# Large-signal and temperature-dependent modeling of heterojunction bipolar transistors for RF and microwave applications

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LARGE-SIGNAL AND TEMPERATURE DEPENDENT MODELING OF HETEROJUNCTION BIPOLAR TRANSISTORS FOR RF AND MICROWAVE APPLICATIONS The members o f the Committee approve the doctoral dissertation o f Aexandru Aurelian Ciubotaru Ronald L. Carter Supervising Professor Kambiz A avi Jonathan W. Bredow W. A an Davis Larry F. Heath Dean o f the Graduate School Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Copyright © by Alexandru Aurelian Ciubotaru 1996 All Rights Reserved Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To my parents Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LARGE-SIGNAL AND TEMPERATURE DEPENDENT MODELING OF HETEROJUNCTION BIPOLAR TRANSISTORS FOR RF AND MICROWAVE APPLICATIONS by ALEXANDRU AURELIAN CIUBOTARU Presented to the Faculty o f the Graduate School o f The University o f Texas at Arlington in Partial Fulfillment o f the Requirements for the Degree o f DOCTOR OF PHILOSOPHY THE UNIVERSITY OF TEXAS AT ARLINGTON December 1996 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9718536 Copyright 1996 by Ciubotaru, Alexandra Aurelian All rights reserved. UMI Microform 9718536 Copyright 1997, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS I would like to address special thanks to my supervising professor. Dr. Ronald Carter, for his inspiration and valuable support that made this work possible. I gratefully acknowledge Dr. Bruce Donecker (HP EEsof Strategic Development) for the financial and technical support given, and the competent and timely assistance o f Ms. Else Schmidt and Mr. Marek Mierzwinski, both from HP EEsof. I also thank Mr. Sherman Reed and Dr. Michal Chwialkowski for their support and friendship. I thank all my professors for their dedication. November 20, 1996 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT LARGE-SIGNAL AND TEMPERATURE DEPENDENT MODELING OF HETEROJUNCTION BIPOLAR TRANSISTORS FOR RF AND MICROWAVE APPLICATIONS Publication N o ._______ Alexandra Aurelian Ciubotaru, Ph.D. The University o f Texas at Arlington, 1996 Supervising Professor: Ronald L. Carter An accurate physics-based model for the heterojunction bipolar transistor (HBT) is developed. The model is large-signal and temperature dependent, and is intended for use in RF and microwave applications. The model is used to represent a planar single-heterojunction structure whose collector area is larger than the emitter area, by means o f an additional basecollector overlap diode in the model o f a one-dimensional transistor structure. An accurate, temperature dependent model for the base spreading resistance is also developed. The temperature dependencies of the transistor model parameters are obtained from the explicit temperature dependencies of the energy gaps o f the emitter, base and collector, from the boundary conditions o f the HBT, and by taking into account the recombination/generation currents in the space-charge regions; an appropriate thermal circuit, that includes the temperature responses o f the emitter, collector and the overlap diode, is used in the transistor model. Compared to the existing models, the proposed model o f the HBT allows a more accurate high-frequency characterization, due to a more appropriate placement o f the junction and diffusion capacitances. The extraction procedures for all the HBT model parameters are vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. developed and described in detail. The model is validated by the excellent agreement between the simulated and measured dc characteristics and S-parameters for two fundamentally different HBT's. vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS ACKNOWLEDGMENTS .......................................................................................................... v ABSTRACT ................................................................................................................................. vi LIST OF ILLUSTRATIONS .................................................................................................... xi LIST OF TABLES ....................................................................................................................... xx CHAPTER 1 INTRODUCTION ........................................................................................... 1 CHAPTER 2 STATIC MODEL FOR THE HBT ............................................................... 5 2.1 Complete Static Model ................................................................................... 10 2.2 Mathematical Model for Base Resistance ................................................... 14 2.3 Approximate Determination o f Base Spreading Resistance through Simulation ................................................................................................. 25 Closed-Form Expression for / bo and Temperature Dependence o f RBmax and /bo ......................................................................................... 30 PARAMETER EXTRACTION PROCEDURE FOR THE STATIC MODEL ............................................................................................................ 33 3.1 Forward Gummel Measurement and Parameter Extraction ..................... 33 3.1.1 Determination of/3/r, o bf and 6b f ............................................................. 37 3.1.2 Determination o f I s , N p, R bviox and R p c ............................................... 40 3.1.3 Determination o f a«B a n d / bo ...................................................................... 44 3.2 Reverse Gummel Measurement and Parameter Extraction ...................... 49 3.2.1 Determination o f I s r o l , N r o l , N r and (3r ............................................. 51 TEMPERATURE DEPENDENT MODEL OF THE HBT .................... 58 4.1 Boundary Conditions and Basic Equations for the HBT .......................... 58 4.2 Temperature Dependence o f I s ................................................................... 65 4.3 Temperature Dependence o f 13p and (3r ..................................................... 69 2.4 CHAPTER 3 CHAPTER 4 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4 Temperature Dependence o f I s e »I s c and I sro l ........................................ 73 4.5 Temperature Dependent Model for the Base Spreading Resistance ........... 74 4.6 Thermal Circuit o f the HBT ............................................................................. 76 CHAPTER 5 EXTRACTION OF PARAMETERS FOR THE TEMPERATURE DEPENDENT MODEL ................................................................................. 86 5.1 Determination o f X t i , X trol and X t b r ..................................................... 87 5.2 Determination o f x (aluminum concentration), /&, 0q and .......................... 90 5.3 Determination o f XT/jsmox ................................................................................... 94 5.4 Determination o f R theo , K th c and K s ...................................................... 95 5.5 Determination o f X ^ rbb ................................................................................. 101 5.6 Determination o f R ............................................................................... 103 CHAPTER 6 CHARGE-STORAGE EFFECTS IN THE HBT ......................................... 107 tholo X j e 6.1 Junction Capacitances o f One-Dimensional HBT ................................... 107 6.2 Diffusion Capacitances o f One-Dimensional HBT ................... 112 6.3 Junction and Diffusion Capacitances o f Overlap Diode .............................. 114 6.4 Expressions for the Transit Time Components o f One-Dimensional HBT ......................................................................................................... 116 Placement o f Capacitances in Complete Model o f One-Dimensional HBT with Non-Zero Base Resistance .................................................. 127 Complete Model o f Intrinsic Planar HBT ..................................................... 142 CHAPTER 7 PARAMETER EXTRACTION PROCEDURE FOR THE HIGH-FREQUENCY MODEL ................................................................... 144 6.5 6.6 7.1 Determination o f the Junction Capacitances o f the HBT ............................ 145 7.1.1 Determination o f the Base-Collector Junction Capacitance ........................ 146 7.1.2 Determination o f the Base-Emitter Junction Capacitance .........~ ............. 153 7.2 Determination o f Pad and Interconnection Parasitics .................................. 160 7.3 Determination o f Forward Transit Time ....................................................... 164 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 8 RESULTS AND DISCUSSION .................................................................. 175 CHAPTER 9 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH .................................................................................................... 229 APPENDIX A MNS CIRCUIT FILE USED IN THE DETERMINATION OF THE TEMPERATURE DEPENDENT MODEL ................................................ 232 APPENDIX B COMPLETE MNS CIRCUIT FILE FOR THE T8 DEVICE MODEL .......................................................................................................... 236 APPENDIX C COMPLETE MNS CIRCUIT FILE FOR THE 78 DEVICE MODEL .......................................................................................................... 241 REFERENCES ......................................................................................................................... 246 x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF ILLUSTRATIONS Fig. 2.1. Schematic cross sections o f various HBT structures ............................................ 6 Fig. 2.2. Development o f a simple circuit modelfor a planar HBT (a) planar HBT structure; (b) equivalent structure o f planar HBT, showing one-dimensional transistor and overlap diode; (c) first-order circuit model o f planar HBT, with the base spreading resistance R b b shown explicitly .......................................................... 8 Fig. 2.3. Complete static model o f the HBT ......................................................................... 12 Fig. 2.4. Circuit diagram o f the model transistor for the analysis o f the base spreading resistance .......................................................................................... 15 Fig. 2.5. Variation o f base resistance with base current, for dominant J s / P f (squares) and dominant J s e (triangles) ....................................................... 20 Fig. 2.6. Variation o f f \ with base current ........................................................................... 21 Fig. 2.7. Variation with current o f base resistance model ((i)— proposed R b b model (eq. (2.36)); (ii)— classical SPICE model [11]; symbols represent the theoretical variation o f R b b for constant beta) ................................................................................................................... 24 Fig. 2.8. Circuit for the simulation o f base spreading effects ............................................. 26 Fig. 2.9. Approximate base spreading resistance obtained by simulation with SPICE ....................................................................................................... 27 Fig. 2.10. as functions of currentI b ................................... 28 Fig. 3.1. Forward Gummel measurement setup ..................................................................... 35 Fig. 3.2. Equivalent circuit o f the HBT in the forward Gummelmeasurement ................. 36 Fig. 3.3. Measured collector current I c m (0 and base current /#„, (ii) o f the HBT, and the estimated local temperature increase A T /ff o f eq. (3.6) (iii) ...................................................................................... 37 Fig. 3.4. Fig. 3.5. I bv/ I b , I r i /I b , and I bz/ I b Ratio o f measured collector and base currents /cm /ffim (i), and Pi (ii) after optimization, as functions of I c ,n. Curve (iii) is the estimated temperature increase A T /g of eq. (3 .6) ....................................... 39 N pe3t (eq. (3.9)) as a function o f the measured base-emitter voltage ............... 41 xi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 3.6. R bbtu (i) and R bbui (ii) after optimization, as functions o f measured base current; curve (iii) is the estimated temperature increase A T fg o f eq. (3.6) ............................................................................................. 46 Fig. 3.7. Measured currents I c m (i) and J s m (ii), and calculated currents I c f (iii) and I b j (iv) in the forward Gummel configuration; curve (v) is the estimated temperature increase A T /g o f eq. (3.6) ........... 49 Fig. 3.8. Reverse Gummel measurement setup ...................................................................... 50 Fig. 3.9. Equivalent circuit o f the HBT in the reverse Gummel measurement ................. 51 Fig. 3.10. Measured base current /gm (i) and emitter current I Em (ii) o f the HBT in the reverse Gummel configuration, and the estimated local temperature increase ATrff o f eq. (3.23) (iii) ....................................... 53 Fig. 3.11. Measured currents (i) and I Em (ii), and calculated currents I bt (iii) and I ev (iv) in the reverse Gummel configuration; curve (v) is the estimated temperature increase A Trg o f eq. (3.23) .......................... 57 Fig. 4.1. One-dimensional npn HBT device structure .......................................................... 59 Fig. 4.2. Simple thermal circuit o f an HBT ........................................................................... 76 Fig. 4.3. M ore accurate thermal circuit o f the planar HBT, which includes the temperature response o f the emitter junction and the overlap diode ........ 78 Fig. 4.4. Simplified circuit for the calculation o f temperatures T je and T jc , assuming that zthol < - thcol .................................................................. 80 Fig. 5.1. I s temp (symbols) and I s (lines) as functions o f temperature ............................... 89 Fig. 5.2. IsROLtemp (symbols) and I srol (lines) as functions o f temperature ................. 91 Fig. 5.3. 0Rtemp (symbols) and (3r (lines) as functions o f temperature ............................. 92 Fig. 5.4. 1/ /3ptemp (symbols) and 1//3p (lines) as functions o f temperature .................... 93 Fig. 5.5. IsEtemp (symbols) and I s e (lines) as functions o f temperature .......................... 94 Fig. 5.6. RBmaxtemp (symbols) and RBmax (lines) as functions o f temperature ................ 96 Fig. 5.7. Setup for the measurement o f the common emitter output characteristics ................................................................................................... 97 xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 5.8. Measured collector current I c m o f the HBT as a function o f the collectoremitter voltage (Vcm — V sm), with the base current as a parameter (common emitter output characteristics). The base current step is 6 fj,A ............................................................................................................... 99 Fig. 5.9. Simulated collector current I c a (lines) and measured collector current Icm (symbols), as functions o f the collector-emitter voltage (Vcm — VEm)- The base current step is 6 (same as in fig. 5.8) ......... 101 Fig. 5.10. Measured currents I c m (0 and Ism (ii), and simulated currents Ic s (iii) and I b 3 (iv) in the forward Gummel configuration, using the parameters o f table 5.10 ................................................................................ 104 Fig. 5.11. Measured currents / g m (i) and I Em (ii), and simulated currents I bs (iii) and I e 3 (iv) in the reverse Gummel configuration, using the parameters o f table 5.11 ................................................................................ 105 Fig. 6.1. Single-heterojunction one-dimensional HBT structure with two setback layers ................................................................................................................ 109 Fig. 6.2. Placement o f depletion capacitances (C j e , C j c ) and diffusion capacitances (C d e , C q c ) in the Ebers-Moll model o f a one dimensional transistor structure with zero base resistance [12] ................ 112 Fig. 6.3. Placement o f junction and diffusion capacitances in the large-signal model o f the overlap diode, by neglecting the spreading effects .......................... 116 Fig. 6.4. Single-heterojunction one-dimensional HBT biased in the forward active region ............................................................................................................... 119 Fig. 6.5. More realistic step-like velocity profile for electrons in the space-charge layer o f the HBT base-collector junction .................................................... 122 Fig. 6.6. Single-heterojunction one-dimensional HBT under reverse bias ...................... 124 Fig. 6.7. Step-like velocity profile for electrons in the emitter space-charge layer o f the HBT under reverse bias ...................................................................... 127 Fig. 6.8. First-order discrete approximation o f one-dimensional HBT with single base contact, using exact models for the elementary transistors; the junction and diffusion capacitances o f each transistor are shown explicitly ........................................................................................................... 128 Dominance o f C d e i over the other capacitances o f fig. 6.8 in forward active bias, for sufficiently large base currents (C d e — TF dlc/dVBE v) ............................................................................... 130 Fig. 6.9. xiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6.10. Further simplification o f model o f fig. 6.9, by replacing the lowfrequency transistor network according to section 2.2 .............................. 130 Fig. 6.11. Classical SPICE Gummel-Poon large-signal model o f the HBT under forward bias, with negligible terminal resistances and negligible junction capacitances, according to [1 1] .................................................................... 131 Fig. 6.12. SPICE ac simulation configurations o f HBT model o f fig. 6.10 and classical SPICE model o f fig. 6.11 (V cci = Vcc-i = 2 V) (a) comparative simulation o f model o f fig. 6.10 and the structure o f fig. 6.8; (b) comparative simulation o f model o f fig. 6.11 (classical SPICE model) and the structure o f fig. 6.8 ................................................. 132 Fig. 6.13. Determination o f small-signal base resistance o f fig. 6.12 (V cci = VcC’2 = 2 V) ................................................................................... 136 Fig. 6.14. Comparative ac simulations o f the reference discrete structure o f fig. 6.8 (i) and proposed model o f fig. 6.10 (ii), obtained by simulating the circuit o f fig. 6 .12(a) (a) magnitude o f input impedance; (b) magnitude o f ac beta .................... 140 Fig. 6.15. Comparative ac simulations o f the reference discrete structure o f fig. 6.8 (i) and classical SPICE model o f fig. 6.11 (ii), obtained by simulating the circuit o f fig. 6 .12(b) (a) magnitude o f input impedance; (b) magnitude o f ac beta ................... 141 Fig. 6.16. Placement o f depletion and diffusion capacitances in the model o f a one-dimensional transistor structure with non-zero base resistance ........ 142 Fig. 6.17. Full large-signal model of the intrinsicplanar HBT ............................................ 143 Fig. 7.1. Layout of the HBT probe pattern .................................................................... 145 Fig. 7.2. HBT base-collector junction capacitance measurement, showing the connection and placement o f the microwave probes ................................. 147 Fig. 7.3. Small-signal equivalent circuit o f the HBT in the measurement o f the basecollector junction capacitance ........................................................................ 148 Fig. 7.4. Magnitude and phase o f the measured S u (Sum ) in the case o f the base-collector junction capacitance measurement, for VB = 0.4 V, Vc = 0 ............................................................................................................ 149 Fig. 7.5.Cjctot (symbols) and C jcth (lines) as functionso f VBC ........................................ 151 Fig. 7.6. HBT base-emitter junction capacitance measurement, showing the connection and placement o f the microwave probes ................................. 154 xiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 7.7. Small-signal equivalent circuit of the HBT in the measurement o f the base-emitter junction capacitance ................................................................. 155 Fig. 7.8. Magnitude and phase o f the measured Soo (So •>,„) in the case o f the base-emitter junction capacitance measurement, for Vg = Vc = 0, VE = - 0 . 4 V ................................................................................................ 156 7.9. CjEtot (symbols) and CjEth (lines) as functions o f Vb e = Vb - V e ............. 158 Fig. 7.10. (a) Equivalent RF circuit model o f the probe pattern o f the microwave HBT; (b) CutofF-mode small-signal equivalent circuit o f the intrinsic HBT for Vbc — 0 and Vbe — VBEmin < 0 ................................................ 161 Fig. 7.11. Measurement configuration for the determination o f the forward transit time, showing the connection and placement o f the microwave probes .............................................................................................................. 166 Fig. 7.12. Equivalent circuit o f the HBT in the forward active region measurement, including the RF probe-pattem parasitics .................................................... 168 Fig. 7.13. Measured (1) and simulated (2) S-parameters o f the HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.435 V, Vc = 2.435 V ( iBm = 20.79/xA, I Cm = 1.4 mA) (a) S u ; (b) Sjo; (c) 5oi; (d) Soo ................................................................... 173 Fig. 8.1. Layout o f the T8 device probe pattern ................................................................ 176 Fig. 8.2. Layout o f the 78 device probe pattern ................................................................ 176 Fig. 8.3. T8 HBT dc measured collector and base currents Icm (i) and Is,n (ii), and corresponding simulated currents I c 3 (iii) and (iv), in the forward Gummel configuration .................................................................... 185 8.4. T8 HBT dc measured base and emitter currents I s m (0 and I Em (ii), and corresponding simulated currents I bs (iii) and I es(iv), in the reverse Gummel configuration ...................................................................... 185 Fig. 8.5. T8 HBT dc measured base and collector voltages (i) and V cm (ii), and corresponding simulated voltages V&, (iii) and Vc3 (iv), in the open collector configuration (VE = 0, I c = 0) ......................................... 186 8.6. T8 HBT dc measured collector current I c m (symbols), and simulated collector current I c 3 (lines), as functions o f the collector-emitter voltage (Vcm ~ VEm). The base current step is 6 n A ....................... 186 Fig. Fig. Fig. xv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 8.7. M easured (1) and simulated (2) S-parameters o f the T8 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.255 V, Vc = 2.255 V Cl Bm = 0.276 /iA, Icm = 3.691 /iA) (a )S n ;( b )S 12;(c )S 21;(d )S 22 .................................................................... 187 Fig. 8.8. M easured (1) and simulated (2) S-parameters o f the T8 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.275 V, Vc = 2.275 V ( I Bm = 0.424 /iA, Icm = 7.518 /iA) ( a ) 5 n ; ( b ) S 12; ( c ) S 21; ( d ) 5 22 .................................................................... 189 Fig. 8.9. M easured (1) and simulated (2) S-parameters o f the T8 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.295 V, Vc = 2.295 V (lBm = 0.656 /iA, I Cm = 15.13 /iA) ( a ) S l l ; ( b ) 5 I2;( c ) 5 2I; ( d ) 5 22 .................................................................... 191 Fig. 8.10. M easured (1) and simulated (2) S-parameters o f the T8 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.315 V, Vc = 2.315 V (lBm = 1-035 jxA, Icm = 30.23 /iA) ( a ) 5 l l ; ( b ) 5 12; ( c ) S 21; ( d ) S ,22 .................................................................... 193 Fig. 8 .11. M easured (I) and simulated (2) S-parameters o f the T8 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.335 V, Vc = 2.335 V (7fim = 1.620/iA, Icm = 59.62 /iA) ^ ) S n - ^ ) S x, -{c) S, u{&)S, , ................................................................. 195 Fig. 8.12. M easured (1) and simulated (2) S-parameters o f the T8 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.355 V, Vc = 2.355 V (I Bm = 2.611/iA, Icm = 115 /iA) ( a ) 5 11;( b ) S ,I2;( c ) 5 2i;(d ) 5 22 .................................................................... 197 Fig. 8.13. M easured (1) and simulated (2) S-parameters o f the T8 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.375 V, Vc = 2.375 V (I Bm = 4.246/iA, Icm = 221 /iA) ( a ) S u ; ( b ) S 12;(c )S 21; ( d ) S 22 ................................................................. 199 xvi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 8.14. Measured (1) and simulated (2) S-parameters o f the T8 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.395 V, Vc = 2.395 V (Iftn = 7.064/xA, Icm = 418 /xA) (a) S n ; (b) Sio', (c) S2l; (d) S22 ................................................................... ^01 Fig. 8.15. Measured (1) and simulated (2) S-parameters o f the T8 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.415 V, Vc = 2.415 V ( I Bm = 11.94/xA, I Cm = 773 /xA) (a) 5 u ; (b) S i2; (c) S2i; (d) S22 ................................................................... 203 Fig. 8.16. Measured (1) and simulated (2) S-parameters o f the T8 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1-435 V, Vc = 2.435 V (lBm = 20.79/xA, Icm = 1.4 mA) ( a ) S „ ; ( b ) S 12;( c ) S 21;( d ) S 22 ................................................................... 