.5 24.2 24.1 24.5 22.3 20.4 17.6 15.4 13.1
7 27.6 26.3 25.4 25.0 23.4 22.2 20.8 19.3 15.6 16.4
8 26.8 25.0 25.0 24.6 24.3 23.1 20.0 19.4 16.7 15.2
9 27.9 26.2 25.5 25.0 24.3 21.6 21.5 18.9 15.9 14.4
10 27.8 25.3 25.1 24.5 23.8 22.5 19.5 17.7 16.0 13.6
Avg. 27.4 25.7 25.0 24.6 23.7 22.1 20.3 18.6 16.4 14.9
Std. 0.46 0.52 0.46 0.29 0.87 0.70 0.88 0.68 0.59 1.02
Similarly, E for FR-406 with temperatures is plotted as shown in Fig. 6.15.
E for FR-406 vs. temperature
10
12
14
16
18
20
22
24
26
28
30
-150 -100 -50 0 50 100 150 200
T (Celsius)
E
(G
Pa
)
Fig. 6.15 - Plot of E for FR-406 versus temperature
Similarly, E for die attachment adhesive (ME 525) are measured over temperatures
ranging from -150 oC to +150 ?C as shown in Table 6.9 and its plot is shown in Fig. 6.16.
111
Table 6.7 - E of ME 525 versus temperature
(Unit: GPa)
T (Celsius) ME 525
-151.0 19.81
-133.4 18.46
-93.2 15.99
-48.2 13.70
0.6 12.00
25.1 10.43
49.9 9.85
75.1 8.75
100.6 7.72
125.9 4.98
151.5 0.89
E for ME525 vs. temperature
0
5
10
15
20
25
-200 -150 -100 -50 0 50 100 150 200
T (Celsius)
E
(G
Pa
)
Fig. 6.16 - Plot of E for ME525 versus temperature
E for the composite materials of the chip-on-beam samples for varying temperatures are
summarized in Table 6.8.
112
Table 6.8 - Summary: Measurement of E vs. temperature (Unit: GPa)
T (Celsius) ME 525 (111)silicon: ]211[ FR-406
-133.4 18.46 172.9 27.41
-93.2 15.99 172.3 25.57
-48.2 13.70 170.9 25.12
0.6 12.00 169.5 24.68
25.1 10.43 169.1 23.73
49.9 9.85 168.5 22.05
75.1 8.75 167.9 20.26
100.6 7.72 167.0 18.55
125.9 4.98 166.6 16.37
151.5 0.98 165.5 14.87
In this work, E values for silicon are calculated analytically and those values are in
agreement with experimental results. The expressions of E and ? for each direction are
summarized in Table 6.9 in which analytic values of E are compared with experimental
values.
113
Table 6.9 - The expressions of E and ? for each direction of silicon
6.4 Summary
In this chapter, it is observed that the cubic nature of the single crystal silicon
lattice leads to orthotropic material properties--that is, E (Young?s modulus) and ?
(Poisson?s ratio) are dependent upon the direction on the silicon surface. For any
crystallographic direction of silicon, E and ? may be expressed by compliance
coefficients (s11, s12, and s44). In this work, E of silicon has been calculated
analytically, and calculations are in agreement with experimental value by using
?Deflection of Beams? method. In addition, E values of the other composite materials
comprising the chip-on-beam specimens were obtained by this method.
114
CHAPTER 7
VAN DER PAUW STRUCTURE
7.1 Van der Pauw?s Theorem
Van der Pauw?s theorem is used to measure the specific resistivity of an arbitrary
shaped sample of constant thickness without isolated holes [78-80]. A flat sample of
conducting material with uniform thickness is shown in Fig. 7.1 where A, B, C, and D are
contacts on the conducting material. Also, a simple structure is shown in Fig. 7.2.
? ?
?
DB
C
A
?
Fig. 7.1 - A flat sample of conducting material with uniform thickness
115
A
D
B
C
Fig. 7.2 - A simple van der Pauw test structure
A current is injected through one pair of the contacts, and the voltage is measured
across another pair of contacts. CDAB,R is defined as the potential difference between the
contacts D and C divided by the current through contacts A and B . For an isotropic
conductor, Van der Pauw [78] demonstrated that two of theses measurement may be related
by
1)??tRexp()??tRexp( DABC,CDAB, =?+? Eq. (7.1.1)
where t is the thickness of the sample and ? is the isotropic resistivity. For an isotropic
conductor, DABC,CDAB, RR = . Hence, Eq. (7.1.1) can be simplified to
CDAB,Rln2?t? = Eq. (7.1.2)
116
The equation of the sheet resistance SR may be calculated using Eq. (7.1.2)
CDAB,S Rln2?t?R == Eq. (7.1.3)
The sheet resistance SR in Eq. (7.1.3) depends on CDAB,R . With no information of t , the
sheet resistance may be calculated. In addition, Van der Pauw [79-80] also extended Eq.
(7.1.1) to the following equation for anisotropic conductors of constant thickness:
1)???tR(exp)???tR(exp
21
BC,DA
21
AB,CD =?+? Eq. (7.1.4)
where 1? and 2? are the components of principal resistivity.
Price [81-82] developed the resistance equations for rectangular isotropic conductors:
]}2?1)(2nABBC{tanh[ln?t8?R
0n
CDAB, +?= ??
=
Eq. (7.1.5)
]}2?1)(2nBCAB{tanh[ln?t8?R
0n
CBAD, +?= ?
?
=
Eq. (7.1.6)
where aAB = and bBC = are the length of the sides of the rectangle shown in Fig. 7.3.
117
D C
A B
b
a
,
1x
,
2x
Fig. 7.3 - Isotropic rectangular VDP structure
For anisotropic conductors, Price [81-82] extended Eq. (7.1.5) and Eq. (7.1.6) as follows:
]}2?1)(2nABBC??{tanh[ln?t??8R
0n
'
11
'
22
'
22
'
11
CDAB, +?= ?
?
=
Eq. (7.1.7)
]}2?1)(2nBCAB??{tanh[ln?t??8R
0n
'
22
'
11
'
22
'
11
BCAD, +?= ?
?
=
Eq. (7.1.8)
where '11? and '22? are resistivity components of the principal axes.
Mian [49-50] demonstrated that the resistance of 0o and 90o VDP can be represented as
]}2?1)(2nABBC??{tanh[ln?t??8R
0n
'
11
'
22
'
12
'
22
'
11
0 +
???= ??
=
Eq. (7.1.9)
]}2?1)(2nBCAB??{tanh[ln?t??8R
0n
'
22
'
11
'
12
'
22
'
11
90 +
???= ??
=
Eq. (7.1.10)
Similarly, for 45=? (+45o/-45o VDP) [49-50],
118
]}2?1)(2nABBC?2?? ?2??{tanh[ln?t??8R
0n
'
12
'
22
'
11
'
12
'
22
'
11
'
12
'
22
'
11
45 +++
?+???= ??
=
Eq. (7.1.11)
]}2?1)(2nBCAB?2?? ?2??{tanh[ln?t??8R
0n
'
12
'
22
'
11
'
12
'
22
'
11
'
12
'
22
'
11
45- +?+
++???= ??
=
Eq. (7.1.12)
For the (001) silicon wafer, in-plane components of resistivity tensor are [49-50]
]?)?(?[?
]???2 ????2 ???1?[?
]???2 ????2 ???1?[?
'
121211
'
12
'
3312
'
22
441211'
11
441211'
22
'
3312
'
22
441211'
11
441211'
11
pi?=
++++?++=
+?+++++=
Eq. (7.1.13)
where 11? , 12? , and 44? are the unique piezoresistive coefficients of silicon. Also, 'ij? are
the stress components in the primed coordinate system. The function of temperature
coefficient of resistance (TCR) is f(?T), and ?T is the temperature change from the
reference temperature where the isotropic resistivity ? is evaluated.
For the (111) silicon wafer in-plane components of resistivity tensor are given by
[49-50]
])?B(B)?B(B22?[?
])?B(B22?B?B?B1?[?
])?B(B22?B?B?B1?[?
'
1221
'
1323
'
12
'
2332
'
333
'
221
'
112
'
22
'
2332
'
333
'
222
'
111
'
11
?+?=
?++++=
??+++=
Eq. (7.1.14)
where 1B , 2B , and 3B are a set of combined piezoresistive parameters that are related with
the on-axis piezoresistive coefficients by
3 ??2?B,6 ??5?B,2 ???B 441211344121124412111 ?+=?+=++= Eq. (7.1.15)
119
The resistivity values change with applied stress in Eq. (7.1.13) and Eq. (7.1.14), which
makes the resistance of a given VDP structure change with applied stress. This fact
indicates that VDP stress sensors may be used as potential sensors.
7.2 Experimental Results for the (111) Silicon
Figure 7.4 shows the layout of the BMW-2.1 test chip in which the resistor rosette
sensors and VDP test structures are fabricated on the (111) silicon surfaces. Rectangular
strips, each strip containing a series of chips, are cut along x?11 axis (??11 direction in Fig.
7.4) from BMW-2.1 test wafers. Measurements are performed by loading a strip in a four-
point bending fixture to apply uniaxial stress. For the VDP sensors as well as resistor
sensors, R0 and R90 are measured at various load conditions and are plotted as a function of
the applied stress. Injected through one pair of electrodes is 100 ?A in Fig. 7.4 in which 4
pads (electrodes) are numbered from 1 to 4. For example, in the case of R0 of the VDP
sensors, a current is injected through 1 and 4 or 2 and 3. Similarly, for R90 of the VDP
sensors, a current is injected through 1 and 2 or 3 and 4. The potential difference between
two electrodes is measured and the VDP resistance R? is calculated by dividing the
potential difference by the injected current. All the steps are controlled by using a
parametric analyzer.
120
Fig. 7.4 - The (111) silicon test chip, BMW-2.1
The resistances of VDP sensors, as well as resistor sensors, are measured under
unstressed and stressed conditions. The normalized resistance changes are evaluated
using
)0,0(R )0,0(R)T,(RR?R
?
??
?
? ???? Eq. (7.2.1)
where )0,0(R ? is the unstressed resistance. For sensors on the (111) surface, the
expression for a resistor sensor at angle ? with respect to the '1x axis is given as follows:
)T(f2sin])?B(B)?B(B22[
sin])?B(B22?B?B?[B
cos])?B(B22?B?B?[BR?R
'
1221
'
1323
2'
2332
'
333
'
221
'
112
2'
2332
'
333
'
222
'
111
?+??+?+
??++++
???++=
Eq. (7.2.2)
121
For uniaxial stress '11?=? with neglect of )T(f ? , Eq. (7.2.2.) simplifies to
? = B
R
R?
1
0
0 Eq. (7.2.3)
? = B
R
?R
2
90
90 Eq. (7.2.4)
in which the stress sensitivity is B1 and B2, respectively. Note that B1 and B2 are calculated
from separate tests for resistor sensors on the same wafer lot. Subtraction of Eq. (7.2.4)
from Eq. (7.2.3) leads to
?)B(B =
R
R? -
R
R?
21
90
90
0
0 ? Eq. (7.2.5)
in which the stress sensitivity is (B1-B2).
Typical results for the normalized resistance changes of R0 and R90 as a function of an
applied stress for p- and n-type resistor sensors are shown in Figs. 7.5 through 7.8. Note
that the stress sensitivity of #1 specimen in Tables 7.1 and 7.2 is plotted in Figs. 7.5 and 7.6
for p-type sensors, whereas Figs. 7.7 and 7.8 are for n-type sensors.
122
?R0/R0 vs. ?
(p-type Resistor Sensors)
y = 4.505E-04x + 8.978E-05
R2 = 9.998E-01
0.00E+00
5.00E-03
1.00E-02
1.50E-02
2.00E-02
2.50E-02
3.00E-02
0 10 20 30 40 50 60 70
? (MPa)
????R
/R
Fig. 7.5 - Typical stress sensitivity of p-type resistor sensors (R0)
?R90/R90 vs. ?
(p-type Resistor Sensors)
y = -1.225E-04x + 1.513E-04
R2 = 9.993E-01
-1.20E-02
-1.00E-02
-8.00E-03
-6.00E-03
-4.00E-03
-2.00E-03
0.00E+00
2.00E-03
0 20 40 60 80 100
? (MPa)
????R
/R
Fig. 7.6 - Typical stress sensitivity of p-type resistor sensors (R90)
123
?R0/R0 vs. ?
(n-type Resistor Sensors)
y = -1.987E-04x - 4.402E-04
R2 = 9.944E-01
-1.40E-02
-1.20E-02
-1.00E-02
-8.00E-03
-6.00E-03
-4.00E-03
-2.00E-03
0.00E+00
0 10 20 30 40 50 60 70
? (MPa)
????R
/R
Fig. 7.7 - Typical stress sensitivity of n-type resistor sensors (R0)
?R90/R90 vs. ?
(n-type Resistor Sensors)
y = 1.923E-04x - 2.140E-04
R2 = 9.957E-01
-2.0E-03
0.0E+00
2.0E-03
4.0E-03
6.0E-03
8.0E-03
1.0E-02
1.2E-02
1.4E-02
0 10 20 30 40 50 60 70
? (MPa)
????R
/R
Fig. 7.8- Typical stress sensitivity of n-type resistor sensors (R90)
124
Note that Table 7.1 represents the values of 10 experiments performed using different
resistor sensors of p-type silicon from the same wafer.
Table 7.1 ? Stress sensitivities of the (111) p-type resistor sensors (Unit: MPa-1)
Specimen Slope of ?R
0/R0 vs. ?
Slope of
?R90/R90 vs. ?
Slope ofuniF020
?R0/R0-?R90/R90 vs. ?
#1 450.5 -122.5 573.0
#2 479.1 -130.7 609.8
#3 442.4 -127.7 570.1
#4 465.4 -117.4 582.8
#5 428.5 -125.4 553.9
#6 487.6 -132.0 619.6
#7 465.5 -115.9 581.4
#8 468.6 -122.5 591.1
#9 465.7 -126.1 591.8
#10 435.1 -128.6 563.7
Average 458.8 -124.9 583.7
Std. Dev. 19.12 5.33 20.23
Similarly, the results of 10 experiments of n-type silicon are presented in Table 7.2.
Table 7.2 ? Stress sensitivities of the (111) n-type resistor sensors (Unit: MPa-1)
Specimen Slope of ?R
0/R0 vs. ?
Slope of
?R90/R90 vs. ?
Slope ofuniF020
?R0/R0-?R90/R90 vs. ?
#1 -198.7 192.3 -391.0
#2 -201.8 200.0 -401.8
#3 -214.7 203.7 -418.4
#4 -201.7 197.2 -398.9
#5 -197.8 201.6 -399.4
#6 -206.6 187.5 -394.1
#7 -198.3 189.6 -387.9
#8 -202.7 199.7 -402.4
#9 -211.0 201.6 -412.6
#10 -197.7 193.8 -391.5
Average -203.1 196.7 -399.8
Std. Dev. 5.89 5.58 9.67
125
Similar tests are performed on several VDP sensors from the same wafer. The measured
stress sensitivities for ten experiments are presented in Tables 7.3 and 7.4. Averages of the
measurements and the corresponding standard deviation are also presented in the tables,
where the magnification factor M is defined as the ratio of the sensitivity of VDP sensors to
the sensitivity of resistor sensors, which will be discussed in the next section.
Table 7.3 - Stress sensitivities of the (111) p-type silicon VDP structures (Unit: MPa-1)
Specimen Slope ofuniF020 ?R
0/R0 vs. ?
Slope of
?R90/R90 vs. ?
Slope of
?R0/R0-?R90/R90 vs. ? M Modified M
#1 1261.5 -869.4 2130.9 3.651 3.202
#2 1227.5 -874.1 2101.6 3.600 3.158
#3 1285.0 -882.3 2167.3 3.713 3.257
#4 1304.0 -870.7 2174.7 3.726 3.268
#5 1277.5 -863.5 2141.0 3.668 3.218
#6 1266.0 -883.0 2149.0 3.682 3.230
#7 1285.5 -865.4 2150.9 3.685 3.232
#8 1329.5 -832.3 2161.8 3.704 3.249
#9 1279.5 -825.6 2105.1 3.606 3.164
#10 1325.5 -843.4 2168.9 3.716 3.259
Average 1284.2 -861.0 2145.1 3.675 3.224
Std. Dev. 30.33 20.24 25.76 0.044 0.039
Table 7.4 - Stress sensitivities of the (111) n-type silicon VDP structures (Unit: MPa-1)
Specimen Slope ofuniF020 ?R
0/R0 vs. ?
Slope of
?R90/R90 vs. ?
Slope of
?R0/R0-?R90/R90 vs. ? M Modified M
#1 -687.1 686.8 -1373.9 3.436 3.014
#2 -680.0 688.4 -1368.4 3.423 3.002
#3 -675.3 704.3 -1379.6 3.451 3.027
#4 -693.8 694.8 -1388.6 3.473 3.047
#5 -703.4 727.5 -1430.9 3.579 3.140
#6 -715.5 694.6 -1410.1 3.527 3.094
#7 -673.0 700.5 -1373.5 3.435 3.014
#8 -690.7 728.2 -1418.9 3.549 3.113
#9 -690.4 700.5 -1390.9 3.479 3.052
#10 -654.5 699.3 -1353.8 3.386 2.970
Average -686.4 702.5 -1388.9 3.474 3.047
Std. Dev. 16.95 14.43 24.31 0.061 0.053
126
Typical results for the normalized resistance changes of R0 and R90 as a function of an
applied stress for p- and n-type VDP sensors are shown in Figs. 7.9 through 7.12. For R0 of
the VDP sensors, we have two pairs (see Fig. 7.4)--that is, a current is injected through 1
and 4 or 2 and 3. Similarly, for R90 of the VDP sensors, a current is injected through 1 and
2 or 3 and 4. The value in Tables 7.3 and 7.4 is an average of two pairs. Figures 7.9 and
7.10 are for p-type VDP sensors, whereas Figs. 7.11 and 7.12 are for n-type VDP sensors.
?R0/R0 vs. ?
(p-type VDP Sensors)
y = 1.281E-03x
R2 = 9.948E-01
Current: 2~3
y = 1.174E-03x
R2 = 9.955E-01
Current: 4~1
0.0E+00
1.0E-02
2.0E-02
3.0E-02
4.0E-02
5.0E-02
6.0E-02
7.0E-02
8.0E-02
9.0E-02
1.0E-01
0 20 40 60 80
? (MPa)
????R
/R
Fig. 7.9 - Typical stress sensitivity of p-type VDP sensors (R0)
127
?R90/R90 vs. ?
(p-type VDP Sensors)
y = -8.545E-04x
R2 = 9.998E-01
Current: 1~2
y = -8.936E-04x
R2 = 9.666E-01
Current:3~4
-7.00E-02
-6.00E-02
-5.00E-02
-4.00E-02
-3.00E-02
-2.00E-02
-1.00E-02
0.00E+00
0 20 40 60 80
? (MPa)
????R
/R
Fig. 7.10 - Typical stress sensitivity of p-type VDP sensors (R90)
?R0/R0 vs. ?
(n-type VDP Sensors)
y = -6.721E-04x
R2 = 9.986E-01
Current: 4~1y = -7.020E-04x
R2 = 9.998E-01
Current: 2~3
-4.50E-02
-4.00E-02
-3.50E-02
-3.00E-02
-2.50E-02
-2.00E-02
-1.50E-02
-1.00E-02
-5.00E-03
0.00E+00
0 10 20 30 40 50 60 70
? (MPa)
????R
/R
Fig. 7.11 - Typical stress sensitivity of n-type VDP sensors (R0)
128
?R90/R90 vs. ?
(n-type VDP Sensors)
y = 6.942E-04x
R2 = 9.983E-01
Current: 3~4
y = 6.794E-04x
R2 = 9.971E-01
Current: 1~2
0.00E+00
5.00E-03
1.00E-02
1.50E-02
2.00E-02
2.50E-02
3.00E-02
3.50E-02
4.00E-02
4.50E-02
0 10 20 30 40 50 60 70
? (MPa)
????R
/R
Fig. 7.12- Typical stress sensitivity of n-type VDP sensors (R90)
As observed in the resistor sensors, the experimental response of the VDP sensors to the
applied stress is also linear. It is apparent that the results of two pairs match well as
expected. As shown in Tables 7.3 and 7.4, p- and n-type sensors, the responses of the
difference of the normalized resistance changes versus stress is several times higher than
the responses experienced by the analogous 0o and 90o resistor sensors. For p-type
sensors, the sensitivity of VDP sensors is 3.68 times higher than that of corresponding
resistor sensors. However, for n-type sensors, it is found to be 3.47 times higher. The
measured magnification of p-type senosrs is observed to be higher than that of n-type
sensors.
129
7.3 Sensitivity Magnification Factor and ?
Mian [50] described how to calculate the modified magnification factor in the
serpentine resistors connected with doped resistive material. Mian [50] also presented the
value of magnification factor by the analytical and numerical methods. In our studies, we
checked and verified the magnification factor to be about 3.157 numerically by using a
MATLAB software program. The following notations are adopted in our work.
WLN = Eq. (7.3.1)
N represents the number of squares and L and W, respectively, represent the length and
width of the resistance R of the rectangular block of uniformly doped material. Naxial and
Ntransverse represent the number of squares in the axial and transverse parts of the resistance
R.
transverseaxial
axial
NN
N
+=? Eq. (7.3.2)
in which ? is the ratio of axial part to the sum of axial part and transverse part. If we
consider the resistor R0_eff, it is actually composed of resistor segments oriented in the 0o
and 90o directions, Hence
900eff_0 R)1(RR ??+?= Eq. (7.3.3)
The normalized change in resistance may be expressed as
T)f()?B(B22?B?B?BR?R
T)f()?B(B22?B?B?BR?R
'
2332
'
333
'
221
'
112
90
90
'
2332
'
333
'
222
'
111
0
0
?+?+++=
?+??++=
Eq. (7.3.4)
130
in which ...TTT)f( 221 +??+??=?
Thus
90
90
0
0
eff_0
eff_0
R
R)1(
R
R
R
?R ???+??= Eq. (7.3.5)
Similar calculation may be performed for the serpentine resistor at 90o orientation with
respect to the x?1 axis. The expression is
90
90
0
0
eff_90
eff_90
R
R
R
R)1(
R
?R ??+???= Eq. (7.3.6)
In our cases, Naxial and Ntransverse are estimated to be 143.2 and 9.37 squares, respectively.
Hence ? = 0.939.
According to Eq. (7.3.5) and Eq. (7.3.6), the results below can be found:
)RRRR)(1-2(R?RR?R
90
90
0
0
eff_90
eff_90
eff_0
eff_0 ????=? Eq. (7.3.7)
)RRRR(R?RR?R
90
90
0
0
eff_90
eff_90
eff_0
eff_0 ?+?=+ Eq. (7.3.8)
Assuming uniaxial stress '11? in Eq. (7.3.4) leads to
'
112
90
90
'
111
0
0
?BRR
?BRR
=?
=?
Eq. (7.3.9)
Substitution of Eq. (7.3.9) into Eqs. (7.3.7) and (7.3.8) yields
'11_eff2_eff1'1121
eff_90
eff_90
eff_0
eff_0 )?B-(B)?-B)(B1-2(
R
?R
R
?R =?=? Eq. (7.3.10)
131
'
11_eff2_eff1
'
1121
_eff90
_eff90
_eff0
_eff0 )?B(B)?B(B
R
?R
R
?R +=+=+ Eq. (7.3.11)
By Eqs. (7.3.10) and (7.3.11),
21_eff1 )B1(BB ??+?= Eq. (7.3.12)
21_eff2 B)B1(B ?+??= Eq. (7.3.13)
Thus
1-2 )B1-(BB _eff2_eff11 ??+?= Eq. (7.3.14)
1-2
)B1-(BB _eff1_eff2
2 ?
?+?= Eq. (7.3.15)
Subtraction and addition of Eq. (7.3.14) and Eq. (7.3.15) yield
_eff2_eff121 BBBB +=+ Eq. (7.3.16)
12 BBBB _eff2_eff121 ???=? Eq. (7.3.17)
Substitution ? = 0.939 into Eq. (7.3.17) yields )B-(B878.0)B-(B 212_eff1_eff = . Therefore,
the measured coefficients must be multiplied by the factor 1/0.878 (= 1.139). In the
previous section, the measured magnification factor is 3.675 for p-type silicon and 3.474
for n-type silicon. The modified magnification factors are 3.224 for the p-type and 3.047
for the n-type coefficients. These results still show discrepancies with the analytical value
132
of 3.157. Reasons for these discrepancies will be discussed in the next section where the
effects of dimensional changes during loading are considered.
7.4 Effects of Dimensional Changes of VDP and Resistor during Loading
Dimensional changes in VDP structures and resistors during loading have been
neglected in the analysis up to here. Strain-effects of VDP structures and resistor sensors
on piezoresistive coefficients and sensitivity magnifications will be discussed and
compared with the cases in which strain-effects are not considered.
7.4.1 Strain-effects of VDP Structures
Consider again the case in which uniaxial stress ? ?'11 = is applied as shown in Fig.
7.13. Application of load induces the change in the side length of the VDP.
Fig. 7.13 - Isotropic rectangular VDP structure under uniaxial stress
The relation between stress and strain is given by
133
)EL?L(?E? == Eq. (7.4.1)
in which L is the unstressed side-length and E is the Young?s modulus. By Eq. (7.4.1),
?LLL' += Eq. (7.4.2)
in which 'L is the stressed side-length. After application of uniaxial stress '11? , the side
length of the VDP structure is given by combining Eqs. (7.4.1) and (7.4.2):
)E?1L()L?L1L(L' +=+= Eq. (7.4.3)
If the sides of the unstressed square VDP structure are LAB and LBC, respectively, the length
of the stressed side AB is defined as AB and is given by
)E?(1LAB AB += Eq. (7.4.4)
in which LAB is the length of the unstressed side AB. Poisson's ratio is a measure of the
simultaneous change in elongation and in the cross-sectional area within the elastic range
during a tensile or compressive test. During a tensile test, the reduction in the cross-
sectional area is proportional to the increase in length in the elastic range by a
dimensionless factor called Poisson's ratio, defined as a ratio of sideways contraction to
length extension (? = - ?22/?11). Therefore, the length of the side BC is expressed as
)E??(1LBC BC ?= Eq. (7.4.5)
in which LBC is the length of the unstressed side BC. Likewise, the thickness 't of VDP
structure under stressed condition is given by
134
)E??-t(1t ' = Eq. (7.4.6)
in which t is the unstressed thickness of VDP structure. Combining Eqs. (7.4.4) and (7.4.5)
yields
)??-E ?E(LLBCAB
BC
AB += Eq. (7.4.7)
Plugging Eqs. (7.4.6) and (7.4.7) into Eqs. (7.1.7) and (7.1.8) yields
]}2?1))(2n?E ??E(LL??{tanh[ln??)-?t(E ??8ER
AB
BC
0n
'
11
'
22
'
22
'
11
CDAB, ++
??= ??
