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. 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C., ?Innovative Sensors for Stress Measurements in Packaged Integrated Circuits,? 99-NJ-459-Third Year Report, SRC Packaging Multiple Contract Review, September 28, 1999. 105. Chen, Y., A Chip-on-Beam Technique for Calibration of Piezoresistive Stress Sensors, M.S. Thesis, Auburn University, Auburn, Alabama, 2003. 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)