205 Fig. 8.17. 78 HBT dc measured collector and base currents I c ,n (0 and I Bm (ii), and corresponding simulated currents Ic* (iii) and I Bj (iv), in the forward Gummel configuration .................................................................... 207 Fig. 8.18. 78 HBT dc measured base and emitter currents I Bm (i) and I Bm (ii), and corresponding simulated currents I Bs (iii) and I e * (iv), in the reverse Gummel configuration ....................................................................... 207 Fig. 8.19. 78 HBT dc measured base and collector voltages VBm (i) and Vc,n (ii), and corresponding simulated voltages VBlj (iii) and V a (iv), in the open collector configuration (VB = 0, I c = 0 ) .................................................... 208 Fig. 8.20. 78 HBT dc measured collector current I c m (symbols), and simulated collector current I c 3 (lines), as functions o f the collector-emitter voltage (Vcm - VEm ). The base current step is 4 /xA .............................. 208 Fig. 8.21. Measured (1) and simulated (2) S-parameters o f the 78 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.181 V, Vc = 2.181 V (.I Bm = 0.121 /xA, I Cm = 1-974 fiA) ( a ) S „ ; ( b ) S , 2 ; ( c ) S 2l;( d ) $ 2 2 ................................................................... 209 Fig. 8.22. Measured (1) and simulated (2) S-parameters o f the 78 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.201 V, Vc = 2.201 V (7fim = 0.185/xA, I Cm = 4.121 /xA) ( a ) 5 u ; ( b ) 5 12; ( c ) 5 21; ( d ) 5 22 ................................................................... 211 XVll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 8.23. Measured (1) and simulated (2) S-parameters o f the 78 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.221 V, Vc = 2.221 V ('/Bm = 0.278 /iA, Icm = 8.560 /iA) (a) ^ n ; (b) ^ i 2; (c) Sbi; (d) 213 Fig. 8.24. Measured (1) and simulated (2) S-parameters o f the 78 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.241 V, Vc = 2.241 V (IBm = 0.421 fiA , I Cm = 17.79 /iA) (a) S i u (b) S 12; (c) $>t; (d) 215 Fig. 8.25. Measured (1) and simulated (2) S-parameters o f the 78 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.261 V, Vc = 2.261 V (jBm = 0.643/iA, Icm = 36.55 /iA) ( a ) 5 u ; ( b ) 5 12 ;(c )S ,2 l; ( d ) 5 a2 ................................................................... 217 Fig. 8.26. Measured (1) and simulated (2) S-parameters o f the 78 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.281 V, Vc = 2.281 V (IBm = 1.008/iA, I Cm = 74.51 /xA) ( a ) 5 u ; ( b ) 5 , o ; ( c ) 5 2 i ; ( d ) 5 22 .......................................................................... 219 Fig. 8.27. Measured (1) and simulated (2) S-parameters o f the 78 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.301 V, Vc = 2.301 V (lam = 1.630/iA, Jem = 149.2 /iA) ( a ) S 1, ; ( b ) S 12; ( c ) S ,2 1; ( d ) S 22 .......................................................................... 221 Fig. 8.28. Measured (1) and simulated (2) S-parameters o f the 78 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1-321 V, Vc = 2.321 V (IBm = 2.724/iA, Icm = 295.6 /iA) (a) 5^li; (b) Sr12; (c) S^i; (d) ^22 .......................................................................... 223 Fig. 8.29. Measured (1) and simulated (2) S-parameters o f the 78 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.341 V, Vc = 2.341 V (IBm = 4.721/zA, I Cm = 578.2 /iA) ( a J S n j O O S r e U c J S ^ C d J S 'a o .......................................................................... xviii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 225 Fig. 8.30. Measured (1) and simulated (2) S-parameters o f the 78 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.361 V, Vc = 2.361 V (IBm = 8.494/xA, Icm = 1-11 mA) (a) S n ; (b) S 12; (c) So u (d) S™ ........................................................ xix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 227 LIST O F TABLES Table 2.1. Parameters o f proposed base resistance model (eq. (2.36)), and o f classical SPICE model [11] after optimization .................................. 25 Table 2.2. SPICE input file for the circuit o f fig. 2.8 ........................................................... 29 Table 3.1. Parameters p F, aBF, bBF after optimization ....................................................... 39 Table 3.2. HP-BASIC transform for Pi o f eq. (3.8), used in the optimization o f p F, aBF, and bBF ................................................................. 39 Table 3 .3. Final values o f parameters Is , N F, R Fc and R Bmax, after repeatedly performing the three optimization steps ..................................... 43 Table 3.4. HP-BASIC transform for Icth. o f eqs. (3 .10)-(3.12), used in the optimization o f I s , N F, R p c and R Bmax ......................................... 43 Table 3.5. HP-BASIC transform for R BBm o f eq. (3.16) .................................................... 45 Table 3.6. Parameters a BB and Iso after optimization ........................................................ 46 Table 3.7. Parameter R p B after optimization ........................................................................ 48 Table 3.8. HP-BASIC transform for I Bf o f eq. (3.20) ........................................................ 48 Table 3.9. HP-BASIC transform for I c / o f eq. (3.21) ........................................................ 48 Table 3.10. Final values o f I sr o l , U rol , H r , Pr , R s , R p c e and R p b c , after repeatedly performing the two optimization steps ................ 56 Table 3.11. HP-BASIC transform for I Br o f eq. (3.24) ....................................................... 56 Table 3.12. HP-BASIC transform for I Fr o f eq. (3.28) ...................................................... 57 Table 5.1. Extracted values o f I s , I s e , I s r o l , P f , P r , and R Bmax (denoted by Istemp, IsROLtemp, PFtemp, PRtemp, and R-Bmaxtemp, respectively) at temperatures between 9.8 °C and 73.6 °C ....................................................................................................... 87 Table 5.2. Parameters o f temperature dependent model for I s (eq. (4.41)) after optimization (To = 300 K) .................................................................... 88 Table 5.3. Parameters o f temperature dependent model for I sro l (eq. (4.63)) after optimization (To = 300 K) ............................................... 90 xx Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 5.4. Parameters o f temperature dependent model for Pr (eq. (4.54)) after optimization (To = 300 K) .................................................................... 90 Table 5.5. Parameters o f temperature dependent model for P f (eq. (4.53)) after optimization ............................................................................................. 92 Table 5.6. Parameters o f temperature dependent model for I s e (eq.(4.61)) after optimization (To = 300 K) .................................................................... 93 Table 5.7. Parameters o f temperature dependent model for RBmax (eq. (4.64)) after optimization (To = 300 K) ............................................... 95 Table 5.8. Parameters R theo , K th c and K s after optimization .................................. 100 Table 5.9. Selective influence o f the junction temperatures on the HBT model parameters ............................................................................................ 102 Table 5.10. Parameters X t r b b and R e after optimization ............................................... 103 Table 5.11. Parameters R tholo and R c after optimization .............................................. 105 Table 6.1. SPICE input file for the circuit o f fig. 6 .12(a) ................................................... 134 Table 6.2. SPICE input file for the circuit o f fig. 6 .12(b) .................................................. 135 Table 6.3. SPICE input file for the circuit o f fig. 6.13 ........................................................ 137 and C qe as functions o f the dc base current I b ....................................... 139 Table 7.1. Final optimization values o f C jctot as a function o f voltage .......................... 151 Table 7.2. Parameters o f base-collector junction capacitance model and Cpb (eq. (7.3)) after optimization .................................................................. 153 Table 7.3. MNS input file for the circuit o f fig. 7.3 ............................................................ 153 Table 7.4. Final optimization values o f CjEtot as a function ofvoltage ......................... 158 Table 7.5. Parameters o f base-emitter junction capacitance model and Cpe (eq. (7.6)) after optimization .................................................................. 159 Table 7.6. MNS input file for the circuit o f fig. 7.7 ............................................................ 160 Table 7.7. Optimized preliminary values o f the model parameters o f the RF probe-pattem o f the measured HBT ..................................................... 164 Table 6.4. Table 7.8. MNS input file for the circuit o f fig. 7 .10(a), with the circuit o f fig. 7 .10(b) as the intrinsic HBT model (Vbc = 0, Vb e = VsEmin < 0) ...................................................................................... xxi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 165 Table 7.9. Dc values o f the applied base and collector voltages (V b and Vc, respectively), and o f the measured base and collector currents (Igm and I c m, respectively) o f the HBT under test, in the forward active region (V e = 0, Vc — Vb = 1 V) ............................ 167 Table 7.10. Parameters Tp, q e , R&>, R thcq , and the probe-pattem parasitics o f the HBT after optimization ...................................................... 172 Table 8.1. Device information for the T8 transistor (npn HBT, common collector, one emitter finger, two base fingers) .......................................... 177 Table 8.2. Device information for the 78 transistor (npn HBT, common collector, one emitter finger, two base fingers) .......................................... 177 Table 8.3. Model parameters for the T8 HBT at Tami, = 23.5 °C ..................................... 178 Table 8.4. Model parameters for the 78 HBT at Tamb = 23.5 °C ..................................... 180 Table 8.5. Applied dc base and collector voltages and measured dc base and collector currents, corresponding to the measured S-parameters o f the T8 device in the forward active region (V e = 0, Vc - VB = 1 V) ............................................................................................ 184 Table 8.6. Applied dc base and collector voltages and measured dc base and collector currents, corresponding to the measured S-parameters o f the 78 device in the forward active region (V e = 0, Vc - VB = 1 V) ............................................................................................ 184 xxii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1 INTRODUCTION The bipolar transistor has played a dominant role in the RF and microwave power applications from the earliest days o f semiconductor electronics, and is still an important device for a variety o f other applications. This type o f device has continuously evolved, the most recent direction being towards the use o f heterojunctions, especially for the emitter junction. This transistor is referred to as heterojunction bipolar transistor (HBT). The progress in the development o f HBT's has imposed a re-examination o f the issues associated with the design and modeling of high-frequency power devices and integrated circuits. Thus, in the case o f a homojunction transistor, once a material system is chosen, the only flexibility one has in the device design is in the doping levels and the device dimensions [1], and the conflicting requirements o f heavy emitter doping, low base doping, and small base width cannot be properly met by a single bandgap structure. In the case o f an HBT, the emitter is made from a wide gap material [2], which enables the use o f very high base doping, allowing low base resistance to be obtained even with small base widths. Also, lowering the emitter doping to moderate levels reduces the base-emitter capacitance while maintaining high current gain. It is also possible to have both the emitter and the collector made from wide gap materials, which has several advantages over a single-heterojunction device, such as the suppression o f hole injection from base into collector under conditions o f saturation, emitter/collector interchangeability, and the possibility o f separately optimizing the base and the collector [2]. 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 With respect to homojunction transistors, HBT's have lower base resistance achievable with relatively large, easily controllable dimensions, higher unity-gain frequencies, and benefit from the availability o f semi-insulating substrates which ensure low-capacitance interconnects. The high values o f power gain attainable with HBT's make them usable for amplification over a frequency range that extends into the millimeter-wave range. HBT's have a number o f advantages for broadband amplifiers [3], wideband operational amplifiers [4], fast A/D conversion circuits [5], delta-sigma (A E ) modulators [6], serrodyne modulators [7], laser driver arrays [8], and flip-flop circuits [9], Among the disadvantages to using HBT's are the large power supply voltages imposed by the large Vb e ( — 1-4 V), and the high power dissipation due to the fact that the best speeds and highest current gain are obtained at the highest allowable current densities [5], Developing an accurate computer model for the HBT is very desirable due to the relative complexity o f the circuits that use HBT's, and due to the fact that in general these devices are not available as discrete components that could be used for breadboarding o r in prototype circuits. The computer model can be used with a circuit simulation program to analyze the circuit and accurately predict its performance in a variety o f conditions. This work is devoted to developing an accurate large-signal, temperature dependent physics-based model for the HBT, and to developing measurement and optimization procedures that allow the extraction o f the equivalent circuit element values, using dc and small-signal S-parameter measurements in a wide frequency range. The HBT is modeled in this work by an improved high-frequency and temperature dependent Ebers-Moll model; in the case o f HBT's, the Ebers-Moll model can be used instead o f the Gummel-Poon model due to the high doping level in the base region and negligible Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 high-injection effects [10]. The developed model is more accurate than the widely-used SPICE Ebers-Moll and Gummel-Poon models o f the transistor [11], [12], for the following reasons: 1) A new accurate representation for the base spreading resistance is used. It is demonstrated in this work that the base spreading resistance model currently implemented in the SPICE Gummel-Poon model [11] is not appropriate and produces large errors. 2) Accurate expressions for the temperature dependencies o f the model parameters, in conjunction with a suitable transistor thermal circuit, are used. The temperature dependencies o f the parameters that appear in the equations o f the HBT are derived starting from the boundary conditions for an arbitrary bias applied to the transistor, and are functions o f the explicit temperature dependencies o f the energy gaps o f the emitter, base and collector regions. These temperature variations o f the transistor model parameters are not accurately represented in the SPICE Gummel-Poon model [11], where the material bandgap is a constant. 3) The capacitances o f the HBT (especially the diffusion capacitances) are placed more appropriately in the transistor model. In the case o f the standard SPICE Ebers-Moll and Gummel-Poon models, the placement o f the diffusion capacitances with respect to the base spreading resistance and the model diodes is shown to produce large errors at high frequencies and high bias currents. The extraction procedures for all the HBT model parameters are developed and described in detail. The HBT model is incorporated into Hewlett-Packard's MNS simulation package [13], which is preferred over other simulators due to the possibility o f implementing high-complexity mathematical functions using symbolically-defined devices. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The validity o f the proposed HBT model and o f the model extraction procedures is demonstrated by the excellent agreement between the simulated and measured dc characteristics, and between the simulated and measured and S-parameters o f the transistor obtained at 10 dc collector currents in a wide range, for tw o essentially different HBT's. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2 STATIC MODEL FOR THE HBT In order to develop an accurate physical static model for the HBT, one has to consider first the physical structure of the transistor and determine the parasitic effects that have an influence on the dc characteristics. HBT structures fall into two main categories—the vertical (one-dimensional) and the planar. In general, for both of these types, the emitter layer is the topmost layer, in which case the structure is commonly referred to as the emitter-up structure. It is possible, however, to have a reversed order o f the transistor's layers, with the collector the topmost and the emitter the bottom layer— this type of structure is commonly referred to as the collector-up structure. The emitter injection efficiency is higher for emitter-up devices; the collector-up devices, although more difficult to fabricate, can be used at higher frequencies due to the smallest possible value o f the base-collector capacitance [14]. Fig. 2.1(a)-(d) shows several typical cross sections o f emitter-up HBT structures [15], The noticeable aspects o f the fabrication processes are the use o f refractory emitter contacts based on InAs cap layers (fig. 2 .1(a), (c) and (d)) and the use o f sidewall spacers at the edges o f the emitter contact (fig. 2.1(c) and (d)). Fig. 2.1(b) illustrates a fabrication technique for the minimization o f the extrinsic base-collector capacitance: oxygen (or protons) is implanted into the regions o f the w-collector layer underneath the base contacts. The structure o f fig. 2 . 1(b) is a vertical structure, because the transistor is defined only by the layers underneath the emitter; all the other structures shown in fig. 2.1 are planar. 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 V—GROOVE ISOLATION OXYGEN—PLA N TED EMITTER BASE / COLLECTOR n + - GoAs SUBCCLLECTCR SL BUFFER S I SUBSTRATE (a) COLLECTOR EMITTER BASE BASE 3710 I— 'I M (b) Fig. 2.1. Schematic cross sections o f various HBT structures (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 Fig. 2.1. Continued (c and d). EMTTTER BASE COLLECTOR (C) BASE EMITTER ^ ^ ,n+ BASE COLLECTOR m r Wa m n i :J---------- -------------------------N - — I (d) The vertical HBT is essentially a one-dimensional transistor whose operation is well understood. Sufficiently accurate models have been developed for this structure [16], [17], ranging from simple static models to large-signal models for high-frequency, high-power applications. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 overbp 1one— | diode ( dimensional transistor (a) planar HBT structure n P B \, n one— dimensional transistor overlap diode C (b) equivalent structure o f planar HBT, showing one-dimensional transistor and overlap diode Fig. 2.2. Development o f a simple circuit model for a planar HBT (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 Fig. 2.2. Continued (c). C B AAAH circuit model o f 1-D transistor BB E (c) first-order circuit model o f planar HBT, with the base spreading resistance R b b shown explicitly The structure that is modeled in this work is a planar HBT structure. In this case (see fig. 2 . 1 (c)), the collector junction area is, in general substantially larger than the emitter area, and an additional diode (known as the overlap diode [18], or external diode [19]) is formed between the base and the collector. Fig. 2.2(a)-(c) shows how the overlap diode appears in the structure, and the way it must be considered in a first-order model o f the HBT. The resistance R bb that appears in fig. 2.2(c) in series with the base o f the one dimensional transistor is the well-known base spreading resistance [20], It is shown in this work that the widely used closed-form expression for the current dependence o f the base spreading resistance [ 1 1 ] is not sufficiently accurate; an accurate model for this resistance will be developed and verified. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 2.1 Complete Static Model The simplified static model o f fig. 2.2(c) is only a first-order model, since the additional components o f the forward current through the overlap diode are not shown explicitly. These currents may be caused by [12]: — the recombination o f carriers at the surface, — the recombination o f carriers in the diode space-charge layer, and — the formation o f diode surface channels. The three components have a similar variation with the applied base-collector voltage which may be expressed by a composite current o f the following form: (2 . 1) where N rol is the low-current emission coefficient, being normally close to 2 [ 1 2 ]. The circuit model o f the one-dimensional structure in one o f its simple forms may be a regular Ebers-Moll model o f the bipolar transistor [12], [21], in which the high-level injection effects can be neglected due to the high doping level in the base [10], [22], The additional recombination components o f the base current o f this one-dimensional transistor may be included in the circuit model by means o f two non-ideal diodes, as in the case o f the relatively complex EM 3 and Gummel-Poon models [12]. The complete static model o f the HBT is shown in fig. 2.3, where the current dependence o f the base spreading resistance must be modeled according to the pattern o f the current flow in the base region [19], [23], For the relatively simple but frequently encountered case o f a single base stripe, an accurate model for the base resistance will be derived in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. next section. The saturation currents and emission coefficients corresponding to the diodes o f fig. 2.3, as well as the forward and reverse beta, will be derived in chapter 4 using the material parameters and the dimensions o f the structure. In fig. 2.3, the following relations hold true [11], [12]: I BE — ^ S £ ( e x p ( j v ^ r ) (2-2) Ib c = / s c ( e x p ( $ i ) - l ) (2 3) Ice = -l) (J4> lEC = I s ( e * P ( $ i ) - l ) (2-5) IoL = IsO L (e x p ( 1/^kT 'j ~ l ) (2-5) IC T ~ IC C ~ I EC (2.7) Resistances R e and R c in fig. 2.3 are the series (ohmic) resistances o f the emitter and collector terminals, respectively, and are assumed to be independent o f the corresponding terminal currents; resistance R b \ is the series ohmic resistances o f the base terminal, and resistance R&> is the ohmic resistance that appears between the overlap diode and the one dimensional transistor structure. Current I rol o f fig. 2.3 is given by eq. (2.1), with Vb c = Vbcol ■In practice, the emission coefficients N r , N p, and N ol are normally close to 1, while N e , N c , and N rol are close to 2 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 ROL 4 ^ 1 — cc CT BEt Fig. 2.3. Complete static model o f the HBT. In the case where the base region o f the structure o f fig. 2.2(a) has a uniform doping and thickness, and the surface effects can be neglected, the saturation currents I s c and I sr o l , and I s and I so l o f eqs. (2.1), (2.3), (2.4), (2.5), (2.6), are not independent, since they can be expressed as functions o f the saturation currents o f similar diodes with different areas only. Thus, if I c s denotes the saturation current o f the base-collector junction o f the one dimensional transistor, then [ 1 2 ]: Ics = <2 '8) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 I f A e and A c denote the base-emitter and base-collector areas, respectively, o f the HBT structure o f fig. 2.2(a) (typically, A c > A e ), then the saturation current o f the overlap diode, I s o l , can be written as: (2.9) Since cxr = (3r /((3r + 1), eq. (2.9) can be rewritten as: (2 1 0 ) I so l = K J s ^ where K a = ( A c / A E) - 1 is a constant for a given structure. If the base is assumed to be uniform and the surface effects are assumed to be negligible, then the recombination components o f the base-collector junction o f the transistor and o f the overlap diode are o f the same nature, and the following relation holds true: ( 2 . 11) ISROL = K aI s c where K a has been defined for eq. (2.10). Eqs. ( 2 . 1 0 ) and ( 2 . 1 1 ) demonstrate the fact that I s c and I s r o l , and I s a and I s , are not independent under the previous assumption. If, as it is sometimes the case, the surface effects dominate over the other effects accounting for the recombination components, then, from the structure o f fig. 2 .2 (a) and the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 model o f fig. 2.3, it is observed that I sro l will be considerably greater than I s c , and the two saturation currents will be virtually unrelated. In practice it is also possible for the saturation currents I sol and I s not to be related by eq. ( 2 . 1 0 ); for example, if the base region o f fig. 2 .2 (a) corresponding to the overlap diode is more heavily doped than the base underneath the emitter (sometimes practiced to reduce the ohmic base series resistance), then the saturation current densities o f the base-collector junction and o f the overlap diode will not be identical, and eq. ( 2 . 1 0 ) will not apply. 2.2 Mathematical Model for Base Resistance Modeling the intrinsic base resistance R bb o f the transistor is needed to accurately reproduce the dc characteristics o f the device [19], [23]. The model presented here uses a distributed base resistance and a distributed transistor, and can be used to represent the transistor characteristics regardless o f the region o f operation. For simplicity, however, it will be assumed that the npn transistor is in the forward active region {Vb e > 0, Vgc < 0 ); the following additional assumptions are made, consistent with [19], [23]: 1) The transistor has a stripe emitter geometry with a single base contact. 