=
Eq. (7.4.8)
]}2?1)(2n)??-E ?E(LL??{tanh[ln??)-?t(E ??8ER
0n BC
AB
'
22
'
11
'
22
'
11
BCAD, +
+?= ??
=
Eq. (7.4.9)
in which '11? and '22? are resistivity components of the principal axes '1x and '2x that are
parallel and perpendicular to the wafer flat of the silicon. The nonzero resistivity
components are given by
])?-B(B)?B(B22?[?
])?B(B22?B?B?B1?[?
])?B(B22?B?B?B1?[?
'
1221
'
1323
'
12
'
2332
'
333
'
221
'
112
'
22
'
2332
'
333
'
222
'
111
'
11
+?=
?++++=
??+++=
Eq. (7.4.10)
For uniaxial stress '11?? = , Eq. (7.4.10) reduces to
0?
?)B?(1?
?)B?(1?
'
12
2
'
22
1
'
11
=
+=
+=
Eq. (7.4.11)
135
Equation (7.4.11) is substituted into Eqs. (7.4.8) and (7.4.9). By an analytic approach, the
normalized change in CDAB,R and AD,BCR may be calculated. In our case, 0AB,CD RR =
and 90AD,BC RR = , and the normalized resistance equation with ?T = 0 is
)0,0(R )0,0(R)0,(R)0,0(R )0,0(R)T,(RR?R
?
??
?
??
?
? ??=???= Eq. (7.4.12)
where R(0,0) and R(?,0) are the unstressed and stressed resistances, respectively. It is
obvious that the normalized resistance may be evaluated without knowing thickness t as
shown in the following equations:
1
]}2?)1n2(LL [tanh{ln
]}2?)1n2)(?E ??E(LL?)B1( ?)B1([tanh{ln)E( ?)B1?)(B1(E
R
?R
AB
BC
0n
AB
BC
0n 1
221
0
0 ?
+
++?++??? ++
=
?
?
?
=
?
= Eq. (7.4.13)
1
]}2?)1n2(LL [tanh{ln
]}2?)1n2)(?E ??E(LL?)B1( ?)B1([tanh{ln)??(E ?)B1?)(B1(E
R
?R
BC
AB
0n
BC
AB
0n 2
121
90
90 ?
+
++?++? ++
=
?
?
?
=
?
= Eq. (7.4.14)
For +45o/-45o VDP, through the proper coordinate transformation described in Section
7.2, the normalized change in resistance may be expressed as the following:
1??)-(E ?)B?)(1B(1E
]}2?1)(2nABBC[{tanh ln
]}2?1)(2nABBC[{tanh ln]}2?1)(2nABBC {tanh[ln??)-(E ?)B?)(1B(1E
R
?R
21
0n
0n0n
21
45
45
?++=
+
+?+++
=
?
??
?
=
?
=
?
=
Eq. (7.4.15)
136
1??)-(E ?)B?)(1B(1E
]}2?1)(2nBCAB[{tanh ln
]}2?1)(2nBCAB[{tanh ln]}2?1)(2nBCAB{tanh[ln??)-(E ?)B?)(1B(1E
R
?R
21
0n
0n0n
21
45-
45-
?++=
+
+?+++
=
?
??
?
=
?
=
?
=
Eq. (7.4.16)
For +45o/-45o VDP, AB = BC because of the symmetrical geometry, and the change in
thickness of the VDP structure is reflected as shown in Eqs. (7.4.15) and (7.4.16). Also, it
is to be emphasized that the normalized change in resistance has the same formula for +45o
VDP and -45o VDP.
For a square of VDP structure, LAB = LBC. Also, B1p = 458.8 TPa-1, B2p = -124.9 TPa-1,
B1n = -203.1 TPa-1, and B2n = 196.7 TPa-1 are substituted into Eqs. (7.4.13) through
(7.4.16). Note that E = 169.1 GPa and ? = 0.262 on the (111) silicon surface. In addition,
? may be determined by the equation in the four-point bending fixture:
? = ?3 2F L d
t h
( ) Eq. (7.4.17)
where uniaxial stress ?=??11 is applied in the 1x? -direction. Hence, all the parameters in
Eqs. (7.4.13) through (7.4.16) are known. The normalized resistance values may be
calculated analytically. The following table presents the analytic value of the magnification
factor M with/without considering the strain effect. It is observed that VDP sensors offer
3.157 times higher sensitivity than their analogous resistor sensors. However, considering
137
the strain effects leads M to 3.238 for p-type sensors and 3.039 for n-type sensors as shown
in Table 7.5.
Table 7.5 - Analytically calculated magnification factor, M
Type M with neglect of E and ? M with consideration of E and ?
p 3.157 3.238
n 3.157 3.039
Table 7.6 - Experimental values of M
Type B1-B2 Slope of ?R0/R0-?R90/R90 vs. ? M Modified M
p 583.7 2145.2 3.675 3.224
n -399.8 -1388.9 3.474 3.047
In Table 7.6, the modified M is calculated by considering ? for the serpentine resistors. In
addition, analytic results are compared with experimental results as shown in Table 7.7.
It is observed that analytic results are now in good agreement with experimental
results. For p-type sensors, M is observed to be higher than 3.157. On the other hand, for n-
type sensors, M shows the result lower than 3.157. Reasons for these discrepancies will be
discussed later.
Table 7.7 - Comparison between Analytic and Experimental M
Type Analytic M Experimental M
p 3.238 3.224
n 3.039 3.047
138
7.4.2 Strain-effects of Resistor Sensors
Before describing the strain-effects of resistor sensors, how ? responds to the
applied stress should be mentioned. When the chip is unstressed ,
transverseaxial
axial
NN
N
+=? Eq. (7.4.18)
In our case, Naxial = 143.2 squares and Ntransverse = 9.37 squares. Therefore ? = 0.939. When
the chip is stressed, ? changes with the applied stress, whereas ? is constant under an
unstressed condition. For R0, ? is given by
transverse
a
t
axial
axial
transversetaxiala
axiala
NCCN
N
NCNC
NC
+
=+=? Eq. (7.4.19)
in which Ca and Ct are modifying coefficients defined as follows:
?)(E ??)(EC,??)(E ?)(EC ta +?=?+= Eq. (7.4.20)
For R90,
transverse
t
a
axial
axial
transverseaaxialt
axialt
NCCN
N
NCNC
NC
+
=+=? Eq. (7.4.21)
139
Table 7.8 - ?0 and ?90 at various stress levels
? (MPa) ca ct ?0 ?90
0 1.0000E+00 1.0000E+00 9.4338E-01 9.4338E-01
20 1.0001E+00 9.9985E-01 9.4340E-01 9.4337E-01
40 1.0003E+00 9.9970E-01 9.4341E-01 9.4335E-01
60 1.0004E+00 9.9955E-01 9.4343E-01 9.4333E-01
80 1.0006E+00 9.9940E-01 9.4345E-01 9.4332E-01
100 1.0007E+00 9.9925E-01 9.4346E-01 9.4330E-01
In Table 7.8, ?0 and ?90 denote ? for 0o and 90o resistors. Further, the plots of ?0 and ?90 are
shown in Fig. 7.14.
? versus ?
? 0= 7.965E-13? + 9.434E-01
R2 = 1.000E+00
? 90 = -7.976E-13? + 9.434E-01
R2 = 1.000E+00
9.432E-01
9.433E-01
9.434E-01
9.435E-01
0.0E+00 2.0E+07 4.0E+07 6.0E+07 8.0E+07 1.0E+08 1.2E+08
? (Pa)
????
Fig. 7.14 - The plot of ?0 and ?90 at various stress levels
As shown in Fig. 7.14, ?0 and ?90 do not change much with varying stress. For instance,
the normalized change in ?0 and ?90 during application of 100MPa uniaxial stress is about
8.4 x 10-3 %. Hence ?0 and ?90 may be assumed to be constant during application of stress.
140
The strain-effects in resistor sensors are described nest. The resistance R of a
rectangular conductor is expressed as
wtl?Al?R == Eq. (7.4.22)
In the formula, ? is the resistivity, and l, w, and t are the length, width, and thickness of the
conductor, respectively. When the resistor is stretched by applying stress, the normalized
change in resistance is given by
???t?tw?wl?lR?R +??? Eq. (7.4.23)
For convenience, the new notations are adopted
???]R?R[t?tw?wl?l]R?R[ resdim =??= , Eq. (7.4.24)
Equation (7.4.24) reflects the normalized change in resistance as follows:
resdim ]R?R[]R?R[R?R += Eq. (7.4.25)
Generally dimensional change is neglected in the calculation of the normalized resistance
change:
???]R?R[R?R res =? Eq. (7.4.26)
If the uniaxial stress )( 11 ?=?? is applied,
?)B1(?B ]R?R[RR 21res
0
0
0
0 ??+?=?? Eq. (7.4.27)
?B?B)-1 (]R?R[RR 21res
90
90
90
90 ?+?=?? Eq. (7.4.28)
141
In addition, if we consider the effects of dimensional change, the results are
E
]1)?1(2[?
)]E??()E?()E??)[(1()]E??()E??()E?[(]R?R[ dim
0
0
?+?=
??????+?????=
Eq. (7.4.29)
E
)]1 (2 )21?[(
)]E??()E??()E?()[-1()]E??()E?()E?? [(]R?R[ dim
90
90
???+??=
?????+?????=
Eq. (7.4.30)
Substitution of E = 169.1 GPa, ? = 0.262, and ? = 0.939 into Eqs. (7.4.29) and (7.4.30)
yields
??= 12-dim
0
0 10102.8 ]
R
?R[ Eq. (7.4.31)
???= ?12dim
90
90 10003.5]
R
?R[ Eq. (7.4.32)
For p-type sensors, the calibration result of the normalized resistance change is
??= ?12
0
0 108.458
R
?R Eq. (7.4.33)
???= ?12
90
90 109.124
R
?R Eq. (7.4.34)
Substitution of the results of Eqs. (7.4.31) and (7.4.32) into Eq. (7.4.25) leads to
??= ?12res
0
0 107.450]
R
?R[ Eq. (7.4.35)
???= ?12res
90
90 109.119]
R
?R[ Eq. (7.4.36)
142
Similarly, for n-type sensors,
???= ?12
0
0 101.203
R
?R Eq. (7.4.37)
??= ?12
90
90 107.196
R
?R Eq. (7.4.38)
Hence
???= ?12res
0
0 102.211]
R
?R[ Eq. (7.4.39)
??= 12-res
90
90 107.211 ]
R
?R[ Eq. (7.4.40)
Tables 7.9 and 7.10 present the values of B1 and B2 with neglect of the dimensional
change of resistors, whereas Tables 7.11 and 7.12 includes the effect of the dimensional
change of resistors. Note that ? is considered in Tables 7.10 and 7.12.
Table 7.9 - The effective B1 and B2 (Unit: TPa-1)
B1_effp B2_effp B1_effn B2_effn
458.8 -124.9 -203.1 196.7
Table 7.10 - Modified B1 and B2 (Unit: TPa-1)
B1p B2p B1n B2n
499.4 -165.5 -230.9 224.5
143
Table 7.11 - The effective B1 and B2 (Unit: TPa-1)
B1_effp B2_effp B1_effn B2_effn
450.7 -119.9 -211.2 201.7
Table 7.12 - Modified B1 and B2 (Unit: TPa-1)
B1p B2p B1n B2n
490.3 -159.5 -240.6 230.4
As compared in Tables 7.9 and 7.11, B1_eff and B2_eff have discrepancies about 2~4 %.
Also, B1 and B2 in Tables 7.10 and 7.12, have discrepancies approximately of 1.6~3.2 %.
Substitution of the modified values of B1 and B2 in Table 7.12 into Eqs. (7.4.13) and
(7.4.14) yields M shown in the last column in Table 7.13.
Table 7.13 - Analytically calculated magnification factor, M, through the use of
modified B1 and B2
Type M with neglect of E and ? M with consideration of E and ?
p 3.157 3.231
n 3.157 3.057
If we compare the values in Tables 7.5 and 7.13, the magnification factor M is almost
constant. The observed discrepancies are less than 0.13 % for all cases. Thus, the
dimensional change of resistors may be neglected. However, in this work, the modified
values of B1 and B2 in Table 7.12 are considered in analytical calculations for
144
completeness and accuracy: B1p = 490.3 TPa-1, B2p = -159.5 TPa-1, B1n = -240.6 TPa-1, and
B2n = 230.4 TPa-1.
In the simulations, 0.9~1.1 times of B1 and B2 is also assumed to consider the errors
induced from measurements. In the cases where the strain effects are neglected, the
magnification factor is close to 3.157 for all cases. In the cases where the strain effects
are considered, the magnification factor is observed to be 3.22~3.29 for p-type sensors
and 3.05~3.08 for n-type sensors, respectively. If we compare these results with the
values in Table 7.13 (3.231 for p-type sensors and 3.057 for n-type sensors), the effects
of aspect ratio, the magnitude of stress, and overestimation/underestimation of B1 and B2
do not seem to significantly affect the magnification factor. On the other hand, the strain
effects should be considered.
7.5 Summary
In summary,
circle6 In this work, the sensitivity of VDP sensors has been calculated/measured both
analytically and experimentally.
circle6 VDP sensors offer 3.157 times higher sensitivity than an analogous two element
resistor sensor rosette. However, considering strain effects leads M to 3.231 for
p-type sensors and 3.057 for n-type sensors.
circle6 Dimensional changes of VDP sensors should be considered in the calculation of
M. However, for resistor sensors, the dimensional changes may be neglected.
145
CHAPTER 8
TRANSVERSE STRESS ANALYSIS AND ERRORS ASSOCIATED WITH
MISALIGNMENT
8.1 Transverse Stress Analysis
8.1.1 Resistor Sensors on the (111) Silicon
The effects of transverse stress on piezoresistive coefficient measurements are
described in the previous chapter in which uniaxial stress is considered in only the (111)
silicon surface. In this chapter two-dimensional states of stress are considered in the (001)
silicon surface as well as the (111) silicon surface. A special (111) silicon test chip JSE-
WB100C is used in the analysis. Each chip incorporates an array of the optimized eight-
element dual polarity measurement rosettes. Experiments are performed with chips cut
from JSE-WB100C test chip wafers. For sensors on the (111) surface, the expression for
a resistor sensor at angle ? with respect to the '1x axis is given by the following formula:
)T(f2sin])BB()BB(2[2
sin])BB(22BBB[
cos])BB(22BBB[RR
'
1221
'
1323
2'
2332
'
333
'
221
'
112
2'
2332
'
333
'
222
'
111
?+???+??+
???+?+?+?+
?????+?+?=?
?
?
Eq. (8.1.1)
Just as for the BMW-2.1 test chip discussed in Chapter 7, the JSE-WB100C sensors
are composed of resistor segments oriented in the 0o and 90o directions because of the
146
serpentine resistor pattern. For the JSE-WB100C, 96.455)WL( axial = squares,
and 10.313)WL( transverse = squares, resulting in ? = 0.9034. For two-dimensional states of
stress on the surface of the die, with neglect of the out-of-plane stresses and temperature
term )T(f ? , Eq. (8.1.1) yields
'
2212
'
1121
'
221
'
112
'
222
'
111
0
0
]?)B1(B[ ]?)B1(B[
)?B?)(B1()?B?(BR?R
??+?+??+?=
+??++?= Eq. (8.1.2)
Similarly, for ? = 90o
'
2212
'
1121
'
221
'
112
'
222
'
111
90
90
]?B )B-1[(]?B )B-1[(
)?B?(B)?B?)(B1(R?R
?+?+?+?=
+?++??= Eq. (8.1.3)
Re-using the notation in Chapter 7,
12_(eff)2
21_(eff)1
)B1( BB
)B1( BB
??+?=
??+?= Eq. (8.1.4)
we arrive at the result below:
'222_(eff)'111_(eff)
0
0 ?B?B
R
?R += Eq. (8.1.5)
'221_(eff)'112_(eff)
90
90 ?B?B
R
?R += Eq. (8.1.6)
From the notation above,
1-2
)B1-(BB
1-2
)B1-(BB
_(eff)1_(eff)2
2
_(eff)2_(eff)1
1
?
?+?=
?
?+?=
Eq. (8.1.7)
147
In Table 8.1, B1 and B2 with varying temperatures are presented without considering the
effect of ?. On the other hand, the effect of ? is considered in the calculation of B1 and B2
in Table 8.2 in which ? = 0.9034 is substituted.
Table 8.1 - Extracted B1_(eff) and B2_(eff) versus temperature
(Unit: TPa-1)
T( oC) B1_(eff)p B2_(eff)p B1_(eff)n B2_(eff)n
-133.4 608.2 -179.6 -166.2 135.3
-93.2 542.1 -157.3 -155.4 127.3
-48.2 447.3 -154.9 -152.7 112.7
-23.6 422.8 -148.6 -146.8 105.7
0.6 398.7 -142.6 -141.0 101.0
25.1 366.2 -133.9 -133.7 97.4
49.9 315.3 -117.3 -114.5 88.1
75.1 271.0 -100.1 -102.5 78.9
100.6 239.6 -92.4 -93.2 70.2
Table 8.2 - Modified B1 and B2 versus temperature
with consideration of ? (Unit: TPa-1)
T( oC) B1p B2p B1n B2n
-133.4 702.5 -273.9 -202.3 171.4
-93.2 625.8 -241.0 -189.2 161.1
-48.2 519.4 -227.0 -184.5 144.5
-23.6 491.2 -217.0 -177.0 135.9
0.6 463.5 -207.4 -170.0 130.0
25.1 426.1 -193.8 -161.4 125.1
49.9 367.1 -169.1 -138.8 112.4
75.1 315.4 -144.5 -124.2 100.6
100.6 279.4 -132.2 -112.8 89.8
As seen in Table 8.1 and Table 8.2, considering ? leads to an increase in temperature
sensitivity of B1 and B2. Also, addition and subtraction of equations in Eq. (8.1.7) yield
148
1?2
BBBB
BBBB
_(eff)2_(eff)1
21
_(eff)2_(eff)121
?
?=?
+=+
Eq. (8.1.8)
in which (B1 + B2) is independent of ?. However, (B1 - B2) depends on ? as shown in
Tables 8.3 and 8.4.
Table 8.3 - Addition and subtraction of B1_(eff) and B2_(eff)
versus temperature (Unit: TPa-1)
T( oC) B1_(eff)p+ B2-(eff)p B1_(eff)p - B2_(eff)p B1_(eff)n + B2_(eff)n B1_(eff)n- B2_(eff)n
-133.4 428.6 787.8 -30.9 -301.5
-93.2 384.8 699.4 -28.1 -282.7
-48.2 292.4 602.2 -40.0 -265.4
-23.6 274.2 571.4 -41.1 -252.5
0.6 256.1 541.3 -40.0 -242.0
25.1 232.3 501.0 -36.3 -231.1
49.9 198.0 432.6 -26.4 -202.6
75.1 170.9 371.1 -23.6 -181.4
100.6 147.2 332.0 -23.0 -163.4
Table 8.4 - Addition and subtraction of B1 and B2
versus temperature (Unit: TPa-1)
T( oC) B1p + B2p B1p - B2p B1n + B2n B1n - B2n
-133.4 428.6 976.5 -30.9 -373.7
-93.2 384.8 866.9 -28.1 -350.4
-48.2 292.4 746.4 -40.0 -329.0
-23.6 274.2 708.2 -41.1 -313.0
0.6 256.1 670.9 -40.0 -300.0
25.1 232.3 619.9 -36.3 -286.4
49.9 198.0 536.2 -26.4 -251.1
75.1 170.9 460.0 -23.6 -224.8
100.6 147.2 411.5 -23.0 -202.5
149
If a sensor is subjected to hydrostatic pressure ( p??? '33'22'11 ?=== ) with
consideration of ?, the normalized resistance change is expressed as shown below:
For 0=? ,
)T(f)pB + B(B
)T(f ?B + ? + B ?B=
)T(]+ f? B+ ?B+ ? B[)-1 (+ ]?B + ?B + ?B [ =R?R
3_(eff)2_(eff)1
' 333' 22_(eff)2'11_(eff)1
'
333
'
22 1
'
112' 333' 222
'
111
0
0
?++?=
?+
???
Eq. (8.1.9)
Similarly, for o90=?
)T( f)pB + B(B
)T( f ?B + ? + B ?B=
)T(]+ f? B+ ?B+ ? B[ + ]?B + ?B + ?B) [-1( =R?R
3_(eff)2_(eff)1
' 333' 22_(eff)1'11_(eff)2
'
333
'
22 1
'
112' 333' 222
'
111
90
90
?++?=
?+
???
Eq. (8.1.10)
It is important to emphasize that pressure coefficient p? is independent of ?
because 2_(eff)1_(eff)21 BBBB +=+ .
During diffusion in the fabrication processes, impurities diffuse laterally as well as
vertically. If lateral diffusion is assumed in this study, we arrive at the following results:
For BMW-2.1 test chip:
Table 8.5 - ? with lateral diffusion (BMW-2.1)
Axial (L/W) Transverse (L/W) ? (2?-1)-1
x = 0 ?m 143.2 9.37 0.939 1.140
x = 1.5 ?m 95.8 7.64 0.926 1.173
x= 2.0 ?m 86.3 7.20 0.923 1.182
150
Similarly, for JSE-WB 100C test chip:
Table 8.6 - ? from lateral diffusion (JSE-WB100C)
Axial (L/W) Transverse (L/W) ? (2?-1)-1
x = 0 ?m 96.46 10.31 0.903 1.239
x = 1.5 ?m 70.89 9.13 0.886 1.296
x = 2.0 ?m 65.16 8.84 0.881 1.314
As shown in Table 8.5 and Table 8.6, ? decreases with increasing lateral diffusion
because the relative rate of number of squares is higher for the part of transverse direction.
8.1.2 Resistor Sensors on the (001) Silicon
circle6 With respect to the unprimed axes
For the unprimed axes, the expression for a resistor sensor at angle ? with respect to the
1x axis is given by
+ ...] ?T?T + [2sin ??
sin)]?(?+ ??+ [?cos)]?(?+ ?? = [?R?R
2
211244
2
3311122211
2
3322121111
??+?+
?+?+
Eq. (8.1.11)
Using ? and neglecting the out-of-plane stresses and temperature terms gives
221112111211
1112221122121111
0
0
]?) ?1( ?[ ]?) ?1([
] ???)[?1(] ???[?R?R
??+?+??+?pi=
+??++?= Eq. (8.1.12)
221112111211
1112221122121111
90
90
]? ?) ?-1[( ]? ?) ?-1[(
] ???[?] ???)[?1(R?R
?+?+?+?=
+?++??= Eq. (8.1.13)
151
in which the new notations are adopted as follows:
1211_(eff)11 ) ?1(?? ??+?= Eq. (8.1.14)
1112_(eff)12 ) ?1(?? ??+?= Eq. (8.1.15)
Then subtraction and addition of two equations above, respectively, yield
121112_(eff)11_(eff) ???? +=+ Eq. (8.1.16)
)?)(?1?2(?? 1211_(eff)12_(eff)11 ??=? Eq. (8.1.17)
From Eq. (8.1.16) and Eq. (8.1.17),
_(eff)12_(eff)1111 ?12 1?12? ????+???= Eq. (8.1.18)
_(eff)12_(eff)1112 ?12?12 1-? ???+???= Eq. (8.1.19)
It can be seen that 1211S ??? += is independent of ?. However, pi11, pi12, and
1211D ??? ?= depend on ?.
circle6 With respect to the primed axes
Similarly, for the primed axes, the expression for a resistor sensor at angle ? with
respect to the '1x axis is given by
+ ...]?T?T + [2sin )?-?(???
sin ? 2 ??? + ?2 ??? +
cos ?2 ??? + ?2 ??? = R?R
2
21
'
1212113312
2'
22
441211'
11
441211
2'
22
441211'
11
441211
??+?++
??
?
?
??
? ?
?
??
?
? ++?
?
??
?
? ?+
??
?
?
??
? ?
?
??
?
? ?+?
?
??
?
? ++
Eq. (8.1.20)
152
Including the effect of ? and neglecting the out-of-plane stresses and temperature terms in
Eq. (8.1.20) yield the following:
'
22 ]?2
)?21(??[]?
2
)?12(??[
]? 2 ? + ? + ? + ? 2 ? - ? + ? )[1(
]? 2 ? - ? + ? + ? 2 ? + ? + ? [R?R
441211'
11
441211
22
441211
11
441211
22
441211
11
441211
0
0
??+++??++=
????????????????+
???????????????=
Eq. (8.1.21)
Similarly, for ? = 90o
'
22
441211'
11
441211
22
441211
11
441211
22
441211
11
441211
90
90
]?2 )?1-2(??[]?2 ?)21(??[
]? 2 ? + ? + ? + ? 2 ? - ? + ? [
]? 2 ? - ? + ? + ? 2 ? + ? + ? )[1(R?R
?+++??++=
???????????????+
????????????????=
Eq. (8.1.22)
Here the new notations are:
2 )?12(??2?? 441211_(eff)44S_(eff) ??++=+ Eq. (8.1.23)
2 ?)?21(??2?? 441211_(eff)44S_(eff) ?++=? Eq. (8.1.24)
Subtraction and addition of both equations above, respectively, yield
SS_(eff) ?? = Eq. (8.1.25)
44_(eff)44 )?12(? ??= Eq. (8.1.26)
By Eq. (8.1.25) and Eq. (8.1.26),
S_(eff)S ?? = Eq. (8.1.27)
153
_(eff)4444 ?)1-2( 1? ?= Eq. (8.1.28)
It is found that S? is independent of ?. However, 44? depends on ?.
