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"STUDIES OF DOUBLE-DIFFUSED TRANSISTOR STRUCTURES" A THESIS presented for the degree of DOCTOR OF PHILOSOPHY of the UNIVERSITY OF LONDON by RAYE EDWARD THOMAS June 1966 2. ABSTRACT The solid-state diffusion process is examined with particular reference to the idealized classical impurity distributions normally assumed to apply in diffused structures. The peculiar properties of the double-diffused structure (graded junctions and a maximum in base doping) are shown to effect an overall improvement in frequency per- formance. Methods used to derive information on the impurity profile both in large area devices (destructive techniques) and in small area devices (physical model derived from terminal measurements) are discussed. Early models are shown to be inadequate and strictly limited in applicability. A physical model (double exponential) is proposed to apply generally to double-diffused transistors. A detailed study of classical distributions establishes that the assumed model not only is a good representation of such distributions in the base region, but also accurately predicts depletion layer and base transport properties. The proper interpretation of terminal measurements allows the constants of the model to be successfully determined for actual transistors. Within the accuracy of the above-mentioned measurements, the derived model is concluded to be a good representation for actual devices. In conclusion, suggestions for further work are offered. 3. ACKNOWLEDGBENTS The author wishes to express his gratitude to his Supervisor, Professor A.R. Boothroyd of The Queen's University of Belfast, (formerly of Imperial College) for his support, guidance and encourangement during the course of the work described in this thesis. Grateful thanks are extended to his fellow research students for friendly and stimulating discussions, in particular, to Viphandh Roengpithya for additional assistance in the reproduction stage of the thesis. The friendly interest of the staff, and the facilities of the Electrical Engineering Department of the Queen's University during the latter stages of the work are appreciated. Financial assistance provided for 1961-1963 by the Board of Trade in the form of an Athlone Fellowship, and for 1963-1966 by the National Research Council of Canada is gratefully acknowledged. Finally, the author pays tribute to Elda, his wife, for her steadfast and affectionate support throughout the study, and for her invaluable assistance in preparing the final product. 4. TABLE OF CONTENTS Page Abstract 2 Acknowledgements 3 Table of Contents 4 Location of Figures 9 Location of Tables 9 List of Principal Symbols 10 1. INTRODUCTION 13 1.1 Historical Background 13 1.2 Formulation of the Problem 17 1.3 Original Contribution 20 2. FORMATION OF THE DOUBLE-DIFFUSED TRANSISTOR AND EARLY CHARACTERIZATION ATTEMPTS 21 2.1 introduction 21 2.2 The Diffusion Equation and its Solution 24 2.2.1 General.. Form 24 2.2.2 Diffusion from Vapour of Constant Impurity Concentration 27 2.2.3 Diffusion from a Planar Source 28 2.2.4 Modifications due to Outward Diffusion and Rate Limitations 31 2.3 Double-Diffused Structures 33 2.3.1 The Planar Process 33 2.3.2 Classical Distributions 36 2.3.3 Practical Device Profiles 39 2.3.4 The Planar Epitaxial Transistor 41 S. Page 2.4 Influence of Diffused Structure on Transistor Performance 45 2.4.1 General Considerations 45 2.4.2 Influence of Base Impurity Profile on Diffusion Coefficient 51 2.4.3 Minority Carrier Density in Base Region 56 2.4.4 Base Transit Time 60 2.4.5 Equivalent Circuit for Transistor 63 2.4.6 Cutoff Frequency Considerations 68 2.5 Basic Approaches to Device Characterization 69 2.6 The Classical Distribution Approach 71 2.7 Equivalent Circuit Approach 78 2.7.1 Direct Derivation of Actual Distributions from Terminal Measurement 78 2.7.2 Single Exponential Mbdel for Base Region 83 2.7.3 Linear Plus Exponential Model for Base Region 86 2.7.3.1 Equations for the Model 86 2.7.3.2 Results and Criticisms 93 2.7.4 Intrinsic Input Admittance Yee 96 3. TREATMENT OF CLASSICAL DISTRIBUTIONS AND APPROXIMATION BY DOUBLE EXPONENTIAL MODEL 98 3.1 Introduction 98 3.2 The Double ERFC Distribution 100 3.2.1 Form of Impurity Distribution 100 3.2.2 Depletion Region Distribution 101 6. Page 3.2.3 