2) The distributed model is considered only for the region under the emitter stripe (one-dimensional transistor). 3) The base conductivity modulation is neglected, owing to the high doping o f the base (negligible forward and reverse Early effects). The base sheet resistance is independent o f the bias conditions, and is constant across the transistor. The base resistance will be determined in two cases, assuming either I b e or I c c /P f o f fig. 2.3 to be negligible. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 Fig. 2.4 shows the circuit configuration o f the distributed transistor; W and L denote the width and length, respectively, o f the emitter stripe, and the base contact is placed at x = 0 . To simplify the analysis, it is assumed that R b i = R m — R e = 0. In fig. 2.4, applying Kirchhoffs current law for the base o f the infinitesimal transistor at coordinate x, we can write: dlb(x) = Je(x)Ldx 4- Jc(x)Ldx J s L d x j^ Q x v (¥ ^ ? ^ j - J s E L d x e x p ( j ^ ^ j ( 2 . 12) where J s = I s / ( L W ) , J s e = I s e / { L W ), VT = k T /q , and the typical values o f I and have been assumed for the emission coefficients N p and N p , respectively. rr eBjdx/L id x /L rBidx/L B o-AA/V !b ( x 0 J U c (x)Ldx \ j f J e (x)Ldx x W Fig. 2.4. Circuit diagram o f the model transistor for the analysis o f the base spreading resistance. If rpi is the base sheet resistance o f the intrinsic base, then: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 16 dVbe(x) = - r Bi^ I b{x) (2.13) Eqs. (2.12) and (2.13) form a system o f differential equations whose unknowns are Ib(x) and V6e(x). Although it is possible to eliminate Ib(x) from these equations (see, for example, [19]), the resulting differential equation in V&e(x) cannot be easily integrated, and obtaining R b b as a function o f the base current is relatively complicated. Instead o f eliminating i*,(x), some simplification will be made to eq. (2.12), and Vbe(x) will be eliminated. The simplification consists o f considering either the term in exp(V6e (x )/V r) or the one in exp(V6e( x ) / 2 Vr) as the dominant term in the right-hand side o f eq. (2 . 1 2 ), which simply means that the transistor either has a constant forward beta (as it is often considered in theoretical analyses [19], [23]), or its base current is dominated by the recombination component (as has been experimentally observed in the case o f some AlGaAs/GaAs HBT's). Since a closed-form solution cannot be obtained for the system (2.12)(2.13), R b b will first be obtained numerically, and then it will be approximated by an analytical function o f I s In view o f the above discussion, the system o f differential equations will be solved in the following two situations: (a) The transistor has a constant forw ard current gain. From eqs. (2.12), (2.13), the system to be solved is: (2.14) (2.15) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 Following the approach described in [23], V ^ x ) can be eliminated from the above equations: + (216) Kl = ^ (2.17) where The differential equation (2.16), with the boundary conditions If,(W ) = 0 and Ib( 0 ) = I b (the entire base current enters the active base at x = 0 ), has the following solution: I b( x ) = 2 § -z ta n (z (l - (2.18) where z tr n z = IB (2.19) By taking the derivative o f h ( x ) and calculating it at x = 0, one obtains 1 ^(0 ) (from eq. (2.14)) as: *W °) = V r* w f c * (M O) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The effect o f the base resistance is modeled by connecting a lumped resistance R b b {I b ) in series with the base o f an ideal transistor whose base current (in this case) is I c c / P f (see eq. (2.4)). The base-emitter voltage K K = V rln j^ o f this ideal transistor is then given by: ( 2 .2 !) From eqs. (2.20), (2.21) and the above discussion, the base resistance is obtained as: R b b (Ib ) = (b) = g ln (# ^ ( 2 .2 2 ) The base current o f the transistor is dominated by the recombination component. From eqs. (2.12), (2.13), the system to be solved is: dJ^ = - J s e L cx p ( !H i ) h(x) = - 4 ^ (2.23) (2.24) Following the approach described in part (a), the current Ib(x) is obtained as: Ib(x) = ^ z ta n (z (l ~ w ) ) <2-25) where: ztanz = 2% I b <2-26) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 K 2 = ^ (2.27) Calculating V ^(0 ) as in the previous case, by taking the derivative o f h ( x ) from eq. (2.25) at x = 0, we have: VUO) = (2.28) The base-emitter voltage of the ideal transistor is obtained as: K = »»> and the base resistance in this case is given by: R b b (Ib ) = = f ln (# (2.30) where the values given by eqs. (2.28) and (2.29) have to be used for V^(0) and V ^. Note that, in either case, R bb is not a function o f saturation currents or beta o f the transistor; the base resistance is only a function o f the parameters o f the base region (sheet resistance, length, width). However, the base resistance depends on the emission coefficients o f the diodes that model the emitter-base junction. This dependence is somewhat hidden by the fact that in our analysis these emission coefficients have been taken to be either beta) or 2 1 (constant (dominant recombination component); the logarithm is multiplied by V t / I b in the former case, and by 2 Vt / I b in the latter, which demonstrates this dependence without resorting to an analysis in which the emission coefficients are considered explicitly. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 At low base currents, in either case, the base resistance has the following approximate value [19], [23]: 3 (2.31) L At relatively high base currents I b , z o f eqs. (2.19) and (2.26) approaches tt/2 , and one can write ztanz ~ z /c o sz [23], As a result, from eqs. (2.19), (2.22), (2.26), (2.30), the following approximations can be made for the base resistance at high base currents: case fa): R BB(IB) = £ l n ( f (2.32) case (b): R BB(IB) ~ (2.33) a * 10 10 - 8 10 - 6 10 - 4 IB 10 - 2 (A ) Fig. 2.5. Variation of base resistance with base current, for dominant J s /P f (squares) and dominant J se (triangles). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 Fig. 2.5 shows the variation o f R b b with the base current in the two situations, considering either a constant forward beta for the transistor or a dominant recombination component. The resistances have been obtained numerically, by solving the transcendental equations (2.19) and (2.26) for c at each value o f the base current I s and using the solutions in eqs. (2.22) and (2.30). A sufficient number o f values I b have been considered in the calculations. To plot the graphs o f fig. 2.5, it was assumed that r BiW/(2>L) = Rsmax = 10 Kfl. The thermal voltage was assumed to be Vt = 25.8 mV. The variations o f R bb with base current shown in fig. 2.5 are very similar and their relative difference is small. Also, the maximum value o f the base resistances in the two cases is the same and equal to rg,W /(3.L ) at low base currents (eq. (2.31)). 10 8 6 4 2 IB (A) Fig. 2.6. Variation o f f \ with base current. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 In order to develop an accurate model for the base resistance, it is desirable to determine a closed-form expression for the above resistance functions. This is possible at least at high base currents, since the logarithms o f eqs. (2.32) and (2.33) vary slightly with I B over several decades. Indeed, fig. 2.6 shows the variation o f / i ( J s ) = ln (W rfll7 e /(2 X V r)) for I b in the range [10 nA, 10 mA], and if the value I s = 100/zA is chosen to be the reference, this function changes by only ± 50% when I b changes by as much as two decades. As a result, at large currents, R bb o f eqs. (2.32) and (2.33) can be approximated by: case (a): R Bb — C \ (2.34) case (b): R bb — ^2 (2.35) where C \ and C-i are constants. Taking into account the fact that the base resistance is constant for I b less than a certain com er value determined by the parameters o f the base region [23], the approximation o f eqs. (2.34) and (2.35), and the fact that only one o f cases (a) and (b) above may be considered as dominant in practice, the dependence o f the base resistance with the base current I b can be written as: R b b (Ib ) (2.36) where R a m a = rB iW /(3 L ) and, in view o f the previous discussion (see eqs. (2.34) and (2.35)), aRB ^ 1. In eq. (2.36), Iso is the com er base current ( R b b ( I bo) = RBmax/2). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 If the npn transistor is assumed to be in the reverse active region (V bc > 0 , Vbe < 0), then eq. ( 2 . 1 2 ) is re-written as: dlb(x) = Je(x)Ldx 4- Jc{x)Ldx = - Js L dxj^Q xp(j^^j where the typical values o f 1 and 2 - JSc L d x e x p (j^ 1') ( 2 . 12') are assumed for the emission coefficients N r and N c , respectively. Following the derivation presented for the forward active region (with Vt,c(x) used instead o f Vbe(x)), the base resistance in the case o f a constant reverse gain is obtained as given by eq. (2 .2 2 ); if the recombination component in the reverse active region is dominant (Isc -Ts), then the base spreading resistance is obtained as given by eq. (2.30). Thus, the model for the base resistance given by eq. (2.36) can be used regardless o f the transistor’s region o f operation. The dependence given by eq. (2.36) has been fitted to the previous values o f R bb obtained numerically in the two cases and plotted in fig. 2.5; the IC-CAP optimizer [13] has been used to automatically adjust the parameters I m , and aRB for minimum error, in a range for I s extending from 10 nA to 10 mA. The optimization process was configured to minimize the error function e given by: N CRBB = H I ln (R b B 2 { Ib ti)) - ln (i2 sB l(-T sn )) | n= l 2 (2.37) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 where R b b \ ( I Bn), R b b i {IBn) are the base resistances obtained from eq. ( 2 .2 2 ) (or eq. (2.30)) and eq. (2.36), respectively, at current I s n ( R bbi is the theoretical variation with current o f the base resistance, while R b b i is the model prediction). 10 a 10 1 0 ~ 8 1 0 “ 6 1 0 - 4 Ib 1 0 “ 2 (a ) Fig. 2.7. Variation with current o f base resistance model ((i)— proposed R b b model (eq. (2.36)), (ii)— classical SPICE model [11]; symbols represent the theoretical variation o f R bb for constant beta). The results given by the optimizer are shown in table 2.1. In order to contrast this model with the well-known model for the base resistance used in SPICE [11], table 2.1 also shows the parameters o f the SPICE model and the RMS error obtained after fitting the latter to the same values o f R bb plotted in fig. 2.5. The RMS error in the case o f the SPICE model is much larger, as can be seen from fig. 2.7 that shows the theoretical variation o f R b b *n the constant beta case, R bb obtained from eq. (2.36) after optimization, and R bb obtained from the classical SPICE model after optimization. The base resistance model given by eq. (2.36) is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 very accurate over several decades, being a much better choice than the model used in SPICE and described in [ 1 1 ]. Table 2 .1. Parameters o f proposed base resistance model (eq. (2.36)), and o f classical SPICE model [11] after optimization eq. (2.36) case (a) 10 Kfi 8.54 (J.A 0.8625 0.245% R-Bmax I BO aRB RMS error case (b) 10 Kfi 17.05 fj.A 0.8615 0.240% Classical SPICE model case (a) 10 Kfi Rb R bm I rb RMS error case (b) 10 Kfi 0 0 1.097 A 7.709% 2.735 y.A 6.397% 2.3 Approximate Determination o f Base Spreading Resistance through Simulation In order to verify the correctness o f the proposed model for the base resistance, and to intuitively understand its current dependence, it is useful at this point to obtain the base spreading resistance by simulation. Note that, although eqs. (2.18) and (2.25) give the spatial dependence o f the base current o f the distributed transistor, there is no way to obtain it directly in terms o f I b , without having to solve a transcendental equation in s. A circuit that provides a reasonable approximation o f the base spreading effects in the transistor is shown in fig. 2 .8 , in which transistors Q i, Qo, ..., Qw are identical, R \, Ro, ..., R l0 are identical resistances that account for the distributed base, and Qj-> is a transistor whose emitter area is 11 times the area o f Q \. The supply voltage V cc is chosen such that all transistors operate in the normal active region. The distributed structure made up o f Q i, Qo, ..., Q ii, and transistor Q\o is attacked by identical currents, I b and I b \, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 W V c c Fig. 2.8. Circuit for the simulation o f base spreading effects. In view o f the previous considerations regarding the base spreading resistance, the following equations hold true: R i = R 2 = ... = Ri o = r B i f i o (2-38) 10 RBmax = | (2.39) i= l The circuit o f fig. 2.8, with i2, = 3 Kf2, i = 1, ..., 10, I s i -2 = ■Tsi = 1 x 10 " 26 A, = ••• = f s n = ^ si 2 /H > P f\ — f i n = ••• = P fu = P fvi = 100, V cc = 2 V, has been dc simulated, with I b being swept from 10 nA to 10 mA. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 The base spreading resistance obtained through simulation is shown in fig. 2.9, being calculated as: (2.40) R b b (I b ) = 1 «3 ft? 1 Ib (A) Fig. 2.9. Approximate base spreading resistance obtained by simulation with SPICE. With the values used in the simulation, the maximum value o f the base spreading resistance is R.Bmax = 10 Kfi, which is correctly obtained in fig. 2.9 at currents on the order o f a few tens o f nA. At very small values o f I b , however, the base spreading resistance calculated according to eq. (2.40) differs slightly from 10 KQ due to the numerical errors (probably due to truncation effects) o f the simulation. At high values o f the attack current, the base resistance displays a variation similar to the theoretical dependence shown in fig. 2.5. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 resistance approaches zero for I b —> oo, due to the fact that the base current o f transistor Q i diverts almost all o f the base drive current I b , as can be inferred from fig. 2 . 1 0 . 1 -3 -2 iB Fig. 2.10. I b i /I b , I b i /I b , (A) and I b s / I b as functions o f current I b As expected, at low values o f Ib , the base currents o f transistors Q t , Q>, ..., Q u are virtually equal, and the base spreading resistance takes on the maximum value R.Bmax- At higher values o f I b , the voltage drops across resistances R i, i = 1, ..., 10, are no longer negligible, and the base resistance exhibits a current dependence similar to the one shown in fig. 2.5; in this case, the first transistor (Q i) conducts most o f the base current. The SPICE input file for the circuit o f fig. 2.8 is given in table 2 .2 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 Table 2.2. SPICE input file for the circuit o f fig. 2.8 * SPICE input file for obtaining the base spreading * resistance through simulation vcc 1 0 2 vl 2 3 0 IB 0 2 le-10 ■A t rl 3 4 3k r2 4 5 3k r3 5 6 3k r4 6 7 3k r5 7 8 3k r6 8 9 3k r7 9 10 3k r8 10 11 3k r9 11 12 3k rlO 12 13 3k ■At ql 1 3 0 qnpnl q2 1 4 0 qnpnl q3 1 5 0 qnpnl q4 1 6 0 qnpnl q5 1 7 0 qnpnl q6 1 8 0 qnpnl q7 1 9 0 qnpnl q8 1 10 0 qnpnl q9 1 11 0 qnpnl qlO 1 12 0 qnpnl qll 1 13 0 qnpnl ql2 1 14 0 qnpn2 ■A t fl 0 14 vl 1 + .model qnpnl n p n (I S = 9 .0909e-28 BF=100) .model qnpn2 npn(IS=le-26 BF=100) * .dc dec IB .probe .end lOO.OOOp .01 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 2.4 Closed-Form Expression for I bo and Temperature Dependence o f RBmax and I bo From eqs. (2.34), (2.35) and (2.36) an approximate expression can be found for the comer current I bo o f the base resistance model. Since a^B — 1 may be imposed as a firstorder approximation, at high currents I b , eq. (2.36) can be re-written as: R bb(Ib) - (2.41) and using eq. (2.34) or (2.35) I bo is obtained as: h o = C x{2)V t r ^ (2-42) Eq. (2.42) gives an expression for the comer current I bo o f the base resistance model; constants C \ approximations or Co, in view o f the discussion made in previous sections, are the of the functions / i (I b ) = In(W rel/f l/( 2 LVr )) and k { l B ) = ln (W rBl/ B/(4 £ V r)), respectively. In order to determine the temperature dependence o f the base resistance, the temperature dependence o f RBmax, I bo, and aRB must be determined. Since a^B — 1 may be assumed for the base resistance model, the temperature dependence o f olbb may be assumed negligible;inaddition, the width and length o f the base stripe (W and L, respectively) may be assumedto betemperature independent, due to the small expansion coefficient o f GaAs (6.86 x 10-6 at 300 K [24]). Using eq. (2.31) for RBmax, and the following expression for the sheet resistance o f the intrinsic base, r Bi (for an npn HBT): Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 (2.43) rB i ~ qNBtiptB where N b , p p, and tg are the base doping concentration, the hole mobility in the p +- GaAs base, and the thickness o f the based region, respectively, the temperature dependence o f RBmax can be expressed as: d d - nB m ax ~ asm 3 W L I W _ l ~~ Z L q N Bt B i_ HP( T ) ( 2 44) K V In the above equation, T is the absolute temperature; N B, t B, W , and L are assumed to be temperature independent. It follows that: R B m a x (T ) = R Bm ax(T0 ) ^ (2.45) where To is the reference temperature. Eq. (2.45) gives the temperature dependence o f RBmax in the case o f an npn transistor. For a pup device, following a similar derivation, the temperature dependence o f RBmax is obtained as: R B m a x(T ) = R Bm ax(T 0 ) ^ (2.46) where p n is the electron mobility in the r f base. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 To obtain the temperature dependence o f Ib o, the closed-form expression for Ibo (eq. (2.42)) and eq. (2.45) (or (2.46)) can be used. Thus, if a generic constant C (which is temperature independent) is used instead o f C x or Co in eq. (2.42), the comer current I bo at temperature T can be written as: I bq ( T ) ~ C y RgJ^iT) (2A1) and using eq. (2.45), it follows that I b o ( T ) ^ Im (Ta) § (2.48) In the case o f a pup transistor with a heavily doped base, the temperature dependence o f I bo is given by (using eq. (2.46)): I m ( T ) =! Ib o (T o ) £ (2 49) The temperature dependencies of RBmax and I bo derived in this section are valid only in the cases where the temperature across the base region is constant, which corresponds to the situation where the device is isotropically heated up to some temperature and self-heating is negligible. If self-heating occurs, then, due to the thermal anisotropy o f the device, the temperature varies across the base layer and the above derivations are not valid. An empirical term, in conjunction with eq. (2.36) and the above temperature dependencies for RBmax and Ibo, will be used in this work to model the base spreading resistance as a function o f temperature under self-heating conditions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3 PARAMETER EXTRACTION PROCEDURE FOR THE STATIC MODEL This chapter describes a procedure for obtaining the set o f model parameters that accurately characterize the static model o f fig. 2.3. The parameter extraction is accomplished through the well-known forward and reverse Gummel measurements [25]. The advanced curve fitting and optimization techniques now available in IC-CAP [13] are employed for the extraction o f the model, using the applicable model equations given in chapter 2. The temperature at which the measurements are taken is assumed to be known accurately; also, the area factor K a o f eqs. (2 . 1 0 ) and (2.11), which is a difficult parameter to obtain experimentally [12], is assumed to be known beforehand for a given transistor. All the device measurements used in this chapter to illustrate the parameter extraction procedure were taken at a temperature o f 23.5 °C. 3.1 Forward Gummel Measurement and Parameter Extraction The forward Gummel measurement can be defined by the following bias conditions: — the base and collector o f the transistor are at the same potential; —the base-emitter voltage is positive for an npn transistor and negative for a pnp transistor. The forward Gummel measurement setup is shown in fig. 3.1, where the dc voltage sources Vb and Vc define the base and collector voltages, respectively (V b = Vc = 0 ); the emitter voltage is defined by source Ve {Ve < 0). An HP4142 dc modular source/monitor controlled by IC-CAP via an HP-IB bus can be used to provide all voltage sources o f fig. 3.1 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 and to sweep source VE while holding Vb and Vc at ground potential; all transistor currents are available as measured currents through the voltage sources. Since Vb = V c, the currents through the overlap diode and base-collector junction o f the transistor model o f fig. 2.3 are negligible, and the model o f fig. 2.3 in the forward Gummel measurement reduces to the circuit shown in fig. 3.2, where R b = R b i + R b -2 Resistance R pb o f fig. 3.2 is the parasitic parallel resistance of the base-emitter junction o f the transistor, resistance R p c is the parasitic output resistance (appearing between the collector and the emitter) o f the device. If the currents through the shunt parasitic resistances are negligible, that is, if iBi » (VB - V e ) / R pb, l a > (VB - VE ) /R p c , then, from the circuit o f fig. 3.2 the measured base and collector currents can be written as, respectively: I Bn, ^ £ ( e x p ( $ f ) - l ) - !) <3 » (3.2) Since the base-emitter junction is forward biased, ex$(V B E il(N f <e -Vt )) 3> 1 , and the following expression is obtained for the ratio o f the above currents: +&BF(/Cm)U^ where Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 aBF _ Np jtE (3-4) (3.5) bBF = D.U.T.J lc B (= 0 ) ~ t r S . ~ ± r (<0 ) Fig. 3.1. Forward Gummel measurement setup. Note that the constant parameters asF ^ bb f defined by eqs. (3.4) and (3.5) are constant at a given temperature, since they are functions o f saturation currents and emission coefficients only. From eq. (3.3) it can be concluded that, in the regions where the influence o f the parasitic parallel resistances o f the transistor is negligible, the ratio o f the measured collector and base currents, Ic m /lB m , is a function o f the collector current only (in the forward Gummel configuration), and— if I c m is considered to be the argument o f Icm i^B m — this function is not affected by the parasitic internal base-emitter voltage, V s E i- s e r ie s resistances R b b , R b > R e or R c , or by the This observation is very important, since in general R bb is a nonlinear resistance (see chapter 2 ) whose current dependence is not known beforehand, which would complicate the procedure for the determination o f the parameters if the internal base-emitter voltage VsEi were to be employed. The parameters a BF, bBF, and (3p o f eq. (3.3) can be found by treating the ratio Icm/lBm as a function o f Ic,n- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 Fig. 3.2. Equivalent circuit o f the HBT in the forward Gummel measurement. Fig. 3.3 shows the measured collector and base currents, plotted as functions o f the measured base-emitter voltage (Vem — Vkm); fig- 3.4 shows the ratio /c m /fs m plotted as a function o f Icm- Parameters /3/r, agp, and bgF o f eq. (3.3) can be determined by using the optimization feature o f IC-CAP, in the range for I c m where the influence o f the parasitic parallel resistances o f the HBT and the self-heating effects are negligible. The local temperature increase A 2 / ff (due to the power dissipation in the device) can be estimated by assuming a typical value for the junction-to-ambient thermal resistance o f the device R th and multiplying it by the maximum instantaneous dissipated power: A Tfg = Rth(VBm - V E m ) ( lB m + I Cm ) (3-6) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 -4 u I? I '-i (iu) -9 0.8 V jB m ~ V E m 00 Fig. 3.3. Measured collector current I c m (0 and base current i # m (ii) o f the HBT, and the estimated local temperature increase A T fg o f eq. (3.6) (iii). 3.1.1 Determination o f P f, a BF and b g p This section presents a procedure for the determination o f parameters Pf , qb f , and bsF o f eq. (3.3). The procedure is optimizer-based, and assumes that the parasitic parallel resistances and the local temperature increase have a negligible effect on the device characteristics. In the case o f the device whose characteristics are shown in fig. 3.3, the range for I c m for which the influence o f the parasitic parallel resistances and the self-heating effects is negligible is approximately [10 ~ 8 A, 5.28 x 10 ~ 5 A] (the value o f 2000 °CAV, provided by the manufacturer for R th, has been used in the calculation o f A T /g shown in fig. 3.3). The optimization process was configured to minimize the error function given by: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 C0F = £ I K A ^ C m n )) “ K ] £ ) (3.7) I2 n=N0Fi where Ipmn and /cmn are the n-th measured base and collector currents (n = N sp i —N sf^), and /3\ is the right-hand side function o f eq. (3.3): M lC m ) = X + i,s f .(JCm, . SF- . <3'8> For the HBT under test, for iV/jpo — N$p\ + 1 = 30 points in the previously determined range for I c m, the values obtained for /?p, a#p, and bBF after optimization and the final errors are given in table 3.1. The final RMS error after optimization is very small, which indicates that P \{Icm ) defined by eq. (3.8) is a good approximation o f Icm /lB m (as can be seen from fig. 3.4). The IC-CAP transform for /3( o f eq. (3.8), written in HP-BASIC, is given in table 3.2. Since agp and bgp define a precise relationship between the emission coefficients N p and N E and between the saturation currents I s and Is e , the number o f variables in the subsequent fitting processes will be reduced (I s e and N p will be written as functions o f I s and N p ), which renders these processes more reliable. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 t rnmi 16 12 0 (Do) *0. B cq 8 m 1 ■■ 10 - 1 0 1 0 -8 1 0 -6 I Cm 10-4 10 - 2 0 (A ) Fig. 3.4. Ratio o f measured collector and base currents Ic,n/lB m OX and 0\ (ii) after optimization, as functions o f I c m- Curve (iii) is the estimated temperature increase A T /g o f eq. (3.6). Table 3.1. Parameters 0 f , a B F , after optimization Pf aBF bBF MAXIMUM error RMS error bb f 109.3 0.5359 198.0 x 10 - 6 0.320% 0.131% Table 3.2. HP-BASIC transform for 0\ o f eq. (3.8), used in the optimization o f 0 f , a B F , and b s F el = 1//BF + b B F * ( (ic.m)A (aBF - 1)) return elA (-l) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 3 . 1 .2 Determination o f Is, Np, RBmax and Rpc This section describes a procedure for the determination o f parameters Is and N p o f eq. (2.4), RBmax o f eq. (2.36), and R p c o f fig. 3.2. The procedure consists o f repeatedly performing three distinct optimizations with the above parameters as variables. In the first step o f the optimization, parameters Is and Np are determined from eq. (3.2) over the range o f the base-emitter voltages where the slope o f ln(Tcm) *s constant [25]; in the following steps, R p c and RBmax are found as optimization variables in the voltage ranges adjacent to the range used in the first step, by considering the additional collector current component due to R p c , and the debiasing effect due to the base spreading resistance R b b • All the above steps are repeated until the convergence o f the variables is achieved, and no further change in their values is observed. The range where the slope o f ln(Jcm) is constant is the range where the current through the shunt parasitic resistance R p c and the debiasing voltage due to resistance R bb are negligible. This range has been determined as the range where the function: T\T " 1 - i V^Em) __ w f W n (3'9) is constant (note that Npe3t is proportional to the reciprocal o f the slope o f ln (/c„J). In the range where it is constant, Npe3t provides an appropriate initial value for the emission coefficient N p , because in that range VpEi — VBm ~ Vpm, yielding N p = Npest from eq. (3.2). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 N F eat o f eq. (3.9) is shown in fig. 3.5 as a function o f the measured base-emitter voltage (Vflm — VEm). From this plot, the range o f constant slope for In(/cm) is found to be approximately [1.085 V, 1.175 V], and the initial value for N p is approximately 1.044. 4 3 2 1 0 0.8 1 1.2 VB m - VB m 1.4 1. 6 00 Fig. 3.5. N p e;it (eq. (3.9)) as a function o f the measured base-emitter voltage. The three steps o f the optimization process were configured to minimize the following error functions, for the determination o f I s and N p (e/5 ), R p c (*r p c ), and Rpmox (z r b m a x Y N isi eiS=T, I W c t h ( V BEn)) - W C m ( V BEn)) I 2 (3.10) n = N is i N r pc t . CRPC = I k l(Ic th (V B E n )) ~ In (IC m {V B E n )) \ " n=N/ipci (3.11) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 N rb MAXI £R B M A X = | ln ( I c th iY B E n ) ) ~ ^ ( / c m ( V b E ti)’) | (3.12) n= N RBM AXl where VsEn >s the n-th value o f the measured base-emitter voltage, and Icth is the theoretical variation o f the collector current with the applied base-emitter voltage around the range o f constant slope for ln(/Cm) (Y be = v Bm - VEm): IctH(VBE) = l £ + /s (e x p (^ - ^ J^ ) - l ) (3.13) In the above equation it has been assumed that the debiasing effect is due mainly to the base spreading resistance R b b > which is constant and equal to Rsmax at sufficiently low values o f the base current (see chapter 2 ). The initial values for RBmax and R p c were 0 and 1 Tfi, respectively, and their choice is not critical to the optimization process. The final values o f I s , N f , R pc and RBmax— along with the corresponding voltage ranges, number o f points, and final optimization errors, after repeatedly performing the optimization steps described above— are given in table 3.3. The final RMS errors are very small, which indicates the correct convergence o f the optimization process. The IC-CAP transform for Icth o f eqs. (3 .10)-(3.12), written in HP-BASIC, is given in table 3.4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.3. Final values o f parameters I s , N p , R p c and RBmax, after repeatedly performing the three optimization steps step 1 (minimization o f e/s) (VBm - VEm) range number o f points Is Np MAXIMUM error RMS error step 2 (minimization o f eRp c ) (VBm - VEm) range number o f points Rpc MAXIMUM error RMS error step 3 (minimization o f er b m a x ) (VBm - VEm) range number o f points R B m ax MAXIMUM error RMS error [1.085 V, 1.175 V] 9 1.506 x 10 " 26 A 1.044 0.049% 0.031% [0.9 V, 1.085 V] 18 53.87 Gfi 1.386% 0.594% [1.175 V, 1.250 V] 8 1 1 . 0 1 KQ 0.069% 0.051% Table 3.4. HP-BASIC transform for Icth o f eqs. (3 .10)-(3.12), used in the optimization o f I s , N p , R p c and R Bmax el = vb.m - ve.m - RBmax*ib.m e2 = (vc.m - ve.m)//RPC e3 = IS*exp(el//(NF*VT)) e4 = e2 + e3 return e4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 3.1.3 Determination o f cirb and I bo In order to model the current dependence o f the base resistance, parameters aBB and I bo o f eq. (2.36) must be determined. From fig. (3.2) and eq. (2.4), the internal base-emitter voltage VBEi is obtained from the measured collector current I c m as: (3.14) where (3.15) By assuming that the debiasing effect is due mainly to the voltage drop across R b b > that is, R BlBm + RE(lBm + Ic m ) < Rbb(IBm)IBm, and having obtained VBEi from eq. (3.14), the base spreading resistance can be calculated as: R s B r , ( h m) = (3.16) where it is understood that both (VBm ~ VEm) and VBEl are obtained (or calculated, respectively) at the corresponding value o f I Bm■ The IC-CAP transform (written in HPBASIC) for the calculation o f R BBm is given in table 3.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 Table 3.5. HP-BASIC transform for R bb ™ o f eq. (3.16) el = ic.m - (ve.m - ve.m)//RPC e2 = NF*VT*log(el//IS) e3 = (vb.m - ve.m - e2) * (ib.m/'-l) return e3 In order to obtain q.rB and Ibo, the dependence given by eq. (2.36) has been fitted to the values o f Rbbth given by eq. (3.16) above. The IC-CAP optimizer has been used to automatically adjust q r b and Ibo, for Ism in the range [0 . 1 /zA, 8.43 n A], The optimization process was configured to minimize the error function £r b b given by: N rbbi tR B B = H | ^ B B t h (I B m n ) I B m n ~ R b B tti{ I B m n ) I B m n | 2 (3.17) n=NRBB\ where I Bmn is the n-th value o f I s m and RsBth is given by eq. (2.36). For the HBT under test, for N r b b -2 ~ values obtained for N rbb\ + 1 cirb and = I[bo 20 points in the previously determined range for / g m, the after optimization and the final errors are given in table 3.6. RBBm and RsBth after optimization, as functions o f Jg m, are shown in fig. 3.6, which reveals some disagreement between RsBth and RsBm at low and high values o f / g m. The effect o f this disagreement at low values o f / g m is minimal, since the voltage drop across R BB is relatively small in that range; the disagreement at high values o f Ib„i >s due to transistor self heating. All the parameters of the equivalent circuit o f fig. 3.2—except the series parasitic resistances R b \, R b 2 , R e , R c and the base parallel resistance R BB— have been extracted in the previous sections. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 Table 3.6. Parameters aRB and I bo after optimization O-RB 0 .8 6 8 6 I bo 1.76 fiA 6.663% 4.295% MAXIMUM error RMS error 16 12 10 (HE) 8 if 0 10 - 1 0 10 - 8 10 - 6 -T sm 10 —4 (A) Fig. 3.6. R BBm (i) and RBBth (ii) after optimization, as functions o f measured base current I Bm\ curve (iii) is the estimated temperature increase A T /g o f eq. (3.6). The correctness o f the extracted parameters can be verified by comparing the measured base and collector currents to the corresponding calculated quantities. Thus, from fig. 3.2, if the internal base-to-emitter voltage VBBi is written as (see eq. (3.16)): (3.18) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 where R B B (lB m ) = , f e f o s 1+l iao ) (3-19) then the calculated base and collector currents can be written respectively as: + / Si r ( e z p ( j ^ : ) - l ) *cf - (3.20) + h ( e x p ( $ % ) - X) (3'21) The only model parameter that has yet to be extracted is R p B- The IC-CAP optimizer can be used to extract a value for this parameter by minimizing the following error function for ( y Bm - VEm) in the range [0.9 V, 1.1 V]: N rpb-i CUPS = i | In(lB f(V B E n ) ) ~ ^ (iB m iy B E n )) I n=Nnpg\ 2 (3 22) where Vbeu is the n-th measured base-emitter voltage. The results given by the optimizer, for N rp b 2 — N rpbx + 1 = 20 points in the previous range for (Vgm — VEm), are shown in table 3.7. Fig. 3.7 shows the measured base and collector currents, I Bm and I c m, and the corresponding calculated currents o f the model o f fig. 3.2, I Bf and /<?/, respectively, as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 functions o f the measured base-emitter voltage (Vfim — VEm)- The agreement between the curves is very good for current variations o f several decades. The IC-CAP transforms for I b / and I c / are given in tables 3.8 and 3.9. Table 3.7. Parameter R p g after optimization R pb MAXIMUM error RMS error 62.51 Gft 0.250% 0.103% Table 3.8. HP-BASIC transform for I b / o f eq. (3.20) NE = NF//aBF ISE = bBF* (IS^aBF) erbl = (1 + ((ib.m//IB0)Aa R B ) )*(-1) rbase = RBmax*erbl vbel = vb.m - ve.m - rbase*ib.m el = (IS//BF)* (exp(vbel//(NF*VT)) - 1) e2 = ISE*(exp(vbel//(NE*VT)) - 1) e3 = (vb.m - ve.m)//RPB return el + e2 + e3 Table 3.9. HP-BASIC transform for I q / o f eq. (3.21) e r b l = (1 + ( ( i b . m / / I B 0 ) Aa R B ) ) * ( - 1 ) rbase = RBmax*erbl vbel = vb.m - ve.m - rbase*ib.m el = IS*(exp(vbel//(NF*VT)) - 1) e2 = (vb.m - ve.m)//RPC return el + e2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 m -6 if 1 0 “ 7 1 n~8 -1 0 -1 1 0.8 1 1.2 VB m ~ Ve m 1.4 1. 6 (V) Fig. 3.7. Measured currents I c m (0 and I qvx (ii), and calculated currents I c f (iii) and I b / (iv) in the forward Gummel configuration; curve (v) is the estimated temperature increase A T /S o f eq. (3.6). 3.2 Reverse Gummel Measurement and Parameter Extraction The reverse Gummel measurement can be defined by the following bias conditions: —the base and emitter o f the transistor are at the same potential; —the base-collector voltage is positive for an npn transistor and negative for a pnp transistor. The reverse Gummel measurement setup is shown in fig. 3.8, where the dc voltage sources Vb and Ve define the base and emitter voltages, respectively; the collector voltage is defined by source Vc, typically assuming negative voltages. As in the case o f the forward Gummel measurement, an HP4142 dc modular source/monitor controlled by IC-CAP via an HP-IB bus can be used to provide all the voltage Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 sources o f fig. 3.8, and to sweep source Vc, while holding Vg and Vg at ground potential; alternatively, Vc may be held at ground potential, while sweeping the positive voltages Vg and Vg. All transistor currents are available as measured currents through the voltage sources. Since Vg = Vg, the current through the base-emitter junction (including the current through the recombination diode) o f the transistor model o f fig. 2.3 is negligible, and the model reduces to the circuit shown in fig. 3.9. Resistance R pgc o f fig. 3.9 is the parasitic parallel resistance o f the base-collector junction o f the transistor; resistance R p c e is the parasitic output resistance o f the device, that appears between the collector and the emitter, equal to R p c o f fig. 3.2. However, R p c e will be determined in this configuration as an independent optimization variable, without using the value o f R pc from the forward Gummel extractions. D.U.T. < 'B vB T Vc (<0) H i VE= VB (= o )4 - Fig. 3.8. Reverse Gummel measurement setup. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 r A /W Bn V V A A /V A ^ V Y R B1 ^B2 RBB I Em w v R PCE Fig. 3.9. Equivalent circuit o f the HBT in the reverse Gummel measurement. 3.2 .1 Determination o f I s r o l -, N rol , N r and (3r This section demonstrates a procedure for the determination o f the parameters o f the recombination component of the overlap diode, I srol and N rol ;N r and /3r (characterizing the one-dimensional transistor) are extracted simultaneously. The procedure is optimizerbased, and assumes that the debiasing effects due to the parasitic series resistances, and the local increase in temperature have a negligible effect on the device characteristics. Fig. 3.10 shows the reverse Gummel characteristics (base and emitter currents as functions o f the applied base-collector voltage) for the device whose forward Gummel characteristics are shown in fig. 3.3. The local temperature increase is also shown in fig. 3.10, and can be estimated as: A Trg = Rth (Vfim —Vcm)(lBm + I Em) (3.23) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 where Rth is the thermal resistance o f the device (assumed here to have the approximate value o f 2000 °C/W). For the device under test, it will be assumed that the surface effects dominate over the other effects accounting for the recombination components (i.e., Is ro l ^ I sc), and Isc will be assumed to be zero. This assumption is important, and its validity will be verified by comparing the measured and simulated characteristics o f the device. Another important consideration with regard to the extraction procedure is that the dependence with current o f the base resistance R bb is known and has been determined using the forward Gummel characteristics o f the device. This considerably simplifies the extraction o f the remaining parameters. From the circuit o f fig. 3.9 (with I s c = 0 from the previous discussion), the calculated base current I bt can be written as: ^ = ^ + “ + /50 i ( e x p ( ^ ) - l ) Isro l In eq. (3.24), I so l ( e x p ( ^ ) - l) + £ ( e x p ^ ) is given by eq. (2 . 1 0 ), and V bcol and V sa - l) (3.24) are the voltages across the overlap diode and across the internal base-collector junction o f the transistor, respectively: V b c o l = Vsm — V c m — R s lB m (3.25) V b C i — V btti — V c m — R b b { I b 2 ) I b 2 ~ R s ^ B m (3.26) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 § -6 10 Q —8 m 1 0 “ 1 4 E__ 0.8 0.6 0.4 0.2 1 00 VB m ~ V Cm Fig. 3.10. Measured base current Ism (i) and emitter current I Em (ii) o f the HBT in the reverse Gummel configuration, and the estimated local temperature increase A Trg o f eq. (3.23) (iii). In the above equations, Rs^Bm accounts for the voltage drop across the ohmic resistances resistance R b\ R s and R c (assuming that R b2 R b b ) ', since, in general, Is m I Em, the can be written as: (3.27) R s — R b i ■+■ R c Current Ib 2 o f eq. (3.26) denotes the internal base current o f the transistor (see fig. 3.9), and from R b b (Ib 2)Ib 2 V bcol is the voltage drop across in order to obtain V bcu R b b ', this voltage drop has to be subtracted as illustrated by eq. (3.26). Resistance R bb o f eq. (3.26) is given by eq. (2.36), where all the parameters have been determined from the forward Gummel characteristics (see tables 3.3 and 3.6). Similarly, the calculated emitter current lEr can be written as: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 (3.28) where V bcx >s defined by eq. (3.26). All the unknown parameters of eqs. (3.24) and (3.28) can be determined simultaneously fitting the calculated currents Ib t and Iet by to the measured base and emitter currents, respectively, I r™. and I Em- The procedure consists of repeatedly performing two distinct optimizations with the above parameters as variables. In the first step of the optimization, resistances R pce and R pbc o f the above equations are found as optimization variables at low values of (Vgm - V cm), where their contribution to the measured base and emitter currents is significant; in the second step, parameters I s r o l, U r o l, P r, H r, and R s are found as optimization variables in the adjacent range for (Vgm — V s m ) corresponding to higher values, in the region where the self-heating effects can still be neglected. The above steps are repeated until the convergence of the variables is achieved, and no further change in their values is observed. The two steps of the optimization process were configured to minimize the following error functions, for the determination of R pce and R pbc (£ r p c e X and I sro l , H RO l , H and P r (e is R O L ): N fip c E l tR P C E = £ | ^ ( I b A V b C ti) ) — ln(-Tem(VsCn)) | 2 U = N r p c EI ^RPCE2 + £ I ^ { lE r{V B C n )) ~ ln(/E;m (V sCn)) | 2 (3-29) n = N fip c E \ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. r , 55 N lSR O L2 ZISROL = 5 2 I M .lB r(V B C n )) ~ n=NisRon V sC n )) | N is r o u i + £ I to (hr(V B C n)) - W E m iV B C n )) I 2 (3.30) n=N[SROLi where Vgcn is the n-th value of the measured base-collector voltage (V bc = VBm — V cm). The final values o f the parameters—along with the corresponding voltage ranges, number o f points, and final optimization errors, after repeatedly performing the optimization steps described above— are given in table 3.10. The final RMS error in the case o f the minimization o f cisrol is very small, which indicates the correct convergence o f the optimization process (the error in the minimization o f crpce is larger due to the measurement noise). Fig. 3.11 shows I Em, iBm, I ev, and I bt as functions o f the applied base-collector voltage. The calculated and measured curves virtually overlap at the values o f the voltage at which the local temperature increase is negligible. The final value for R s given by the optimizer is close to zero, which indicates the fact that the influence o f the series ohmic resistances R b and R c is negligible in the voltage range under consideration. The IC-CAP transforms (written in HP-BASIC) for I bt and Igr are given in tables 3.11 and 3.12. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.10. Final values o f I s r o l , U r o l , H r , P r , R s , R p c e and R p b c , after repeatedly performing the two optimization steps step 1 (minimization o f crpce ) (VBm ~ VEm) range number o f points R pce R p bc MAXIMUM error RMS error step 2 (minimization o f e i s r o (Vsm - VEm) range number o f points I sro l N rol N r Pr Rs MAXIMUM error RMS error [0.45 V, 1.00 V] 55 27.93 Gfi 311.6 M fi 20.51% 3.018% l) [1.00 V, 1.23 V] 23 21.86 fA 1.977 1.027 1.451 0.023 f t 0.708% 0.189% Table 3.11. HP-BASIC transform for I Er o f eq. (3.24) iei = ie.m - (ve.m - vc.m)//RPCE ib2 = iei//BR erbl = (1 + ((ib2//IB0)~aRB))A (-1) rbase = RBmax*erbl vbcol = v b . m - vc.m - rs*ib.m vbci = v b . m - vc.m - rbase*ib2 - rs*ib.m ISOL = Ka*IS*(BR + 1)//BR el = vbcol//RPBC e2 = ISOL*(exp(vbcol//(NR*VT)) - 1) e3 = ISROL*(exp(vbcol//(NROL*VT)) - 1) e4 = (IS//BR)* (exp(vbci//(NR*VT)) - 1) return el + e2 + e3 + e4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 Table 3.12. HP-BASIC transform for I e t o f eq. (3.28) iei = ie.m - (ve.m - vc.m)//RPCE ib2 = iei//BR erbl = (1 + ((ib2//IB0)"aRB))~ (-1) rbase = RBmax*erbl vbci = vb.m - vc.m - rbase*ib2 - rs*ib.m el = (ve.m - vc.m)//RPCE e2 = IS* (exp (vbci// (NR*VT) ) - 1) return el + e2 -2 15 (y) -4 * 3 j <*Bj* m 3j * -6 U < 1 0 -1 0 0.2 0.4 0.6 0.8 VB m ~ V Cm 00 Fig. 3.11. Measured currents Ism (i) and I Em (ii), and calculated currents I bt (iii) and I et (iv) in the reverse Gummel configuration; curve (v) is the estimated temperature increase A Trg o f eq. (3.23). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4 TEMPERATURE DEPENDENT MODEL O F THE HBT Based on the static model o f fig. 2.3, a temperature dependent model for the HBT is developed in this chapter. For the accurate temperature characterization o f the model parameters, the physics-based, mathematical expressions o f these parameters are used. This approach is justified by the accurate expressions for the one-dimensional HBT model parameters now available [2 1 ], [26], and because empirical models without physical meaning generally produce larger errors than the physics-based models (see [27] for a comparison of MOSFET analog models). The new temperature dependent model developed in this work takes into account the explicit temperature dependencies o f the energy gaps o f the emitter, base, and collector regions o f the transistor. The temperature dependencies o f the saturation currents that appear in the equations o f the HBT are derived starting from the boundary conditions for an arbitrary bias applied to the transistor, and by taking into account the recombination/generation currents o f the space-charge regions. Using the boundary conditions and the Ebers-Moll model o f the HBT, the forward and reverse beta o f the transistor as functions o f temperature are determined. 4 .1 Boundary Conditions and Basic Equations for the HBT Assuming a one-dimensional npn transistor structure as in fig. 4.1, the charge-control equation can be written as in [26], where J $ is the electron current density in the base o f the transistor: 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 J n = qDn (pn)x^ FC(ln)-x^ (4.1) f Pdx XpE B B n N H 1----- NE W, H------> v ^ PE WB PC y A NC W. Fig. 4.1. One-dimensional npn HBT device structure. Assuming that the base-emitter junction is a heterojunction, and that the carrier flow across the interface is only by thermionic emission, the pn product at the space-charge region boundary is obtained as: p ( X p E) n ( X p E) = - ( g j ) p ( X > * ) + „2s e x p (2 £ f? ) (4.2) In eq. (4.2), F e n is the thermionic electron flux at the emitter-base junction, and S e n is the effective interface carrier velocity, defined as: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 S e n where — ^exp [(qVjp —A E c) / k T \ , the component of the electron thermal (4.3) velocity in the x direction is vx = y /'fcT/(27rm*), the sum o f the electrostatic potential difference across the p type and n type parts o f the space-charge region is V j = V j^ + V jp , and A E c is the difference between the electron afBnities o f the base and the emitter, respectively. With similar definitions, the same product evaluated at the boundary o f the collector space-charge region is: p { X p C ) n ( X p c ) = - ( jj“ ) p { X p c ) + n&exp ( (4.4) Using eqs. (4.1), (4.2) and (4.4) and integrating over the cross-sectional area A o f a one-dimensional structure, the expression for the collector-emitter linking current in the lowinjection case, with the base more heavily doped than the emitter and the collector, can be written (following [26]) as: _ <lA n h N ab [exp(gKgg/feT ) -exp(qVBC/kT)} wb/D b +1/S en + I/S cx ( , where w b is the thickness of the quasi-neutral base, assumed constant. (It should be noted that a typical heterojunction bipolar transistor has a heavily doped base.) In an abrupt-junction HBT, S e n will be small and 1 / S e n will dominate over w b /D b in the denominator o f the above expression for the collector current. However, in most HBT's, the composition o f the emitter (the aluminum concentration) is varied over some distance from the metallurgical junction, in order for the bandgap o f the emitter to match the bandgap o f the GaAs base region; in these cases S e n will be large and w b /D b will be the dominant term. Consequently, in the graded emitter, graded collector (or Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. single heterojunction) device, w b / D b 3> 1 / S e n + l/^ c r v and eq. (4.5) reverts to the form for the homojunction device. For the single heterojunction device l/Scw ~*0. According to the Ebers-Moll model o f the HBT derived in [21], the forward and reverse injection efficiencies are, respectively: IF ~ \ ' D b nQB vjb 'Yp — / 1 - 1 1 R — T- D b ^ M 1 + 1 /S'EP HS L i1-/ w c i + i /S 'c p / ?CK \ J C4 7-v < ,« ■ ') where: S e n = S e n /{ D b / ^ b ) (4 -8) S'c n = S c n / ( D b / w b ) (4.9) S'e p = S e p / ( D e / w e ) (4 1 0 ) S 'c p = S c p / ( D c / w c ) (4.11) In the above equations S e p and S e p are defined similarly to S e n and S c n in eqs. (4.4) and (4.5), w e , w b , wc are the widths o f the quasi-neutral emitter, base, and collector regions, respectively, D e (b ,C) is the minority diffusion coefficient in the emitter (base, collector), and po e , n ob , and poc are the equilibrium minority carrier concentrations in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 emitter, base, and collector, respectively. As above, S 'CN, S'EPl and S'c p have large values and can be neglected in eqs. (4.6) and (4.7). From eqs. (4.6) and (4.7) we obtain the expressions for the forward and reverse beta o f the HBT— (3p and /3r , respectively, o f fig. 2.3— by recognizing that: I f or Pf = 1 - 7 Far 0R = 1 -iRCtr (4.12) (4.13) where a p is the base transport factor o f the HBT [28] (w E L r , L r = diffusion length o f electrons in the base): cosh[wB/Lb] l+xiig/(2L^b) (4.14) From the above equations it then follows that these parameters are identified as (4.15) (4.16) where the contributions o f 1 /S'CN and 1 /S'CP have been neglected, assuming a homojunction at the collector; 1 /S 'EP is neglected since in general there is no notch in the valence band. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 At low biases, the surface and space-charge recombination o f the carriers is the source for additional junction currents. It is important to include these additional currents in a realistic model o f the HBT, such as the one in fig. 2.3. With the notations o f chapter 2 and fig. 2.3, the following expressions can be written for the recombination currents: ib e = o t i r + is o E ) (exp ( s ? ) ~ x) (4-i7 > /Sc = ^ ( e x p ( j 0 f ) - l ) Irol = + '* * ) ( “ * ( * £ & ) - (4.18) 0 <4 1 9 > In the above equations, x'E and x'c represent portions o f the emitter and collector space-charge regions across which the recombination-generation rate is maximum, and toe and Toe are the lifetimes associated with the recombination o f excess carriers in the emitter and collector (assuming that the doping concentration in the base is much higher than the doping concentration in the emitter or collector). I so e and I s o c o f eqs. (4.17) and (4.19) are the saturation currents o f the surface recombination currents o f the emitter and collector junctions, respectively [29], Since terms like tqe or roc are not known with high precision (following the approach o f [30]), x'E and x'c can be approximated by the widths o f the spacecharge regions o f the emitter and collector junctions, respectively. By identifying eqs. (4.17), (4.18) and (4.19) with eqs. (2.2), (2.3) and (2.1), respectively, the following expressions can be written for the saturation currents Ise , Isc Isrol- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and 64 J « (4.20) = * & * + /* » (4.21) Isrol = (4.22) h o c N ote that I s c is proportional to A g , not A c , due to the fact that it corresponds to the one-dimensional structure o f area A g . From [31], the saturation currents I soe and I s o e can be written as: Use (4.23) —X p c)L scU sc (4.24) I so e — q(X p E - Isoe = q( X nc X n e )L se where L s e and L s c are the perimeters o f the emitter and collector, respectively, and U s e and U se are the steady-state recombination rates per unit surface area. These recombination rates are complicated functions o f the surface-recombination velocities, the electron and hole quasi-Fermi levels, junction bias voltage, and the energy level o f the recombination centers measured from the intrinsic Fermi level. An acceptable approximation is to assume the recombination rates constant regardless o f bias [31 ], in which case, taking into account the fact that the depletion regions extend mainly into the emitter and collector because o f the high base doping, Us e and U se are proportional to the corresponding intrinsic carrier concentrations [31]: Use °c n iE (4.25) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 (4.26) U s e ex n iC A model for the first term in eq. (4.20) as applied to the graded heterojunction is reported in [32], This result for the recombination at the emitter-base junction with a grading distance greater than 30 nm is: •free OC n i g A (4.27) s /T where r = y/Tporn0, and rpo and rn0 are the hole and electron lifetimes in the emitter. Since the currents o f eqs. (4.17), (4.18) and (4.19) are recombination currents, the emission coefficients N e , N c and N rol are typically equal to 2, but in general may assume values between 1 and 2 [12], [29], [30], [31], 4.2 Temperature Dependence of I s Comparing eqs. (4.5), (2.4), (2.5) and (2.7), we identify I s with the expression that multiplies the difference o f exponentials in eq. (4.5). The saturation current I s o f the linking current (assuming graded bandgaps) is identified as: In order to obtain the temperature dependence of I s , the temperature dependence o f all the parameters that appear in the above equation is required. From the Einstein relation, the diffusion coefficient D can be written as: (4.29) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 where n is the mobility. In [33], the electron drift and Hall mobilities in GaAs at temperatures close to 300 K are modeled by the following empirical expressions, respectively, in the case o f relatively low doping concentrations: ltn d ( T ) = a i ( 3 0 0 / T ) 6 (4.30) VmH{T) = a 2( Z 0 0 / T ) b (4.31) where the mobilities are expressed in cn r/V s, T is in K, a\ = 8000,ao = 9400, and b = 2.3. However, from experimental data taken from highly-doped GaAs samples [34], a t , ao, and b in eqs. (4.30) and (4.31) are functions o f doping, and 6 can be as low as 1.2, for doping concentrations on the order o f 1016 cm- 3 . Using the above equations, the overall low-field electron mobility is estimated to be [35]: MT) = { w r r 1 + (Mn/ftr)]-1} ' 1 (4.32) Also from [33], the total low-field mobility o f holes as a function o f temperature and concentration o f impurities, at temperatures close to 300 K, has the following empirical expression ( jj,p fjLp( T ) is expressed in cm2 /Vs): = [0.0025(r/300)2-3 + 4 x ^ ^ ^ /(S O O /r )1-5]"1 (4.33) where N r is the concentration o f the ionized impurities, expressed in cm-3, and T is in K. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 From the above equations we infer the approximate temperature dependence o f the diffusion coefficients o f minority carriers in the emitter, base and collector. Assuming that the mobility in the AlGaAs emitter is the same as in GaAs, we obtain: D E(T) = ^ E ( T ) f (4.34) D B (T )=^B (T )f (4.35) D c (T )= M p c (T ) f (4.36) where the subscripts E, B and C indicate the emitter, base and collector regions, respectively, and the mobilities as functions o f temperature are given by eqs. (4.32) and (4.33). Since the temperature dependence o f n ^ c ) and E gB(C) can be expressed as in [36]: n u x o ( T ) OC T3/2e x p ( - E gB fC )(T ) = BgOB(C) - f f j <4-37> (4.38) where EgOB(C) = 1.519 eV, a = 5.405 x 10 - 4 eV/K, /? = 204 K, the intrinsic carrier concentration as a function o f temperature can be written as: n w (o ( T ) = niB(c,(r 0) ( | ) 3/2exp( - ( ^ § 2 2 - ^ P ) ) (4.39) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 where rc,B(C)Cfo) is the value o f the intrinsic concentration at the reference temperature 7o, and E 9b {c ) (T ) is given above; the subscripts B and C in the above equations refer to the base and collector regions, respectively (GaAs material). If the temperature effect on A s , N ab and w b is neglected, then from eqs.(4.28) and (4.39), I s as a function o f temperature is obtained as: I s ( T ) = / s ( T o ) a g J ( £ ) ’e«p( - ( 5 # - S fg a )) (4.40) For modeling purposes, taking into account the temperature dependence o f fj,nB in eq. (4.35) (see eqs. (4.30), (4.31) and (4.32)), the above expression for I s as a function of temperature can be re-written as (a similar expression has been used in [17]): I s i T ) = / S(T o )(§ );C"exp( - _ S g a )) (4.41) X t i o f eq. (4.41) (temperature exponent o f I s , similar to the one used in SPICE [11]) is a parameter that can easily be incorporated in the proposed model o f the HBT, and takes on values on the order o f 3, if b o f eqs. (4.30) and (4.31) is assumed to be approximately equal to 1, due to the high doping o f the base. In practical cases where b ^ 1, different values will be obtained for X t i ■ Eq. (4.41) corrects the form given in [11], where in the standard SPICE BJT model E 3b (T ) is replaced with E 9b {To)\ in this way, the variation o f the base bandgap with temperature is overlooked in [11] (see [17] and [37] for correct expressions). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 4.3 Temperature Dependence o f f3p and (3R From eqs. (4.15) and (4.16), assuming a graded emitter bandgap and a high emitter efficiency, the same considerations apply with respect to the same value o f 1 / S e n compared to w b /D b , and (3p and (3R are approximately: / ^ ( ( f t s ? * - ( ( § s ^ r + ^ r r + & 1 <4 -42> <4-«> y Taking into account the fact that the minority carrier concentrations in the emitter and base are Po e ^ ^ e / ^ de and n QB = njB/ N AB, respectively, and that nqEe x p ( A E g/ k T ) ~ n~iB (by neglecting the changes in the density o f states in the emitter and base [26]), where A E g = E gE - E gs , from eq. (4.42) l//3 p is obtained as: JrfX ) = f bQX p ( j + k In the above equation, f, — He Eab. m Jb — Da N de w e (4 45\ ^ ^ and k (4-46) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 Eq. (4.44) is similar to the temperature dependent expression for the forward beta o f a heterojunction bipolar transistor given in [38], with the exception o f A E g which in our case replaces A E v, the valence bandgap difference o f the base-emitter heterojunction. The difference between the two expressions is explained by the fact that in [38] the heterojunction was abrupt, whereas in this case the emitter o f the device is assumed to have a graded bandgap. Due to the high base doping concentration, P r o f eq. (4.43) is determined by the relatively low value o f the injection efficiency j r compared to the base transport factor Q r, and eq. (4.43) can be re-written as: fl • = Dr no£W£ T)r* P rk\r> d Dc OC tJI U)B Using tiq r = n~iB / N ar and poc = ti^ / N rc (where (4.47) t i zr = n ,c —the base and the collector are o f the same material), P r as a function o f temperature is obtained as: (4.48) Note that P r is a weak function o f temperature owing to the base-collector homojunction and the similar temperature dependence o f diffusion coefficients D r and D e l i the conduction band edges o f the emitter and the base are not matched at the metallurgical junction, then 1 / S e n will dominate over w r / D r in eqs. (4.15) and (4.16), and the expressions for 1/P f and P r become, respectively: 1 0f _ D e N ab WBVZ N q e (4.49) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 (4.50) In eq. (4.49) A E v is the previously-mentioned valence bandgap difference o f the baseemitter heterojunction; A E c o f eq. (4.50) is the difference between the electron affinities o f the base and the emitter. In the above equations, V jp o f eq. (4.3) has been neglected (V j p ~ 0), since from [39] we have: VJP = K 1( V t e - V BE) (4.51) where V^e is the built-in voltage o f the heterojunction, Vgg is the bias voltage applied to the heterojunction, and *eN De ^b N ab^ eN de (4.52) due to the high doping o f the base compared to the doping o f the emitter; in the above relation, e# and eg are the permittivities o f the emitter and base regions, respectively. In this work we assume a graded heterojunction at the emitter and a homojunction at the collector, so the following expressions are used to model the temperature dependence o f 13p and (3r (see eqs. (4.44) and (4.48)): (4.53) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 (4.54) In eq. (4.53) /3b and /& are temperature independent parameters; eq. (4.54) is similar to the function used in SPICE [II], [17] for the temperature modeling o f /3r . Due to the weak temperature variation o f /3r in the graded heterojunction case, X tb r is expected to take on values close to zero. The exact expression for A E g as a function o f temperature (for use in eq. (4.53)) can be found using eq. (4.38) for E 3b ( T ) , and one o f the following expressions for E 3e ( T ) , depending on the value x o f the aluminum concentration in the AlGaAs emitter material (following the information given in [40]): E 9e {T) = (1 —x ) E9b ( T ) + x E VMAs {T) + 0.37a;(l —x), 0 < x < 0.4242 (4.55) E 9e {T) = (1 —x) ExGaAs { T) + x E X aias (T) + 0.245rr(l —x), 0.4242 < x < 1 (4.56) where 5.41xl0~4r 2 r+ 204 (4.57) E xG a A s(T ^T2 ) = 1.982 — 4.6x10 T+204 (4.58) E x a ia s { T ) x io ^ r 2 = 2.239 - e .oT+408 (4.59) E ta ia s { T ) = 3.114 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 4.4 Temperature Dependence of I s e , I s c and I s r o l The temperature dependence o f I s e , I s c , and I s r o l can be inferred from eqs. (4.20)(4.27), (4.37), and from the temperature dependence o f the intrinsic carrier concentration in the emitter: T3/2exp( - n iE{ T ) <x (4.60) where E 9e ( T ) is given by eq. (4.55) or (4.56). Using eq. (4.60), the recombination saturation currents as functions of temperature can be written as (similar expressions have been used in [17]): Ise(t ) = _ ^ g a )) (4.6I) i s c ( T ) = / s c ( r , ) ( s ) J5' « p ( - (& g p - % ® ) ) ( 4 , 2, W ( r ) /,* ( r 0) ( $ ) x" « p ( - = W ( r . ) ( S ) W exp( - ( S g p - i g g l) ) (4.63) where X t e = X t c x 'e (C)> = X t r o l = 3/2, if the temperature dependence of Ae(C), I>se(C), r oE ( C) , and o f the widths of the depletion regions is negligible. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 4.5 Temperature Dependent Model for the Base Spreading Resistance If the self-heating effects are negligible, in view o f the results derived in section 2.4 (eq. (2.46)), the temperature dependence o f -Rsmax can be modeled as: . / <r R B m a x (T aTnb) = -RsmaxC^o) \ X trbi (4.64) where To is the reference temperature and Xj'RBmax accounts for the temperature variation o f the hole mobility in the /?+-base o f an npn transistor, or the electron mobility in the /i+-base o f a pnp transistor. From eqs. (2.47) and (4.64), the temperature dependence o f I bo can be modeled as: (4.65) lB o{T am b) — -Tbo(?o) Thus, in the region where the self-heating effects are negligible, exponent X trbttuh describes the temperature variation o f both Remax and I b q ', temperature Tam& in eqs. (4.64) and (4.65) is the ambient (substrate) temperature. If the self-heating effects are not negligible, then, from the thermal circuit o f the HBT (to be described in the next section), the emitter and collector junctions will have different temperatures - T j E and Tjc, respectively— and because o f the physical placement o f the base between these two junctions, a thermal gradient will exist in this region. As a result, the derivation o f the base resistance model described in chapter 2 will no longer apply, because the resistivity varies across the base region according to the temperature distribution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 To account for the self-heating effects and the thermal gradient in the base, an empirical term will be used in this work in conjunction with the base spreading resistance model o f eq. (2.36), and the temperature dependencies for RBm ax and /so , expressed by eqs. (4.64) and (4.65). This term assumes a mean temperature (equal to (Tje + T j c )/2) for the base region under self-heating conditions, and a power-law dependence for the base resistance as a function o f this temperature. Thus, the proposed temperature dependent model for the base spreading resistance is: D R B m a x jT g m b ) f / ; £ + 2jC ^ T nr \ ___ j J-am b 5J - jE i J -jC ) — ^ | / iR y i w ^ 2Tamb ) \ amb)) (T T TRBB (4.66) where RBmax(Tamb) and /so (T Qm6) are given by eqs. (4.64) and (4.65), respectively, T je and T jc are the temperatures o f the emitter and collector junctions, respectively, and X t r b b is an empirical parameter. If self-heating is negligible, then T je = Tjc = Tamb and the model o f eq. (4.66) reverts to the model described by eq. (2.36); the term in T je and T jc affects the base spreading resistance only in cases where a temperature gradient exists in the base due to self heating. The extraction o f parameters Xj'RBmax and X t r b b will be described in chapter 5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 4.6 Thermal Circuit o f the HBT The previous temperature dependent expressions derived for the saturation currents, the forward and reverse beta's o f the device, and the base spreading resistance, are used in the complete temperature dependent model o f the HBT. The simplest thermal circuit o f the transistor can be obtained by assuming that the structure has isothermal properties; the circuit is shown in fig. 4.2 [22], [41]. R t h and C t h o f fig. 4.2 are the thermal resistance and the thermal capacitance, respectively, o f the transistor (both assumed constant), and Tamb and T are the ambient temperature and the device temperature, respectively. -F I VC8 — -F VBE — amb v- ambient --------- 11 C il E V BE E transistor > Fig. 4.2. Simple thermal circuit o f an HBT. Although the circuit of fig. 4.2 is attractive due to its simplicity, it should be used with caution because, in general, the transistor cannot be approximated by an isothermal structure. Moreover, the thermal conductivity o f the transistor material may be a function o f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 temperature, in which case the thermal resistance o f the device is a function o f its geometry, the dissipated power, and the ambient temperature [42], It should be noted that the thermal conductivity o f GaAs (which is the substrate material for the HBT's modeled in this work) is indeed a function o f temperature and, depending on the doping concentration, may drop by as much as 85% (relative to the value at 300 K) for a temperature increase o f 600 K [43], In view o f the above considerations, and taking into account the physical structure o f the device (shown in fig. 2 .2 ) and the relatively low value o f the thermal conductivity o f GaAs, a more accurate thermal circuit o f the planar HBT is obtained as shown in fig. 4.3. The circuit is similar to the thermal circuit proposed in [44], where the power dissipated in the emitter junction is accounted for, and different temperatures are calculatedfor the emitter and collector junctions. In addition to the circuit givenin [44], the proposed circuit o f fig. 4.3 also includes the thermal circuit o f the overlap diode, according to the physical structure o f the transistor. In the circuit o f fig. 4.3, T je , Tjc , and T ol are the temperatures o f the emitter junction, collector junction, and overlap diode, respectively; the dissipated powers in each o f these junctions are calculated separately, according to the model o f fig. 2.3: PdE — VBE (j'BE + ijf o + IC t ) (4.67) P dc — v b c ( j’BC + (4.68) ~ * c t) PdOL — v B ci}R O L + i o i ) (4.69) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 o ~r PdC R th c o l " C thcol __ C-rHjE > J _ C THjC R THOL R THjC S Tam b Fig. 4.3. More accurate thermal circuit o f the planar HBT, which includes the temperature response o f the emitter junction and the overlap diode. The thermal impedances o f the circuit o f fig. 4.3 are modeled using single-pole RC subcircuits connected between the nodes; for better accuracy, RC ladder subcircuits may be used [44], [45], with the drawback o f increased complexity. Resistances R t h j e , R thj C, and R thol are the junction-to-ambient thermal resistances o f the emitter junction, collector junction, and the overlap diode, respectively; resistances R t h e c and R t h c ol are the thermal resistances between the emitter and the collector, and between the collector and the overlap diode, respectively. Each thermal resistance is connected in parallel with its corresponding thermal capacitance. The calculation o f temperatures TjE and Tjc from fig. 4.3 can be simplified by assuming that zthol < zthcol due to the relatively large area o f the overlap diode, and the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 relatively poor thermal contact between this diode and the collector junction ( z t h o l and zthcol are the thermal impedances R th o l \\C t h o l , and R th co l \\C t h c o l , respectively). With this assumption, the equivalent thermal circuit for the calculation o f temperatures TjE and T jc is obtained from fig. 4.3, and is shown in fig. 4.4. In general, the dissipated powers pdE and pdc o f eqs. (4.67) and (4.68) are functions o f time; denoting by PdE^s) and Pdc(s) the Laplace transforms o f PdEit) and Pdc(t), respectively (PdE(s) = C(pdE(t)), Pdc(s) = £(P dc(t))), from fig. 4.4 TjE and TjC are obtained as: TjE = Tamb + C - 1(A T jE( s )) (4.70) T j c = Tamb + C - l ( A T j C ( s ) ) (4.71) where C ~l is the inverse Laplace transform operator, Tj e (0) = T jc(0 ) = TaTnb, and & T j e { s ) = K £ Z T H jE P d E (s ) + (1 — K e ) z t h C i P(1c { s ) (4.72) & T j c { s ) = (1 “ K c ) z T H jE P d E ( s ) + K c Z T H C \P d c { s ) (4.73) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ? TiC TiE ? C T 1 R thcol C thcol < -) T a r n b Fig. 4.4. Simplified circuit for the calculation o f temperatures T je and TjC, assuming that z T hol < - thcol In the above equations, T H jE Rtw e l+ s C r H jE R -T H jE T H jC 1+ sCthjcR thjC R „ t (4.74) H jC (4.75) R thcol_ THCOL ZTHEC= (4.76) 1+ sCthcolR thcol (4.77) 1+ sCtI ecR thec zTHCl = zTHjc\\zTHCOL = ZTHjCZTHCOL ZTHjC+ZTHCOL (4.78) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 R (4 79 ) ZTHEC+VMCl = b ZT H jE + Z T H E C + Z T H C l ^E + Z T H E C = ^ . (4 g0) ZT H jE + Z T H E C + Z T H C l I f the following notations are made: ZTHE = K Zt H j E (4.81) ZTHC = K c Z T H C l (4.82) K tbC (4.84) e then the Laplace transforms A Tj e (s ) and A Tjc(s) o f eqs. (4.72) and (4.73) can be re-written as: ^ T j-e ( s ) = Zt h e [P(1e ( s ) + K t h c P<1c { s )] (4.85) A T j c ( s ) = Z T H c [ P d c ( s ) + K t h e P < 1e { s ) ] ( 4 .86 ) In the above equations, z t h e and ? th c (defined by eqs. (4.81) and (4.82)) are the effective thermal impedances o f the emitter and collector junctions, respectively; coefficients K t h e and K th c (defined by eqs. (4.83) and (4.84)) are the effective thermal coupling coefficients from the emitter junction to the collector junction, and from the collector junction to the emitter junction, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 It is important to note that knowing K th e is useful only in cases where the power dissipated in the emitter junction is comparable to the pow er dissipated in the collector junction. In the reverse active region the emitter current and the power dissipated in the em itter junction are both relatively small due to the small values o f /3r , and eq. (4.86) can be approximated by: A T j c ( s ) ~ z THc P d c ( s ) (4.87) If the transistor is in the forward active region, the power dissipated in the emitter junction is no longer negligible and, although the effect o f T jc on the linking and saturation currents o f the one-dimensional transistor is inconsequential, knowing K th e is important because T jc explicitly appears in the temperature dependent model for the base spreading resistance (eq. (4.66)). However, because the temperature dependent model for the base spreading resistance given by eq. (4.66) is only an empirical approximation, eq. (4.87) will be adopted in this work for obtaining Tjc regardless o f the bias conditions o f the transistor. Using the assumption that z th o l C - t h c o l , from fig. 4.3 the temperature o f the overlap diode can be calculated as: TOL — Tamb + £ w here £ 1 A T 1{ A T q l ( s )) (4.88) is the inverse Laplace transform operator, Tq l (0) = Tamb, and q l {s ) — zth o l PdO L{s) (4.89) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 In the above equation, P doi{s) is the Laplace transform o f PdOL(t) o f eq. (4.69) (PdOL (s ) = and z th o l is the thermal impedance o f the overlap diode: Z™ ° L = (4-50) Determining the thermal capacitances for the thermal circuits o f figs. 4.3 and 4.4 is beyond the scope o f this work. Two approaches for the extraction of the thermal impedances o f bipolar transistors are described in [44] and [45], If the time dependent components are small and can be neglected in the dissipated powers o f eqs. (4.67)-(4.69), or if their frequency is sufficiently small, the thermal capacitances can be removed from the thermal circuits o f figs. 4.3 and 4.4, in which case the thermal impedances o f eqs. (4.74)-(4.77) and (4.90) revert to the corresponding thermal resistances. It has been stated in the beginning o f this section, and it has been assumed throughout all previous derivations, that the thermal resistances o f the thermal equivalent circuit are functions o f geometry, dissipated power, and ambient temperature. If all these dependencies are considered lumped into one single temperature dependence, a generic thermal resistance R t h can be approximated by the following temperature function (see [2 2 ] for the temperature dependence o f the thermal resistance): R t h { T ) = R t HQ + K r & T (4.91) where R tho = RTH (Tamb) and A T = T - Tamb. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 Since A T can be calculated as the product o f the thermal resistance defined above, and a generic dissipated power pd (assuming an isolated device whose thermal resistance is R thY A T = R TH{T ) p d = { R t h o + K r A T )pd (4.92) it follows that A T can be calculated as: A T = R TH 0 T Z ^ fd (4.93) From eq. (4.93) it follows that the temperature increase A T can be calculated by assuming the reference value for the thermal resistance, and multiplying it by a modified dissipated power, p«f/(l - K r Ps) . By defining a thermal coefficient o f the thermal resistance, K s , as: and assuming that it is constant, from eq. (4.91) K r is calculated as: K r = K s R th o (4.95) Using eq. (4.95), eq. (4.93) can be re-written as: A T = R t h o i - KsPRTmPd (4-96) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 W ith the result expressed by eq. (4.96), under the assumption o f constant K s, and using eqs. (4.70), (4.71), (4.85), (4.87), (4.88) and (4.89), the junction temperatures in the case o f negligible time dependent components in the dissipated powers, or low frequency o f these components, can be calculated as: T jE = T amb + R t HEQP^E (4 -97) T j c = T amb + R rH C o tfd c (4 -98) T q l — T amb + RTHOmrfdOL (4 " ) where, according to eq. (4.96), A e = A c = (4.100) i - k M 'hcmc <4 I 0 1 > A oL = i <4 1 0 2 ) In eqs. (4.97)-(4.102), parameters R th e o , R th c o , R th o lo , K th c , and K s are constant and can be determined as variables in optimization processes, as will be illustrated in the next chapter. In most cases R th c o can be ignored, since T jc — Tamb due to the negligible values o f Pdc- Eqs. (4.97)-(4.102) have been incorporated in the proposed HBT model written for the MNS simulator [13] (see appendices B and C); the results are presented in chapter 8 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5 EXTRACTION OF PARAMETERS FOR THE TEMPERATURE DEPENDENT MODEL A procedure for the parameter extraction o f the temperature dependent model o f the HBT is described in this chapter. The procedure relies on the extraction o f the static model (described in chapter 3), performed using data taken at different temperatures. The determination o f the parameters o f the functions that model the temperature dependencies o f the saturation currents and beta's o f the device is accomplished in a straightforward manner by using the curve fitting and optimization techniques available in IC-CAP [13]. If possible, the same device whose static model has been extracted at the temperature o f interest (usually the room temperature) should have its forward and reverse Gummel characteristics measured at different temperatures, and its static model extracted at each o f these temperatures. From the variation o f the model parameters (saturation currents, beta's, etc.) with temperature, the parameters o f the corresponding temperature functions can be extracted using simple numerical optimization procedures, to be described in the following sections. In this work, only an HBT o f the same type and from the same wafer as the HBT characterized in chapter 3 has been available for the temperature measurements. The static model parameters o f this device, extracted at several temperatures according to the procedure described in chapter 3, are given in table 5.1. 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 Table 5 . 1 . Extracted values o f I s , I s e , I s r o l , P f , Pr , and R Bmax (denoted by Istem p, ISEtemp, ISROHemp, PFtemp, PRtemp, and RBmaxtemp, respectively) at temperatures between 9.8 °C and 73.6 °C temperature (°C) 9.8 15.1 23.5 31.4 42.3 53.3 62.8 73.6 Istem p (A) I s Etemp (A) 9.9 x 10 ~ 28 3.9 x 10 " 27 2.8 x lO " 26 1 . 