8.2 Off-Axis Alignment Error on the (001) Silicon Plane
The rosette configuration on the (001) plane silicon consists of 0o, 90o, 45o, and -
45o resistors relative to the 1x axis [100] (or '1x axis [110]). The strip is cut along "" y x ?
axes counter-clockwise rotated ? from yx ? axes [100] (or '' yx ? axes [110]). If we use
the double-primed notation instead of the unprimed (or primed) notation, the equations
change as discussed below.
circle6 With respect to the unprimed axes
For the unprimed axes on the (001) plane silicon, the general equation for counter-
clockwise rotation of the "" y x ? axes by an angle of ? from the x ? y axes, is given as
follows:
154
... ]?T?T[
m]} l? ?2cos ) ? (?4 ?4sin[ -?
]? ?2sin ) ? (?4 ?4sin[ -?
]? ?2sin ) ? (?4 ?4sin[ {?2
)nm?nl(? ?2
] n ?2 ?4sin)-?? ?4sin21?( ??)? ?4sin21( ?[?
]} m ?2 ?4sin? ?2sin21? ?2sin21[ -?
] ?2 ?4sin?? ?2sin21? ) ?2sin21-1[ (?
] ?2 ?4sin? ) ?2sin211(? ?2sin21[{?
]} l ?2 ?4sin? ?2sin21? ?2sin21[ ?
] ?2 ?4sin?? )?2sin211(? ?2sin21[ ?
] ?2 ?4sin? ?2sin21? ) ?2sin211[({?R?R
2
21
"""
12
2"
11
"
2244
"
12
2"
11
"
2212
"
12
2"
11
"
2211
"""
23
"""
1344
2" "
1244
"
12
"
22
"
1112
"
12
"
3311
2" "
12
"
22
2"
11
2
44
"
12
"
33
"
22
2"
11
2
12
"
12
"
22
2"
11
2
11
2" "
12
"
22
2"
11
2
44
"
12
"
33
"
22
2"
11
2
12
"
12
"
22
2"
11
2
11
+?+?+
+?+
??+
+?+
++
?++++
?++
?+++
+?++
+?+
++?++
?+?=
Eq. (8.2.1)
In the case of ? = 0, the double-primed axes are aligned with the unprimed axes, and Eq.
(8.2.1) simplifies to
] ...TT[
)mnlnlm(2 )]n( +[ +
)]m( +[ +)]l( +[ = R?R
2
21
23131244
2
2211123311
2
3311122211
2
3322121111
+??+??+
?+?+?pi+?+?pi?pi
?+?pi?pi?+?pi?pi
Eq. (8.2.2)
By introducing 0n and , sinm , cosl =?=?= , Eq. (8.2.2) is expressed as
] ...TT[
sin cos ? 2?
sin )]?(???[?
cos )]?(???[?R?R
2
21
1244
2
3311122211
2
3322121111
+??+??+
??+
?+++
?++=
Eq. (8.2.3)
155
Considering ? and assuming uniaxial stress "11? with neglect of temperature terms gives
"
1111
"
11
2441211
11
"
11
2
44
2
12
2
11
0
0
??
?] ?2sin)2 ???([?
?]?2sin21??2sin21?? )2sin211(?[R?R
?=
???=
++?=
Eq. (8.2.4)
"
1112
"
11
2441211
12
"
11
2
44
2
12
2
11
90
90
?
?]?2sin)2 ???(?[
?]?2sin)21( -?? )2sin21-1(??2sin21?[R?R
?pi=
??+=
++=
Eq. (8.2.5)
?] 2cos?2sin)2 ???()2[(
?)]2 2cos2sin( ? )2 2cos2sin1(?)2 2cos2sin1(?[R?R
"
11
4412111211
"
11441211
45
45
????pi+pi=
??+??++???=
Eq. (8.2.6)
?] 2cos?2sin)2 ???()2[(
?)]2 2cos2sin( ? )2 2cos2sin1(?)2 2cos2sin1(?[R?R
"
11
4412111211
"
11441211
45
45
???+pi+pi=
??????+??+=
?
?
Eq. (8.2.7)
From these equations, we find that the 0o/90o and ? 45o pairs are insensitive to rotational
alignment error. For ? = 0, assuming uniaxial stress "11? with neglect of temperature
terms gives
156
11"11
0
0 ??
R
?R = Eq. (8.2.8)
??R?R 11"12
90
90 = Eq. (8.2.9)
)?2 ??(R?RR?R 11"1211
45-
45-
45
45 +== Eq. (8.2.10)
where 11"11 ?? = since the doubled primed axes are aligned with the unprimed axes for
? = 0. In Eq. (8.2.4), ?11? is defined as
? 2sin)2 ???( ?? 244121111?11 ???= Eq. (8.2.11)
Similarly, in Eq. (8.2.5), ?12? is defined as
2?)sin2 ???( ?? 24412111212 ??+=? Eq. (8.2.12)
For 0=? and o90=? , the normalized error in ?R/R induced by ? is given by
? 2sin)?2 ??(
?2sin)?2 ??-?(
]R?R[
]R?R- []R?R[
2
11
44D
2
11
441211
0?
0
0
0?
0
0
?
0
0
??=
++=
=
=
Eq. (8.2.13)
?2sin)?2 ??(
?2sin)?2 ???(
]R?R[
]R?R- []R?R[
2
12
44D
2
12
441211
0?
90
90
0?
90
90
?
90
90
?=
??=
=
=
Eq. (8.2.14)
157
where 1211D pi?pi=pi . For o45?=? , the normalized error in ?R/R induced by ? is given
by
?4sin)?2 ??(
2cos2sin)? ???(
]R?R[
]R?R- []R?R[
S
D44
S
441211
0?
45
45
0?
45
45
?
45
45
?=
?????=
=
=
Eq. (8.2.15)
?4sin)?2 ??(
2cos?2sin)? ???(
]R?R[
]R?R- []R?R[
S
44D
S
441211
0?
45
45
0?
45
45
?
45
45
?=
???=
=
?
?
=
?
?
?
?
Eq. (8.2.16)
For p-type silicon, |?| |?| 1144 >> and |?| |?| 1244 >> give [6], [98]
2?sin2??
]R?R[
]R?R[ -]R?R[
2
11
44
0?
0
0
0?
0
0
?
0
0
?
=
=
Eq. (8.2.17)
2?sin2??
]R?R[
]R?R[ -]R?R[
2
12
44
0?
90
90
0?
90
90
?
90
90
??
=
=
Eq. (8.2.18)
?4sin??21
]R?R[
]R?R- []R?R[
S
44
0?
45
45
0?
45
45
?
45
45
?
=
=
Eq. (8.2.19)
158
?4sin)??(21
]R?R[
]R?R- []R?R[
S
44
0?
45
45
0?
45
45
?
45
45
??
=
?
?
=
?
?
?
?
Eq. (8.2.20)
For n-type silicon, the approximation 1211 ?2? ?? [23] yields
2?)sin4? ?3?(
]R?R[
]R?R[ -]R?R[
2
12
4412
0?
0
0
0?
0
0
?
0
0
+??
=
=
Eq. (8.2.21)
2?)sin2? ??3(
]R?R[
]R?R[ -]R?R[
2
12
4412
0?
90
90
0?
90
90
?
90
90
+??
=
=
Eq. (8.2.22)
?4sin)? ?3?(21
]R?R[
]R?R- []R?R[
12
1244
0?
45
45
0?
45
45
?
45
45
+??
=
=
Eq. (8.2.23)
?4sin)? 3?(21
]R?R[
]R?R- []R?R[
12
1244
0?
45
45
0?
45
45
?
45
45
pi+?
=
?
?
=
?
?
?
?
Eq. (8.2.24)
In the equations, for p-type sensors, the normalized % error in ?R/R induced by ? can
have a considerable value because of |?| |?| 1144 >> and |?| |?| 1244 >> . For 0=? , ? =
90o, and o45?=? , the normalized % error in ?R/R versus ? is presented in Tables 8.7
and 8.8, whose plots are shown in Figs. 8.1 through 8.4.
159
Table 8.7 - Normalized % error in ?pi11 and ?pi12 versus ?
pi11? pi12?
? P-type sensors N-type sensors P-type sensors N-type sensors
-10 1.16E+02 -8.13E+00 6.94E+02 -1.56E+01
-9 9.44E+01 -6.63E+00 5.66E+02 -1.27E+01
-8 7.50E+01 -5.27E+00 4.50E+02 -1.01E+01
-7 5.79E+01 -4.07E+00 3.47E+02 -7.79E+00
-6 4.27E+01 -3.00E+00 2.56E+02 -5.75E+00
-5 2.98E+01 -2.09E+00 1.79E+02 -4.01E+00
-4 1.91E+01 -1.35E+00 1.15E+02 -2.58E+00
-3 1.08E+01 -7.63E-01 6.45E+01 -1.46E+00
-2 4.85E+00 -3.42E-01 2.91E+01 -6.55E-01
-1 1.21E+00 -8.81E-02 7.27E+00 -1.69E-01
0 0.00E+00 0.00E+00 0.00E+00 0.00E+00
1 1.21E+00 -8.81E-02 7.27E+00 -1.69E-01
2 4.85E+00 -3.42E-01 2.91E+01 -6.55E-01
3 1.08E+01 -7.63E-01 6.45E+01 -1.46E+00
4 1.91E+01 -1.35E+00 1.15E+02 -2.58E+00
5 2.98E+01 -2.09E+00 1.79E+02 -4.01E+00
6 4.27E+01 -3.00E+00 2.56E+02 -5.75E+00
7 5.79E+01 -4.07E+00 3.47E+02 -7.79E+00
8 7.50E+01 -5.27E+00 4.50E+02 -1.01E+01
9 9.44E+01 -6.63E+00 5.66E+02 -1.27E+01
10 1.16E+02 -8.13E+00 6.94E+02 -1.56E+01
Normalized % Error in pi11? versus ?
-20
0
20
40
60
80
100
120
140
-15 -10 -5 0 5 10 15
?
% p type
n type
Fig. 8.1 - Normalized % error in pi11? versus ?
160
Normalized % Error in pi12? versus ?
-100
0
100
200
300
400
500
600
700
800
-15 -10 -5 0 5 10 15
?
% p type
n type
Fig. 8.2 - Normalized % error in pi12? versus ?
Table 8.8 - Normalized % error in ?R45/R45 and ?R-45/R-45 versus ?
? = 45o ? = -45o
? P-type sensors N-type sensors P-type sensors N-type sensors
-10 -7.62E+02 9.35E+01 7.62E+02 -9.35E+01
-9 -6.97E+02 8.55E+01 6.97E+02 -8.55E+01
-8 -6.28E+02 7.71E+01 6.28E+02 -7.71E+01
-7 -5.57E+02 6.83E+01 5.57E+02 -6.83E+01
-6 -4.82E+02 5.92E+01 4.82E+02 -5.92E+01
-5 -4.05E+02 4.98E+01 4.05E+02 -4.98E+01
-4 -3.27E+02 4.01E+01 3.27E+02 -4.01E+01
-3 -2.46E+02 3.02E+01 2.46E+02 -3.02E+01
-2 -1.65E+02 2.02E+01 1.65E+02 -2.02E+01
-1 -8.27E+01 1.01E+01 8.27E+01 -1.01E+01
0 0.00E+00 0.00E+00 0.00E+00 0.00E+00
1 8.27E+01 -1.01E+01 -8.27E+01 1.01E+01
2 1.65E+02 -2.02E+01 -1.65E+02 2.02E+01
3 2.46E+02 -3.02E+01 -2.46E+02 3.02E+01
4 3.27E+02 -4.01E+01 -3.27E+02 4.01E+01
5 4.05E+02 -4.98E+01 -4.05E+02 4.98E+01
6 4.82E+02 -5.92E+01 -4.82E+02 5.92E+01
7 5.57E+02 -6.83E+01 -5.57E+02 6.83E+01
8 6.28E+02 -7.71E+01 -6.28E+02 7.71E+01
9 6.97E+02 -8.55E+01 -6.97E+02 8.55E+01
10 7.62E+02 -9.35E+01 -7.62E+02 9.35E+01
161
Normalized % Error in ?R45/R45 versus ?
-1000
-800
-600
-400
-200
0
200
400
600
800
1000
-15 -10 -5 0 5 10 15
?
% p type
n type
Fig. 8.3 - Normalized % error in ?R45/R45 versus ?
Normalized % Error in ?R-45/R-45 versus ?
-1000
-800
-600
-400
-200
0
200
400
600
800
1000
-15 -10 -5 0 5 10 15
?
%
p type
n type
Fig. 8.4 - Normalized % error in ?R-45/R-45 versus ?
162
Through Figs. 8.1 and 8.4, it can be seen that p-type sensors have larger
normalized % error compared with n-type sensors as described above.
circle6 With respect to the primed axes
For the case of ? = 45o, the double-primed axes are aligned with the primed axes,
and Eq. (8.2.1) simplifies to
] ...TT[sin2 )-(? ?
sin 2 + 2 +
cos 2 + 2 = R?R
2
21
'
1212113312
2'
22
441211'
11
441211
2'
22
441211'
11
441211
+??+??+??pipi++
??
?
?
??
? ??
?
??
?
? pi+pi+pi??
?
??
?
? pi?pi+pi
??
?
?
??
? ??
?
??
?
? pi?pi+pi??
?
??
?
? pi+pi+pi
Eq. (8.2.25)
where 0, and nsin, mcosl ''' =?=?= has been introduced, and ? is the angle between
the 1x? -axis and the resistor orientation. The stress components are now measured in the
double-primed coordinate system, and 0?? '33"33 == has been assumed.
For the primed axes on the (001) plane silicon, the general equation for counter-
clockwise rotation of the "" y x ? axes by an angle of ? from the '' yx ? axes [110], is
given as follows:
163
] ...TT[
ml ]}? 2?sin )?(? 4sin4? [?
]? 2?cos )?(? 4sin4? [?
]? 2?cos )?(? 4sin4? [2{?
)nm?nl(? ?2
n ]? 2sin4??)? sin4?21?? (?)? sin4?21? ([?
m ]}? 2sin4?? 2?cos21? 2?cos21- [?
]? 2sin4??? 2?cos21? ) 2?cos21-(1 [?
]? 2sin4?? ) 2?cos211(? 2?cos21[{?
l ]}? 2sin4?? 2?cos21? 2?cos21 [?
]? 2sin4??)? 2?cos21(1? 2?cos21 [?
]? 2sin4?? 2?cos21? ) 2?cos21[(1{?R?R
2
21
"""
12
2"
11
"
2244
"
12
2"
11
"
2212
"
12
2"
22
"
1111
"""
23
"""
1344
2 ""
1244
"
12
"
22
"
1112
"
12
"
3311
2 ""
12
"
22
2"
11
2
44
"
12
"
33
"
22
2"
11
2
12
"
12
"
22
2"
11
2
11
2 ""
12
"
22
2"
11
2
44
"
12
"
33
"
22
2"
11
2
12
"
12
"
22
2"
11
2
11
+??+??+
+?+
??+
+?+
++
++++?+
+++
++++
??++
??+
?+?++
++?=
Eq. (8.2.26)
In the formula, (? - 45)o instead of ? is substituted into Eq. (8.2.1) in order to consider the
fact that the [110] axis is counter-clockwise rotated by 45o from the [100] axis. In the case
of ? = 0, the double-primed axes are aligned with the primed axes, and Eq. (8.2.26)
simplifies to
164
] ...TT[
ml )??-2(?)nm ?nl (?2?
n )]?(???[?
m )]?21?21(-?)??21?21(?)?21?21([?
l )]?21?21(?)??21?21(?)?21?21([?R?R
2
21
''12'
1211
''23'''13'
44
2 '22'11'
1233
'
11
2 '22'11'
4433
'22'11'
1222
'11'
11
2 '22'11'
4433
'22'11'
1222
'11'
11
+??+??+
+++
+++
+++++++
?+++++=
Eq. (8.2.27)
By introducing 0n and , sinm , cosl ''' =?=?= , Eq. (8.2.27) is expressed as
] ...TT[
sin cos ? )?-2(? ??
sin ])?2 ???()?2 ???[(
cos ])?2 ???()?2 ???[(R?R
2
21
12'121133'12
222'44121111'441211
222'44121111'441211
+??+??+
??++
?+++?++
??++++=
Eq. (8.2.28)
It may be noted that Eq. (8.2.28) is the same as Eq. (8.2.25). Considering ? and assuming
uniaxial stress "11? in Eq. (8.2.26) with neglect of temperature terms gives
"
11
2442122
11
0
0 ? ] 2?cos
2
? 2?cos
2
?) 2?cos
2
1(1?[
R
?R ++?= Eq. (8.2.29)
"
11
2442
12
211
90
90 ? ] 2?cos
2
? ) 2?cos
2
1-(1? 2?cos
2
?[
R
?R ?+= Eq. (8.2.30)
"
1144D
S
45
45 ] ?
4
4sin)(
2[R
?R ?pi?pi+pi= Eq. (8.2.31)
"
1144D
S
45
45 ] ?
4
4sin)(
2[R
?R ?pi?pi?pi=
?
? Eq. (8.2.32)
165
It may be stressed that the 0o/90o and ? 45o pairs are insensitive to rotational alignment
error. In addition, Eqs. (8.2.6) and (8.2.7) are equal to Eqs. (8.2.32) and (8.2.31),
respectively. For ? = 0, assuming uniaxial stress "11? with neglect of temperature terms
gives
11"441211
0
0 )?
2
???(
R
?R ++= Eq. (8.2.33)
)?2 ???(R?R 11"441211
90
90 ?+= Eq. (8.2.34)
)?2 ??(R?RR?R 11"1211
45-
45-
45
45 +== Eq. (8.2.35)
In the equations, 11"11 ?? = because the doubled primed axes are aligned with the primed
axes for ? = 0.
Subtraction of Eq. (8.2.30) from Eq. (8.2.29) leads to
"
11
?
44
"
11
2
44
2
D
90
90
0
0
??
? ] ?2cos? )2cos1([?RRR?R
=
pi+?=??
Eq. (8.2.36)
In Eq. (8.2.36), ??44 is defined as
? 2cos )2cos1(?? 2442D?44 pi+??= Eq. (8.2.37)
Combining Eqs. (8.2.29) and (8.2.30) yields the normalized % error in ?pi44 versus ? as
presented in Table 8.9, whose plots are shown in Fig. 8.5.
166
Table 8.9 - Normalized % error in
?pi
44 versus ?
? P-type sensors N-type sensors
-10 -1.10E+01 1.22E+02
-9 -9.02E+00 9.97E+01
-8 -7.17E+00 7.93E+01
-7 -5.53E+00 6.11E+01
-6 -4.08E+00 4.51E+01
-5 -2.85E+00 3.15E+01
-4 -1.83E+00 2.02E+01
-3 -1.03E+00 1.14E+01
-2 -4.59E-01 5.08E+00
-1 -1.15E-01 1.27E+00
0 0.00E+00 0.00E+00
1 -1.15E-01 1.27E+00
2 -4.59E-01 5.08E+00
3 -1.03E+00 1.14E+01
4 -1.83E+00 2.02E+01
5 -2.85E+00 3.15E+01
6 -4.08E+00 4.51E+01
7 -5.53E+00 6.11E+01
8 -7.17E+00 7.93E+01
9 -9.02E+00 9.97E+01
10 -1.10E+01 1.22E+02
Normalized % Error in pi44? versus ?
-20
0
20
40
60
80
100
120
140
-15 -10 -5 0 5 10 15
?
% p typen type
Fig. 8.5 - Normalized % error in pi44? versus ?
167
For 0=? , ?=? 90 , and ??=? 45 , the normalized % error in ?R/R versus ? is given by
?2sin)?? ??(
?2sin)??? ???(
]R?R[
]R?R- []R?R[
2
44S
44D
2
441211
441211
0?
0
0
0?
0
0
?
0
0
+
?=
++
??=
=
=
Eq. (8.2.38)
?2sin)?? ??(
?2sin)??? ??-?(
]R?R[
]R?R- []R?R[
2
44S
44D
2
441211
441211
0?
90
90
0?
90
90
?
90
90
?
??=
?+
++=
=
=
Eq. (8.2.39)
? 4sin)?2 ??(
4sin)(2 )(
]R?R[
]R?R- []R?R[
S
44D
1211
441211
0?
45
45
0?
45
45
?
45
45
?=
?pi+pi pi?pi?pi=
=
=
Eq. (8.2.40)
?4sin)?2 ??(
?4sin)??(2 ??-?
]R?R[
]R?R- []R?R[
S
44D
1211
441211
0?
45
45
0?
45
45
?
45
45
??=
+
++=
=
?
?
=
?
?
?
?
Eq. (8.2.41)
Assuming |?| |?| 1144 >> and |?| |?| 1244 >> for p-type silicon [6], [98] yields
168
?2sin
]R?R[
]R?R- []R?R[
2
0?
0
0
0?
0
0
?
0
0
??
=
=
Eq. (8.2.42)
?2sin
]R?R[
]R?R- []R?R[
2
0?
90
90
0?
90
90
?
90
90
??
=
=
Eq. (8.2.43)
?4sin?2?
]R?R[
]R?R- []R?R[
S
44
0?
45
45
0?
45
45
?
45
45
??
=
=
Eq. (8.2.44)
?4sin?2?
]R?R[
]R?R- []R?R[
S
44
0?
45
45
0?
45
45
?
45
45
?
=
?
?
=
?
?
?
?
Eq. (8.2.45)
For n-type silicon, assuming the approximation 1211 ?2? ?? [23] yields
?2sin)?? ??3(
]R?R[
]R?R- []R?R[
2
4412
4412
0?
0
0
0?
0
0
?
0
0
?
+?
=
=
Eq. (8.2.46)
?2sin)
??
??3(
]R?R[
]R?R- []R?R[
2
4412
4412
0?
90
90
0?
90
90
?
90
90
+
+??
=
=
Eq. (8.2.47)
4sin2 )3(
]R?R[
]R?R- []R?R[
12
4412
0?
45
45
0?
45
45
?
45
45
?pi pi+pi=
=
=
Eq. (8.2.48)
169
?4sin?2 )??3(-
]R?R[
]R?R- []R?R[
12
4412
0?
45
45
0?
45
45
?
45
45
+=
=
?
?
=
?
?
?
?
Eq. (8.2.49)
For 0=? , o90=? , and 45?=? , the normalized % error in ?R/R versus ? is presented
in Tables 8.10 and 8.11, whose plots are shown in Figs. 8.6 through 8.9.
Table 8.10 - Normalized % error in ?R/R versus ? for 0=? and o90=?
0o sensors:
2?)sin??? ???( 2
441211
441211
++
??
90o sensors:
2?)sin??? ???-( 2
441211
441211
?+
++
? P-type sensors N-type sensors P-type sensors N-type sensors
-10 -10.6 13.3 -11.5 -47.2
-9 -8.7 10.9 -9.4 -38.5
-8 -6.9 8.6 -7.5 -30.6
-7 -5.3 6.7 -5.8 -23.6
-6 -3.9 4.9 -4.3 -17.4
-5 -2.7 3.4 -3.0 -12.2
-4 -1.8 2.2 -1.9 -7.8
-3 -1.0 1.2 -1.1 -4.4
-2 -0.4 0.6 -0.5 -2.0
-1 -0.1 0.1 -0.1 -0.5
0 0.0 0.0 0.0 0.0
1 -0.1 0.1 -0.1 -0.5
2 -0.4 0.6 -0.5 -2.0
3 -1.0 1.2 -1.1 -4.4
4 -1.8 2.2 -1.9 -7.8
5 -2.7 3.4 -3.0 -12.2
6 -3.9 4.9 -4.3 -17.4
7 -5.3 6.7 -5.8 -23.6
8 -6.9 8.6 -7.5 -30.6
9 -8.7 10.9 -9.4 -38.5
10 -10.6 13.3 -11.5 -47.2
170
Normalized % Error in ?R0/R0 versus ?
-15
-10
-5
0
5
10
15
-15 -10 -5 0 5 10 15
?
% p typen type
Fig. 8.6 - Normalized % error in ?R0/R0 versus ?
Normalized % Error in ?R90/R90 versus ?
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
5
-15 -10 -5 0 5 10 15
?
% p typen type
Fig. 8.7 - Normalized % error in ?R90/R90 versus ?
171
Table 8.11 - Normalized % error in ?R45/R45 and ?R-45/R-45 versus ?
? = 45o ? = -45o
? P-type sensors N-type sensors P-type sensors N-type sensors
-10 7.62E+02 -9.35E+01 -7.62E+02 9.35E+01
-9 6.97E+02 -8.55E+01 -6.97E+02 8.55E+01
-8 6.28E+02 -7.71E+01 -6.28E+02 7.71E+01
-7 5.57E+02 -6.83E+01 -5.57E+02 6.83E+01
-6 4.82E+02 -5.92E+01 -4.82E+02 5.92E+01
-5 4.05E+02 -4.98E+01 -4.05E+02 4.98E+01
-4 3.27E+02 -4.01E+01 -3.27E+02 4.01E+01
-3 2.46E+02 -3.02E+01 -2.46E+02 3.02E+01
-2 1.65E+02 -2.02E+01 -1.65E+02 2.02E+01
-1 8.27E+01 -1.01E+01 -8.27E+01 1.01E+01
0 0.00E+00 0.00E+00 0.00E+00 0.00E+00
1 -8.27E+01 1.01E+01 8.27E+01 -1.01E+01
2 -1.65E+02 2.02E+01 1.65E+02 -2.02E+01
3 -2.46E+02 3.02E+01 2.46E+02 -3.02E+01
4 -3.27E+02 4.01E+01 3.27E+02 -4.01E+01
5 -4.05E+02 4.98E+01 4.05E+02 -4.98E+01
6 -4.82E+02 5.92E+01 4.82E+02 -5.92E+01
7 -5.57E+02 6.83E+01 5.57E+02 -6.83E+01
8 -6.28E+02 7.71E+01 6.28E+02 -7.71E+01
9 -6.97E+02 8.55E+01 6.97E+02 -8.55E+01
10 -7.62E+02 9.35E+01 7.62E+02 -9.35E+01
Normalized % Error in ?R45/R45 versus ?