Base Region Properties 108 3.2.4 Emitter Doping 110 3.2.5 Representative Example 112 3.3 Double Gaussian Distribution 118 3.3.1 Impurity Distribution 118 3.3.2 Depletion Region Equations 119 3.3.3 Base Region Properties 122 3.3.4 Emitter Region 123 3.3.5 Representative Example 123 3.4 Planar Epitaxial Transistor 129 3.4.1 Collector Substrate 129 3.4.2 Extensions to Junction Equations 130 3.4.3 Base Region Properties 137 3.4.4 Representative Example 138 3.5 Double Exponential Model to Approximate Base 144 3.5.1 Justification of the Model and Choice of Exponentials 144 3.5.2 Collector Junction 149 3.5.3 Emitter Junction 152 3.5.4 Minority Carrier Density in the Base Region 158 3.5.5 Base Transit Time 161 3.5.6 Model Applied to Representative Double ERFC and Double Gaussian Examples 163 3.5.7 Model Extended to Epitaxial Structure 166 3.6 Numerical Techniques 169 7. Pave 4. TERMINAL CHARACTERISTICS AND CIRCUIT ELEMENTS OF ACTUAL TRANSISTORS BASED ON THE DOUBLE EXPONENTIAL MODEL 173 4.1 Introduction 173 4.2 Collector and Emitter Junctions 174 4.2.1 Collector Depletion Region 174 4.2.2 Emitter Depletion Region 178 4.2.3 Measurement of Transition Capacitances 183 4.2.4 Emitter and Collector Areas 192 4.2.5 Estimate of NB from Collector Breakdown 199 4.3 Emitter Diode Characteristics 205 4.3.1 General Equations 205 4.3.2 DR and DA Approximation 207 4.3.3 Approximation Based on BR only 208 4.3.4 Measurement of Diode Characteristic 209 4.4 Base Transit Time 212 4.4.1 General Equations 212 4.4.2 Simplification when 1 B is Negligible in the Base 217 4.4.3 Transit Time Measurement 223 5. DETERMINATION OF THE PHYSICAL CONSTANTS OF THE DOUBLE EXPONENTIAL MODEL FOR ACTUAL DEVICES 230 5.1 Introduction 230 5.2 Evaluation of Constants of Model 231 5.2.1 Emitter and Collector Areas Known 231 8. Page 5.2.2 Known from r' and Measurement 238 Ae/Ac bb r'bbCtci 5.2.3 Checks on Parameters Determined 240 5.2.4 Discussion of Results 244 5.3 Evaluation from Emitter Capacitance Data 249 5.3.1 Derivation of Parameters 249 5.3.2 Discussion of Results 251 5.4 Numerical Techniques 256. 6. CONCLUSIONS 259 6.1 General Conclusions 259 6.2 Suggestions for Further Work 264 REFERENCES 267 APPENDICES A.1 Determination of Parameters of Linear-Exponential Model 274 A.2 Attempt to Measure Yee 277 A.3 Determination of L and L when both Exponentials are Significant at Collector Junction 282 A.4 Program for Double ERFC 285 A.5 Program for Double Gaussian 297 A.6 Double Exponential for Double ERFC 304 A.7 Detailed Equations for Input and Output Admittance 320 A.8 Evaluation of Integral in Transit Time Expression 322 A.9 Program to Determine Double Exponential Nbdel 324 9. Location of Figures Fig. Page Fla. Page Fig. Page 2.1 29 3.3 115 4.7 194 2.2 30 3.4 115 4.8 198 2.3 35 3.5 116 4.9 200 2.4 35 3.6 117 4.10 200 2.5 37 3.7 117 4.11 201 2.6 37 3.8 125 4.12 211 2.7 40 3.9 126 4.13 215 2.8 40a 3.10 126 4.14 215 2.9 44 3.11 127 4.15 221 2.10 44 3.12 127 4.16 221 2.11 53 3.13 128 4.17 222 2.12 64 3.14 141 4.18 222 2.13 64 3.15 142 4.19 224 2.14 73 3.16 143 4.20 224 2.15 75 3.17 143 4.21 228 2.16 77 3.18 155 4.22 228 2.17 80 3.19 155 5.1 241 2.18 80 4.1 174 5.2 253 2.19 84 4.2 174 A.1 279 2.20 84 4.3 184 A.2 281 2.21 90 4.4 184 A.3 285 3.1 102 4.5 190 A.4 304 3.2 114 4.6 194 A.5 324 Location of Tables Table Page Table Page Table Page 2.1 92 4.3 191 5.3 237 2.2 92 4.4 195 5.4 241 2.3 92 4.5 213 5.5 252 2.4 92 5.1 235 5.6 252 4.1 187 5.2 236 5.7 253 4.2 189 10. LIST OF PRINCIPAL MBOLS A A Emitter and Collector areas. es c Arg = Argument or phase-angle of complex quantity. a = Common-base intrinsic short-circuit current gain. a Low-frequency asymptote of a. o Extrinsic (measured) common-base short- circuit current gain. a a Depletion layer semi-widths on emitter and ne base sides respectively of emitter. junction. Depletion layer semi-widths on base and aica2c collector sides respectively of collector junction. C C Emitter and collector transition te' tc capacitances. C C Fractions of collector transition tc.t ' tc2 capacitance under the emitter area and outside this area respectively. C Emitter diffusion capacitance. de Stray capacitances between emitter-base, Cseb)Csec'Cscb emitter-collector, and collector-base leads respectively. D Impurity diffusion coefficient.
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