6 x 1 (T 25 1.5 x 10 " 24 1 . 2 x 1 0 " 23 7.5 x 10 " 23 4.6 x 10 " 22 4.7 x 1 .1 x 3.8 x 1 .1 x 4.7 x 1 .8 x 5.5 x 1.7 x 10 " 19 1 (T 18 10 " 18 1 (T 17 1(T 17 1 0 ~ 16 1(T 16 1(T 15 temperature (°C) 9.8 15.1 23.5 31.4 42.3 53.3 62.8 73.6 IsROLtemp (A) PFtemp PRtemp 2 .2 x 4.6 x 1 .2 x 3.1 x 1 .0 x 3.0 x 7.5 x 2 .0 x 127.7 124.0 121.3 116.1 108.4 4.8 0.7 2.9 13 1 0 0 .2 13 93.8 84.2 3.0 0.7 2.3 1 0 " 15 10 “ 15 1 0 " 14 10" 14 1 0 -1 3 10 " 10 " 1 0 - 12 2 .0 0 .8 RBmaxtemp (Kf 2) 3.67 2.54 3.12 3.16 2.29 1 .6 6 1.63 1.48 The forward emission coefficient N F has been extracted only at the lowest temperature (9.8 °C); due to its constancy with temperature [22], N p was allowed to assume this low-temperature value throughout all subsequent extractions. 5 .1 Determination o f X t i , X trol and The temperature exponents X X tbr t i , X trol , and X tbr o f eqs. (4.41), (4.63), and (4.54), respectively, can be easily determined from the corresponding temperature variations of I s , I srol »and P r , by using simple computer optimization procedures. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 With the temperature dependence o f E s b given by eq. (4.38), X can be determined ti as an optimization parameter by fitting I g ( T ) o f eq. (4.41) to the extracted values IstemP(T ) o f table 5.1. By configuring the optimization process to minimize the error function eistemp given by: Nt ^istem p = £ n=l where N t = 8, I ^ ( I s i T n ) ) ~ ^ ( I s te m p ( T n )) \ 2 (5-1) Tn is the n-th temperature o f table 5.1, and Tq = 300 K in eq. (4.41), the values o f table 5.2 have been obtained for X t i and Is(T o). Using the parameters o f table 5.2, I s o f eq. (4.41) is shown in fig. 5.1 as a function o f 1/ T , along with Istem p o f table 5.1. The agreement between I s and Istemp is very good, confirmed by the small value o f the final RMS error. Table 5.2. Parameters o f temperature dependent model for I s (eq. (4.41)) after optimization (To = 300 K) X ti I s (To) MAXIMUM error RMS error In a similar way, I sro l (T X trol 6 .1 1 1 5.789 x 10 - 2 6 A 0.129% 0.064% can be determined as an optimization parameter by fitting ) o f eq. (4.63) to the extracted values I s R O L t e m P( T ) o f table 5.1, by configuring the optimization process to minimize the error function eisROLtemp given by: Nt eiSROLtemp = £ | In { I S R O L { T n ) ) ~ ^ (IsR O L tem p(T n ) ) | 2 (5.2) n=l Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 3.0 2.8 3.4 3.2 1 I T ( x 1 0 “ 3 K_ 1 ) 3.6 Fig. 5.1. Istemp (symbols) and I s (lines) as functions o f temperature. The exponent X tb r can also be determined by fitting (3 r(T ) o f eq. (4.54) to PRtemp(T) o f table 5.1, by configuring the optimization process to minimize the error function ^QRtemp below. Nt ^PRtemp = Z ) | ln ( /3 f l( r „ ) ) - fo{(3RtemP(T n )) | n=l The extracted values o f X trol and X tbr 2 (5.3) are given in tables 5.3 and 5.4, respectively, along with the remaining optimization parameters and final errors. Figs. 5.2 and 5.3 show the temperature variation o f Isro l (T ) and (3r ( T ), using the parameters o f tables 5.3 and 5.4, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 Table 5.3. Parameters o f temperature dependent model for I s r o l (eq. (4.63)) after optimization (To = 300 K) X I trol srol (T o) MAXIMUM error RMS error 4.242 18.51 £A 0.079% 0.042% Table 5.4. Parameters o f temperature dependent model for Pr (eq. (4.54)) after optimization (To = 300 K) X tbr M To) MAXIMUM error RMS error - 1.993 1.898 116.2% 75.65% 5.2 Determination o f x (aluminum concentration), /&, (3q and X t e Due to the fact that the emitter bandgap is a fUnction o f the aluminum concentration x (see eqs. (4.55) and (4.56)), which is not known at this point, the parameters o f /3p as a function o f temperature (eq. (4.53)) will have to be determined before the extraction o f X t e is attempted. In the extraction o f the parameters o f P f(T ), x is also determined as a parameter o f the bandgap difference, A E g(T); with this extracted value o f x, E 9e is written as a function o f temperature according to eqs. (4.55) or (4.56), and X t e o f eq. (4.61) can be found using an optimization procedure similar to the ones previously described. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 10“ n 2 8 io - 12 * | 1 0 “ 13 ■w I •"i i o - 1* 1 0 " 15 2. 8 Fig. 5.2. I s R O L te m p 3.0 3.2 3.4 1 J T ( x 1 0 ~ 3 K- 1 ) (symbols) and I sro l 3.6 (lines) as functions o f temperature. In the extraction o f the parameters o f /3f (T ), by configuring the optimization process to minimize the error function epFtemp given by: epFtemp = 5 3 | l n ( ^ r y ) - ^ ( 0PtJ - (r „ ) ) I 2 n=l (5-4) where N t = 8 and Tn is the n-th temperature, the values o f table 5.5 have been obtained for x , f b, and /%. Using the parameters o f table 5.5, 1/ftp ° f eq- (4.53) is shown in fig. 5.4 as a function o f 1 /T , along with 1 /(If temp obtained from table 5.1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 Table 5.5. Parameters o f temperature dependent model for Pr (eq. (4.53)) after optimization X h Po MAXIMUM error RMS error 0.1173 9.128 144.0 0.198% 0 .1 2 1 % ft. E Ol 3.0 3.2 3.4 1 / T ( x 1 0 ~ 3 K- 1 ) 3.6 Fig. 5.3. PRtemp (symbols) and Pr (lines) as functions o f temperature. Using the previously determined value o f x which is used in the expression o f E 3e { T ) (eq. (4.55), with x Ise(T q ). < 0.4242), the exponent X te is extracted as an optimization parameter, along By configuring the optimization process to minimize the error function e/5 E t e m p given by: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 Nt (5.5) e/S Etemp = 53 I ^ ( I s E i T n ) ) — In{ I SE tem p(Tn )) n= 1 the values o f table 5.6 have been obtained for X t e and I s e {T$). 12 I O X 11 10 9 8 7 2.8 3 .0 3 .2 3 .4 3 .6 1 / T ( x 1 0 - 3 K " 1) Fig. 5.4. 1 /PFtemp (symbols) and 1//3p (lines) as functions o f temperature. Table 5.6. Parameters o f temperature dependent model for I s e (eq. (4.61)) after optimization (To = 300 K.) X Te I s e (Tq) MAXIMUM error RMS error 7.218 6.052 x 10~ 18 A 0 .1 2 1 % 0.055% With the parameters o f table 5.6 and x from table 5.5, I s e o f eq. (4.61) along with I s Etemp o f table 5.1 are shown in fig. 5.5 as functions o f temperature. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 -------- -■ 1 0 “ 14 !" : i1 !I1 ! ti !1 ii 1 0 - 15 1 s . c *1 05 10 - 1 8 1 0 ~ 19 - 28 = :z : 1 ' * ■1 1 1 1 mu 1 0 ~ 1 7 ft, E u 1 E -- Hill 11 1 1 0 - 16 mnn —1 11Mill a. 05 ! i ii! 1 1 i !! I? ! :t i _____________ i_____________ 3 0 | N i ! t: ? i______________ 3.2 3.4 1 / T ( x 1 0 - 3 K_ 1 ) 3.6 Fig. 5.5. I s Etemp (symbols) and I s e (lines) as functions o f temperature. The values o f the temperature exponents X t i , X trol , X tb r . and X t e do not depend on the choice o f the reference temperature To; the values extracted in this chapter for these exponents, as well as the values for x , /&, and (3q, will be used in the full temperature dependent model o f the HBT. 5.3 Determination o f XTRBmax The temperature exponent XTRBmax of eq. (4.64) can be easily determined from the temperature variation o f R.Bmax, by using a simple computer optimization procedure. XTRBmax can be determined as an optimization parameter by fitting R sm axiT) o f eq. (4.64) to the extracted values RBmaxtemP(T ) o f table 5.1 (T here is the ambient temperature Tam* o f eq. (4.64)). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 By configuring the optimization process to minimize the error function CRBmaxtemp given by: Nt €RBmaxtemp — 2 \to (R B m a x (T n )) - In( RBmaxtemp( ) ) | 2 (5.6) n=l where N t = 8, Tn is the n-th temperature o f table 5.1, and To = 300 K in eq. (4.64), the values o f table 5.7 have been obtained for XTRBmax and RBmax (To). Using the parameters o f table 5.2, RBmax o f eq. (4.64) is shown in fig. 5.6 as a function o f T, along with RBmaxtemp o f table 5.1. Table 5.7. Parameters o f temperature dependent model for RBmax (eq. (4.64)) after optimization (To = 300 K) XTRBmax RBmax (To) MAXIMUM error RMS error - 4.321 2.733 KQ 3.139% 1.665% 5.4 Determination o f R theq -, K t h c and K s In order to determine the parameters o f the thermal circuit o f the HBT, it is necessary to use the set o f measured device characteristics on which these characteristics are likely to have an effect. One such set o f characteristics which is extensively used in the extraction o f the temperature dependent model o f the HBT is made up of the dc common emitter output characteristics (I c -V c e ) measured at different base currents [2 2 ], [46], In the measurement o f these characteristics, in the region where the transistor is active, the powers dissipated in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 emitter and collector junctions are relatively large, which allows for the extraction o f R K th c and K s, th eo , o f eqs. (4.97), (4.99), (4.100) and (4.102), provided that sufficiently wide ranges are chosen for the collector-emitter voltage and for the base current. 1 280 300 320 340 T <K) Fig. 5.6. RBmaxtemp (symbols) and RBmax (lines) as functions o f temperature. In the region where the device is active, both the base-collector junction and the overlap diode are reverse biased, which makes pdOL o f eq. (4.102) (the dissipated power in the overlap diode) negligible. As a result, the thermal resistance using the common emitter output characteristics; R th o lo R th o lo cannot be extracted will be determined using the reverse Gummel characteristics, where the power dissipated in the overlap diode is relatively large. The bias conditions that define the measurement o f the common emitter output characteristics are: — the emitter is held at ground potential; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 — a current source provides one or several values for the base current o f the transistor (positive values for npn transistors, negative values for pnp transistors); — a voltage source sweeps the collector voltage from zero to a maximum value that is well beyond saturation (positive for npn transistors, negative for p n p transistors). The setup for the measurement o f the common emitter output characteristics is shown in fig. 5.7, where the dc voltage sources Vg and Vc define the emitter and collector voltages, respectively (Vg = 0, Vc > 0); the base current is defined by source Ig- An HP4142 dc modular source/monitor controlled by IC-CAP via an HP-EB bus can be used to provide all voltages and currents o f fig. 5.7, and to sweep Vc (sweep order 1) and I b (sweep order 2), while holding Ve at ground potential. The collector current is available as the measured current through source Vq . D.U.T. (>0 ) (>0) Fig. 5.7. Setup for the measurement o f the common emitter output characteristics. Fig. 5.8 shows the measured collector current o f the HBT under test, plotted as a function o f the measured collector-emitter voltage (Vcm - Vgm )\ 10 values have been used for the applied base current, in the range [0, 54 pA], with 6 -p A steps. The apparent negative Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 slope o f the characteristics is entirely due to the transistor self-heating, since the forward Early voltage is typically greater than 1000 V [22], Parameters R theo, K th c and K g can be determined using the optimization feature o f IC-CAP, in the region where the power dissipated in the overlap diode is negligible (outside saturation, in the forward active region); from the characteristics o f fig. 5.8, this region corresponds to collector-emitter voltages greater than approximately 0.8 V. In this range, the influence o f the parasitic series resistances o f the transistor is negligible and, if the temperature dependence o f the model parameters is considered, the collector current depends on R theo , K th c and K s and the powers dissipated in the transistor junctions. The optimization consists o f simultaneously fitting the MNS simulated collector currents o f the model o f fig. 2.3 (whose temperature dependent parameters are modeled according to eqs. (4.41), (4.53), (4.54), (4.61), (4.63), (4.97), (4.99), (4.100) and (4.102)) to the measured collector currents o f fig. 5.8, with R theo , K t h c and K s as variables, by minimizing the error function e rtheo given below: N/b Ncbz ^R T H E O ^Y l 53 \IC sk{V cE n ) ~ IC m k{V cE n )\ fc=1 n=NcE\ (5.7) In eq. (5.7) VcEn denotes the n-th measured collector-emitter voltage {V cm ~ Vsm), and le a is the collector current obtained by simulation; subscript k refers to the fc-th value o f the base current, and N ib is the number o f base currents used in the measurement o f the output characteristics. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 4 3 2 1 0 1 0 2 1 V Cm ~ VE m 3 00 Fig. 5.8. Measured collector current I c m o f the HBT as a function o f the collectoremitter voltage (Vcm ~ VEm), with the base current as a parameter (common emitter output characteristics). The base current step is 6 fi A. The results given by the optimizer, for N ib = 10 base current values (corresponding to the measurement of figs. 5.7 and 5.8), and N c e -2 — N c e i + 1 = 39 points in the previously determined range for (Vcm — Vsm), are shown in table 5.8. The values RTHEOi = 2000 °CAV (provided by the manufacturer for the thermal resistance o f the transistor), KxHCi = 0.5, and K s i = 2.8 x 10 ~ 3 °C " 1 (calculated using the results given in [2 2 ]), have been used as initial values in the optimization. The small value of the RMS error indicates the correct convergence o f the parameters. With the parameters o f table 5.8, R e , R c from tables 5.10 and 5.11, respectively, and X trbb from table 5.10, the simulated collector current I c 3 is shown in fig. 5.9 as a function o f (Vcm — VEm), along with the measured current I c ,n- The agreement between the curves is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 very good, which indicates the correctness o f the model. It is important to note that R e , R c , and X t r b b have a dominant effect only in the saturation region o f the HBT. Table 5.8. Parameters R t h e o , K and K s after optimization R theo K thc Ks MAXIMUM error RMS error t h c 3354 °C/W 0.5114 11.16 x 10~ 3 °C ~ 1 0.954% 0.377% The MNS circuit file used in the above optimization is given in appendix A, where the temperature dependence o f the parameters has been modeled according to eqs. (4.41), (4.53), (4.54), (4.61), (4.63), (4.97), (4.99), (4.100) and (4.102); parameters X 7 7 , X t b r , X t e and X trol o f eqs. (4.41), (4.54), (4.61) and (4.63), respectively, have been read in from tables 5.2, 5.4, 5.6 and 5.3, respectively, while I s ( T 0), P r (T q), I s e (T0) and I s r o l (Tq) o f the same equations (T0 = are the extracted values Is, Pr , I se and I sro l from chapter 3 T E M P = 23.5 + 273.15 = 296.65 K). The temperature dependence o f j.dp has been implemented as: a (rp\ P f {± ) - a /rp \ P f ( I q) fbexp(-AEg(To)/(kTo))+l/@o '/6exp'(-Ais(r)7(fcf)j+i/A . (58) where ft,, /3b and x (the aluminum concentration in the emitter region, appearing explicitly in the expression o f A E g(T )) have been read in from table 5.5, and P f {Tq) is the extracted value P f from chapter 3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 '^ o o o c p x * * o»»o-»-»»oo V C m ~ VE m 00 Fig. 5.9. Simulated collector current I c 3 (lines) and measured collector current Icm. (symbols), as functions o f the collector-emitter voltage (V c,n — VE,n). The base current step is 6 p A (same as in fig. 5.8). Due to the manner in which the temperature dependence o f the parameters is modeled, these parameters assume their room-temperature values in the regions where the self-heating effects are negligible. For the MNS circuit given in appendix A, the temperature o f the emitter and collector junctions and the overlap diode selectively affect the parameters o f the model o f fig. 2.3, according to the correspondence given in table 5.9. 5.5 Determination o f X Parameter tr b b X trbb of eq. (4.66) can be determined using the optimization feature o f IC-CAP in the forward Gummel configuration (fig. 3.1), in the region where self-heating occurs. In this configuration, no power is dissipated in the collector junction or the overlap Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 diode, and T jc = Tamb can be assumed. Only the emitter junction temperature T je increases due to self-heating, and this is sufficient for the extraction o f X t r b b - Table 5.9. Selective influence o f the junction temperatures on the HBT model parameters temperature parameter affected term TjE T je T je T jc T je T ol T ol T ol Is ISE Pf Is Pr Is ISROL Pr Ic c I be I cc/P f I ec I ec/P r I sol I rol I sol equation (2.4) ( 2 .2 ) (2.5) (2 . 1 0 ) ( 2 . 1) (2 . 1 0 ) The optimization consists of simultaneously fitting the MNS simulated collector and base currents o f the model o f fig. 2.3 (whose parameters are written as functions o f temperature and dissipated powers according to eqs. (4.41), (4.53), (4.61), (4.66), (4.97) and (4.100)) to the measured collector and base currents o f fig. 3.3, with X t r b b and R e as variables, by minimizing the error function c x t r b b given below: tX T R B B = 2 m-fCs(VB£n)) “ n=1 + D M i .( W Vb Eti)) |2 + ) - K W V be ,,) ) ! 2 (5.9) n=l In eq. (5.9), VsEn denotes the n-th measured base-emitter voltage (Vsm — Vsm), and I c 3 and I bs are the collector and base currents, respectively, obtained by simulation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 The results given by the optimizer, for N fg = 61 points for (Vjjm — V sm) in the entire measured range [0.895 V, 1.495 V], are shown in table 5.10. The RMS error is very small, which indicates the correct convergence o f the optimization. Table 5.10. Parameters X t r b b and R e after optimization X trbb Re MAXIMUM error RMS error 8.351 7.422 Q 4.512% 0.741% With the parameters o f table 5.10, the simulated collector and base currents, I c 3 and I b s , respectively, are shown in fig. 5.10 as functions o f (Vflm - Vew) , along with the corresponding measured currents, I c m and respectively. The agreement between the curves is excellent over the entire range for (Vem - Vsm)- The MNS circuit used in the above optimization is given in appendix A. 5.6 Determination o f R tholo As mentioned in section 5.4, R tholo can be determined from the reverse Gummel characteristics using the optimization feature o f IC-CAP, since the power dissipated in the overlap diode is relatively large in this configuration. The power dissipated in the collector junction (pdc o f eq. (4.68)) can be neglected in this configuration due to the fact that the currents through this junction are relatively small. The optimization consists o f simultaneously fitting the MNS simulated base and emitter currents o f the model o f fig. 2.3 (whose parameters are written as functions o f temperature and dissipated powers according to eqs. (4.41), (4.54), (4.63), (4.99) and (4.102)) to the measured base and emitter currents o f fig. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 3.10, with R th o lo and R c as variables, by minimizing the error function £r t h o l o given below: N ,rg tR T H O L O = Y1 n=l m^Bs(VsCn)) “ ^ { 1 B m iV B C n ))\ + N ,rg +n=1 H \^{IEs{VbCti)) - \^lE m {V B C n))\4 (5.10) -2 -3 01 -4 14 -6 B -r -8 -9 0.8 VBm ~ VEm 00 Fig. 5.10. Measured currents icm (0 and Ism (ii), and simulated currents I c 3 (iii) and I bs (iv) in the forward Gummel configuration, using the parameters of table 5.10. In eq. (5.10), VBc n denotes the n-th measured base-collector voltage (Vflm — V cm ), and I bs and I rs are the base and emitter currents, respectively, obtained by simulation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 The results given by the optimizer, for N rg = 95 points for (Vsm - Vcm) in the entire measured range [0.389 V, 1.330 V], are shown in table 5.11. The RMS error is relatively large due to the noise in the measured emitter current at low values o f (Vgm — Vcm). The final value for R tholo was practically zero, which does not have to be interpreted as a physical value; it can be stated, however, that R tholo has a sufficiently low value such that the effect o f R c is dominant at high currents. Table 5.11. Parameters R tholo and R c after optimization R tholo Rc MAXIMUM error RMS error 0 3.413 ft 23.09% 3.068% (4 63 <0 03 6 63 IQ ' 10 B 0.2 0.4 0.6 0.8 VB m ~ VCm 00 Fig. 5.11. Measured currents Ism (i) and I Em 00, and simulated currents Ib 3 (iii) and Ie s (iv) in the reverse Gummel configuration, using the parameters o f table 5.11. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. With the parameters o f table 5.11, the simulated base and emitter currents, I bs and I e s , respectively, are shown in fig. 5.11 as functions o f (Vsm — Vc,n), along with the corresponding measured currents, Ism and I Em, respectively. The curves are in close agreement over the entire range for (Vgm — Vcm). The MNS circuit used in the above optimization is given in appendix A. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6 CHARGE-STORAGE EFFECTS IN THE HBT In order to accurately model the high-frequency response o f the HBT, the chargestorage effects in the device, along with the parasitics associated with the transistor contact pads and interconnections, will have to be evaluated. The charge storage in the one-dimensional HBT is modeled by the introduction o f two types o f capacitors: tw o nonlinear junction capacitors and two nonlinear diffusion capacitors. In the case o f a planar HBT (see fig. 2.2), two nonlinear junction and diffusion capacitors, respectively, model the charge-storage effects in the overlap diode. 6 .1 Junction Capacitances o f One-Dimensional HBT The two junction capacitances model the incremental fixed charges stored in the transistor’s space-charge layers for incremental changes in the associated junction voltages [12]. Each junction capacitance is a nonlinear function o f the voltage across the junction with which it is associated. In the case o f a single-heterojunction HBT, an intrinsic setback layer (or spacer) is frequently used in the abrupt heterojunction to improve the emitter injection efficiency, and to reduce the impurity out-diffiision from the heavily doped base to the emitter [47], Because the electron velocity can overshoot its steady-state value until intervalley scattering occurs, it is important to design a collector potential profile in which electrons can travel a certain length without gaining enough energy for intervalley scattering. Thus, in addition to the emitter setback layer, two stacked layers o f lightly doped p-GaAs and normally doped n-GaAs 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 sometimes replace the conventional n-type collector o f an npn device, to improve the highfrequency performance [48], Therefore, in the general case, the single-heterojunction HBT structure may have two setback layers associated with each junction, as shown in fig. 6 . 1 . With the notations o f fig. 6.1, the thickness o f the emitter-base depletion region is [47]: A X e = X \ 4- X 2 (6.1) where y ( rN bX r 1 . 6e N e + £ b N b X 2 = ( (.r N r X r \ . 2(.e ^ b N b { ^ e —V b e ) Y \ zeN e + zbN b ) + Xe q N e {£e N e + * b N b ) (6-3> In the above equations, ^ e (B) and N e {B) denote the permittivities and doping concentrations o f the emitter and base, respectively. In eq. (6.2), Vb e is the applied base-emitter voltage and $ £ denotes the base-emitter junction built-in potential [47]: $ £ = where A E g = - Q . l A E G/ q + VT l n % ^ E qe (6.4) ~ E g b is the base-emitter energy bandgap difference, n ^ B ) denotes the intrinsic carrier concentrations o f the emitter and base, respectively, and Vp = k T /q . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 w-x Fig. 6.1. Single-heterojunction one-dimensional HBT structure with two setback layers. Using eqs. (6.1)-(6.4), the depletion capacitance o f the base-emitter junction o f a one dimensional structure o f area A e can be written as: C je (6.5) = Taking into account the fact that for an npn HBT the base is heavily doped (N b » N e ), and that the permittivities o f the base and emitter are comparable, from eqs. (6.1)-(6.3) the width o f the emitter-base depletion region as a function o f V be is obtained as: a + (6 -6> and using eq. (6.5) it follows that the depletion capacitance o f the heterojunction can be modeled as: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 C j e (V b e ) = ~ ^ j wg In the above expression, Vbe = 0 {C jeq = (6-7) is the value o f the emitter-base junction capacitance at C jeq £e A e / & x e (0 )), m g is the emitter-base capacitance gradient factor (m g = 0.5 in the case o f the abrupt heterojunction considered in this analysis), and $'E is the effective emitter-base built-in potential: + A |f f (6 .8 ) where $ g is given by eq. (6.4). Similarly, in the case o f the base-coHector homojunction (eg = ec), the thickness o f the depletion region can be written as: A xC = X 2 + X 4 (6.9) where X i — ^3 ~ X t = $c = _____ . / ( N c+ N b ^ X + y\N 3 + Xc V r ln ^ N S XC c+ Nb ) \ J_ 2^ b N b { ^ C -V b c ) ^ q N c { N c + N B) , qx (6 . 