-1000
-800
-600
-400
-200
0
200
400
600
800
1000
-15 -10 -5 0 5 10 15
?
%
p type
n type
Fig. 8.8 - Normalized % error in ?R45/R45 versus ?
172
Normalized % Error in ?R-45/R-45 versus ?
-1000
-800
-600
-400
-200
0
200
400
600
800
1000
-15 -10 -5 0 5 10 15
?
%
p type
n type
Fig. 8.9 - Normalized % error in ?R-45/R-45 versus ?
As depicted in Figs. 8.6 through 8.9, n-type sensors have larger normalized % error
than p-type sensors.
Besides, it should be noted that Eq. (8.2.3) and Eq. (8.2.28) can be also derived
using the standard equations for transforming the in-plane stress components from one
coordinate system to another [12-13]:
circle6 With respect to the unprimed coordinate system
For ? , the angle of counter-clockwise rotation of the "" y - x coordinate system with
respect to the y - x coordinate system, the double-primed stress componenets are given as
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???????
????
????
=
?
?
?
?
?
?
?
?
?
?
?
?
?
12
22
11
22
22
22
"
12
"
22
"
11
sincos cos sin cos sin-
cos 2sin- cos sin
cos 2sin sin cos
Eq. (8.2.50)
173
The stress components are now measured in the new double-primed coordinate system
instead of the unprimed coordinate system.
circle6 With respect to the primed coordinate system
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???????
????
????
=
?
?
?
?
?
?
?
?
?
?
?
?
?
'
12
'
22
'
11
22
22
22
"
12
"
22
"
11
sincos cos sin cos sin-
cos 2sin- cos sin
cos 2sin sin cos
Eq. (8.2.51)
Likewise, ? represents the angle of counter-clockwise rotation of the "" y - x coordinate
system with respect to the 'y - 'x coordinate system. The stress components are measured
in the new double-primed coordinate system instead of the primed coordinate system. In
addition, it maybe be noted that Eq. (8.2.4) and Eq. (8.2.5) can be derived by combining
the inverse of Eq. (8.2.50) with Eq. (8.2.3) for ?=? and , 90+?=? respectively.
Similarly, Eq. (8.2.29) and Eq. (8.2.30) can be obtained by combining the inverse of Eq.
(8.2.51) with Eq. (8.2.28) for ?=? and , 90+?=? respectively. Calibration of the off-
axis rosettes can be accomplished by using a uniaxial stress applied along the "1x or "2x
axis.
8.3 Off-Axis Error on the (111) Silicon Plane
On the (111) plane silicon, the rosette configuration consists of 0o, 90o, 45o, and -45o
resistors relative to the ]0 1 1[ axis. The strip is cut along the axis "" yx ? counter-
clockwise rotated ? from the primed axis ]0 1 1[ . The off-axis piezoresistive coefficients
can be determined by using the transformation
174
?? ?? ?? ??pi pi? = T T-1 Eq. (8.3.1)
The appropriate direction cosines for the (111) silicon wafer is given as follows:
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+?
++
=
??
?
?
?
?
??
?
?
?
?
31 31 31
6
cos? 2-
2
sin?
6
cos?
2
sin?
6
cos?
6
sin? 2-
6
sin?
2
cos?-
6
sin?
2
cos?
nml
nml
nml
333
222
111
Eq. (8.3.2)
If 0? = ,
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
??
?
?
?
?
??
?
?
?
?
31 31 31
6
2-
6
1
6
1
0 21- 21
nml
nml
nml
333
222
111
Eq. (8.3.3)
and pi??? is given as follows:
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+???
+??????
??+?
?+?+?+
???+++?+
++??+?+++
?
3
?2?? 0
3
)??(?2 0 0 0
0 3 ??2?2 0 0 6 )??(?2 6 )??(?2
3
)??(?2 0
3
??2?2 0 0 0
0 0 0 3 ??2? 3 ??2? 3 ??2?
0 3 )??(?2 0 3 ??2? 2 ??? 6 ??5?
0 3 )???(2 0 3 ??2? 6 ??5? 2 ???
= ?
441211441211
441211441211441211
441211441211
441211441211441211
441211441211441211441211
441211441211441211441211
??
Eq. (8.3.4)
As described in Chapter 3, the normalized change in resistance can be expressed as
175
. . . ]?T?T[2sin])?B(B)?B(B22[
sin])?B(B22?B?B?[B
cos])?B(B22?B?B?[BR?R
2
21
'
1221
'
1323
2'
2332
'
333
'
221
'
112
2'
2332
'
333
'
222
'
111
+?+?+??+?+
??++++
???++=
Eq. (8.3.5)
In addition, the normalized change in resistance can be expressed in terms of the off-
axis stress components using
... ]?T?T[
m )l? ?(2n )m? ?(2n )l? ?(2+
n)?"?" + (m)?"?" + (l)?"?" = (R?R
2
21
"""
?
"
?6
"""
?
"
?5
"""
?
"
?4
" 2??3" 2??2" 2??1
+?+?+
++ Eq. (8.3.6)
Assuming )0??(? 0??? "12"23"13"6"5"4 ====== in Eq. (8.3.6) yields
] ...TT[
m)l ????? ?2(+
n")?????(?
m")?????(? +
l")?????(? = R?R
2
21
"""
33
"
63
"
22
"
62
"
11
"
61
2"
33
"
33
"
22
"
32
"
11
"
31
2"
33
"
23
"
22
"
22
"
11
"
21
2"
33
"
13
"
22
"
12
"
11
"
11
+??+??+
++
+++
++
++
Eq. (8.3.7)
Introducing , sinm , cosl "" ?=?= and 0n" = yields
176
] ...TT[
sin )cos ????? ?2(+
sin)?????(? +
)cos?????(? = R?R
2
21
"
33
"
63
"
22
"
62
"
11
"
61
2"
33
"
23
"
22
"
22
"
11
"
21
2"
33
"
13
"
22
"
12
"
11
"
11
+??+??+
??++
?++
?++
Eq. (8.3.8)
and then combining Eq. (8.3.1) and Eq. (8.3.2) yields
2 ???
?2 )sin(cos ?)]6 ?sin 2 ?cos( ?sin38)6sin?2cos?()6sin?2cos?[2(
??]sin94)6sin?2cos?()6sin?2cos?[(?
441211
44
222
12
22
222
11
444"
11
++=
?+?+++?++
+?++=
Eq. (8.3.9)
6
?5??
?6 )sin(cos}? 9 ?sin ?4cos?)sin?6(cos
])6sin?2cos?(?)sin 3(cos?)6sin?2cos?() sin?3-[(cos?61{
}? ?sin ?cos94])6sin?2cos?(?)sin 3(cos?)6sin?2cos?(?)sin 3-[(cos?61{?
441211
44
222
12
2244
2222
11
222222"
12
?+=
?+??+++
+++?+
+?+++=
Eq. (8.3.10)
3 ? ?2?)? ?2(? 3 )sin(cos ? 441211441211
222
"
13
?+=?+?+?= Eq. (8.3.11)
177
6
?5??
?6 )sin(cos ]? 9 ?sin ?4cos?)sin?6(cos
)6sin?2cos?()2sin?6cos?()6sin?2cos?()2sin?6cos?[(
]? ?sin ?cos94])6sin?2cos?()2sin?6cos?()6sin?2cos?()2sin?6cos?[(?
441211
44
222
12
2244
2222
11
222222"
21
?+=
?+??+++
??++++
+?+++?=
Eq. (8.3.12)
2 ???
?2 )sin(cos)]?2 ?sin6 ?cos(cos38)2 ?sin6 ?cos[2(
??]cos94)2sin?6cos?()2sin?6cos?[(?
441211
44
222
12
22
22
22
11
444"
22
++=
?+?++?+?+
+?++=
Eq. (8.3.13)
3 ? ?2? )? ?2(?3 )sin(cos? 441211441211
222
"
23
?+=?+?+?= Eq. (8.3.14)
0??? "63"62"61 === Eq. (8.3.15)
It is apparent that all the pi-coefficients ( ,? " 11 " 22? , " 62" 61" 23" 21" 13" 12 ?, ?, ?, ?, ?,? and " 63? ) are
independent of ? . For any ? the following findings are yielded:
178
2
???? 441211"
11
++= ,
6
?5??? 441211"
12
?+= ,
3
?2??? 441211"
13
?+=
6
?5??? 441211"
21
?+= ,
2
???? 441211"
22
++= ,
3
?2??? 441211"
23
?+= Eq. (8.3.16)
0 ?0, ?0,? "63"62"61 ===
A set of linearly independent temperature dependent combined piezoresistive
parameters, B1, B2, and B3 [26] are given as follows:
2
???B 441211
1
++= ,
6
?5??B 441211
2
?+= and
3
?2??B 441211
3
?+= Eq. (8.3.17)
Hence any? using the notations above gives
B?? 1" 22" 11 ==
B?? 2" 21" 12 ==
B???? 3" 32" 23" 31" 13 ==== Eq. (8.3.18)
0 ?0, ?0,? "63"62"61 ===
Substitution of Eq. (8.3.18) into Eq. (8.3.8) with neglect of the out-of-plane normal stress
"
3? and the terms of temperature coefficients of resistance leads to
?+?+ sin]?B?[B+cos]?B? = [BR?R 2"21"122"22"11 Eq. (8.3.19)
For 0=? and o90=? , the expressions for the stress-induced resistance changes
"22"12"11"11
0
0 ??? = ?
R
?R + Eq. (8.3.20)
"22"22"11"21
90
90 ??? = ?
R
?R + Eq. (8.3.21)
179
It is to be emphasized that, for the (111) silicon surface, normalized % error in ?R/R
versus ? is zero due to the isotropic characteristics of the (111) silicon. Note that Eqs.
(8.3.20) and (8.3.21) may be derived also combining the inverse of Eq. (8.2.51) with Eq.
(8.3.5) for ?=? and , 90+?=? respectively. The stress components are measured in the
new double-primed coordinate system instead of the primed coordinate system.
In addition, the errors in piezoresistive coefficients induced by the rotational
misalignment of strip on the supports by an angle ? with respect to the ideal longitudinal
axis of strip will be explained in Appendix D.
8.4 Summary
Parameter ? is defined as the ratio of the axial portion to the sum of axial and
transverse portion of the diffused serpentine resistor. On the (111) silicon surface, (B1 +
B2) is constant with ?. However, (B1 - B2) depends on ?. Likewise, for sensors on the
(001) surface, S? is constant with ?. On the other hand, 44? depends on ?.
In addition, errors in misalignment with the given crystallographic axes are
described and analyzed. Precise determination of the crystallographic orientation in (001)
silicon wafers is found to be essential for extraction of piezoresistive coefficients of
silicon. However, for the (111) silicon wafers, errors associated with misalignment are
observed to have no effect on the determination of piezoresistive coefficients of silicon
due to the isotropic characteristics of the (111) silicon.
180
In order to extract a complete set of pi-coefficients (pi11, pi12, and pi44) for both p- and
n-type sensors, hydrostatic tests are required for the (111) silicon. On the other hand,
those tests are not needed for stress sensors on the (001) silicon. Instead, cutting the (001)
wafers along two directions (e.g., the unprimed axis [100] and the primed axis [110]) and
then combining both can give a complete set of pi-coefficients. In addition, by using off-
axis sensor rosettes [29], experimental calibration results for the piezoresistive
coefficients of silicon may be determined.
181
CHAPTER 9
(001) TEST CHIP DESIGN AND CALIBRATION
9.1 Mask Alignment Using Wet Anisotropic Etching
The main objective of this work is to detect the orientation with the highest precision
possible. As described in Chapter 8, the normalized error induced by rotational
misalignment is zero for the (111) silicon surface. On the other hand, rotational
misalignment can be a very important source of error on the (001) surface.
For precise determination of the crystallographic orientation in the (001) silicon
wafers, anisotropic wet etching is used. It may be noted that our proposed design takes
advantage of the symmetric under-etching behavior around the [110] direction. Also, it
includes alignment marks for aligning subsequent masks. In our cases, mask alignment of
the (001) silicon needs high precision because off-axis induced error cannot be neglected,
as described in Chapter 8. To determine the crystallographic directions, x-ray diffraction
is commonly used to determine the crystallographic properties with very high precision.
However, it would be difficult to put x-ray equipment into a mask aligner. Since the eye
is very sensitive to symmetries, it is not difficult to find two points symmetrically
distributed around a [110] direction. As shown in Fig. 9.1, we introduced alignment forks
that are repeated with an angular increment of 0.25o in the range of ?? 5 from the
182
presumed [110] direction. In addition, alignment marks for aligning subsequent masks
(active region, contact, and metal) with the correct crystallographic direction are included
in the design (see Fig. 9.2).
Fig. 9.1 - The alignment forks of both sides on silicon surface
Fig. 9.2 - Alignment marks for subsequent masks
183
To determine the [110] direction, the etched structure of alignment forks formed using
wet anisotropic etching is checked. In principle, the alignment fork is etched
symmetrically when perfectly aligned with the [110] direction. In this work, KOH
(Potassium Hydroxide) is used as an etchant. When the etch rate is plotted in regard to
the degree from the [110] direction, the minimum occurs for the [110] direction. The
exact behavior at the minimum has not been fully investigated, but a linear dependence
somewhat off the minimum is observed. A symmetric etch rate is observed from the
[110] direction, and it monotonically rises with increasing rotation from the [110]
direction. If alignment forks are patterned symmetrically from the [110] direction as
shown in Fig. 9.1, the under-etching will cut equal length. Asymmetric under-etch, on the
other hand will give different lengths. Hence, among the pair of alignment forks, it is
easy to select the one that is symmetrically etched. In our layout, the angular increment of
alignment fork is 0.25o. In order to re-align subsequent masks, every alignment mark
should be affixed with an alignment mark on the previous mask as shown in Fig. 9.2.
In this work, we limit ourselves to stating a precision of ?? 125.0 , which leads to a
negligible error. For instance, ?? 125.0 misalignment produces the following % error:
? With respect to the unprimed axes on the (001) silicon surface
For 0=? and ?=? 90 , the normalized error induced by ? is given by
) 125.0(2sin)?2 ???(
]R?R[
]R?R- []R?R[
2
11
441211
0?
0
0
0?
0
0
125.0?
0
0
???=
=
==
Eq. (9.1.1)
184
= 1.88 ?10-2 % for p-type sensors
= -1.32 ?10-3 % for n-type sensors
2(0.125))sin2? ???(
]R?R[
]R?R[ -]R?R[
2
12
441211
0?
90
90
0?
90
90
0.125?
90
90
??=
=
==
Eq. (9.1.2)
= 1.13 ?10-1 % for p-type sensors
= -2.53 ?10-3 % for n-type sensors
? With respect to the primed axes on the (001) silicon surface
For 0=? and ?=? 90 , the normalized error induced by ? is given by
2(0.125))sin??? ???(
]R?R[
]R?R[ -]R?R[
2
441211
441211
0?
0
0
0?
0
0
0.125?
0
0
++
??=
=
==
Eq. (9.1.3)
= -1.73 ?10-3 % for p-type sensors
= 4.33 ?10-3 % for n-type sensors
2(0.125))sin??? ???-(
]R?R[
]R?R[ -]R?R[
2
441211
441211
0?
90
90
0?
90
90
0.125?
90
90
?+
++=
=
==
Eq. (9.1.4)
= -1.87 ?10-3 % for p-type sensors
= -7.68 ?10-3 % for n-type sensors
185
where the normalized errors induced by 0.125o misalignment with the appropriate axes
are negligible in all cases. It should be noted that pi-coefficients in this calculation are
from Smith [6]:
6611 =pi (1/TPa), 1112 ?=pi (1/TPa), and 1381 44 =pi (1/TPa) for p-type silicon
102211 ?=pi (1/TPa), 53412 =pi (1/TPa), and 136- 44 =pi (1/TPa) for n-type silicon
In the case of KOH etching, temperature affects the etch time, and the etched
cutback rises with an increasing etch temperature for fixed duration. In our process, the
etch time is about 45 minutes at 65oC to obtain comparable etch depths. Inspection of the
etching is made in an optical microscope. A typical example of etched test structure is
shown in Fig. 9.3. The easiest way to find the alignment is to use human eye.
As seen in Fig. 9.3, the asymmetric etched structures are observed in the upper and
lower parts since two ridges are patterned symmetrically with respect to an axis which is
off the [110] direction. On the other hand in the central part, we can see almost
symmetric etching because two ridges are patterned symmetrically near the [110]
direction and the under-etching will cut the arms to equal length.
(Upper Part) (Central Part) (Lower Part)
Fig. 9.3 - An example of an etched structure of alignment forks in one wafer
186
9.2 The (001) Silicon Test Chips
A special test chip was designed and fabricated. The test chip contains p-type and n-
type sensor sets, each with resistor elements making angles of ? = 0, o45? , and 90o with
respect to the 1x (or '1x ) axis. The layout of the test chip in which the resistor rosette
sensors and VDP test structures are fabricated on the (001) silicon surfaces is shown in Fig.
9.4. It is noteworthy that three different cells are repeated in the layout of our test chip.
Fig. 9.4(a) - The test chip on the (001) silicon surfaces
Fig. 9.4(b) - Microphoto of the test chip on the (001) silicon surfaces
187
Each test chip has only one type (p-type or n-type sensors). Resistors are often designed
with relatively large meandering patterns to achieve acceptable resistance levels for
measurement. However, as discussed in Chapter 8, they suffer from transverse sensitivity
which is difficult to estimate because of the lateral diffusion that occurs during the
fabrication process. In order to minimize transverse stress sensitivity, resistor legs are
interconnected with metal links, but interconnections require additional contacts that
further increase resistor size. For the comparison of transverse stress sensitivity, one pair
of 0o, 90o, +45o, and -45o resistors without metal link are also contained in the layout. The
large pads in the left hand cell are designed for more convenient calibration of the
sensitivity. The metallurgical junction depth at which the impurity profile intersects the
background concentration is approximately 1.7 ?m for p-type sensors and 1.2 ?m for n-
type sensors.
As displayed in Table 9.1, the sheet resistance measured by Van der Pauw?s
method is about 211.5 ohms per square for p-type sensors and 122.8 ohms per square for
n-type sensors, respectively. Since 5.105WL = in the layout of our test chip, the
unstressed resistance is expected to be 22.3 ?k and 13.0 ?k for p- and n-type sensors,
respectively (see Table 9.2), which are in good with the calibration results of resistors
(22.8 ?k for p-type resistors and 13.2 ?k for n-type resistors). It is noteworthy that no
lateral diffusion is assumed in the calculation of WL . The expected resistance should be
smaller with consideration of the lateral diffusion.
188
Table 9.1 - Sheet resistance measured by Van der Pauw?s method
(Unit: ohms per square)
VDP-type P-type P-type N-type N-type
Injection pair #1 #2 #1 #2
1~2 205.7 206.0 122.0 116.6
1~4 216.5 213.6 137.6 129.4
2~3 214.7 211.8 108.4 111.0
3~4 213.4 210.6 124.3 133.0
Average 212.5 210.5 123.1 122.5
Std.Dev 4.7 3.2 12.0 10.4
Table 9.2 - Expected resistance (Unit: ohm)
VDP-type P-type P-type N-type N-type
Injection pair #1 #2 #1 #2
1~2 21699 21735 12867 12298
1~4 22836 22531 14522 13656
2~3 22646 22345 11433 11715
3~4 22511 22217 13116 14029
Average 22423 22207 12985 12925
Std.Dev 501 340 1265 1097
Ion implantation is used as the method of introducing impurities such as boron and
phosphorous into the surface of silicon target wafers, followed by drive-in step used to
move the diffusion front to the desired depth. After annealing, I-V characteristics for p-
and n-type silicon are tested and shown in Figs. 9.5 and 9.6.
189
P-type Resistor
y = 4.282E+01x
R2 = 1.000E+00
-100
-80
-60
-40
-20
0
20
40
60
80
100
-3 -2 -1 0 1 2 3
Applied Voltage (V)
(E
-6)
A
Fig. 9.5 - I-V characteristics of a p-type resistor after annealing
N-type Resistor
-80
-60
-40
-20
0
20
40
60
80
100
-3 -2 -1 0 1 2 3
Applied Voltage (V)
(E
-6)
A
N-type Resistor (PtSi)
-200
-150
-100
-50
0
50
100
150
200
-3 -2 -1 0 1 2 3
Applied Voltage (V)
(E
-6)
A
N-type Resistor
(PtSi and diffusion)
y = 7.610E+01x
R2 = 9.993E-01
-200
-150
-100
-50
0
50
100
150
200
-3 -2 -1 0 1 2 3
Applied Voltage (V)
(E
-6)
A
Fig. 9.6 - I-V characteristics of an n-type resistor after annealing
190
Obviously, it is desirable to form ohmic contacts between the metal and semiconductor.
As shown in Fig. 9.5, p-type resistor exhibits a straight line I-V characteristic. However,
a problem arises in trying to contact n-type silicon as shown in Fig. 9.6, since aluminum
may form a metal-semiconductor Schottky diode rather than an ohmic contact. In order to
resolve the problem for n-type silicon, PtSi (platinum silicide) contact metallurgy is used.
In the process, a 200 nm Pt film is deposited by electron-beam evaporation onto the (001)
Si substrates which have been pre-cleaned in buffered HF. The wafer is then sintered at
400oC for 10 minutes in order to form a high quality layer of PtSi. Next, the Pt film is
stripped off with aqua regia (a mixture of nitric and hydrochloric acids that dissolves gold
or platinum) in which silicon, silicon dioxide, and nitride will not be etched. Furthermore,
in order to form desirable ohmic contacts, heavy impurities of phosphorous are
introduced into the surface of the contacts by diffusion before forming the layer of PtSi.
For electrical testing, connections between the strip and the PC board utilize wire-
bonding between inner pads on the board and the pads on the silicon strip, as shown in
Fig. 9.7.
circle6 [100] strip-on-beam specimen
circle6 [110] strip-on-beam specimen
Fig. 9.7 - The [100] and [110] strip-on-beam specimens
191
9.3 Resistance Equations for the (001) Silicon
The piezoresistive coefficients for the (001) surface are 11? , 12? , and 44? . A wafer plot
showing the two directions that are cut from the (001) wafer is presented in Fig. 9.8. If a
wafer strip along the 1x direction (or 1x? direction) is subjected to four-point-bending, and a
known uniaxial stress ?=?11 (or ?=??11 ) is applied in the 1x (or 1x? ) direction on the
(001) silicon surface, the simplified normalized equations are expressed as follows:
Fig. 9.8 - Two directions cut from the (001) silicon wafer
? With respect to the unprimed axes
...]TT[sin2
)]sin( +[ +)]cos( +[ = R?R
2
211244
2
3311122211
2
3322121111
+??+??+??pi+
??+?pi?pi??+?pi?pi
Eq. (9.3.1)
where ? is the angle between the 1x axis and the resistor orientation. The normalized
resistance equations for 0o and 90o sensors can determine 11? and 12? directly.
192
? ?= R?R 1111
0
0 Eq. (9.3.2)
1112
90
90 ? ?
R
?R = Eq. (9.3.3)
111211
45-
45-
45
45 ? )
2
?(
R
?R
R
?R +pi== Eq. (9.3.4)
Addition of Eqs. (9.3.2) and (9.3.3) yields
)R?R(2)R?R(2) ??(?R?RR?R
45-
45-
45
45
111211
90
90
0
0 ==+=+ Eq. (9.3.5)
? With respect to the primed axes
...]TT[sin2 )-(??
sin 2 + 2 +
cos 2 + 2 = R?R
2
21
'
1212113312
2'
22
441211'
11
441211
2'
22
441211'
11
441211
+??+??+??pipi++
??
?
?
??
? ??
?
??
?
? pi+pi+pi??
?
??
?
? pi?pi+pi
??
?
?
??
? ??
?
??
?
? pi?pi+pi??
?
??
?
? pi+pi+pi
Eq. (9.3.6)
where ? is the angle between the 1x? axis and the resistor orientation. The normalized
resistance equations are given by
'11441211
0
0 ? )
2
???(
R
?R ++= Eq. (9.3.7)
? )2 ???(R?R '11441211
90
90 ?+= Eq. (9.3.8)
'111211
45-
45-
45
45 ? )
2
?(
R
?R
R
?R +pi== Eq. (9.3.9)
Subtraction Eq. (9.3.8) from Eq. (9.3.7) determine 44? as follows:
193
'1144
90
90
0
0
R
?R
R
?R ?pi=?
'
11?
Eq. (9.3.10)
Addition of Eqs. (9.3.7) and (9.3.8) yields
)R?R(2)R?R(2) ??(?R?RR?R
45-
45-
45
45'
111211
90
90
0
0 ==+=+ Eq. (9.3.11)
Through the use of four-point bending tests, all pi-coefficients may be determined without
hydrostatic tests. If we consider two-dimensional states of stress, the normalized resistance
equations are expressed as follows:
? With respect to the unprimed axes
22121111
0
0 ??? = ?
R
?R + Eq. (9.3.12)
22111112
90
90 ??? ?
R
?R += Eq. (9.3.13)
)? (? 2 ?? R?R 124422111211
45
45 ?pi+++= Eq. (9.3.14)
)? (? 2 ?? =R?R 124422111211
45-
45- ?pi?++ Eq. (9.3.15)
It is apparent that 11? and 12? can be determined by adding and subtracting Eqs. (9.3.12)
and (9.3.13). In order to express two-dimensional states of stress as a function of force F,
we adopt the notations as cited previously in Chapter 4:
F?F, ??? F2222F1111 ?? , and )R?R(dFdS
?
?
? ? Eq. (9.3.16)
The results are
194
2
F22
2
F11
90F220F11
11 )(?)(?
S?S??
?
?= Eq. (9.3.17)
2
F22
2
F11
0F2290F11
12 )(?)(?
S?S??
?
?= Eq. (9.3.18)
In the equations, 11F? and 22F? may be obtained by the finite element simulation, in which
11? and 22? are the same as 11F? and 22F? for a 1-N force, respectively. In addition, Spi can
be determined as follows:
? SS ? SS =
22F11F
45-45
22F11F
900
S ?+
+=
?+
+pi Eq. (9.3.19)
? With respect to the primed axes
'22441211' 1 1441211
0
0 ? )
2
???(? )
2
???(
R
?R ?++++= Eq. (9.3.20)
? )2 ???( ? )2 ???(R?R '22441211'11441211
90
90 +++?+= Eq. (9.3.21)
)-()? (? )2 ??(R?R '121211'22'111211
45
45 ?pipi+++= Eq. (9.3.22)
)-()? (? )2 ??(R?R '121211'22'111211
45-
45- ?pipi?++= Eq. (9.3.23)
Note that 44? can be determined by adding and subtracting Eqs. (9.3.20) and (9.3.21). The
results are
'
F22
'
F11
900
44 ??
SS?
?
?= Eq. (9.3.24)
'
F22
'
F11
900
S ??
SS?
+
+=
Eq. (9.3.25)
195
In the equations, '11F? and '22F? may be obtained by the finite element simulation, in which
'
11? and
'
22? are the same as
'
11F? and
'
22F? for a 1-N force, respectively. In addition, in
the cases of the primed axes, 4545900 SS SS ?+=+ . Hence Spi may be given by
'
22F
'
11F
45-45
'
22F
'
11F
900
S ??
SS
??
SS?
+
+=
+
+= Eq. (9.3.26)
By Eqs. (9.3.17), (9.3.18), and (9.3.24), all the piezoresistive coefficients 11? , 12? , and 44?
can be determined for the (001) silicon surface.
The details of errors in piezoresistive coefficients induced by the rotational
misalignment of strips on the supports by an angle ? with respect to the ideal longitudinal
axis of the strips are analyzed in Appendix D. It is noteworthy that Spi is not influenced
by any rotational error only in the form of ( 900 SS + ) or ( 4545 SS ?+ ). On the other hand,
rotational error affects all individual pi-coefficients ( 11? , 12? , and 44? ) on the (001)
surface.
9.4 Strip-on-beam Test Samples
Our first attempt at building strip-on-beam samples used a single strip on one side of
the PCB beam. However, as shown in Fig. 9.9, the single-sided silicon strip-on-beam
samples were significantly warped after cooling from their assembly temperature,
resulting from the mismatch in the coefficients of expansion of the various packaging
materials. The warpage shifts the initial resistance values due to the induced stresses. In
order to minimize the warpage, the double-sided silicon strip-on-beam samples were used
(see Fig. 9.10). A second dummy strip was mounted on the back of the beam resulting in
196
a symmetrical structure. Theses samples achieve an almost ?stress free? condition before
applying the force in 4PB apparatus because of their symmetrical structure.
From the simulation results in the next section, it can be seen that the stress in the
direction of the beam is uniform between the inner 4PB supports, and the transverse
component is negligible. Thus, from a stress uniformity point of view, the double-sided
strip-on-beam technique is very similar to just having a silicon beam directly in the 4PB
fixture.
Fig. 9.9 - The obvious warp of a single-sided silicon strip-on-beam sample after cooling
from 150oC to room temperature
Fig. 9.10 - The almost warp-free double-sided silicon strip-on-beam sample after cooling
from 150oC to room temperature
197
9.5 Simulation Results for the (001) Silicon Test Chips
Meshes for the [100] and [110] silicon strip-on-beam sample are shown in Figs.
9.11 through 9.14. Double-sided silicon strip-on-beam samples are used in order to
minimize the deformation that occurs upon cooling from the assembly temperature and to
maintain an almost ?stress free? condition before applying the force in 4PB apparatus.
The central quarter-die size part of silicon strip was meshed into 12 x 12 x 5
elements for the [100] directional strip-on-beam (8 x 8 x 5 for the [110] directional strip-
on-beam). Note that quarter-model was used due to the limitations of number of elements
and the duration of simulations. The other parts were meshed less densely compared with
the central part. The meshes for both types ([100] and [110] silicon strip-on-beam) are
very similar other than the number of elements.
Fig. 9.11 - Mesh plots of the [100] silicon strip-on-beam sample (quarter model)
198
Fig. 9.12 - Mesh plots of the [100] silicon strip (central part)
Fig. 9.13 - Mesh plots of the [110] silicon strip-on-beam sample (quarter model)
199
Fig. 9.14 - Mesh plots of the [110] silicon strip (central part)
The calculated contour plots of ?11 and ?22 (?'11 and ?'22) in the double-sided silicon
strip-on-beam are presented for a 1-N force in Figs. 9.15 through 9.18. Note that the
direction of ?11 (or ?'11) is parallel to the direction of the beam in our tests.
200
Fig. 9.15 - Contour plot of ?11 on [100] silicon strip-on-beam at 25oC
Fig. 9.16 - Contour plot of ?22 on [100] silicon strip-on-beam at 25oC
201
Fig. 9.17 - Contour plot of ?'11 on [110] silicon strip-on-beam at 25oC
Fig. 9.18 - Contour plot of ?'22 on [110] silicon strip-on-beam at 25oC
202
Stresses, ?11 and ?22 ( ?'11 and ?'22 ) at the location of sensor over the temperature
range -150oC to +125oC appear in Table 9.3.
Table 9.3 - ?11 and ?22 (?'11 and ?'22 ) at the sensor location
with temperature (Unit: MPa)
T(oC) ?11 ?22 ?'11 ?'22
-151.0 3.7727 0.00534 5.3222 -0.00669
-133.4 3.7784 0.00510 5.3303 -0.00629
-113.4 3.7822 0.00495 5.3355 -0.00602
-93.2 3.7865 0.00481 5.3413 -0.00564
-71.4 3.7881 0.00481 5.3432 -0.00541
-48.2 3.7897 0.00480 5.3440 -0.00540
-23.6 3.7913 0.00477 5.3472 -0.00500
0.6 3.7923 0.00478 5.3486 -0.00485
25.1 3.7968 0.00462 5.3547 -0.00450
49.9 3.8029 0.00429 5.3635 -0.00423
75.1 3.8098 0.00395 5.3748 -0.00387
100.6 3.8164 0.00360 5.3836 -0.00354
125.9 3.8164 0.00306 5.3986 -0.00322
The primary normal stress component ?11 (and/or ?'11) is almost constant with varying
temperatures. The second normal stress component ?22 (and/or ?'22 ) is negligible over
the whole temperature range because of their symmetrical structure. Note that they are
relatively small compared with the chip-on-beam cases in Chapter 4. In the simulations,
the mechanical properties (E) of composite materials are reflected in Table 9.4. Also, the
generally accepted values of ? are 0.278 and 0.062 respectively for the [100] silicon and
[110] silicon over the whole range of temperature.
203
Table 9.4 - Measurements of E with temperature (Unit: GPa)
T(oC) ME525 Silicon [100] Silicon [110] FR-406
-151.0 19.81 131.6 172.9 28.82
-133.4 18.46 131.4 172.4 27.41
-113.4 17.53 131.3 172.0 26.49
-93.2 15.99 131.2 171.5 25.57
-71.4 14.90 131.0 171.0 25.35
-48.2 13.70 130.8 170.6 25.12
-23.6 12.85 130.5 170.0 24.80
0.6 12.00 130.2 169.7 24.68
25.1 10.43 130.1 169.1 23.73
49.9 9.85 130.0 168.4 22.05
75.1 8.75 129.6 167.9 20.26
100.6 7.72 129.4 167.3 18.55
125.9 4.98 129.0 166.8 16.37
Tables 9.5 and 9.6 give the dimensions of materials of strip-on-beam structure
representing the average values of 10 specimens. These were obtained by microscope
measurement with a resolution of 0.05 mil.
Table 9.5 - Dimensions of composite materials of [100] silicon strip-on-beam
(Unit: mil)
Beam material (001) silicon: [100] Adhesive material
Length (l) 3400 3400 3400
Width (b) 650 226 226
Thickness (h) 22.67 20 2.5
Table 9.6 - Dimensions of composite materials of [110] silicon strip-on-beam
(Unit: mil)
Beam material (001) silicon: [110] Adhesive material
Length (l) 3400 3400 3400
Width (b) 650 160 160
Thickness (h) 22.67 20 2.5
204
9.6 Sensor Calibration for the (001) Silicon Test Chips
The 4PB apparatus has been used to generate the required stress. However, due to
the mismatch of mechanical properties such as E and ? among silicon, die attachment
adhesive (ME 525), and PCB material (FR-406), two-dimensional states of stress are
induced. For the double-sided silicon strip-on-beam cases, considering only ?11 (and/or
?'11) is enough for determination of pi-coefficients because the second normal stress
component ?22 (and/or ?'22 ) is negligible. However, in this work, a two-dimensional state
of stress is still considered for completeness and accuracy. The characterization results
are displayed with respect to applied force F instead of uniaxial stress. As discussed in
section 9.3, the stress-induced resistance changes for the (001) silicon are given as
follows:
? With respect to the unprimed axes
2
F11
2
F22
90F110F22
122
F11
2
F22
0F1190F22
11 )(?)(?
S?S? , ?
)(?)(?
S?S??
?
?=
?
?= Eq. (9.5.1)
?? SS ?? SS= ?
F22F11
45-45
F22F11
900
S +
+=
+
+ Eq. (9.5.2)
? With respect to the primed axes
'
F22
'
F11
900
44 ??
SS?
?
?= Eq. (9.5.3)
'
F22
'
F11
45-45
'
F22
'
F11
900
S ??
SS
??
SS?
+
+=
+
+= Eq. (9.5.4)
The slopes of the resistance versus force curves, S0, S90, S45, and S-45 for the (001) silicon
have been measured over temperature, and the average of 10 specimens is presented in
205
Tables 9.7 through 9.10. Temperature was measured directly using a Type-T (-270oC to
+300oC) thermocouple inserted into the Delta Design 2850 test chamber. The details of
S0, S90, S45, S-45, and pi-coefficients for the (001) silicon over temperature are presented
in Appendix E.
Table 9.7 - S0, S90, S45, and S-45 for [100] p-type
silicon with temperature (Unit: 10-6 N-1)
T (oC) S0 S90 S45 S-45
-151.0 203.8 -55.7 79.6 78.5
-133.4 184.7 -46.5 77.3 73.5
-113.4 168.7 -47.7 60.9 65.6
-93.2 163.6 -44.2 63.0 60.8
-71.4 150.2 -36.7 53.5 56.4
-48.2 136.7 -35.7 54.4 48.9
-23.6 123.8 -29.2 48.6 47.7
0.6 116.2 -27.3 39.2 53.5
25.1 111.5 -23.4 40.4 37.1
49.9 98.2 -21.6 36.0 35.7
75.1 83.5 -17.1 30.3 24.2
100.6 70.8 -10.1 31.1 28.5
125.9 68.3 -9.7 32.6 22.6
Table 9.8 - S0, S90, S45, and S-45 for [110] p-type
silicon with temperature (Unit: 10-6 N-1)
T (oC) S0 S90 S45 S-45
-151.0 3831 -3598 134.2 131.3
-133.4 3602 -3377 126.8 124.4
-113.4 3369 -3171 111.8 123.5
-93.2 3193 -3005 111.9 122.7
-71.4 3094 -2894 102.9 108.0
-48.2 2891 -2726 91.3 100.5
-23.6 2737 -2612 86.0 79.5
0.6 2603 -2481 77.9 89.0
25.1 2456 -2366 69.5 72.5
49.9 2298 -2199 69.8 64.4
75.1 2077 -1984 54.2 55.5
100.6 1938 -1859 44.7 46.9
125.9 1777 -1731 44.5 43.2
206
Table 9.9 - S0, S90, S45, and S-45 for [100] n-type
silicon with temperature (Unit: 10-6 N-1)
T (oC) S0 S90 S45 S-45
-151.0 -4383 2443 -993 -975
-133.4 -4171 2333 -930 -900
-113.4 -3914 2210 -878 -857
-93.2 -3726 2103 -822 -821
-71.4 -3493 2001 -776 -795
-48.2 -3294 1887 -702 -707
-23.6 -3054 1732 -667 -667
0.6 -2856 1600 -627 -626
25.1 -2622 1478 -578 -569
49.9 -2456 1346 -529 -520
75.1 -2299 1258 -472 -484
100.6 -2067 1159 -438 -424
125.9 -1858 1032 -388 -401
Table 9.10 - S0, S90, S45, and S-45 for [110] n-type
silicon with temperature (Unit: 10-6 N-1)
T (oC) S0 S90 S45 S-45
-151.0 -1623 -983 -1322 -1284
-133.4 -1536 -918 -1267 -1166
-113.4 -1423 -849 -1142 -1134
-93.2 -1337 -810 -1088 -1051
-71.4 -1256 -736 -962 -1011
-48.2 -1185 -699 -941 -911
-23.6 -1117 -664 -858 -873
0.6 -1053 -632 -782 -806
25.1 -973 -577 -718 -766
49.9 -903 -523 -701 -686
75.1 -804 -464 -630 -634
100.6 -744 -422 -589 -560
125.9 -646 -356 -521 -519
Combining Eqs. (9.5.1) and (9.5.3) with the FEM results, we can determine all the pi-
coefficients (pi11, pi12, and pi44) for p- and n-type material, and the extracted values appear
207
in Table 9.11 and Figs. 9.19 through 9.22. Our experimental results show that the
magnitudes of the pi-coefficients, pi11, pi12, and pi44 for p- and n-type silicon decrease
monotonically with increasing temperature over the temperature range -150oC to +125oC.
The four coefficients exhibit an approximately linear variation with temperature over the
full range.
Table 9.11 - pi11, pi12, and pi44 for (001) p- and n-type
silicon with temperature (Unit: TPa-1)
P-type silicon N-type silicon
T(oC) pi11 pi12 pi44 pi11 pi12 pi44
-151.0 54.0 -14.8 1395.5 -1162.7 649.1 -120.1
-133.4 48.9 -12.4 1309.2 -1104.7 618.8 -115.7
-113.4 44.6 -12.7 1225.6 -1035.5 585.8 -107.5
-93.2 43.2 -11.7 1160.3 -984.8 556.5 -98.4
-71.4 39.7 -9.7 1120.8 -922.8 529.3 -97.2
-48.2 36.1 -9.5 1051.1 -869.9 498.9 -90.9
-23.6 32.7 -7.8 1000.2 -806.2 457.9 -84.6
0.6 30.7 -7.2 950.5 -753.6 422.7 -78.7
25.1 29.4 -6.2 898.5 -691.0 390.0 -73.9
49.9 25.8 -5.7 838.0 -646.2 354.7 -70.8
75.1 21.9 -4.5 755.0 -603.7 330.9 -63.3
100.6 18.5 -2.7 704.7 -541.9 304.2 -59.7
125.9 17.9 -2.5 649.0 -485.8 270.2 -53.7
208
pi 44 with temperature (p-type silicon)
y = -2.550E+00x + 9.541E+02
R2 = 9.903E-01
0
200
400
600
800
1000
1200
1400
1600
-200 -100 0 100 200
T (Celsius)
1/T
Pa
Figure 9.19 - pi44 for the (001) p-type silicon with temperature
pi 11 and pi 12 with temperature (p-type silicon)
pi11 = -1.257E-01T + 3.162E+01
R2 = 9.830E-01
pi12 = 4.270E-02T - 7.419E+00
R2 = 9.812E-01
-20
-10
0
10
20
30
40
50
60
-200 -100 0 100 200
T (Celsius)
1/T
Pa
p11
p12
Figure 9.20 - pi11 and pi12 for the (001) p-type silicon with temperature
209
pi 11 and pi 12 with temperature (n-type silicon)
pi12 = -1.366E+00T + 4.322E+02
R2 = 9.968E-01
pi11 = 2.387E+00T - 7.689E+02
R2 = 9.939E-01-1400-1200
-1000
-800
-600
-400
-200
0
200
400
600
800
-200 -100 0 100 200
T (Celsius)
1/T
Pa p11
p12
Figure 9.21 - pi11 and pi12 for the (001) n-type silicon with temperature
pi 44 with temperature (n-type silicon)
y = 2.331E-01x - 8.113E+01
R2 = 9.873E-01
-140
-120
-100
-80
-60
-40
-20
0
-200 -100 0 100 200
T (Celsius)
1/T
Pa
Fig. 9.22 - pi44 for the (001) n-type silicon with temperature
Figure 9.23 plots the ratio pi11/pi12 versus temperature. The ratio is observed to be
constant with 1211 8.1 pi??pi . This result is in good agreement with the theoretical
210
prediction that 1211 2 pi??pi based on the electron-transfer mechanism in n-type silicon
[96].
pi 11/pi 12 (n-type silicon)
Average = - 1.78
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
-200 -150 -100 -50 0 50 100 150
T (Celsius)
pipipipi1
1/
pipipipi1
2
Fig. 9.23 - pi11/pi12 for the (001) n-type silicon with temperature
The value of the combined coefficient 1211S pi+pi=pi may be determined from both
the 45? o and 0o/90o resistor pairs for the unprimed and/or primed coordinate system as
represented in Eqs. (9.5.2) and (9.5.4). The results appear in Tables 9.12 and 9.13 and
Figs. 26 and 27. Both the 45? o and 0o/90o pairs are insensitive to rotational alignment
error [58], and should yield the most precise measurements.
211
Table 9.12 - piS for the (001) p-type silicon with temperature (Unit: TPa-1)
T (oC) [110]: (+45,-45) [110]: (0,90) [100]: (+45,-45) [100]: (0,90) Average Std.Dev
-151.0 50.0 44.0 41.9 39.2 43.8 4.6
-133.4 47.3 42.3 39.9 36.5 41.5 4.5
-113.4 44.2 37.3 33.4 32.0 36.7 5.5
-93.2 44.0 35.2 32.7 31.5 35.9 5.7
-71.4 39.6 37.5 29.0 29.9 34.0 5.3
-48.2 36.0 31.0 27.2 26.6 30.2 4.3
-23.6 31.0 23.4 25.3 24.9 26.2 3.4
0.6 31.3 22.9 24.4 23.4 25.5 3.9
25.1 26.6 16.9 20.4 23.2 21.8 4.1
49.9 25.1 18.5 18.8 20.1 20.6 3.0
75.1 20.5 17.4 14.3 17.4 17.4 2.5
100.6 17.1 14.7 15.6 15.9 15.8 1.0
125.9 16.3 8.5 14.4 15.3 13.6 3.5
Table 9.13 - piS for the (001) n-type silicon with temperature (Unit: TPa-1)
T (oC) [110]: (+45,-45) [110]: (0,90) [100]: (+45,-45) [100]: (0,90) Average Std.Dev
-151.0 -490.3 -490.3 -521.1 -513.5 -503.8 15.9
-133.4 -456.8 -460.8 -483.7 -485.9 -471.8 15.1
-113.4 -427.0 -426.2 -458.0 -449.8 -440.2 16.1
-93.2 -400.9 -402.3 -433.5 -428.2 -416.2 17.0
-71.4 -369.7 -373.3 -414.1 -393.5 -387.6 20.5
-48.2 -347.0 -352.8 -371.5 -371.0 -360.6 12.5
-23.6 -324.0 -333.4 -351.2 -348.3 -339.2 12.8
0.6 -297.2 -315.2 -329.9 -330.8 -318.3 15.8
25.1 -277.3 -289.6 -301.9 -300.9 -292.4 11.5
49.9 -258.9 -266.1 -275.6 -291.5 -273.0 14.1
75.1 -235.3 -236.1 -250.6 -272.8 -248.7 17.6
100.6 -213.6 -216.6 -225.6 -237.7 -223.4 10.8
125.9 -192.7 -185.7 -205.9 -215.6 -200.0 13.4
212
pi s with temperature (p-type silicon)
y = -1.080E-01x + 2.578E+01
R2 = 9.859E-01
0
20
40
60
80
-200 -100 0 100 200
T (Celsius)
1/T
Pa
[110]:(+45,-45) [110]:(0,90) [100]:(+45,-45)
[100]:(0,90) Average Linear (Average)
Fig. 9.24(a) - piS for the (001) p-type silicon with temperature
pi S with temperature (p-type silicon)
y = -1.080E-01x + 2.578E+01
R2 = 9.859E-01
0
10
20
30
40
50
-200 -100 0 100 200
T (Celsius)
1/T
Pa
Fig. 9.24(b) - piS for the (001) p-type silicon with temperature. Fit to the average values
from Fig. 9.24(a).
213
pis with temperature (n-type silicon)
y = 1.051E+00x - 3.235E+02
R2 = 9.899E-01
-600
-500
-400
-300
-200
-100
0
-200 -150 -100 -50 0 50 100 150
T (Celsius)
1/T
Pa
[110]:(+45,-45) [110]:(0,90) [100]:(+45,-45)
[100]:(0,90) Average Linear (Average)
Fig. 9.25(a) - piS for the (001) n-type silicon with temperature
pi s with temperature (n-type silicon)
y = 1.051E+00x - 3.235E+02
R2 = 9.899E-01
-600
-500
-400
-300
-200
-100
0
-200 -100 0 100 200
T (Celsius)
1/T
Pa
Fig. 9.25(b) - piS for the (001) n-type silicon with temperature. Fit to the average values
from Fig. 9.25(a).
214
The temperature variation of the three coefficients in p-type material is linear and
monotonically decreases in magnitude over the full temperature range studied, -150oC to
+125oC. pi44 is large and positive in p-type material. For the small coefficients in p-type
material, pi11 is found to be positive and pi12 is negative over the complete temperature
range. These results are consistent in sign and magnitude with the room temperature
results originally presented by Smith [6].
The three pi-coefficients are larger and more easily measured in n-type material. The
variation of the coefficients in n-type material is also linear and monotonically decreases
in magnitude over the full temperature range as for the p-type coefficients. pi11 and pi44 are
clearly negative, and pi12 is positive over the measured temperature range. Our test results
of pi44 for p-type silicon and pi11 for n-type silicon versus temperature are compared with
the collected data from the literature as shown in Figs. 9.26 and 9.27, respectively.
215
pi44p Vs. Temperature (experimental)
500
700
900
1100
1300
1500
1700
-200 -150 -100 -50 0 50 100 150
T (Celsius)
(1
/T
Pa
)
C = 8.0E17 C = 3.0E18
C = 8.2E18 C = 9.0E18
C = 5.0E19 [NA]
(001): C = 2.0E18 [CHO]
Fig. 9.26 - Experimental data for pi44 versus temperature with different doping
concentration for p-type silicon
pi11n Vs. Temperature (experimental)
-3500
-3000
-2500
-2000
-1500
-1000
-500
0
-250 -200 -150 -100 -50 0 50 100 150
T (Celsius)
(1/
TP
a)
C = 1.3E16 C = 1.8E18 C = 8.8E18
C = 5.0E19 C = 5.2E19 C = 9.0E19
C = 2.1E20 (001): C = 4.0E18 [CHO]
Fig. 9.27 - Experimental data for pi11 versus temperature with different doping
concentration for n-type silicon
216
9.7 Summary
The flat on silicon wafers that is normally used for alignment purposes can be off
by 1-2 degrees, and any misalignment of the resistors with the crystallographic axes will
lead to errors in the measured piezoresistive coefficients. In this work, a method for
precise determination of the crystallographic orientation [110] in the (001) silicon wafer
is used. The design takes advantage of the symmetric KOH under-etching behavior
around [110] direction, and the resulting misalignment obtained is less than 0.125o.
Piezoresistive coefficients for the (001) silicon can be determined by direct
calibration. Through the use of four-point bending tests, all pi-coefficients may be
determined without hydrostatic tests. On the other hand, (111) silicon sensors need
hydrostatic tests in order to extract a complete set of pi-coefficients ( 441211 ?and , ?,? ).
By using the sensors on the (001) silicon strip cut along the unprimed axis [110], pi44
can be determined by combining the normalized resistance equations for 0o and 90o sensors.
Also, it is apparent that 11? and 12? can be determined by the equations from the sensors on
the (001) silicon strip cut along the unprimed axis [100]. The value of the combined
coefficient 1211S pi+pi=pi may be determined from both the 45? o and 0o/90o resistor
pairs for the unprimed and/or primed coordinate system. Both the 45? o and 0o/90o pairs
are insensitive to rotational alignment error.
In order to get data, experimental apparatus has been built for characterizing the
temperature dependencies of the piezoresistive coefficients of silicon. A four-point-
bending fixture has been configured to operate over a wide temperature range. In our
work, we used the double-sided silicon strip-on-beam samples because the primary
217
normal stress component ? 11 (and/or ?uniF020'11) is uniform and independent of the location as
long as the sensor is located between the inner supports and their secondary normal stress
components are negligible because of their symmetrical structure. Experimental
measurements have been combined with finite element simulations to produce the
temperature dependence of the piezoresistive coefficients.
Our test results show that the pi-coefficients in both p- and n-type silicon exhibit an
approximately linear variation with temperature over the measurement range. All the pi-
coefficients decrease with rising temperatures.
218
CHAPTER 10
SUMMARY AND CONCLUSIONS
This work presents an extensive experimental study of the temperature dependence
of the piezoresistive coefficients of silicon. Measurements were performed using stress
sensors fabricated on both (001) and (111) silicon mounted on PCB material including
both chip-on-beam and strip-on-beam mounting techniques. Four-point bending (4PB)
was used to generate the required stress, and finite element simulations have been used to
determine the actual states of stress applied to the calibration samples. Stress sensors
fabricated on the surface of the (111) silicon wafers offer the advantage of being able to
measure the complete stress state on the silicon surface, but they require use of
hydrostatic measurement of the silicon ?pressure? coefficients for calibration. On the
other hand, all these coefficients can be measured on the (001) surface using judicious
application of uniaxial stress. Hydrostatic experiments were performed on the test chips
over the temperature range of -25oC to 100oC. By subtraction of the temperature-induced
resistance change f(?T) from the total resistance change at each data point, adjusted
resistance versus pressure data are obtained. The pressure coefficients of p- and n-type
silicon versus temperature are calibrated. For n-type silicon, ppi is very small, as
expected from the approximation pi11 ? -2pi12 [96], so direct measurement of these values
219
is quite difficult. For both p- and n-type sensors, ppi decreases in magnitude with rising
temperature.
A special four-point bending (4PB) apparatus has been constructed and integrated
into an environmental chamber capable of temperatures from -155 to +300oC. Force
generated by a vertical translation stage is applied to the four-point bending fixture inside
the chamber through a ceramic rod penetrating the bottom side of the chamber. During
experiments, a chip-on-beam (and/or a double-sided strip-on-beam) specimen is placed
on the bottom supports of the four-point bending fixture.
The hydrostatic pressure apparatus has been developed to make measurements at
elevated and reduced temperature. In order to increase the temperature of fluid, a
resistance heater is used inside the pressure vessel. To lower the temperature of fluid,
liquid nitrogen is injected into a specially designed box surrounding the pressure vessel.
E (Young?s modulus) and ? (Poisson?s ratio) of silicon are dependent on direction
because the anisotropic nature of the single crystal silicon. For any crystallographic
direction of silicon, the expression of E and ? by compliance coefficients (s11, s12, and
s44) were presented. In this work, E of silicon and the other composite materials of chip-
on-beam samples were calculated analytically and found to be in good agreement with
experimental values obtained by using the ?Deflection of Beams? method. For the (111)
silicon surface, E and ? were observed to be isotropic.
The VDP sensor has been identified to have higher sensitivity than a conventional
resistor sensor [49-50]. In this work, the effects of dimensional changes during loading
were considered. It was observed that VDP sensors offer 3.157 times higher sensitivity than
220
an analogous two element resistor sensors. On the other hand, considering strain effects
changes the magnification factor M to 3.23 for p-type sensors and 3.06 for n-type sensors.
However, the strain effects were observed to be negligible for resistor sensors, so
dimensional changes should be considered in the calculation of M for the VDP sensors.
In most prior investigations, calibration of the piezoresistive coefficients has been
performed neglecting the error in misalignment. Literature values exhibit wide
discrepancies in magnitude as well as disagreement in signs. Thus the literature data
limits the accuracy of test chip stress measurements. From the analysis of off-axis sensors,
misalignment with the crystallographic axes may lead to an enormous error in
determining the pi-coefficients for the (001) silicon surface, whereas misalignment has no
effect on the pi-coefficients for the (111) silicon surface because of its isotropic
characteristics. In this work, for precise determination of the crystallographic orientation
in silicon wafers, anisotropic wet etching is used. Furthermore, experimental calibration
results for the piezoresistive coefficients of silicon as a function of temperature are
presented and compared and contrasted with existing values from the literature. Our test
results show that the temperature variation of the three coefficients in p- and n-type
material is linear and monotonically decreases in magnitude over the full temperature
range studied, -150oC to 125oC. pi44 is large and positive in p-type material. For the small
coefficients in p-type silicon, pi11 is found to be positive and pi12 is negative over the
complete temperature range. The three pi-coefficients are larger and more easily
measured in n-type material. pi11 and pi44 are clearly negative, and pi44 is positive over the
measured temperature range. These results are consistent in sign and magnitude with the
room temperature results originally presented by Smith [6].
221
The double-sided silicon strip-on-beam method was developed to eliminate curvature
problems. By this technique, the samples achieve an almost ?stress free? condition before
applying the force in 4PB apparatus because of their symmetrical structure. In addition,
this technique offers stress uniformity in the 4PB fixture.
222
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232
APPENDICES
233
APPENDIX A
TYPICAL RESULTS OF S0 and S90 FOR THE (111) SILICON AT DIFFERENT
TEMPERATURES
Calibration results of S0 and S90 for the (111) silicon for each temperature are
displayed in this section. As mentioned in Chapter 4, the following notation is adopted:
)R?R(dFdS
?
?
? ? Eq. (A.1)
Through Figs. A.1 and A.16, typical results of S0 and S90 for the (111) silicon are shown
for each temperature of calibration.
234
S0 and S90 (p-type sensors)
y = -2.344E-03x
(0 deg.)
y = 4.988E-03x
(90 deg.)
-2.E-03
-1.E-03
0.E+00
1.E-03
2.E-03
3.E-03
4.E-03
0.0 0.2 0.4 0.6 0.8
F (N)
????R
/R
Fig. A.1 - S0 and S90 for the (111) silicon at -133oC (p-type sensors)
S0 and S90 (n-type sensors)
y = 1.302E-03x
(0 deg.)
y = -1.587E-03x
(90 deg.)-1.E-03-9.E-04
-6.E-04
-3.E-04
0.E+00
3.E-04
6.E-04
9.E-04
1.E-03
0.0 0.2 0.4 0.6 0.8
F (N)
????R
/R
Fig. A.2 - S0 and S90 for the (111) silicon at -133oC (n-type sensors)
235
S0 and S90 (p-type sensors)
y = -2.070E-03x
(0 deg.)
y = 4.488E-03x
(90 deg.)
-2.E-03
-1.E-03
0.E+00
1.E-03
2.E-03
3.E-03
4.E-03
0.0 0.2 0.4 0.6 0.8
F (N)
????R
/R
Fig. A.3 - S0 and S90 for the (111) silicon at -93oC (p-type sensors)
S0 and S90 (n-type sensors)
y = 1.250E-03x
(0 deg.)
y = -1.517E-03x
(90 deg.)
-2.E-03
-1.E-03
-5.E-04
0.E+00
5.E-04
1.E-03
0.0 0.2 0.4 0.6 0.8
F (N)
????R
/R
Fig. A.4 - S0 and S90 for the (111) silicon at -93oC (n-type sensors)
236
S0 and S90 (p-type sensors)
y = -1.974E-03x
(0 deg.)
y = 3.854E-03x
(90 deg.)
-2.E-03
-1.E-03
-5.E-04
0.E+00
5.E-04
1.E-03
2.E-03
2.E-03
3.E-03
0.0 0.2 0.4 0.6 0.8
F (N)
????R
/R
Fig. A.5 - S0 and S90 for the (111) silicon at -48oC (p-type sensors)
S0 and S90 (n-type sensors)
y = 1.173E-03x
(0 deg.)
y = -1.440E-03x
(90 deg.)
-1.E-03
-8.E-04
-6.E-04
-4.E-04
-2.E-04
0.E+00
2.E-04
4.E-04
6.E-04
8.E-04
0.0 0.1 0.2 0.3 0.4 0.5 0.6
F (N)
????R
/R
Fig. A.6 - S0 and S90 for the (111) silicon at -48oC (n-type sensors)
237
S0 and S90 (p-type sensors)
y = -1.780E-03x
(0 deg.)
y = 3.524E-03x
(90 deg.)
-2.E-03
-1.E-03
-5.E-04
0.E+00
5.E-04
1.E-03
2.E-03
2.E-03
3.E-03
0.0 0.2 0.4 0.6
F (N)
????R
/R
Fig. A.7 - S0 and S90 for the (111) silicon at 0oC (p-type sensors)
S0 and S90 (n-type sensors)
y = 1.065E-03x
(0 deg.)
y = -1.308E-03x
(90 deg.)-1.E-03-8.E-04
-6.E-04
-4.E-04
-2.E-04
0.E+00
2.E-04
4.E-04
6.E-04
8.E-04
0.0 0.2 0.4 0.6 0.8
F (N)
????R
/R
Fig. A.8 - S0 and S90 for the (111) silicon at 0oC (n-type sensors)
238
S0 and S90 (p-type sensors)
y = -1.859E-03x
(0 deg.)
y = 3.317E-03x
(90 deg.)
-2.E-03
-1.E-03
-5.E-04
0.E+00
5.E-04
1.E-03
2.E-03
2.E-03
3.E-03
0.0 0.2 0.4 0.6 0.8
F (N)
????R
/R
Fig. A.9 - S0 and S90 for the (111) silicon at 25oC (p-type sensors)
S0 and S90 (n-type sensors)
y = 9.261E-04x
(0 deg.)
y = -1.220E-03x
(90 deg.)-1.E-03
-8.E-04
-6.E-04
-4.E-04
-2.E-04
0.E+00
2.E-04
4.E-04
6.E-04
8.E-04
0.0 0.2 0.4 0.6 0.8
F (N)
????R
/R
Fig. A.10 - S0 and S90 for the (111) silicon at 25oC (n-type sensors)
239
S0 and S90 (p-type sensors)
y = -1.510E-03x
(0 deg.)
y = 3.080E-03x
(90 deg.)
-2.E-03
-1.E-03
-5.E-04
0.E+00
5.E-04
1.E-03
2.E-03
2.E-03
0.0 0.2 0.4 0.6 0.8
F (N)
????R
/R
Fig. A.11 - S0 and S90 for the (111) silicon at 50oC (p-type sensors)
S0 and S90 (n-type sensors)
y = 9.522E-04x
(0 deg.)
y = -1.073E-03x
(90 deg.)
-8.E-04
-6.E-04
-4.E-04
-2.E-04
0.E+00
2.E-04
4.E-04
6.E-04
0.0 0.2 0.4 0.6
F (N)
????R
/R
Fig. A.12 - S0 and S90 for the (111) silicon at 50oC (n-type sensors)
240
S0 and S90 (p-type sensors)
y = -1.271E-03x
(0 deg.)
y = 2.607E-03x
(90 deg.)
-1.E-03
-5.E-04
0.E+00
5.E-04
1.E-03
2.E-03
2.E-03
0.0 0.2 0.4 0.6 0.8
F (N)
????R
/R
Fig. A.13 - S0 and S90 for the (111) silicon at 75oC (p-type sensors)
S0 and S90 (n-type sensors)
y = 9.302E-04x
(0 deg.)
y = -1.117E-03x
(90 deg.)
-8.E-04
-6.E-04
-4.E-04
-2.E-04
0.E+00
2.E-04
4.E-04
6.E-04
8.E-04
0.0 0.2 0.4 0.6 0.8
F (N)
????R
/R
Fig. A.14 - S0 and S90 for the (111) silicon at 75oC (n-type sensors)
241
S0 and S90 (p-type sensors)
y = -1.425E-03x
(0 deg.)
y = 2.573E-03x
(90 deg.)
-1.E-03
-5.E-04
0.E+00
5.E-04
1.E-03
2.E-03
2.E-03
0.0 0.1 0.2 0.3 0.4 0.5 0.6
F (N)
????R
/R
Fig. A.15 - S0 and S90 for the (111) silicon at 100oC (p-type sensors)
S0 and S90 (n-type sensors)
y = 9.163E-04x
(0 deg.)
y = -1.023E-03x
(90 deg.)
-8.E-04
-6.E-04
-4.E-04
-2.E-04
0.E+00
2.E-04
4.E-04
6.E-04
0.0 0.1 0.2 0.3 0.4 0.5 0.6
F (N)
????R
/R
Fig. A.16 - S0 and S90 for the (111) silicon at 100oC (n-type sensors)
242
APPENDIX B
DETERMINATION OF PIEZORESISTIVE COEFFICIENTS
A general plot of resistance with varying temperatures and applied force is shown
in Fig. B.1, whose graphs may vary with the doping concentrations and the
crystallographic orientations of sensors.
Fig. B.1 - Resistance change with temperature and stress
In Fig. B.1, ?F1 and ?F2 at a given temperature denote the corresponding stresses for F =
F1 and F = F2, respectively. Also, it may be noted that TR is defined as the resistance
value at temperature T. Then we derive the relationship of piezoresistive coefficients
between different temperatures as follows:
243
circle6 The cases for ?T = 0
At reference temperature A, the general expression for resistance discussed in Chapter 4
?(?T)]?}[?)T(f1){0,0R(T)R( ++?+=?? , can be expressed for two different stress
states ( 1F? and 2F? ) with ?T = 0:
1FA
A
A1FA ??
)0,0(R
)0,0(R)0,(R =?? Eq. (B.1)
2FA
A
A2FA ??
)0,0(R
)0,0(R)0,(R =?? Eq. (B.2)
where A? is defined as ? at temperature A. Subtraction of Eq. (B.1) from Eq. (B.2)
leads to
)??(?R )0,(R)0,(R 1F2FA
A
1FA2FA ?=??? Eq. (B.3)
Hence, at reference temperature A, the slope of R?R versus stress )??( 1F2F ? is A? as
shown in Fig. B.2
244
Fig. B.2 - R?R versus stress at reference temperature A
Similarly, at reference temperature B,
)??(?)0,0(R )0,(R)0,(R 1F2FB
B
1FB2FB ?=??? Eq. (B.4)
For ?T= 0, ? can be calculated directly.
circle6 The cases for 0T ??
By using ?(?T)]?}[?)T(f1){0,0R(T)R( ++?+=?? , , BR at reference temperature
A can be expressed for two different stress states ( 1F? and 2F? ) as the following:
1FAAA
A
A1FB A)]?(B?[?A)(Bf
)0,0(R
)0,0(R)0,(R ?++?=?? Eq. (B.5)
2FAAA
A
A2FB A)]?(B?[?A)(Bf
)0,0(R
)0,0(R)0,(R ?++?=?? Eq. (B.6)
Subtraction of Eq. (B.5) from Eq. (B.6) yields
245
)?A)](?(B?[?)0,0(R )0,(R)0,(R 1F2FAA
A
1FB2FB ??+=??? Eq. (B.7)
Fig. B.3 - R?R versus stress at reference temperature A
For ? 0T ? , direct calculation of ? is not possible. However, combining Eqs. (B.4)
and (B.7), the relationship of ? can be determined between different temperatures.
Hence it can be a proper method for relating coefficients at different temperatures.
Equation (B.4) and Eq. (B.7) yielded the following result:
)?A)](?(B?[?)0,0(R)??(?)0,0(R)0,(R)0,(R 1F2FAAA1F2FBB1FB2FB ??+=?=???
Hence,
]A)(B?[?)0,0(R?)0,0(R AAABB ?+=
246
APPENDIX C
TYPICAL RESULTS FOR THE PRESSURE COEFFICIENT OF (111) SILICON AT
DIFFERENT TEMPERATURES
The pressure coefficients for the (111) silicon at different temperatures are shown in
this section. As discussed in Chapter 5, subtraction of the effect of temperature from the
resistance change determines the pressure coefficient. Through Figs. C.1 and C.6,
adjusted hydrostatic calibrations for the (111) silicon are shown for each temperature of
calibration. The slope of the curve corresponds to the piezoresistive coefficient p? as
presented in Chapter 5:
p?)pBB(B)p?2(?)T(fR?R p3211211 =++?=+?=?? Eq. (C.1)
Adjusted Hydrostatic Calibration
y = 1.723E-04x
(p-type)
y = 4.467E-05x
(n-type)
0.0E+00
5.0E-04
1.0E-03
1.5E-03
2.0E-03
2.5E-03
0 2 4 6 8 10 12 14
Pressure, p (MPa)
????R
/R
-f(
????T
)
Fig. C.1 - Adjusted hydrostatic calibration for the (111) silicon at -25oC
247
Adjusted Hydrostatic Calibration
y = 1.697E-04x
(p-type)
y = 4.178E-05x
(n-type)
0.0E+00
5.0E-04
1.0E-03
1.5E-03
2.0E-03
2.5E-03
0 5 10 15Pressure, p (MPa)
????R
/R
-f(
????T
)
Fig. C.2 - Adjusted hydrostatic calibration for the (111) silicon at 0oC
Adjusted Hydrostatic Calibration
y = 1.613E-04x
(p-type)
y = 3.789E-05x
(n-type)
0.0E+00
5.0E-04
1.0E-03
1.5E-03
2.0E-03
2.5E-03
0 5 10 15Pressure, p (MPa)
????R
/R
-f(
????T
)
Fig. C.3 - Adjusted hydrostatic calibration for the (111) silicon at 25oC
248
Adjusted Hydrostatic Calibration
y = 2.578E-05x
(n-type)
y = 1.330E-04x
(p-type)
0.0E+00
5.0E-04
1.0E-03
1.5E-03
2.0E-03
0 5 10 15
Pressure, p (MPa)
????R
/R
-f(
????T
)
Fig. C.4 - Adjusted hydrostatic calibration for the (111) silicon at 50oC
Adjusted Hydrostatic Calibration
y = 2.177E-05x
(n-type)
y = 1.152E-04x
(p-type)
0.0E+00
4.0E-04
8.0E-04
1.2E-03
1.6E-03
0 5 10 15
Pressure, p (MPa)
????R
/R
-f(
????T
)
Fig. C.5 - Adjusted hydrostatic calibration for the (111) silicon at 75oC
249
Adjusted Hydrostatic Calibration
y = -1.221E-05x
(n-type)
y = 1.137E-04x
(p-type)
-2.0E-04
0.0E+00
2.0E-04
4.0E-04
6.0E-04
8.0E-04
1.0E-03
1.2E-03
0 2 4 6 8 10 12
Pressure, p (MPa)
????R
/R
-f(
????T
)
Fig. C.6 - Adjusted hydrostatic calibration for the (111) silicon at 100oC
250
APPENDIX D
BEAM ROTATIONAL ERROR
In Chapter 8, errors in misalignment with the given crystallographic axes are
described and analyzed. Bittle, et al. [51] calculated percent error in the axial normal
stress at the midpoint between the supports versus the angle of misalignment of the wafer
strip in a four-point bending test fixture. In order to calculate error in the axial normal
stress at the midpoint between the supports, a finite element numerical simulation was
performed [51]. In addition, Bittle, et al. [51] showed that the error will be less than
approximately 1% if the wafer strip can be aligned to within 5o. However, in our work,
the errors in piezoresistive coefficients induced by the rotational misalignment of the strip
on the supports by an angle ? with respect to the ideal longitudinal axis of the strip are
explained:
circle6 With respect to the unprimed coordinate system
Fig. D.1 - Misalignment of the wafer strip in a four-point bending test fixture
251
For ? , the angle of counter-clockwise rotation of the "" y - x coordinate system relative
to the y - x coordinate system, the double-primed stress componenets are given as [12-
13]
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???????
????
????
=
?
?
?
?
?
?
?
?
?
?
?
?
?
12
22
11
22
22
22
"
12
"
22
"
11
sincos cos sin cos sin-
cos 2sin- cos sin
cos 2sin sin cos
Eq. (D.1)
Hence
sin2- sin cos "12"222"11211 ????+??=?
sin2 cos sin "12"222"11222 ??+??+??=? Eq. (D.2)
"
12
"
22
"
1112 2cos 2sin2
1 2sin
2
1 ??+?????=?
Addition of the first two equations yields the finding below:
"22"111211 ?+?=?+? Eq. (D.3)
The stress components are now measured in the new double-primed coordinate system
instead of the unprimed coordinate system. The general resistance equation is given by
...]TT[
sin cos ? 2?
sin ]???[?
cos ]???[?R?R
2
21
1244
2
11122211
2
22121111
+??+??+
??+
?++
?+=
Eq. (D.4)
252
in which only in-plane stresses are assumed. If we consider ? with neglect of temperature
terms, ?R/R induced by ? relative to the actual longitudinal axis of strip
for 0=? , o90=? , and o45?=? is presented below:
"
121112
"
22
2
12
2
11
"
11
2
12
2
11
"
12
"
22
2''
11
2
12
"
12
"
22
2''
11
2
11
22121111
0
0
2sin)()cossin()sincos(
) sin2 cos (sin?) sin2- sin (cos?
????R?R
??pi?pi+??pi+?pi+??pi+?pi=
??+??+??+????+??=
+=
Eq. (D.5)
"
121211
"
22
2
11
2
12
"
11
2
11
2
12
"
12
"
22
2''
11
2
11
"
12
"
22
2''
11
2
12
22111112
90
90
2sin)()cossin()sincos(
) sin2 cos (sin?) sin2- sin (cos?
????R?R
??pi?pi+??pi+?pi+??pi+?pi=
??+??+??+????+??=
+=
Eq. (D.6)
"
1244
"
22
441211"
11
441211
"
12
"
22
''
1144
"
22
''
11
1211
12442211
1211
45
45
2cos)2sin22()2sin22(
) cos2 sin221 sin221(?))(2(
))(2(R?R
??pi+??pi?pi+pi+??pi+pi+pi=
??+?????+?+?pi+pi=
?pi+?+?pi+pi=
Eq. (D.7)
"
1244
"
22
441211"
11
441211
"
12
"
22
''
1144
"
22
''
11
1211
12442211
1211
45-
45-
? ?2cos??)?2sin2?2 ??(?)?2sin2?2 ??(
)? ?2cos? ?2sin21? ?2sin21(?) ?)(?2 ??(
??)?)(?2 ??(R?R
?+++?+=
+??++=
?++=
Eq. (D.8)
Through the use of the equations,
253
]? ?2sin2)??(?2cos)[?(?
)S? ?2sin? ?cos? ?sin()S? ?2sin-? ?sin? ?cos(
)(?)(? S?S??
"
F12
"
F22
"
F11
"
F22
"
F11
90
"
F12
"
F22
2"
F11
2
0
"
F12
"
F22
2"
F11
2
2
F22
2
F11
90F220F11
11
??+
++?+=
?
?=
Eq. (D.9)
]? ?2sin2)??(?2cos)[?(?
)S? ?2sin? ?cos? ?sin()S? ?2sin-? ?sin? ?cos(
)(?)(?
S?S??
"
F12
"
F22
"
F11
"
F22
"
F11
0
"
F12
"
F22
2"
F11
2
90
"
F12
"
F22
2"
F11
2
2
F22
2
F11
0F2290F11
12
??+
++?+=
?
?=
Eq. (D.10)
"
F22
"
F11
45-45
"
F22
"
F11
900
F22F11
45-45
F22F11
900
S ? ?
SS
? ?
SS=
??
SS
??
SS= ?
+
+=
+
+
+
+=
+
+ Eq. (D.11)
Note that Eq. (D.3) is used in the calculation of Spi in Eq. (D.11)
circle6 With respect to the primed coordinate system
Fig. D.2 - Misalignment of the wafer strip in a four-point bending test fixture
254
By [12-13],
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???????
????
????
=
?
?
?
?
?
?
?
?
?
?
?
?
?
'
12
'
22
'
11
22
22
22
"
12
"
22
"
11
sincos cos sin cos sin-
cos 2sin- cos sin
cos 2sin sin cos
Eq. (D.12)
Hence
sin2- sin cos "12"222"11211' ????+??=?
sin2 cos sin "12"222"11222' ??+??+??=? Eq. (D.13)
"
12
"
22
"
1112
' 2cos 2sin
2
1 2sin
2
1 ??+?????=?
Addition of the first two equations yields the finding below:
"22"11'22'11 ?+?=?+? Eq. (D.14)
In Eq. (D.13), ? represents the angle of counter-clockwise rotation of the
"" y - x coordinate system with respect to the 'y - 'x coordinate system. The stress
components are measured in the new double-primed coordinate system instead of the
primed coordinate system. The general resistance equation is expressed as the following:
...]TT[
sin cos ? )?-2(?
sin ])?2 ???()?2 ???[(
cos ])?2 ???()?2 ???[(R?R
2
21
12'1211
222'44121111'441211
222'44121111'441211
+??+??+
??+
?+++?++
??++++=
Eq. (D.15)
in which only in-plane stresses are assumed. If we consider ? with neglect of temperature
terms, ?R/R induced by ? with respect to the ideal longitudinal axis of strip
for 0=? , o90=? , and o45?=? is presented below:
255
"
1244
"
22
441211"
11
441211
"
12
"
22
2"
11
2441211
"
12
"
22
2"
11
2441211
22'44121111'441211
0
0
2sin)2 2cos???()2 2cos???(
) sin2 cos )(sin2 ???(
) sin2- sin )(cos2 ???(
)?2 ???()?2 ???(R?R
??pi????++??++=
??+??+???++
????+??++=
?++++=
Eq. (D.16)
"
1244
"
22
441211"
11
441211
"
12
"
22
2"
11
2441211
"
12
"
22
2"
11
2441211
22'44121111'441211
90
90
2sin)2 2cos???()2 2cos???(
) sin2 cos )(sin2 ???(
) sin2- sin )(cos2 ???(
)?2 ???()?2 ???(R?R
??pi+??+++???+=
??+??+??+++
????+???+=
+++?+=
Eq. (D.17)
"
121211
"
221211
1211"
111211
1211
"
12
"
22
"
11121122
"11"1211
12'121122'11'1211
45
45
2cos)?-(?
)]?-(? 2sin21)2 ??([ )]?-(? 2sin21)2 ??([
) 2cos 2sin21 2sin21)(?-(?)?)(?2 ??(
? )?-(?)?)(?2 ??(R?R
??+
???++??++=
??+?????+++=
+++=
Eq. (D.18)
256
"
121211
"
221211
1211"
111211
1211
"
12
"
22
"
11121122
"11"1211
12'121122'11'1211
45-
45-
2cos)?-(?
)]?-(? 2sin21)2 ??([ )]?-(? 2sin21)2 ??([
) 2cos 2sin21 2sin21)(?-(?)?)(?2 ??(
? )?-(?)?)(?2 ??(R?R
???
??+++???+=
??+??????++=
?++=
Eq. (D.19)
Through the use of the equations,
"
F12
"
F22
"
F11
900
'
F22
'
F11
900
44 ? ?2sin2)-?? (?2cos
SS
-??
SS?
?
?=?= Eq. (D.20)
"
F22
"
F11
45-45
"
F22
"
F11
900
'
F22
'
F11
45-45
'
F22
'
F11
900
S ??
SS
??
SS
??
SS
??
SS?
+
+=
+
+=
+
+=
+
+= Eq. (D.21)
Note that Eq. (D.14) is used in the calculation of Spi in Eq. (D.21). As presented in Eqs.
(D.11) and (D.21), the 0o/90o and o45? pairs are insensitive to beam rotational error and
should yield the precise measurement. It addition, the 0o/90o and o45? pairs are
insensitive to rotational alignment error.
257
APPENDIX E
TYPICAL RESULTS FOR S0, S90, S45, AND S-45 FOR THE (001) SILICON VERSUS
TEMPERATURE
Typical results of S0, S90, S45, and S-45 for the (001) silicon with temperature are
displayed in this section. It may be noted that 10 samples were used for these calibrations.
circle6 P-type: [100] direction
Table E.1 - S0 for [100] p-type silicon with temperature (Unit: N-1)
T (oC) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std.Dev
-151.0 244.4 162.1 205.8 173.4 160.4 237.6 239.5 258.1 187.8 169.0 203.8 38.1
-133.4 197.1 171.4 179.3 198.2 149.6 187.9 219.2 147.3 193.4 203.7 184.7 23.1
-113.4 156.5 130.2 192.5 126.9 112.6 166.3 203.9 210.1 177.7 210.1 168.7 36.3
-93.2 168.6 162.0 200.2 149.1 145.0 157.4 130.3 146.8 150.2 226.5 163.6 28.8
-71.4 133.4 136.9 175.1 172.1 123.5 117.9 132.1 145.5 182.7 183.2 150.2 25.4
-48.2 111.4 156.3 163.0 177.0 131.4 105.9 107.0 95.2 138.5 180.8 136.7 31.4
-23.6 103.2 135.6 135.0 134.4 115.0 110.2 110.6 131.9 110.1 151.6 123.8 15.9
0.6 89.3 119.2 94.4 164.4 128.2 85.9 86.7 114.9 130.8 148.5 116.2 27.3
25.1 99.7 121.9 131.0 123.1 82.8 94.6 115.6 111.0 112.1 123.0 111.5 15.0
49.9 68.1 133.7 111.5 102.7 67.0 80.0 97.3 121.8 97.9 101.7 98.2 21.7
75.1 71.7 81.4 60.2 88.7 69.8 76.8 104.4 102.2 104.2 75.6 83.5 15.7
100.6 42.8 71.1 65.8 65.9 60.9 88.2 74.9 77.5 75.0 85.5 70.8 13.0
125.9 47.7 75.4 78.4 42.0 54.6 57.9 86.9 88.2 85.8 66.4 68.3 17.0
258
Table E.2 - S90 for [100] p-type silicon with temperature (Unit: N-1)
T (oC) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std.Dev
-151.0 -69.6 -48.2 -53.6 -43.7 -50.2 -67.9 -67.2 -62.7 -50.7 -42.9 -55.7 10.3
-133.4 -46.7 -10.2 -49.8 -50.4 -31.4 -61.6 -47.7 -44.9 -56.0 -66.6 -46.5 16.0
-113.4 -42.7 -50.6 -58.4 -30.9 -20.3 -71.4 -55.6 -38.6 -46.6 -61.8 -47.7 15.2
-93.2 -53.8 -34.4 -48.1 -42.3 -29.0 -58.6 -38.4 -42.9 -36.2 -58.1 -44.2 10.2
-71.4 -42.8 -46.3 -60.4 -45.4 -13.0 -25.7 -32.1 -20.6 -35.0 -45.8 -36.7 14.2
-48.2 -33.9 -58.5 -27.2 -42.2 -38.3 -15.9 -36.7 -35.3 -28.5 -40.0 -35.7 11.1
-23.6 -32.0 -13.7 -30.9 -35.2 -31.0 -23.1 -35.2 -32.5 -32.6 -26.3 -29.2 6.6
0.6 -34.0 -35.0 5.1 -31.1 -40.0 -28.6 -29.1 -26.3 -25.4 -29.0 -27.3 12.2
25.1 -24.7 -30.9 -17.8 -25.5 -24.5 -19.4 -27.5 -18.6 -22.6 -22.9 -23.4 4.1
49.9 -43.3 -17.1 -25.7 -22.1 -16.7 -16.0 -19.6 -21.6 -12.5 -21.6 -21.6 8.5
75.1 -22.6 -31.3 -27.1 -11.8 -13.4 -14.9 -14.3 -10.7 -10.9 -14.0 -17.1 7.3
100.6 -9.4 -10.4 3.6 -7.5 -10.9 -8.0 -15.5 -15.7 -15.0 -11.7 -10.1 5.7
125.9 -10.7 -7.6 8.1 -12.1 7.0 -17.5 -18.1 -12.8 -18.6 -14.4 -9.7 9.7
Table E.3 - S45 for [100] p-type silicon with temperature (Unit: N-1)
T (oC) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std.Dev
-151.0 64.4 67.2 66.8 40.7 70.7 118.2 93.6 115.4 82.5 76.8 79.6 23.9
-133.4 66.8 74.2 70.3 94.0 49.1 96.5 75.2 91.9 60.5 94.7 77.3 16.4
-113.4 47.2 55.0 63.7 28.6 52.4 69.6 91.6 64.4 49.4 87.3 60.9 18.9
-93.2 61.2 46.7 43.7 64.2 61.3 66.1 80.0 69.3 64.5 73.6 63.0 11.1
-71.4 45.7 60.3 50.8 54.1 42.7 45.5 44.3 69.1 54.6 67.5 53.5 9.6
-48.2 38.8 63.2 51.6 63.6 53.1 36.3 62.9 73.3 31.9 68.9 54.4 14.5
-23.6 31.3 59.6 48.1 35.2 54.5 47.2 47.8 72.3 42.9 46.8 48.6 11.7
0.6 46.4 42.5 16.6 48.2 36.7 44.8 46.7 33.7 33.1 43.4 39.2 9.6
25.1 54.6 54.2 15.7 40.3 47.7 43.7 43.7 30.2 37.9 35.8 40.4 11.6
49.9 39.4 42.9 12.9 55.6 39.5 29.9 29.9 30.4 32.9 46.3 36.0 11.6
75.1 24.1 27.4 12.0 32.3 32.8 39.4 39.7 30.9 27.4 36.8 30.3 8.3
100.6 29.3 30.4 33.2 27.1 22.6 33.3 37.0 33.6 34.4 29.8 31.1 4.1
125.9 23.7 33.9 47.9 29.0 25.1 35.4 36.2 36.1 26.8 32.4 32.6 7.1
259
Table E.4 - S-45 for [100] p-type silicon with temperature (Unit: N-1)
T (oC) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std.Dev
-151.0 81.1 122.1 61.6 50.2 59.3 83.8 112.9 84.4 47.2 82.6 78.5 25.0
-133.4 85.8 50.9 59.0 58.8 67.2 101.8 106.4 66.8 66.9 70.9 73.5 18.6
-113.4 61.1 32.6 59.8 61.3 40.0 78.1 91.2 85.3 71.9 74.7 65.6 18.7
-93.2 54.0 57.0 71.6 38.4 46.4 85.8 64.0 80.1 49.2 61.9 60.8 15.0
-71.4 48.3 40.7 60.8 40.0 64.9 58.4 82.1 68.7 44.3 55.5 56.4 13.5
-48.2 34.1 37.3 49.0 34.5 67.8 61.5 38.0 57.9 56.1 52.7 48.9 12.2
-23.6 48.3 52.0 21.9 67.1 38.8 50.6 51.0 50.3 30.6 66.2 47.7 14.2
0.6 38.0 54.5 62.3 79.6 43.6 58.9 58.6 55.2 33.5 51.0 53.5 13.2
25.1 41.1 37.4 25.8 47.8 34.0 32.3 32.3 48.1 29.1 43.5 37.1 7.7
49.9 30.4 39.2 35.9 27.6 29.3 43.1 43.1 43.6 30.6 34.3 35.7 6.2
75.1 -20.9 18.4 51.4 41.6 24.8 29.9 5.2 33.7 18.9 38.7 24.2 20.6
100.6 7.6 26.2 27.1 24.8 35.7 36.6 33.3 35.7 25.2 32.6 28.5 8.7
125.9 18.6 24.7 8.7 19.9 17.0 27.8 28.1 29.8 29.0 22.9 22.6 6.7
circle6 P-type: [110] direction
Table E.5 - S0 for [110] p-type silicon with temperature (Unit: N-1)
T (oC) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std.Dev
-151.0 4216 4149 3963 3592 4119 4069 3686 3456 3590 3468 3831 300.9
-133.4 3865 3702 3727 3365 3955 3768 3466 3358 3409 3401 3602 225.8
-113.4 3456 3452 3311 3284 3539 3400 3369 3236 3354 3293 3369 93.4
-93.2 3188 3160 3180 3237 3223 3187 3298 3173 3237 3042 3193 66.8
-71.4 3093 2874 3063 3180 3209 2982 3270 3064 3226 2980 3094 126.9
-48.2 2811 2719 2846 3029 2905 2688 3213 2892 2953 2858 2891 151.7
-23.6 2687 2685 2642 2820 2733 2461 3133 2723 2740 2744 2737 168.1
0.6 2632 2583 2418 2627 2550 2485 3045 2497 2518 2677 2603 173.9
25.1 2462 2419 2362 2306 2496 2340 2882 2376 2410 2507 2456 163.4
49.9 2348 2234 2197 2216 2295 2305 2520 2349 2353 2166 2298 103.3
75.1 2128 2130 1923 2138 2129 2141 2219 2263 2179 1519 2077 215.1
100.6 2047 1969 1806 1875 1984 2023 2091 2129 2025 1431 1938 202.2
125.9 1964 1903 1640 1638 1823 1870 1845 1829 1930 1324 1777 193.1
260
Table E.6 - S90 for [110] p-type silicon with temperature (Unit: N-1)
T (oC) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std.Dev
-151.0 -3930 -4038 -3804 -3360 -3868 -3812 -3541 -3259 -3345 -3019 -3598 339.7
-133.4 -3630 -3546 -3735 -3048 -3653 -3559 -3360 -3120 -3233 -2886 -3377 291.8
-113.4 -3276 -3330 -3220 -2966 -3362 -3200 -3265 -3047 -3190 -2854 -3171 164.3
-93.2 -2947 -3019 -2987 -2921 -3057 -3006 -3253 -3046 -3084 -2731 -3005 132.7
-71.4 -2834 -2770 -2842 -2895 -2981 -2759 -3188 -2880 -3074 -2720 -2894 147.9
-48.2 -2690 -2637 -2670 -2794 -2668 -2442 -3150 -2804 -2835 -2571 -2726 189.3
-23.6 -2516 -2591 -2500 -2618 -2589 -2440 -3005 -2711 -2695 -2459 -2612 165.6
0.6 -2457 -2484 -2261 -2462 -2604 -2320 -2923 -2457 -2439 -2404 -2481 180.8
25.1 -2247 -2293 -2150 -2240 -2420 -2232 -2848 -2400 -2376 -2451 -2366 195.4
49.9 -2217 -2204 -2017 -2094 -2247 -2203 -2465 -2283 -2299 -1964 -2199 145.3
75.1 -2113 -2105 -1824 -2049 -2015 -1902 -2227 -2106 -1992 -1504 -1984 203.6
100.6 -3930 -4038 -3804 -3360 -3868 -1938 -2076 -1980 -1865 -1337 -1859 205.7
125.9 -3630 -3546 -3735 -3048 -3653 -1800 -1923 -1847 -1899 -1181 -1731 225.2
Table E.7 - S45 for [110] p-type silicon with temperature (Unit: N-1)
T (oC) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std.Dev
-151.0 133.1 126.9 145.1 108.0 94.9 141.1 156.7 129.4 138.0 168.8 134.2 21.6
-133.4 107.9 110.9 135.1 117.5 83.0 139.3 136.6 132.1 126.1 179.1 126.8 25.2
-113.4 79.2 106.0 94.5 115.3 97.6 120.3 133.0 111.5 101.2 159.2 111.8 22.3
-93.2 95.4 96.7 114.8 102.9 98.0 125.1 123.4 113.7 101.6 147.8 111.9 16.6
-71.4 83.2 35.3 116.0 105.7 89.5 108.9 129.6 98.8 117.7 144.3 102.9 29.8
-48.2 67.7 90.4 84.6 82.9 87.6 115.8 108.2 71.3 94.2 109.9 91.3 16.1
-23.6 82.3 85.4 93.7 93.9 55.5 90.1 103.4 86.3 84.9 84.2 86.0 12.4
0.6 83.0 47.4 74.1 71.2 64.6 94.1 89.5 78.6 110.6 65.7 77.9 17.7
25.1 62.3 58.6 65.6 67.6 56.8 80.8 81.5 62.9 87.4 71.8 69.5 10.5
49.9 69.7 59.3 60.7 61.6 51.9 71.9 80.4 76.7 95.0 71.1 69.8 12.4
75.1 56.9 39.9 48.9 47.6 44.5 41.2 43.3 71.7 84.0 63.7 54.2 14.7
100.6 27.9 48.0 23.5 45.7 27.2 56.5 50.2 60.9 46.7 60.9 44.7 13.9
125.9 39.9 36.5 40.0 40.1 38.0 34.1 55.4 53.9 63.2 43.8 44.5 9.6
261
Table E.8 - S-45 for [110] p-type silicon with temperature (Unit: N-1)
T (oC) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std.Dev
-151.0 121.6 66.3 135.9 122.9 121.6 144.4 173.2 111.3 113.8 202.3 131.3 36.8
-133.4 113.5 114.7 128.6 115.8 110.1 132.2 146.2 110.0 114.3 158.7 124.4 16.8
-113.4 90.0 111.4 121.8 123.2 112.2 119.6 154.3 105.7 128.4 168.0 123.5 22.8
-93.2 106.4 118.4 119.6 141.4 106.4 133.0 138.5 106.5 116.9 140.2 122.7 14.4
-71.4 91.2 76.9 118.4 98.6 95.8 110.1 139.1 109.7 99.0 141.7 108.0 20.5
-48.2 83.6 75.8 103.4 95.1 104.4 117.2 106.9 101.0 89.4 128.4 100.5 15.5
-23.6 78.7 34.1 85.3 87.0 85.9 93.2 80.0 76.2 76.7 98.3 79.5 17.5
0.6 73.7 73.0 93.3 85.6 96.8 101.2 99.1 77.9 85.2 103.9 89.0 11.5
25.1 70.0 65.8 69.7 62.6 52.9 72.8 93.6 84.8 85.7 66.9 72.5 12.2
49.9 72.0 53.1 63.0 54.9 61.1 64.8 69.5 78.0 75.2 52.2 64.4 9.2
75.1 46.5 38.1 66.9 43.0 47.3 54.7 66.2 61.3 79.1 52.2 55.5 12.7
100.6 48.8 51.5 30.4 29.0 38.5 44.9 63.1 74.5 39.4 49.0 46.9 14.0
125.9 45.9 41.5 39.8 45.9 34.2 51.7 33.3 44.9 53.0 41.6 43.2 6.5
circle6 N-type: [100] direction
Table E.9 ? S0 for [100] n-type silicon with temperature (Unit: N-1)
T (oC) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std.Dev
-151.0 -4550 -4648 -4120 -4547 -4232 -4318 -4367 -4735 -4262 -4050 -4383 228.7
-133.4 -4312 -4758 -3970 -4104 -3923 -3937 -4223 -4582 -3947 -3953 -4171 297.2
-113.4 -4105 -4229 -3851 -3922 -3697 -3668 -3804 -4319 -3753 -3788 -3914 227.4
-93.2 -3846 -4088 -3634 -3644 -3303 -3516 -3857 -4171 -3623 -3579 -3726 264.8
-71.4 -3583 -3717 -3332 -3438 -3200 -3327 -3613 -3949 -3451 -3322 -3493 224.6
-48.2 -3128 -3542 -3061 -3255 -3000 -3194 -3561 -3691 -3232 -3279 -3294 229.6
-23.6 -2766 -3256 -2899 -3101 -2874 -3091 -3291 -3322 -2937 -3006 -3054 190.8
0.6 -2626 -2896 -2825 -2903 -2697 -2928 -2987 -3130 -2809 -2756 -2856 146.3
25.1 -2399 -2600 -2717 -2723 -2433 -2606 -2716 -2642 -2734 -2647 -2622 119.2
49.9 -2201 -2570 -2552 -2540 -2270 -2502 -2455 -2518 -2518 -2432 -2456 124.4
75.1 -2094 -2327 -2356 -2362 -2121 -2315 -2331 -2414 -2342 -2326 -2299 104.8
100.6 -1858 -2241 -2039 -2060 -1823 -2124 -1991 -2244 -2163 -2128 -2067 144.4
125.9 -1671 -1920 -1826 -1857 -1690 -1874 -1840 -2006 -1955 -1939 -1858 108.8
262
Table E.10 ? S90 for [100] n-type silicon with temperature (Unit: N-1)
T (oC) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std.Dev
-151.0 2337 2716 2157 2540 2270 2391 2606 2865 2384 2162 2443 234.9
-133.4 2323 2625 2087 2264 2108 2215 2512 2782 2161 2248 2333 232.3
-113.4 2222 2512 1973 2209 1957 2057 2334 2581 2105 2153 2210 211.4
-93.2 2035 2478 1937 2108 1814 1972 2214 2352 2069 2046 2103 197.8
-71.4 1911 2214 1873 2030 1805 1857 2236 2173 1978 1929 2001 156.5
-48.2 1701 2092 1690 1936 1675 1805 2082 2061 1917 1907 1887 163.0
-23.6 1442 1927 1592 1821 1607 1758 1782 1911 1733 1750 1732 149.4
0.6 1364 1577 1510 1682 1466 1708 1657 1780 1636 1615 1600 123.8
25.1 1310 1455 1485 1474 1310 1434 1671 1623 1512 1504 1478 115.1
49.9 1198 1403 1298 1289 1208 1345 1437 1503 1323 1456 1346 102.9
75.1 1152 1353 1284 1196 1146 1264 1248 1406 1250 1285 1258 81.8
100.6 1035 1281 1163 1077 1040 1226 1130 1273 1154 1212 1159 89.6
125.9 912 1058 1027 1071 894 1034 1032 1147 1040 1107 1032 77.8
Table E.11 ? S45 for [100] n-type silicon with temperature (Unit: N-1)
T (oC) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std.Dev
-151.0 -1103 -1162 -936 -1069 -1187 -980 -851 -1008 -856 -783 -993 137.8
-133.4 -958 -804 -935 -924 -1022 -933 -973 -1092 -775 -881 -930 94.0
-113.4 -862 -814 -898 -867 -825 -944 -878 -923 -886 -879 -878 39.6
-93.2 -878 -887 -765 -833 -781 -925 -849 -732 -745 -832 -822 64.8
-71.4 -784 -725 -748 -782 -826 -767 -835 -777 -726 -788 -776 36.7
-48.2 -673 -740 -676 -699 -669 -714 -739 -692 -743 -678 -702 29.7
-23.6 -646 -689 -641 -663 -636 -664 -660 -690 -669 -708 -667 23.1
0.6 -634 -641 -612 -628 -606 -617 -655 -655 -611 -609 -627 18.8
25.1 -531 -582 -547 -592 -527 -614 -620 -591 -629 -550 -578 37.3
49.9 -487 -512 -534 -513 -513 -569 -538 -525 -561 -541 -529 24.6
75.1 -489 -513 -426 -454 -491 -490 -506 -468 -446 -440 -472 29.7
100.6 -417 -468 -422 -435 -448 -463 -462 -416 -430 -416 -438 20.8
125.9 -402 -435 -396 -358 -383 -412 -388 -366 -368 -370 -388 23.8
263
Table E.12 ? S-45 for [100] n-type silicon with temperature (Unit: N-1)
T (oC) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std.Dev
-151.0 -1290 -964 -1039 -913 -943 -1006 -949 -930 -794 -925 -975 127.9
-133.4 -1148 -1036 -810 -965 -927 -859 -733 -870 -924 -732 -900 130.2
-113.4 -923 -1044 -728 -963 -951 -882 -773 -853 -683 -768 -857 116.3
-93.2 -889 -827 -810 -845 -829 -820 -817 -893 -732 -749 -821 51.5
-71.4 -829 -971 -810 -729 -746 -840 -787 -837 -741 -659 -795 84.5
-48.2 -761 -696 -698 -666 -726 -671 -734 -766 -645 -714 -707 40.3
-23.6 -674 -660 -690 -641 -663 -633 -716 -682 -633 -674 -667 26.5
0.6 -579 -630 -623 -626 -647 -655 -599 -637 -649 -617 -626 23.5
25.1 -535 -588 -525 -514 -579 -631 -597 -617 -565 -543 -569 39.6
49.9 -516 -543 -473 -531 -496 -555 -519 -555 -509 -502 -520 26.6
75.1 -467 -508 -456 -444 -477 -521 -490 -505 -464 -506 -484 25.9
100.6 -437 -462 -368 -389 -420 -450 -413 -471 -402 -428 -424 32.3
125.9 -385 -468 -388 -376 -377 -441 -323 -444 -398 -404 -401 41.8
circle6 N-type: [110] direction
Table E.13 ? S0 for [110] n-type silicon with temperature (Unit: N-1)
T (oC) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std.Dev
-151.0 -1782 -1726 -1715 -1440 -1439 -1696 -1719 -1457 -1728 -1530 -1623 138.8
-133.4 -1620 -1770 -1599 -1367 -1391 -1522 -1603 -1403 -1613 -1467 -1536 128.4
-113.4 -1422 -1568 -1474 -1335 -1338 -1432 -1473 -1315 -1493 -1378 -1423 81.5
-93.2 -1375 -1439 -1334 -1275 -1266 -1323 -1386 -1249 -1406 -1312 -1337 63.7
-71.4 -1234 -1432 -1196 -1182 -1229 -1269 -1275 -1190 -1313 -1240 -1256 74.3
-48.2 -1173 -1343 -1127 -1144 -1169 -1215 -1112 -1126 -1236 -1203 -1185 69.2
-23.6 -1114 -1197 -1095 -1088 -1110 -1084 -1076 -1090 -1158 -1158 -1117 40.3
0.6 -1033 -1093 -1019 -1042 -1048 -1033 -1012 -1019 -1150 -1079 -1053 43.0
25.1 -941 -965 -952 -957 -981 -977 -946 -969 -1035 -1004 -973 28.8
49.9 -875 -910 -878 -904 -891 -876 -863 -915 -969 -952 -903 34.6
75.1 -771 -835 -744 -826 -774 -801 -771 -842 -813 -864 -804 38.5
100.6 -728 -746 -699 -759 -720 -737 -702 -776 -768 -800 -744 32.7
125.9 -618 -691 -583 -645 -634 -646 -618 -684 -679 -664 -646 34.2
264
Table E.14 ? S90 for [110] n-type silicon with temperature (Unit: N-1)
T (oC) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std.Dev
-151.0 -1157 -1079 -1007 -827 -859 -928 -1023 -916 -1068 -968 -983 103.4
-133.4 -1005 -1129 -861 -789 -833 -865 -943 -882 -972 -901 -918 98.5
-113.4 -840 -879 -885 -745 -808 -890 -836 -813 -922 -867 -849 51.3
-93.2 -844 -836 -830 -717 -780 -818 -812 -775 -886 -805 -810 45.8
-71.4 -727 -854 -682 -668 -736 -777 -743 -703 -718 -755 -736 52.8
-48.2 -704 -778 -670 -689 -706 -721 -639 -681 -678 -720 -699 37.3
-23.6 -705 -708 -636 -654 -683 -627 -652 -647 -649 -683 -664 28.5
0.6 -660 -671 -608 -619 -628 -612 -619 -593 -653 -652 -632 25.9
25.1 -559 -572 -580 -583 -587 -583 -563 -568 -586 -587 -577 10.4
49.9 -501 -533 -529 -539 -514 -517 -502 -525 -528 -542 -523 14.2
75.1 -451 -479 -424 -501 -444 -469 -450 -481 -443 -495 -464 25.0
100.6 -417 -433 -402 -410 -421 -432 -424 -418 -416 -446 -422 12.5
125.9 -345 -391 -337 -333 -349 -358 -334 -378 -369 -364 -356 19.6
Table E.15 ? S45 for [110] n-type silicon with temperature (Unit: N-1)
T (oC) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std.Dev
-151.0 -1705 -1227 -1258 -1059 -1044 -1644 -1243 -1310 -1423 -1304 -1322 217.8
-133.4 -1551 -1121 -1244 -1091 -1014 -1401 -1360 -1256 -1366 -1261 -1267 161.3
-113.4 -1357 -1093 -1186 -992 -886 -1254 -1222 -1224 -1089 -1117 -1142 136.5
-93.2 -1290 -1251 -934 -1008 -850 -1118 -1137 -1140 -1170 -979 -1088 141.1
-71.4 -1085 -1135 -928 -932 -863 -931 -900 -958 -934 -955 -962 83.3
-48.2 -945 -1018 -875 -910 -847 -1124 -874 -883 -966 -973 -941 83.7
-23.6 -962 -1011 -853 -811 -831 -848 -816 -753 -846 -848 -858 74.7
0.6 -859 -764 -790 -730 -778 -881 -791 -722 -725 -784 -782 53.4
25.1 -716 -719 -724 -759 -712 -692 -708 -711 -703 -737 -718 18.9
49.9 -699 -740 -709 -727 -713 -666 -715 -691 -671 -679 -701 24.2
75.1 -585 -624 -615 -600 -635 -616 -586 -717 -685 -638 -630 42.2
100.6 -518 -593 -578 -579 -609 -568 -539 -638 -632 -637 -589 41.2
125.9 -514 -533 -534 -485 -583 -453 -445 -586 -541 -536 -521 48.1
265
Table E.16 ? S-45 for [110] n-type silicon with temperature (Unit: N-1)
T (oC) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std.Dev
-151.0 -1318 -1491 -1399 -1173 -1197 -1206 -1432 -1164 -1309 -1155 -1284 123.3
-133.4 -1156 -1450 -1332 -930 -933 -1223 -1223 -1188 -1140 -1082 -1166 160.9
-113.4 -1122 -1310 -1242 -956 -978 -1171 -1026 -1122 -1238 -1170 -1134 117.4
-93.2 -986 -1003 -1137 -935 -876 -1263 -967 -1198 -1069 -1080 -1051 121.5
-71.4 -1128 -940 -992 -919 -923 -1161 -932 -999 -1054 -1064 -1011 87.5
-48.2 -901 -1038 -904 -869 -830 -900 -823 -1024 -903 -921 -911 71.0
-23.6 -838 -886 -883 -872 -816 -909 -850 -860 -910 -909 -873 32.4
0.6 -772 -781 -852 -759 -808 -787 -799 -833 -833 -834 -806 31.3
25.1 -739 -776 -781 -698 -769 -782 -737 -809 -769 -799 -766 32.9
49.9 -620 -644 -674 -647 -664 -724 -706 -763 -718 -705 -686 44.0
75.1 -610 -641 -602 -649 -629 -598 -579 -704 -670 -653 -634 37.9
100.6 -509 -581 -472 -594 -553 -518 -521 -634 -608 -611 -560 53.4
125.9 -472 -602 -465 -527 -447 -547 -479 -581 -567 -500 -519 53.9
Typical results of S0, S90, S45, and S-45 for the (001) p-type and n-type silicon at 25oC
are shown in Figs.E.6 through Fig.E.13.
[100]: S0 and S90 at 25oC (p type)
S0 = 1.219E-04x
R2 = 9.881E-01
S90 = -3.089E-05x
R2 = 9.763E-01
-1.0E-04
-5.0E-05
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
2.5E-04
3.0E-04
0.0 0.5 1.0 1.5 2.0 2.5
F(N)
????R
/R
Fig. E.6 - S0 and S90 for [100] p-type silicon at 25oC (Unit: N-1)
266
[100]: S45 and S-45 at 25oC (p type)
S45 = 4.027E-05x
R2 = 9.972E-01
S-45 = 4.776E-05x
R2 = 9.126E-01
0.E+00
2.E-05
4.E-05
6.E-05
8.E-05
1.E-04
1.E-04
0.0 0.5 1.0 1.5 2.0 2.5F(N)
????R
/R
Fig. E.7 - S45 and S-45 for [100] p-type silicon at 25oC (Unit: N-1)
[110]: S0 and S90 at 25oC (p type)
S0 = 2.496E-03x
R2 = 9.978E-01
S90 = -2.420E-03x
R2 = 9.998E-01-6.E-03-4.E-03
-2.E-03
0.E+00
2.E-03
4.E-03
6.E-03
0.0 0.5 1.0 1.5 2.0 2.5
F(N)
????R
/R
Fig. E.8 - S0 and S90 for [110] p-type silicon at 25oC (Unit: N-1)
[110]: S45 and S-45 at 25oC (p type)
S45 = 5.682E-05x
R2 = 9.895E-01
S-45 = 5.293E-05x
R2 = 9.983E-01
0.E+00
5.E-05
1.E-04
2.E-04
2.E-04
3.E-04
0 1 2 3 4 5F(N)
????R
/R
Fig. E.9 - S45 and S-45 for [110] p-type silicon at 25oC (Unit: N-1)
267
[100]: S0 and S90 at 25oC (n type)
S0 = -2.395E-03x
R2 = 9.989E-01
S90 = 1.293E-03x
R2 = 9.966E-01
-5.0E-03
-4.0E-03
-3.0E-03
-2.0E-03
-1.0E-03
0.0E+00
1.0E-03
2.0E-03
3.0E-03
0.0 0.5 1.0 1.5 2.0
F(N)
????R
/R
Fig. E.10 - S0 and S90 for [100] n-type silicon at 25oC (Unit: N-1)
[100]: S45 and S-45 at 25oC (n type)
S45 = -5.309E-04x
R2 = 9.930E-01
S-45 = -5.350E-04x
R2 = 9.964E-01
-1.E-03
-1.E-03
-8.E-04
-6.E-04
-4.E-04
-2.E-04
0.E+00
0.0 0.5 1.0 1.5 2.0
F(N)
????R
/R
Fig. E.11 - S45 and S-45 for [100] n-type silicon at 25oC (Unit: N-1)
[110]: S0 and S90 at 25oC (n type)
S0 = -9.648E-04x
R2 = 9.974E-01
S90 = -5.722E-04x
R2 = 9.996E-01
-2.0E-03
-1.5E-03
-1.0E-03
-5.0E-04
0.0E+00
0.0 0.5 1.0 1.5 2.0 2.5
F(N)
????R
/R
Fig. E.12 - S0 and S90 for [110] n-type silicon at 25oC (Unit: N-1)
268
[110]: S45 and S-45 at 25oC (n type)
S45 = -7.190E-04x
R2 = 9.963E-01
S-45 = -7.759E-04x
R2 = 9.998E-01-1.4E-03-1.2E-03
-1.0E-03
-8.0E-04
-6.0E-04
-4.0E-04
-2.0E-04
0.0E+00
0.0 0.5 1.0 1.5 2.0
F(N)
????R
/R
Fig. E.13 - S45 and S-45 for [110] n-type silicon at 25oC (Unit: N-1)
269
APPENDIX F
THE EFFECTS OF ERRORS ASSOCIATED WITH INITIAL RESISTANCE ON THE
DETERMINATION OF PI-COEFFICIENTS
The correct calibration of initial resistance is essential for the accurate
determination of pi-coefficients. The incorrect initial values of resistance occur from
uneven expansions and contractions during thermal expansion of the various materials
which expand and contract at different rates and have different elastic moduli. Under
heating and cooling of such assemblies of materials, the coefficient of thermal expansion
mismatches lead to thermal stresses.
Figure F.1. is presented in order to help with the understanding of the effects of the
incorrect initial resistance, as shown below.
Fig. F.1 - Resistance at various stress levels
In Fig. F.1, ?w denotes the stress induced by any warpage, for example, resulting
from the mismatch in the coefficients of expansion of the various packaging materials,
and ?p denotes the stress that we want to apply on the sample, respectively. In Fig. F.1,
270
the linear relationship between the resistance and stress is established because the
piezoresistive effect has been observed to be linear in the applied stress. Through the use
of pi?=?RR , the resistance equation becomes:
circle6 From A to P
P
0
0P0 ? ?
R
R)R(R =?+ Eq. (F.1)
Hence
0
P
P R
R1
?=pi Eq. (F.2)
circle6 From A? to P?
)? (??)RR( )RR()RR(R WPW'
W0
W0PW0 ??+=
+
+?++ Eq. (F.3)
Hence
)RR R(1
W0
P
P
'
+?=pi Eq. (F.4)
Dividing Eq. (F.2) by Eq. (F.4) yields
W0
0
' RR
R
+=pi
pi Eq. (F.5)
Then it leads to
pi+=pi )R RR(
0
W0' Eq. (F.6)
in which WR is the resistance induced by ?w . The initial resistance value is shifted from
0R to W0 RR + due to the induced stresses W? . If 0W RR << , Eq. (F.6) becomes
271
pi?pi' Eq. (F.7)
However, as WR becomes more comparable with 0R , the discrepancy between pi and
'pi is exacerbated.
In Figs. F.2 and F.3, the contour plots of ?11 and ?22 in the single-sided silicon strip-
on-beam from thermal simulations (from 150oC to 25oC) are presented. The colors of
contour represent the stress value.
Fig. F.2 - Thermal simulation of ?11 for [100] strip-on-beam
272
Fig. F.3 - Thermal simulation of ?22 for [100] strip-on-beam
Similarly, contour plots of ?'11 and ?'22 in the [100] double-sided silicon strip-on-
beam from thermal simulations (from 150oC to 25oC) are presented as shown in Figs. F.4
and F.5.
273
Fig. F.4 - Thermal simulation of ?'11 for [110] strip-on-beam
Fig. F.5 - Thermal simulation of ?'22 for [110] strip-on-beam
274
In addition, thermal simulations of z-directional displacement are performed from 150oC
to 25oC as shown in Figs. F.6 and F.7.
Fig. F.6 - Thermal simulation of z-displacement for [100] strip-on-beam
275
Fig. F.7 - Thermal simulation of z-displacement for [110] strip-on-beam
For both [100] and [110] strip-on-beam structures, the maximum change in z-directional
displacement occurs at the central part. The maximum displacement of simulation is
about 1.36~1.37 mm for both cases. The simulation result for the maximum change in z-
directional displacement is in good agreement with the experimental result (about 1.3~1.5
mm) as shown in Fig. F.8. It can be seen that a single-sided silicon strip-on-beam sample
was significantly warped after cooling from their assembly temperature, resulting from
the mismatch in the coefficients of expansion of the various packaging materials.
276
Fig. F.8 - The obvious warp of a single-sided silicon strip-on-beam sample after cooling
from 150oC to room temperature
From thermal simulations, the in-plane stresses at sensor location in [100] and [110]
silicon strip-on-beam samples are displayed in Tables F.1 and F.2, respectively.
Table F.1 - In-plane stresses at sensor location in
[100] silicon strip-on-beam (Unit: MPa)
?11 ?22 ?12
57.63 44.85 -1.00E-08
Table F.2 - In-plane stresses at sensor location in
[110] silicon strip-on-beam (Unit: MPa)
?'11 ?'22 ?'12
60.31 34.64 3.81E-09
Pi-coefficients reflecting the average values of 10 specimens, as discussed in
Chapter 9, are shown in Table F.3
Table F.3 - pi11, pi12, and pi44 of (001) p- and n-type silicon at 25oC
(Unit: TPa-1)
Type pi11 pi12 pi44
p 29.4 -6.2 898
n -691 390 -73.9
277
The normalized change in resistance induced by thermal stresses, arising from the
assembly operations, can be calculated by substitution of the values in Tables F.1 through
F.3 into the equations below:
circle6 With respect to the unprimed axes
...]TT[sin2
)]sin( +[ +)]cos( +[ = R?R
2
211244
2
3311122211
2
3322121111
+??+??+??pi+
??+?pi?pi??+?pi?pi
Eq. (F.8)
For p- and n-type sensors,
sensors type-nfor 102.23
sensors type-pfor 101.42
??? ?= R?R
2
3
22121111
0
0
?
?
??=
?=
+
Eq. (F.9)
sensors type-nfor 108.52
sensors type-pfor 109.61
??? ? R?R
3
4
22111112
90
90
?
?
??=
?=
+=
Eq. (F.10)
sensors type-nfor 10-1.54
sensors type-pfor 101.19
)? (? )2 ??( R?R
2-
3
2211
1211
45
45
?=
?=
++=
? Eq. (F.11)
sensors type-nfor 10-1.54
sensors type-pfor 101.19
)? (? )2 ??( R?R
2-
3
2211
1211
45
45
?=
?=
++=
?
?
?
Eq. (F.12)
278
circle6 With respect to the primed axes
...] + T? + T?[sin2 )-(??
sin 2 + 2 +
cos 2 + 2 = R?R
2
21
'
1212113312
2'
22
441211'
11
441211
2'
22
441211'
11
441211
??++??pipi++
??
?
?
??
? ??
?
??
?
? pi+pi+pi??
?
??
?
? pi?pi+pi
??
?
?
??
? ??
?
??
?
? pi?pi+pi??
?
??
?
? pi+pi+pi
Eq. (F.13)
For p- and n-type sensors,
sensors type-nfor 10-1.52
sensors type-pfor 10 1.26
? )2 ???(? )2 ???(R?R
2
2
'
22
441211'
1 1
441211
0
0
?
?
?=
?=
?++++=
Eq. (F.14)
sensors type-nfor 101.33
sensors type-pfor 101.04
? )2 ???( ? )2 ???(R?R
2
2
'
22
441211'
11
441211
90
90
?
?
??=
??=
+++?+=
Eq. (F.15)
sensors type-nfor 101.43
sensors type-pfor 101.10
)? (? )2 ??(R?R
2
3
'
22
'
11
1211
45
45
?
?
??=
?=
++=
Eq. (F.16)
sensors type-nfor 101.43
sensors type-pfor 101.10
)? (? )2 ??(R?R
2
3
'
22
'
11
1211
45-
45-
?
?
??=
?=
++=
Eq. (F.17)
279
For p-type sensors, the normalized % changes in initial resistance for [100] and [110]
silicon strip-on-beam compared with the corresponding silicon strip are summarized in
Tables F.4 and F.5.
Table F.4 - Typical results of % change in initial resistance of [100]
p-type silicon at 25oC
Direction
Strip
(Unit:kohm)
Single-sided strip-on-beam
(Unit: kohm)
% change
(Experimental)
% change
(Analytic)
? = 0 22.351 22.387 0.16 0.14
? = 90 22.305 22.322 0.08 0.10
? = +45 22.325 22.332 0.03 0.12
? = -45 22.312 22.334 0.10 0.12
Table F.5 - Typical results of % change in initial resistance of [110]
p-type silicon at 25oC
Direction
Strip
(Unit:kohm)
Single-sided strip-on-beam
(Unit: kohm)
% change
(Experimental)
% change
(Analytic)
? = 0 23.051 23.396 1.49 1.26
? = 90 22.946 22.651 -1.29 -1.04
? = +45 23.084 23.098 0.06 0.11
? = -45 23.022 23.084 0.27 0.11
As shown in Table F.4, the normalized % change in initial resistance for [100] p-type
sensors is negligible because all the related pi-coefficients 11pi , 12pi , and Spi for p-type
silicon are very small. For the same reason, the last two rows in Table F.5 show the
small % changes in initial resistance.
280
Similarly, for n-type sensors, the normalized % changes in initial resistance for [100]
and [110] silicon strip-on-beam compared with the corresponding silicon strip are
summarized in Tables F.6 and F.7.
Table F.6 - Typical results of % change in initial resistance of [100]
n-type silicon at 25oC
Direction
Strip
(Unit:kohm)
Single-sided strip-on-beam
(Unit: kohm)
% change
(Experimental)
% change
(Analytic)
? = 0 13.265 12.898 -2.77 -2.23
? = 90 13.351 13.135 -1.62 -0.85
? = +45 13.442 13.152 -2.16 -1.54
? = -45 13.368 13.138 -1.72 -1.54
Table F.7 - Typical results of % change in initial resistance of [110]
n-type silicon at 25oC
Direction
Strip
(Unit:kohm)
Single-sided strip-on-beam
(Unit: kohm)
% change
(Experimental)
% change
(Analytic)
? = 0 13.110 12.968 -1.08 -1.52
? = 90 12.883 12.663 -1.71 -1.33
? = +45 12.978 12.573 -3.12 -1.43
? = -45 13.345 13.184 -1.21 -1.43
Compared with p-type sensors, the normalized % changes in initial resistance for n-
type sensors are relatively large because of the large pi-coefficients. It is observed that
analytic results are in good agreement with experimental results for both p- and n-type
sensors. In order to resolve the problems concerning the initial resistance, we use the
double-sided silicon strip-on-beam samples.
281
APPENDIX G
THE COMPARISONS OF PI-COEFFICIENTS BETWEEN STRIPS AND DOUBLE-
SIDED SILICON STRIP-ON-BEAM SAMPLES
Typical results of pi-coefficients, which reflect the average values of 10 specimens,
are presented in Chapter 9. For comparison purpose, stress test strips are calibrated and
characterized. As discussed in Chapter 4, a rectangular strip containing a row of chips is
cut from a wafer and is loaded in a four-point bending fixture to generate uniaxial stress
state. Hence the variation of the resistance of sensors with applied uniaxial stress has
been measured. Through the use of four-point bending test, we may determine all pi-
coefficients and the values will be compared with those from double-sided silicon strip-
on-beam samples. As discussed in Chapter 4, the induced uniaxial stress is given by
2bh D)3F(L? ?= Eq. (G.1)
where F = 1N, (L-D) = 2.4 x 10-2 m and the dimensions of the (001) silicon strips are
shown in Table G.1.
Table G.1 - Dimensions of the (001) silicon strips (Unit: mil)
(001) silicon:[100] (001) silicon: [110]
Length (l) 3400 3400
Width (b) 226 160
Thickness (h) 20 20
Note that the dimensions of composite materials of strip-on-beam samples are presented
in Tables 9.5 and 9.6.
282
?R0/R0
y = 3.626E-05x
R2 = 9.902E-01
0.0E+00
1.0E-04
2.0E-04
3.0E-04
4.0E-04
5.0E-04
6.0E-04
7.0E-04
0 5 10 15 20
Stress (MPa)
????R
0/
R0
Fig. G.1 - Typical stress sensitivity of p-type resistors on the [100] silicon strip (R0)
?R90/R90
y = -7.388E-06x
R2 = 9.839E-01
-1.0E-04
-9.0E-05
-8.0E-05
-7.0E-05
-6.0E-05
-5.0E-05
-4.0E-05
-3.0E-05
-2.0E-05
-1.0E-05
0.0E+00
0 2 4 6 8 10 12 14
Stress (MPa)
????R
90
/R
90
Fig. G.2 - Typical stress sensitivity of p-type resistors on the [100] silicon strip (R90)
283
?R45/R45
y = 1.853E-05x
R2 = 9.953E-01
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
2.5E-04
3.0E-04
3.5E-04
0 5 10 15 20
Stress (MPa)
????R
45
/R
45
Fig. G.3 - Typical stress sensitivity of p-type resistors on the [100] silicon strip (R45)
?R-45/R-45
y = 1.666E-05x
R2 = 9.806E-01
0.0E+00
2.0E-05
4.0E-05
6.0E-05
8.0E-05
1.0E-04
1.2E-04
0 1 2 3 4 5 6 7
Stress (MPa)
????R
-4
5/
R-
45
Fig. G.4 - Typical stress sensitivity of p-type resistors on the [100] silicon strip (R-45)
284
?R0/R0
y = 5.073E-04x
R2 = 9.938E-01
0.0E+00
1.0E-03
2.0E-03
3.0E-03
4.0E-03
5.0E-03
6.0E-03
7.0E-03
8.0E-03
0 5 10 15 20
Stress (MPa)
????R
0/
R0
Fig. G.5 - Typical stress sensitivity of p-type resistors on the [110] silicon strip (R0)
?R90/R90
y = -4.804E-04x
R2 = 9.974E-01
-6.0E-03
-5.0E-03
-4.0E-03
-3.0E-03
-2.0E-03
-1.0E-03
0.0E+00
0 2 4 6 8 10 12 14
Stress (MPa)
????R
90
/R
90
Fig. G.6 - Typical stress sensitivity of p-type resistors on the [110] silicon strip (R90)
285
?R45/R45
y = 1.934E-05x
R2 = 9.868E-01
0.0E+00
1.0E-05
2.0E-05
3.0E-05
4.0E-05
5.0E-05
6.0E-05
7.0E-05
8.0E-05
9.0E-05
0 1 2 3 4 5
Stress (MPa)
????R
45
/R
45
Fig. G.7 - Typical stress sensitivity of p-type resistors on the [110] silicon strip (R45)
?R-45/R-45
y = 1.847E-05x
R2 = 9.855E-01
0.0E+00
2.0E-05
4.0E-05
6.0E-05
8.0E-05
1.0E-04
1.2E-04
1.4E-04
1.6E-04
0 2 4 6 8
Stress (MPa)
????R
-4
5/
R-
45
Fig. G.8 - Typical stress sensitivity of p-type resistors on the [110] silicon strip (R-45)
286
?R0/R0
y = -6.945E-04x
R2 = 9.995E-01
-2.0E-02
-1.5E-02
-1.0E-02
-5.0E-03
0.0E+00
0 5 10 15 20 25
Stress (MPa)
????R
0/
R0
Fig. G.9 - Typical stress sensitivity of n-type resistors on the [100] silicon strip (R0)
?R90/R90
y = 3.607E-04x
R2 = 9.999E-01
0.0E+00
1.0E-03
2.0E-03
3.0E-03
4.0E-03
5.0E-03
6.0E-03
7.0E-03
8.0E-03
0 5 10 15 20 25
Stress (MPa)
????R
90
/R
90
Fig. G.10 - Typical stress sensitivity of n-type resistors on the [100] silicon strip (R90)
287
?R45/R45
y = -1.597E-04x
R2 = 9.989E-01
-3.5E-03
-3.0E-03
-2.5E-03
-2.0E-03
-1.5E-03
-1.0E-03
-5.0E-04
0.0E+00
0 5 10 15 20 25
Stress (MPa)
????R
45
/R
45
Fig. G.11- Typical stress sensitivity of n-type resistors on the [100] silicon strip (R45)
?R-45/R-45
y = -1.527E-04x
R2 = 9.984E-01
-4.0E-03
-3.5E-03
-3.0E-03
-2.5E-03
-2.0E-03
-1.5E-03
-1.0E-03
-5.0E-04
0.0E+00
0 5 10 15 20 25
Stress (MPa)
????R
-4
5/
R-
45
Fig. G.12- Typical stress sensitivity of n-type resistors on the [100] silicon strip (R-45)
288
?R0/R0
y = -2.018E-04x
R2 = 9.774E-01
-1.2E-02
-1.0E-02
-8.0E-03
-6.0E-03
-4.0E-03
-2.0E-03
0.0E+00
0 10 20 30 40 50 60
Stress (MPa)
????R
0/
R0
Fig. G.13- Typical stress sensitivity of n-type resistors on the [110] silicon strip (R0)
?R90/R90
y = -9.707E-05x
R2 = 9.949E-01
-7.0E-03
-6.0E-03
-5.0E-03
-4.0E-03
-3.0E-03
-2.0E-03
-1.0E-03
0.0E+00
0 10 20 30 40 50 60 70
Stress (MPa)
????R
90
/R
90
Fig. G.14- Typical stress sensitivity of n-type resistors on the [110] silicon strip (R90)
289
?R45/R45
y = -1.507E-04x
R2 = 9.992E-01
-1.0E-02
-8.0E-03
-6.0E-03
-4.0E-03
-2.0E-03
0.0E+00
0 10 20 30 40 50 60
Stress (MPa)
????R
45
/R
45
Fig. G.15- Typical stress sensitivity of n-type resistors on the [110] silicon strip (R45)
?R-45/R-45
y = -1.468E-04x
R2 = 9.984E-01
-8.0E-03
-7.0E-03
-6.0E-03
-5.0E-03
-4.0E-03
-3.0E-03
-2.0E-03
-1.0E-03
0.0E+00
0 10 20 30 40 50
Stress (MPa)
????R
-45
/R
-45
Fig. G.16- Typical stress sensitivity of n-type resistors on the [110] silicon strip (R-45)
290
The comparisons of pi-coefficients between stress test strips and double-sided silicon
strip-on-beam are summarized in Tables G.2 through G.5. It may be noted that 5 samples
are reflected in the values of strip. Both have an agreement in sign for all pi-coefficients.
For p-type silicon, large spreads in magnitude are observed for 11pi and 12pi (and then
Spi ) because the coefficients are small. For n-type silicon, the smallest pi-coefficient 44pi
shows a relatively large spreads in values. On the other hand, the large pi-coefficients
11pi , 12pi , and Spi are close for both cases.
circle6 [100]
Table G.2 - Comparisons of pi-coefficients for p-type [100] silicon
(Unit: TPa-1)
p-type pi11 pi12 pis (0, 90) pis (+45, -45)
strip 36.1 -7.9 28.2 30.2
strip-on-beam 29.4 -6.2 23.2 20.4
Table G.3 - Comparisons of pi-coefficients for n-type [100] silicon
(Unit: TPa-1)
n-type pi11 pi12 pis (0, 90) pis (+45, -45)
strip -673 361 -312 -324
strip-on-beam -691 390 -301 -302
291
circle6 [110]
Table G.4 - Comparisons of pi-coefficients for p-type [110] silicon
(Unit: TPa-1)
p-type pi44 pis (0, 90) pis (+45, -45)
strip 965 26.6 27.3
strip-on-beam 898 12.2 26.6
Table G.5 - Comparisons of pi-coefficients for n-type [110] silicon
(Unit: TPa-1)
n-type pi44 pis (0, 90) pis (+45, -45)
strip -105 -295 -302
strip-on-beam -73.9 -290 -277
292
APPENDIX H
DETERMINATION OF THE STIFFNESS COEFFICIENT MATRIX FOR THE
UNPRIMED/PRIMED COORDINATE SYSTEM
In Chapter 6, the transformation relations for the reduced index stress and strain
components were discussed as repeated below:
'?1??? ?][T? ?= Eq. (G.1)
'?t??? ]?[T? = Eq. (G.2)
Inverting Eq. (G.2) leads to
?][T? 1t' ?= Eq. (G.3)
Through the use of klijklij ?S? =
[S]?][T? 1t' ?= Eq. (G.4)
Finally, substitution of Eq. (G.1) into Eq. (G.4) yields the relations between stress and
strain in a rotated primed coordinate system as follows:
'11t' ?[S][T]][T? ??= Eq. (G.5)
If an unprimed coordinate system is assumed, 1t ]T[ ? and 1[T]? in Eq. (G.5) simplify to
unit matrices.
11t' [S][T]][T][S ??= Eq. (G.6)
By 1'' ]C[]S[ ?= , Eq. (G.6) becomes
11t1' [S][T]][T][C ??? = Eq. (G.7)
Inverting Eq. (G.7) gives
293
][T][C][T][C t' = Eq. (G.8)
Many calculations may be solved with matrix algebra.
circle6 With respect to the unprimed coordinate system
[ ]
c 0 0 0 0 0
0 c 0 0 0 0
0 0 c 0 0 0
0 0 0 c c c
0 0 0 c c c
0 0 0 c c c
c
44
44
44
111212
121112
121211
??
??
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
= Eq. (G.9)
circle6 With respect to the primed coordinate system
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
++?+
++++
=
2 CC 0 0 0 0 0
0 C 0 0 0 0
0 0 C 0 0 0
0 0 0 C C C
0 0 0 CC2 CC C2 CC
0 0 0 CC2 CC C2 CC
][C
1211
44
44
111212
1244
1211
44
1211
1244
1211
44
1211
'
Eq. (G.10)
where C11=165.7 GPa, C12=63.9 GPa, and C44=79.6GPa [90].
294
APPENDIX I
THE PROFILES OF CARRIER CONCENTRATION VERSUS DEPTH IN SILICON
The profiles of carrier concentration vs. depth in silicon are provided using Spreading
Resistance Analysis (SRA) as presented in the following figures. For the p- and n-type
samples prepared for this research, the impurity concentration at the wafer surface (N0) is
2.0x1018/cm3 and 4.0x1018/cm3, respectively. The metallurgical junction depth at which
the impurity profile intersects the background concentration is approximately 1.7 ?m for
p-type sensors and 1.2 ?m for n-type sensors.
295
Fig. I.1 - The profiles of carrier concentration vs. depth in n-type silicon (sample #1)
296
Fig. I.2 - The profiles of carrier concentration vs. depth in n-type silicon (sample #2)
297
Fig. I.3 - The profiles of carrier concentration vs. depth in p-type silicon (sample #1)
298
Fig. I.4 - The profiles of carrier concentration vs. depth in p-type silicon (sample #2)