1 1 ) (6.12) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill Because the base is heavily doped (N b N c), from eqs. (6.9)-(6.12), the depletion capacitance o f the base-collector homojunction can be written as: (6.13) C j c (V „ c ) = where C jc o is the value o f the base-collector junction capacitance at Vb c = 0, m e is the base-collector capacitance grading factor (m e = 0.5 for an abrupt base-collector junction), and $'c is the effective base-collector built-in voltage. In the above equation, c jc o = (6.14) + A xc (0) ~ s j \ h + 2ji $ r (6.15) <616> Note that eqs. (6.7) and (6.13) have similar forms despite the fact that they model the depletion capacitances o f a heterojunction and a homojunction, respectively. A linearly graded doping profile corresponds to a grading coefficient o f 0.333 and produces the smallest variation o f capacitance with the applied voltage [11]. According to the previous derivations, abrupt junctions have grading coefficients o f 0.5. It is possible, however, to have grading coefficients greater than 1, in the case o f hyper-abrupt junctions; for such doping profiles, these coefficients are typically less than 4 [49], Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 6.2 Diffusion Capacitances o f One-Dimensional HBT The two diffusion capacitances model the charge associated with the mobile carriers in the transistor. For a one-dimensional device, if the effect o f the base resistance is neglected, this charge is divided into two components [ 1 2 ]: one associated with the reference collector source current, I c c , and the other one associated with the emitter source current, component being modeled by a capacitance, C de and C qc, o Collector Iec, each respectively, as shown in fig. 6 .2 , where I c t = I c c — I e c - BC Base 'EC o •b E ,c c / £ f Emitter Fig. 6.2. Placement o f depletion capacitances ( C j e , C j c ) and diffusion capacitances { C d e , C d c ) in the Ebers-Moll model o f a one-dimensional transistor structure with zero base resistance [ 1 2 ]. For the one-dimensional transistor structure o f fig. 2.2, the total mobile charge associated with I c c , for the base-emitter junction forward biased and Vb c = 0, can be written as the sum o f the individual minority charges: Q d E = Te I c C + Te b I c C + Tb I c C + Tb c I c C (6.17) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 113 In the above equation, rB space-charge layer transit time, t b is the emitter region delay time, is the base transit time, and t e b r BC is the emitter-base is the base-collector space-charge layer transit time. From eq. (6.17), Q d e can be re-written as: Q de — tf (6 1 8 ) Icc where tf = te + teb + tb + tbc (6 1 9 ) The quantity rp defined by eq. (6.19) is the total forward transit time o f the one dimensional transistor. Similarly, the total mobile charge associated with I Ec , for the base-collector junction forward biased and VBB = Q d c 0, can be written as: = T cIeC + t b c I e c + T brIeC + t e b I e c where r c is the collector region delay time and t Br (6.20) is the reverse base transit time. Eq. (6.20) can be re-written as: Q dc = trIec ( 6 .2 1 ) where tr = tc + tbc + tbr + teb (6.22) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The quantity r R defined by eq. (6.22) is the total reverse transit time o f the one dimensional transistor. The nonlinear capacitances C ue and C u e that model the charges Q u e and Q u c o f eqs. (6.18) and (6 .2 1 ), are given by [ 1 1 ]: (6.23) (6.24) where Vb e and Vbc denote the forward base-emitter and base-collector bias voltages, respectively, o f the one-dimensional structure, as shown in fig. 6 .2 . 6.3 Junction and Diffusion Capacitances o f Overlap Diode Since an overlap diode has to be included in the transistor model in cases where the area o f the collector junction is larger than the area of the emitter junction (see figs. 2 .2 and 2.3), it becomes important to determine the charge-storage effects in this diode. Following a derivation similar to the one for the case o f the one-dimensional transistor structure o f area A e , the charge storage in the depletion region o f the overlap diode can be modeled by the following capacitance: C j o l (V b c ) = - 1 ^ C j c (V b c ) (6.25) where A c is the area o f the collector junction and C j c i Y s c ) has been defined by eq. (6.13); eq. (6.25) implicitly assumes that the doping concentrations o f the base and the collector are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 the same in the regions o f the overlap diode and the one-dimensional structure, so that the corresponding depletion capacitances differ only by a factor o f proportionality [11]. In addition to the charge storage in the depletion region o f the overlap diode, the charge storage due to the minority carrier charges injected into the neutral regions also has to be considered. Because the minority carrier concentrations reach high levels in forward bias and are negligible in reverse bias, the charge stored in the neutral regions can be found in the same way as the mobile charge Q dc o f eq. (6.20), with the exception o f the term containing the emitter-base space-charge layer transit time teb - Thus, according to [11], the charge stored due to the minority carriers in the overlap diode can be written as: Q d o l = Tq I o L + Tb c I o L + Tb r I o L = Td Ol I o L where and tq Iol , tb c , tbr are the transit times o f eq. (6.20), tq ol (6.26) is the overlap diode transit time, is the current through the overlap diode shown in fig. 2.3. From eq. (6.26), the nonlinear diffusion capacitance C qol that models the stored charge in the overlap diode is given by: C ool = where Vbcol (6.27) denotes the forward bias voltage across the diode (see fig. 2.3); the large-signal model o f the overlap diode, containing the capacitances C jo l and C qo l, is shown in fig. 6.3. In an accurate model o f the overlap diode, however, the spreading effects due to the nonzero resistance o f the base layer underneath the base contact, should also be considered. From the analysis o f chapter 2, it can be expected that a current-dependent resistance should be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 connected in the series with the overlap diode; the placement o f capacitances in this diode model is also an important issue (see the following sections in this chapter). In order to keep the size o f the HBT model within reasonable limits, the spreading effects in the overlap diode will not be modeled in this work; the simplified model o f fig. 6.3 is an acceptable approximation for the overlap diode, in cases where sufficiently small forward bias voltages are applied across it. Collector BCCL RCL overlap diode dimensional transistor Fig. 6.3. Placement o f junction and diffusion capacitances in the large-signal model o f the overlap diode, by neglecting the spreading effects. 6.4 Expressions for the Transit Time Components o f One-Dimensional HBT The diffusion capacitances o f the HBT and the overlap diode can be calculated from eqs. (6.23), (6.24) and (6.27). For the accurate modeling o f these capacitances, it is important to derive theoretical expressions for the components o f the forward and reverse transit times, which are defined by eqs. (6.19) and (6.22). (a) The emitter region delay time (t ^ J : When the HBT is operated in the forward active region, the emitter-base junction is in forward bias. If a current flows into the emitter Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117 terminal, it will divide between the capacitance o f the emitter-base junction and the spacecharge resistance, and only the current flowing through this resistance will be amplified [50], The emitter-region delay time te o f eqs. (6.17) and (6.19) is associated with this phenomenon; in the case o f HBT's, r E can be calculated accurately as described in [51]. Thus, for an npn single-heterojunction one-dimensional HBT with no setback layer in the emitter junction, the following general expression can be found for the emitter region delay time: rE = G % g (6.28) where G= ft ((! + t + fc M 1 + £ + t ) ) exP( - A^ / fcT) (6.29) In the above equations, N e and N b are the doping concentrations o f the emitter and base, respectively, D ub is the diffusion coefficient o f electrons in the p-type base, and A E g is the bandgap difference between the emitter and base. The emitter thickness and the base width are denoted by W e and W b , respectively, according to fig. 6.4, which shows a one dimensional npn HBT under active bias (V e b < 0, Vc b = 0)- Fig- 6.4 also shows the excess minority charges in the emitter and base, Q ve and Q ub , respectively. Quantities Sdp and £>*, o f eq. (6.29) are the diffusion velocities o f holes in the emitter and o f electrons in the base, respectively, in the quasi-neutral regions (Sdp = D pE/ W e , Sdn = D n s /W s , D pe = diffusion coefficient o f holes in the emitter). S ip and S{n are the effective interface velocities o f the holes and electrons, and S ep and are the drift and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 diffusion (emission) velocities o f the holes and electrons across the space-charge region, respectively. For an HBT with an abrupt emitter-base junction, the interface velocity is much larger than the diffusion velocity, the electron current is controlled by the drift and diffusion velocity o f the carriers across the space-charge-region barrier (S en <C S</n), and the hole current is limited by diffusion velocity within the quasi-neutral base (S ep SdP) [52]. Under these conditions, the coefficient G o f eq. (5.29) becomes: G ~ gj ex p ( - A E g/ k T ) (6.30) In the case o f a graded heterojunction, the hole and electron currents are controlled by their respective diffusion velocities [51], and the coefficient G o f eq. (6.29) becomes: G ~ jjfj ex p ( - A E g/ k T ) (6.31) For typical values o f N b / N e and A E g/k T , G o f eq. (6.31) has very small values (G 1). The emitter delay time is much less than the base delay time o f HBT's with graded heterojunction, and can therefore be neglected [51]. In the case o f HBT's with abrupt emitter-base junctions, G is typically greater than 1, and the emitter delay time te given by eq. (6.28) must be included in the calculation o f the forward transit time o f the transistor. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 Emitter Base Colector nB Fig. 6.4. Single-heterojunction one-dimensional HBT biased in the forward active region. (b) The emitter-base space-charge layer transit time ( teb )• In the case o f emitter- base homojunctions, t e b can be found by solving Poisson's equation in the presence o f free carriers in the junction, and obtaining the normalized potential difference between the metallurgical junction and the boundary [53], It has been found in [53] that for small metallurgical basewidths and low emitter doping gradients (in the case where the emitter is heavily doped compared to the base), teb decreases with increasing gradient, and may attain values lower than r B for sufficiently high emitter doping gradients. From the modeling point o f view, however, the mobile minority charge in the emitterbase space-charge layer associated with I c c (see eq. (6.17)) is normally considered to be zero [11], [12], regardless o f the doping gradients o f the homojunction transistor, resulting in r EB — 0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 In the case o f HBT's, the analysis o f the free carriers in the space-charge layer o f the emitter-base junction is more complicated than in the case o f homojunction BJT's due to the dissimilar em itter and base materials. However, the assumption o f negligible contribution o f these carriers to the total transit time o f the device is used for HBT's as well [35], and in this work t e b will be considered to be negligible (rEB = 0). (c) (t B c ) : The base transit time ( tb) and the base-collector space-charge layer transit time In the case o f an HBT, for the base transit time and base-collector depletion layer transit time calculations, the nature o f the carrier charge distributions in these regions must be considered [54], According to [54], the following expressions are obtained for rg and r Bc o f eq. (6.19), using the charge-control definition o f the carrier transit time in the base region and the collector depletion region, and taking into account the fact that the finite background saturated electron concentration is determined by the carrier saturation velocity v3at and the collector current density: (6.32) (6.33) In the above equations, W B is the width o f the ungraded and uniformly doped base, and Xdc is the width o f the base-collector depletion region. Eq. (6.33) assumes a constant velocity profile for the electrons in the base-collector space-charge layer. In the case o f an HBT, however, this approximation is not acceptable due to the presence o f velocity overshoot in the region [55], Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 By using a more realistic step-like velocity profile for the electrons in the basecollector space-charge layer (which can be assumed to extend entirely in the collector region due to the high doping concentration in the base), the base-collector space-charge layer transit time is obtained as [55]: (6.34) where Wo, vc, vsat are as shown in fig. 6.5. A more accurate expression for tbc can be obtained by considering a piecewise-linear drift velocity profile as in [55]; however, due to its complexity, this expression is somewhat impractical for modeling purposes. The transit times given by eqs. (6.32) and (6.34) are not independent o f the bias conditions because the velocity profile is determined by the electric field and thickness o f the depletion layer [54], the latter being determined by the base-collector voltage. However, a recent measurement technique which has been proposed for the extraction o f the base transit time shows that r# and tqc are practically independent o f the bias currents in a sufficiently wide range [56], (d) The collector region delay time (tc ): This transit time can be considered in a manner similar to the one in which the expression for the emitter region delay time has been developed. In order to find an expression for r c , the HBT is considered to be biased in the reverse active region (V cb < 0), with no bias voltage applied across the emitter-base junction (V e b = 0), according to fig. 6.6. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 O O ■5 ^ Vs a t base base-collector sp ace-ch arge layer collector Fig. 6.5. More realistic step-like velocity profile for electrons in the space-charge layer o f the HBT base-collector junction. The current flowing into the collector terminal will divide between the capacitance o f the collector-base junction (which is a homojunction in the case o f a single-heterojunction HBT) and the base-collector space-charge resistance; like in the case o f the forward active bias, only the current flowing through this resistance will be amplified in the device. The collector region delay time tq o f eqs. (6.20) and (6.22) for a single-heterojunction HBT can be calculated according to [51], by recognizing that the analysis done for the emitter delay time o f a heterojunction transistor applies also to the case of the collector delay time o f the HBT. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 In the case o f the base-collector homojunction, both the hole and electron currents are controlled by their respective diffusion velocities in the quasi-neutral regions near the junction, that is, referring to eq. (6.29), S m » S dn, S ep » S dp, S in » S dn, 5 ip » S dp [51]. Thus, with the notations o f fig. 6.6, the collector region delay time o f an HBT whose base-collector junction is graded, can be written as: r n - r C - . p°( 4 c) . W cW n n0(-x^B) 2D n s - W e W ff - 2D nB f6 3 5 ) ^ because the base-collector space-charge layer follows the symmetrical linearly graded law and P o « c ) = rc o (-Z p s ). For an abrupt base-collector junction, the collector delay time becomes [51]: < * * > and due to the normally high doping in the base compared to the collector, this delay time may not be negligible. (e) The base-collector space-charge layer transit lime (tbc ) associated with I ec ' In the case o f the reverse active bias, the base-collector space-charge layer transit time tbc o f eqs. (6.20) and (6.22) is not identical to tbc o f eqs. (6.17) and (6.19), because it is associated with a different current (I g c rather than Ic c ), and the base-collector junction is now forward biased, rather than reverse biased. As a consequence, the expression derived for tbc and given by eq. (5.33) is not valid for the HBT in reverse active bias. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Emitter Base Colector nB -W nC Fig. 6.6. Single-heterojunction one-dimensional HBT under reverse bias. In order to find an accurate estimate for tbc o f eqs. (6.20) and (6.22), the analysis described in [53] can be employed for the forward-biased base-collector homojunction. Thus, by solving Poisson's equation in the presence o f free carriers in the base-collector junction, and obtaining the normalized potential difference between the metallurgical junction and the boundary, the following expression can be found for the base-collector transit time for the HBT in reverse bias [53]: tbc = 2G b ^ (6.37) where ac is the collector doping gradient and G B is the base Gummel number (G b = fba3ep ( x ) d x /D nB). In the above equation, U l is the normalized potential drop between the metallurgical junction and the boundary o f the collector space-charge layer averaged over the range o f currents considered: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 U l = j ± jJ UL( J ) d J (6.38) Jl t b c o f eq. (6.37) decreases with increasing collector doping gradient, and under certain conditions may take on values larger than the other components o f the reverse transit time t r . F or practical modeling purposes, however, the mobile minority charge in the basecollector space-charge layer associated with I e c is normally considered to be zero [11], [12], yielding tb c (f) = 0. The reverse base transit time (t Br) : The base layer design in conventional bipolar transistors usually involves an impurity grading to achieve a built-in drift field. In the case o f ultrasubmicrometer-basewidth HBT structures, however, the carrier diffusion velocity can become sufficiently high, so that the presence o f a base drift field is not critical for highfrequency operation [54]. Thus, the base region o f the one-dimensional HBT can be assumed to be uniformly doped; in this way, the carriers injected either from the emitter side or from the collector side o f the device do not encounter an electric field (low injection conditions exist due to the high doping in the base), and travel with the same velocity in either direction. Therefore, when no drift field exists in the base, the forward and reverse base transit times are identical, and the following expression can be given for t Br [54] (see eq. (6.32)): (6.39) (g) The emitter-base space-charge layer transit lime (te b ) associated with I e c • In the case o f the reverse active bias, the emitter-base space-charge layer transit time t Bb o f eqs. (6.20) and (6.22) is not identical to t Eb o f eqs. (6.17) and (6.19), because, like tBc , it is associated with a different current (I e c , rather than I c e ) , and the emitter-base junction is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 now reverse biased, rather than forward biased. As a consequence, te b can no longer be assumed negligible (see previous discussion on why te b = 0 is a reasonable assumption in forward active bias), and an estimate for this transit time in reverse active bias must be found. If the base-emitter space-charge layer is assumed to extend entirely in the emitter layer due to the high doping concentration in the base, and taking into account the fact that the velocity profiles o f electrons in AlGaAs and GaAs are similar [24], then the base-emitter space-charge layer transit time is obtained as [55]: where x * is the width o f the base-emitter space-charge layer, and W q, i/c, i/3at refer to the AlGaAs emitter and are as shown in fig. 6.7; a step-like velocity profile has been assumed for the electrons in the emitter. In the case o f AlGaAs, however, the velocity overshoot is in general less pronounced than in the case o f GaAs [24], and the following approximation can be used for t e b by assuming a constant velocity vfsat for electrons: Te b - (6.41) Thus, from the previous discussions and derivations, if transistor anomalies such as base pushout, space-charge-limited current flow, and quasi-saturation [11] are neglected, the forward and reverse transit times r F and t r can be written as the sum o f the appropriate components, and can be modeled as two constant quantities. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 A\ o o * sat base i— b ase-em itter sp a ce-ch a rg e layer emitter Fig. 6.7. Step-like velocity profile for electrons in the emitter space-charge layer o f the HBT under reverse bias. The previously-mentioned transistor anomalies have the tendency o f increasing the transit times tf and tr at high terminal currents; if these anomalies are not negligible, multiplying empirical functions with a value o f 1 at low currents can be employed for increased accuracy [11], 6.5 Placement o f Capacitances in Complete Model o f One-Dimensional HBT with Non-Zero Base Resistance In the case o f a planar transistor structure with a stripe emitter geometry, whose base resistance is not negligible, the one-dimensional HBT can be approximated by a cascade of identical transistors with constant base resistances, as shown in fig. 6.8. In fig. 6.8, the number o f transistors, n, is sufficiently large. A similar structure with n = 11 has been used to illustrate how the base resistance can be obtained by simulation (see section 2.3 and fig. 2.8). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 Collector C jc i C jc 2 C dci Ri ^1 4= 4= C jC n C dc 2 4= 4= I I l wv- x - i J ^Rn -1 _L _L n i Qi cJE1 ^DE1 'JE2 C oen ^D E2 ^JEn r J 'Q n ^-'DEn Emitter Fig. 6.8. First-order discrete approximation o f one-dimensional HBT with single base contact, using exact models for the elementary transistors; the junction and diffusion capacitances o f each transistor are shown explicitly. The junction and diffusion capacitances o f each elementary transistor Q {, i = 1 ,..., n, o f fig. 6.8 are represented separately; these capacitances are connected according to fig. 6.2, and, in view of eqs. (6.7), (6.13), (6.23) and (6.24), their values are: (6.42) C JE %— ^ C ja = (6.43) 1 = (6.44) Q — OlZlECi) — TD ME£L — dVfic. 2 — 1 — TR rlVr,^: <% cr’ 1 — 72 (6.45) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 In eqs. (6.42)-(6.45), the quantities having the subscript i are associated with transistor Q i, i = 1 In eqs. (6.44) and (6.45) it is assumed that the transit times r p and Tft are constant. The accuracy o f the first-order HBT model o f fig. 6.8 increases as the number n o f the elementary devices increases. It has been found in section 2.3 that as few as 11 identical transistors allow the modeling o f the base spreading effects with reasonable accuracy. However, even for a reduced number o f transistors, the model o f fig. 6.8 is inordinately complex, and its use is not viable especially in cases where circuits containing several transistors are to be simulated. The model o f fig. 6.8 can be greatly simplified by noticing that in the active region, for sufficiently large base currents, the first transistor o f the network (<Qi) diverts almost all o f the base current (and consequently conducts almost all o f the total collector current), according to fig. 2.10. From eq. (6.44) it follows that the diffusion capacitance C q ei is much larger than (C de 2 + C d e 3 + ••• + C deti), because the collector current I c c i dominates over the sum o f the other elementary collector currents, and Tp can be assumed to be constant (see previous section). Since in forward bias the effect o f the base-emitter diffusion capacitance dominates, in general, over the effect o f the depletion capacitances, the only important capacitance in fig. 6.8 is C q e i , ail the other capacitances having negligible effect, as shown in fig. 6.9. Since the low-frequency transistor network o f fig. 6.9 can be modeled by a regular transistor with a nonlinear base resistance, as shown in sections 2.2 and 2.3, the model o f fig. 6.9 can be simplified further as shown in fig. 6.10, where R b b (I b ) is the nonlinear base resistance, and the area o f Q is n times the area o f Q i . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 lJ c Base 1 ao- ■ |y —> a Qi Collector i i i C M' a M'' w - C — P DE1 DE— <1 Emitter diffusion capacitance low—frequency transistor network Fig. 6.9. Dominance of C qei over the other capacitances o f fig. 6.8 in forward active bias, for sufficiently large base currents (C qe — T p d lc / dVsEx)- The model o f fig. 6.10 is in obvious disagreement with the classical SPICE GummelPoon large-signal model described in [11] and reproduced for convenience in fig. 6.11, under the same simplifying assumptions as the device o f fig. 6.9. Collector static model of transistor Base Emitter Fig. 6.10. Further simplification o f model o f fig. 6.9, by replacing the low-frequency transistor network according to section 2.2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131 Collector static model o f transistor Base DE Emitter Fig. 6.11. Classical SPICE Gummel-Poon large-signal model o f the HBT under forward bias, with negligible terminal resistances and negligible junction capacitances, according to [1 1], To illustrate the validity of the model o f fig. 6.10 and the inaccuracy o f the classical SPICE model at sufficiently large dc base currents, the models of figs. 6.8 and 6.10, and o f figs. 6.8 and 6.11, have been ac simulated with SPICE, and the input impedances and ac beta's have been compared. Note that both the model o f fig. 6.10 and the model o f fig. 6.11 have identical performance as static models, and their accuracy has been demonstrated in chapters 2 and 3. The circuit o f fig. 6.8 is assumed to be the reference for the transistor behavior, for n sufficiently large. As in section 2.3, the following parameters have been used in the SPICE simulations: n = 11, R i = 3 kfi, i = l,...,10, i s = 1 x 10-2G A, I SI = Is* = ... = I s n = I s / l l , Pf \ = P f 2 = ••• = P f u = P f = 100, Vccx = V cci = 2 V. I s and P r are the saturation current and forward beta, respectively, o f transistor Q o f figs. 6.10 and 6.11, and Is i and are the corresponding quantities for elementary transistors <3„ i = 1 o f fig. 6.8. In addition to the above parameters, a forward transit time rp = 0.1 nsec has been assumed for the elementary transistors o f fig. 6.8. The junction capacitances which appear explicitly in fig. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 132 6.8 assumed their default values (zero); the diffusion capacitances were determined by the non-zero value o f rp. r CC2 bbac /K DE dc-hac (a) comparative simulation o f model o f fig. 6.10 and the structure o f fig. 6.8 Fig. 6.12. SPICE ac simulation configurations o f HBT model o f fig. 6.10 and classical SPICE model o f fig. 6.11 (V cci = Vcca = 2 V) (a). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 Fig. 6.12. Continued (b). Fig. 6.8 r d c+ ac cci CC2 m ^ bbac -v w /K DE dc-bac (b) comparative simulation o f model o f fig. 6.11 (classical SPICE model) and the structure of fig. 6.8 The circuits used in the ac simulations are shown in fig. 6.12; the SPICE input files are given in tables 6 .1 and 6.2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 134 Table 6.1. SPICE input file for the circuit of fig. 6 .12(a) * SPICE input file for comparing the reference structure * and proposed model (CDE outside RBB) vccl 1 0 2 vcc2 nvcc2 0 2 vl 2 B1 0 ib 0 2 + dc 100.00e-6 + ac 1.000e-9 0 + s i n ( 0 1.000e-9 1.000e3 0 0 0 ) rbbac B2 15 39.876 CDE B2 0 3 8.7 5p * rl B1 4 3k r2 4 5 3k r3 5 6 3k r4 6 7 3k r5 7 8 3k r6 8 9 3k r7 9 10 3k r8 10 11 3k r9 11 12 3k rlO 12 13 3k ql 1 B1 0 qnpnl q2 1 4 0 qnpnl q3 1 5 0 qnpnl q4 1 6 0 qnpnl q5 1 7 0 qnpnl q6 1 8 0 qnpnl q7 1 9 0 qnpnl q8 1 10 0 qnpnl q9 1 11 0 qnpnl qlO 1 12 0 qnpnl qll 1 13 0 qnpnl ql2 nvcc2 15 0 qnpn2 ■ir fl 0 B2 vl 1 .model qnpnl n p n (I S = 9 .0909e-28 BF=100 TF=0.1n) .model qnpn2 npn(IS=le-26 BF=100) •At .ac dec 21 .probe .end l.OOOmeg lOO.OOOg Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 Table 6.2. SPICE input file for the circuit o f fig. 6 .12(b) * SPICE input file for comparing the reference structure * a nd classical SPICE model (CDE inside RBB) vccl 1 0 2 vcc2 nvcc2 0 2 vl 2 B1 0 ib 0 2 + dc 100.00e-6 + ac 1.000e-9 0 + s i n ( 0 1.000e-9 1.000e3 0 0 0 ) rbbac B2 15 39.876 CDE 15 0 38.75p ■k rl r2 r3 r4 r5 r6 B1 4 3k 4 5 3k 5 6 3k 6 7 3k 7 8 3k 8 9 3k r l 9 10 3k r8 10 11 3k r9 11 12 3k rlO 12 13 3k •k ql 1 B1 0 qnpnl q2 1 4 0 qnpnl q3 1 5 0 qnpnl q4 1 6 0 qnpnl q5 1 7 0 qnpnl q6 1 8 0 qnpnl q7 1 9 0 qnpnl q8 1 10 0 qnpnl q9 1 11 0 qnpnl qlO 1 12 0 qnpnl qll 1 13 0 qnpnl ql2 nvcc2 15 0 qnpn2 * fl 0 B2 vl 1 .model qnpnl n p n (I S = 9 .0909e-28 BF=100 TF=0.1n) .model qnpn2 npn(IS=le-26 BF=100) + .ac dec 21 .probe .end l.OOOmeg lOO.OOOg Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 Because the simulation o f the circuits o f fig. 6.12 is small-signal, the large-signal base resistance R b b base resistance {Ib ) o f figs. 6.10 and 6.11 must be replaced by the corresponding small-signal which can be found as illustrated in fig. 6.13, where a low-frequency ac simulation is performed using SPICE. As in fig. 6.12, the current source %b o f fig. 6.13 contains both a dc and a small-signal component; the SPICE input file for the circuit o f fig. 6.13 is given in table 6.3. Fig. 6,8 r dc-f-ac cci T dc+ac Fig. 6.13. Determination o f small-signal base resistance Tbbac o f fig. 6.12 (Y c c i = ^CC2 = 2 V). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 Table 6.3. SPICE input file for the circuit o f fig. 6.13 * SPICE input file for comparing the small-signal base * spreading resistance as a function of the dc base * current vcc 1 0 2 vl 2 3 0 * ib 0 2 + dc 100.00e-6 + ac 1.000e-9 0 + s i n ( 0 1.000e-9 1.000e3 0 0 0 ) * rl r2 r3 r4 r5 r6 3 4 3k 4 5 3k 5 6 3k 6 7 3k 7 8 3k 8 9 3k r l 9 10 3k r8 10 11 3k r9 11 12 3k rlO 12 13 3k ql 1 3 0 qnpnl q2 1 4 0 qnpnl q3 1 5 0 qnpnl q4 1 6 0 qnpnl q5 1 7 0 qnpnl q6 1 8 0 qnpnl q7 1 9 0 qnpnl q8 1 10 0 qnpnl q9 1 11 0 qnpnl qlO 1 12 0 qnpnl qll 1 13 0 qnpnl ql2 1 14 0 qnpn2 * fl 0 14 vl 1 ★ .model qnpnl n p n (IS=9.0909e-28 BF=100 TF=0.1n) .model qnpn2 npn(IS=le-26 BF=100) ■ k .ac dec 21 l.OOOmeg lOO.OOOg .probe .end Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 With the notations o f fig. 6.13, r»ac where is obtained as: = (6.46) is the small-signal component o f the base current source (ie = I b + h ), and v ( N\ ) and u(iVo) are the small-signal voltages o f nodes N \ and N-i. The diffusion capacitance C de o f fig. 6.12(a)-(b) has been calculated according to the equation [11]: C d e = Tf (3f v ± (6 4 7 ) where I b is the dc base current. Table 6.4 lists a series of values for and C de obtained through SPICE simulation at different dc base currents using eq. (6.46), and calculated from eq. (6.47), respectively. From fig. 2.10, which shows the current distribution in the discrete reference circuit, the base current o f transistor Q \ is the dominant base current for currents Ib larger than approximately 10 fiA. To ensure the dominance o f the base current o f Qi, a dc base current I b = 100 y.A has been chosen, and the circuits o f fig. 6.12 have been ac simulated using a frequency sweep between 1 MHz and 100 GHz. Fig. 6.14 shows the magnitude o f the input impedance and the magnitude of the ac beta for the reference structure and the proposed model o f fig. 6.10, obtained by simulating the circuit o f fig. 6 .12(a). Both sets o f curves are in good agreement over the entire frequency Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 range (a similar agreement has been observed for the phase-frequency curves o f the input impedance and ac beta). Table 6.4. and C d e as functions o f the dc base current I b Ib I'bbac C de 100 nA 300 nA 1 fj, A 3 fj , A 10 f i A 3 0 /xA 100 /xA 300 f i A 1 mA 3 mA 10 mA 9.24 Kft 8.74 Kft 7.19 Kft 4.38 Kft 1.38 Kfl 288.1 ft 39.87 ft 5.92 f t 0.707 ft 0.097 ft 0.0108 ft 38.75 fF 116.66 fF 387.59 fF 1.166 pF 3.87 pF 11.66 pF 38.75 pF 0.116 nF 0.387 nF 1.16 nF 3.87 nF Fig. 6.15 shows the magnitude o f the input impedance and the magnitude o f the ac beta for the reference structure and the classical SPICE model o f fig. 6.11, obtained by simulating the circuit o f fig. 6 .12(b). Although the magnitudes o f the ac beta o f the reference structure and o f the classical SPICE model are in excellent agreement over the entire frequency range (a similar agreement was observed for the phase-frequency curves o f the ac betas), the input impedances are in obvious disagreement at high frequencies, due to the way in which the base resistance is connected. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140 0.001 0.01 0-1 10 100 10 0.1 1 frequency (GHz) 100 1 frequency (GHz) (a) magnitude o f input impedance 100 0 .0 0 1 0 .0 1 (b) magnitude o f ac beta Fig. 6.14. Comparative ac simulations o f the reference discrete structure o f fig. 6.8 (i) and proposed model o f fig. 6.10 (ii), obtained by simulating the circuit o f fig. 6 .12(a) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 N i o ~ 2 ; --------------- o .o o i o .o i 0.1 1 10 100 10 100 frequency (GHz) (a) magnitude o f input impedance 100 0 . 01 + -------------- r „ 0 .0 0 1 0 .0 1 0.1 1 frequency (GHz) (b) magnitude o f ac beta Fig. 6.15. Comparative ac simulations o f the reference discrete structure o f fig. 6.8 (i) and classical SPICE model o f fig. 6.11 (ii), obtained by simulating the circuit o f fig. 6 .12(b) (a and b). From the above results it can be concluded that the base-emitter diffusion capacitance has to be placed outside the current-dependent base resistance o f the transistor model, as shown in fig. 6.10. In a similar manner, it can be shown that the base-collector diffusion Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 142 capacitance must be connected outside the current-dependent resistance o f the transistor model. Thus, in the case o f a one-dimensional HBT with a non-zero base resistance, the model o f fig. 6.16 is obtained, where the depletion capacitances C je and C jc are connected in parallel with the corresponding diffusion capacitances C d e and C dc- Collector Cdc Cjc !b fe e fe c /P j —> ^ *" Base _ p T 7^ h e * VvCv ] — CDE bbO b ) ^ C je W/ 'CT zls zl •be \q c / P <j Emitter Fig. 6.16. Placement o f depletion and diffusion capacitances in the model o f a one-dimensional transistor structure with non-zero base resistance. In fig. 6.16, and C dc C je and C jc are given by eqs. (6.7) and (6.13), respectively, and C de by eqs. (6.23) and (6.24), respectively. 6.6 Complete Model o f Intrinsic Planar HBT In view o f the model o f a one-dimensional transistor structure with a non-zero base resistance shown in fig. 6.16, and o f the model o f the model o f the overlap diode shown in fig. 6.3, the full large-signal model o f an intrinsic planar HBT (excluding the package and terminal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 143 parasitics) can be obtained as shown in fig. 6.17, where the diodes modeling the overlap diode are connected as in fig. 2.3. In fig. 6.17, resistances R b i and R b 2 are the series parasitic resistances o f the base region; resistances R e and R c are series parasitic resistances o f the emitter and collector, respectively, o f the intrinsic structure. DC DOL 'CL VOL -r R JOL Fig. 6.17. Full large-signal model o f the intrinsic planar HBT. If the capacitances are removed, the model o f fig. 6.17 is identical to the static model o f fig. 2.3, with I r o l , I , I o l b e , h e , h e , h e , and I q t having the same expressions as for the model o f fig. 2.3, and R b b being defined by eq. (2.36). Capacitances C jol respectively; capacitances C dc and C je C qol and Cjc o f fig. 6.17 are given by eqs. (6.25) and (6.27), are given by eqs. (6.7) and (6.13), while C de are given by eqs. (6.23) and (6.24). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and CHAPTER 7 PARAMETER EXTRACTION PROCEDURE FOR THE HIGH-FREQUENCY MODEL This chapter describes a procedure for obtaining the junction capacitances, the pad and interconnection parasitics, the forward transit time, and the remaining parameters o f the model o f fig. 6.17. The complete high-frequency model o f the HBT is obtained using S-parameter measurements on the device using on-wafer RF probing over a range o f frequencies and biases. For the on-wafer probing, the additional RF probe pads and interconnections have to be considered in the high-frequency simulations o f the device, because they are present in the measurement. The experimental determination o f the parasitics is o f primary importance, because it allows for the reduction o f the parameter space dimension, and for the avoidance o f non physical local minimae which occur in subsequent computer optimizations at high frequencies [57]. Appropriate models for the pads and interconnections are given, and a procedure for their extraction is described. The junction capacitances o f the HBT are determined using a combination o f 'cold' and two-port low-frequency measurements at various reverse-bias voltages applied across the junction o f interest; using the models for the pad and interconnection parasitics, the baseemitter diffusion capacitance and forward transit time o f the transistor is determined from the high-frequency measurements at several collector currents, with the device biased in the forward active region. 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 145 The advanced curve fitting and optimization techniques o f IC-CAP [13] are extensively used for the extraction o f the high-frequency model o f the HBT, by fitting a series o f simulation curves o f the appropriate equivalent circuits to the measured values. 7.1 Determination o f the Junction Capacitances o f the HBT Fig. 7.1 shows the layout o f the probe pattern for on-wafer measurements o f the HBT whose model parameters have been extracted in the previous chapters. Collector HBT Base Emitter Collector Fig. 7.1. Layout o f the HBT probe pattern. This layout is intended to be used in conjunction with three-conductor microwave probes whose tips are normally placed in the center o f the corresponding pads. In the measurement o f the junction capacitances o f the transistor, the measurement frequencies are normally low enough, so that the ohmic series resistances have a negligible effect. Also, if a probe is offset from the intended geometrical position, the effect on the measured Sparameters is negligible due to the large wavelength o f the injected signal, relative to the dimensions o f the layout. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 146 7.1.1 Determination o f the Base-Collector Junction Capacitance The total base-collector junction capacitance (C j o l + Cjc ) (see fig. 6.17) o f a planar HBT whose layout is shown in fig. 7.1 can be measured first. Referring to the layout o f fig. 7.1, it is possible to connect the base and collector terminals to one port o f a network analyzer and to ground respectively, and leave open the emitter contact, as in a normal 'cold' measurement [12], Fig. 7.2 illustrates the measurement scheme for the base-collector junction capacitance, with a bias tee ensuring that a dc voltage Vb is applied to the base terminal (Vc = 0). The range o f the dc base-collector voltage is between the junction breakdown voltage and a forward bias o f about 0.4 V. A larger forward bias must be avoided, because the diffusion capacitance would otherwise become important [58], The probe on the right that normally ensures the emitter contact is placed in such a way that only the 'sense' terminal o f voltage source Vc makes contact to the collector. In this way, both the base and collector terminals are connected to the 'force' and 'sense' terminals of voltage sources Vb and Vc, respectively, and the emitter is left open. As previously mentioned, the maximum frequency o f the measurement signal is low enough so that the displacement o f the emitter probe has negligible effect on the measured data. The basecollector junction capacitance is measured at several dc bias voltages, and the parameters o f its voltage dependence are determined by a computer optimization procedure. From the diagram o f fig. 7.2 and the intrinsic HBT model o f fig 6.17, if the basecollector bias voltage and maximum frequency satisfy the previous requirements, the smallsignal equivalent circuit seen from port 1 o f the network analyzer is as shown in fig. 7.3. Capacitances C jc and C jol o f fig. 7.3 are the junction capacitances o f the one-dimensional HBT and o f its overlap diode, respectively; resistance Rb accounts for the inherent series resistances o f the pad and interconnections, and capacitance Cpb models the parasitic parallel Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 147 capacitance which appears across the base-collector junction in this measurement. Due to the low values o f the measurement frequencies, the base spreading resistance o f the HBT, resistance R m o f fig. 6.17, and the parasitic inductances o f the pad and interconnections [57] can be neglected altogether. Vc = 0 FORCE COLLECTOR DC BAS SENSE D. U. T. BAS FORCE/SENSE BASE DC BAS RF PORT 1 Fig. 7.2. HBT base-collector junction capacitance measurement, showing the connection and placement o f the microwave probes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 148 RF HBT port 1 R b -AAAr <> >------------------= C JOL c pb = = II O >■ Fig. 7.3. Small-signal equivalent circuit o f the HBT in the measurement o f the base-collector junction capacitance. It is clear from fig. 7.3 that the parasitic capacitance Cpb is always present in the measurement and cannot be physically separated. It is possible, however (and this will be done in the last stage o f the parameter optimization procedure), to mathematically extract the value o f C p b from the overall dependence with voltage o f the total capacitance Cpb + C jol + C jc ■ Fig. 7.4 shows the measured S-parameters (S n ) in the case o f the base-collector capacitance measurement o f fig. 7.2, for Vb = 0.4 V and Vc = 0 (slight forward bias), for the frequency / in the range [46 MHz, 25.87 GHz], From fig. 7.4 it can be seen that the measured circuit behaves like a pure capacitance ( | S n m | ~ 1, arg(Sum) < 0) at frequencies less than approximately 10 GFIz. The total capacitance C jc to t = Cpb + C jo l + C jc can be obtained directly in this frequency range from Sum, by taking into account the fact that the input impedance o f the equivalent circuit o f fig. 7.3 is practically imaginary at sufficiently low frequencies: Cjctot = + C jol + C j c ^ (71) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 149 B H <0 0 .9 0 1 i I— rT T T T T i T m i i i i in i i — i— r r r 111 0 .4 - 0.81— 0.01 0.1 10 1 100 /(GHz) Fig. 7.4. Magnitude and phase o f the measured S n ( 5 n m) in the case o f the base-collector junction capacitance measurement, for Vb = 0.4 V, Vc = 0. Applying eq. (7.1) for finding C jc to t measurement noise o f 5 i lm to the final result; has the drawback o f propagating the C jc is subject to further errors due to resistance Rb o f fig. 7.3 which is not considered in the calculations. An accurate optimizer-based procedure for the determination o f C jc to t has been preferred in this work over the direct application of eq. (7.1). The optimization was performed over a low-frequency range, which eliminates in part the inherent measurement noise o f S n . The values o f S u obtained by simulating the circuit o f fig. 7.3 (with a capacitance C jc to t replacing Cpb, C j o l and C jc ) using the MNS simulator, have been fitted to the measured values by letting the IC-CAP optimizer [13] adjust the circuit parameters ( C j c t o t Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 and/or Rb) for minimum least-squares error. The optimization process was configured to minimize the error function e s c jc given by: N scjci tS C J C = E | S iim (fn ) \ I I S lls (/n )| ~ \ n=Nscjci Nscjcn +E Ia r g ( S n s ( / n )) - a rg (5 n m(/„ )) | (7.2) n = N scjc\ where f n is the n-th measurement frequency, and S \ \ 3 and S n m denote the simulated and measured values, respectively, o f S u . For the HBT under test, the values obtained for Rb and C jc to t at different base- collector dc voltages are listed in table 7.1 along with the corresponding final RMS optimization errors. The optimizations were performed using N s c j c z — N s c j c i + 1 = 37 points in the range [46 MHz, 3 GHz], Resistance Rb was an optimization parameter (along with C jc to t) only in the case o f Vbc = 0.4 V; for the other values o f Vbc, Rb was allowed to keep the value obtained for VBc = 0.4 V, while C jc to t was the only optimization parameter. The final RMS errors after optimization are very small, which indicates that the circuit o f fig. 7.3 is a valid low-frequency model o f the open-emitter HBT. The capacitance 7.1 is plotted as a function o f the dc voltage VBc C jc to t , as shown in fig. 7.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o f table 151 Table 7.1. Final optimization values o f C j c t o t as a function o f voltage VBC (V) 0.4 0 - 0 .4 - 0 .8 - 1.2 - 1.6 - 2 .0 - 2 .4 C jctot (fF) 40.60 37.57 35.96 35.21 34.81 34.58 34.43 34.32 RbV) 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 RMS error (%) 0.098 0.104 0.102 0.101 0.092 0.093 0.088 0.090 42 o s o 34 L_ - 0.5 0 0.5 1.5 1 - VB C 2 2.5 00 Fig. 7.5. C jctot (symbols) and C jcth (lines) as functions o f VBc- Using eqs. (6.13) and (6.25) for C j c (V Bc ) and C j o l ( V b c ), respectively, the total measured capacitance can be modeled by the following function: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 152 (7.3) CjcthiVBc) = Cjt 4- where C jcoth = (A c / A e )C jcq = (K a 4- l)C jc o (parameter K a— the area factor—has been defined for eqs. (2.10) and (2.11)), with C jcq being the value o f the base-collector junction capacitance o f the one-dimensional HBT at Vbc = 0- Parameters $'c and m e o f eq. (7.3) have been defined for eq. (6.13) and represent the effective base-collector built-in potential and the base-collector capacitance grading factor, respectively. The parameters o f eq. (7.3) can be determined using a computer optimization procedure, by fitting C jc th iV B c ) to the measured C jc to t (V b c ) as described in [12]. In this way, Cpb can be determined as an optimization parameter, despite the fact that it cannot be physically removed from the device under test. By configuring the optimization process to minimize the error function z-cjc given by: Ncjc ZC JC = £ I fa (C jc th {V B C n )) ~ fo (C jC to t(V B C n )) where VEcn is the n-th dc base-collector voltage and N c jc | 2 (7-4) = 8 points in the range [ — 2.4 V, 0.4 V], the values o f table 6.2 have been obtained for the parameters o f the base-collector capacitance. Using the parameters o f table 7.2, C jc th is also shown in fig. 7.5 as a function o f Vb c - Capacitances respectively, with Cjc C jco and C jol = (l/(K a o f fig. 7.3 are defined by eqs. (6.13) and (6.25), 4- l ) ) C j c o t h ' , for the transistor under test, K a = A c / A e — 1 = 4.164. The MNS input file for the circuit o f fig. 7.3 is given in table 7.3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 153 Table 7.2. Parameters o f base-collector junction capacitance model and Cpb (eq. (7.3)) after optimization C jc m m.c Cpb MAXIMUM error RMS error 3.471 fF 2.513 V 3.712 34.01 fF 0.247% 0.119% Table 7.3. MNS input file for the circuit o f fig. 7.3 define hbt (C E B) Rb = 1 Cjctot = 40f r:rrl B B1 r=Rb c:ccl B1 C c=Cjctot end hbt 7.1.2 Determination o f the Base-Emitter Junction Capacitance The base-emitter junction capacitance C je (see fig. 6.17) o f a planar HBT whose layout is shown in fig. 7.1, can be determined in a manner similar to the one employed in the determination o f the base-collector capacitance. Fig. 7.6 illustrates the measurement scheme for C je , with two bias tees ensuring the application o f the dc bias voltages. The two microwave probes are placed normally onto the corresponding base, emitter, and collector pads o f the layout, ensuring that the 'force' and 'sense' terminals o f the dc voltage sources are both connected to the corresponding transistor terminals. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 154 FORCE Vc = □ COLLECTOR DC BIAS D. U. T. BIAS BIAS BASE • DC BIAS > RF PORT 2 RFPORT1 Fig. 7.6. HBT base-emitter junction capacitance measurement, showing the connection and placement of the microwave probes. The placement o f the microwave probes is the regular placement used in full two-port measurement o f the transistor. In this case, however, only the measured S oo is used in the extraction o f the capacitance. A zero bias voltage is applied across the base-collector junction (Vu = 0, V c = 0). A variable dc voltage Ve is applied to the emitter terminal (port 2 in fig. 7.6). As in the case of the base-collector capacitance measurement, the range o f the dc base-emitter voltage is between the junction breakdown voltage and a sufficiently small forward bias, which ensures that the diffusion capacitance is negligible. Cje is measured at several dc bias voltages o f the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 155 emitter junction, and the parameters o f its voltage dependence are determined by a computer optimization procedure. From the diagram o f fig. 7.6 and the intrinsic HBT model o f fig. 6.17, the small-signal equivalent circuit seen from port 2 o f the network analyzer is as shown in fig. 7.7, where it was assumed that the base resistance in series with the 50-Ohm loading o f port I is much smaller than the reactance o f the emitter capacitance. RF HBT port 2 4> - R6 A W ---------- - •------- c pe = Fig. 7 .7. Small-signal equivalent circuit o f the HBT in the measurement o f the base-emitter junction capacitance. Capacitance C je of fig. 7.7 is the base-emitter junction capacitance o f the HBT; resistance Re accounts for the series resistance o f the pad and interconnections, and capacitance C ^ models the parasitic parallel capacitance which is reflected between the emitter and collector terminals in this measurement. As in the case o f the base-collector junction capacitance measurement, the base resistance o f fig. 6.17 and the inductances o f the pad and interconnections can be neglected. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 156 Capacitance Cpe is always present in the measurement and cannot be physically separated; the value o f Cpe will be extracted from the overall dependence with voltage o f the total capacitance Cpe + C j e - Fig. 7.8 shows the measured S-parameters (Soo) in the case o f the base-emitter junction capacitance measurement o f fig. 7.6, for Vb = Vc — 0, V e — — 0.4 V (slight forward bias), for the frequency / in the range [46 MHz, 25.87 GHz], ]_ ( CN CN 0.9 0 - 0 .4 0.01 0.1 1 10 10 0 /(GHz) Fig. 7.8. Magnitude and phase o f the measured S-v> (S22 m) in the case o f the baseemitter junction capacitance measurement, for Vb = Vc = 0, V e = — 0.4 V. Like the base-collector capacitance measurement, since the measured circuit behaves like a capacitance (in fig. 7.8, | S22m I — 1, arg(.S22m) < 0), it is possible to determine the total capacitance CjEtot = Cpe + C j e directly, by applying a formula similar to eq. (7.1), with Soom replacing S u m. This approach is not used in this work, however, because a large amount o f the measurement noise o f Soom propagates to the final result. Instead, an Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 157 optimizer-based procedure similar to the one used for the base-collector junction capacitance will be used for finding C j e ■ Thus, the values o f So2 obtained by simulating the circuit o f fig. 7.7 (with one capacitance CjEtot replacing Cpe and C je ) have been fitted to the measured values, by letting the IC-CAP optimizer [13] adjust the circuit parameters (CjEtot and/or Re) for minimum least-squares error. The optimization process was configured to minimize the error function c sc je given by: N'sCJETl tSC JE = £ I I $22 s(/n) | ~ | S 22 m ( f n )| | n —N s c j E i N s c je i + £ I arg(S22s(/n)) - arg(S22m(/n)) I (7.5) n = N Sc j E \ where /„ is the n-th measurement frequency, and Soo3 and Soom denote the simulated and measured values, respectively, o f Soo. For the HBT under test, the values obtained for R e and CjEtot at different baseemitter dc voltages are listed in table 7.4 along with the corresponding final RMS optimization errors. The optimizations were performed using N s c je z — ^ s c j e i + 1 = 37 points in the range [46 MHz, 3 GHz]. Resistance R e was an optimization parameter (along with CjEtot) only in the case o f Vb e — 0-4 V; for the other values o f Vb e , Re was allowed to keep the value obtained for Vb e = 0-4 V, while CjEtot was the only optimization parameter. The final RMS errors after optimization are very small, which indicates that the circuit o f fig. 7.7 is a valid low-frequency model o f the HBT in this measurement. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 158 Table 7.4. Final optimization values o f CjEtot as a function o f voltage Vb e OH 0.4 0.2 0 - 0 .2 - 0 .4 - 0 .6 - 0 .8 - 1 - 1.2 CjEtot (fF) 20.11 19.63 19.08 18.98 18.89 18.72 18.62 18.49 18.44 R em 18.56 18.56 18.56 18.56 18.56 18.56 18.56 18.56 18.56 RMS error (%) 0.071 0.070 0.081 0.079 0.074 0.074 0.071 0.071 0.071 The capacitance CjEtot o f table 7.4 can be plotted as a function o f the dc voltage Vb e , as shown in fig. 7.9. * 20 3 o e. o 0.5 - 0.5 - VBE 0 0 Fig. 7.9. CjEtot (symbols) and C jsth (lines) as functions o f VBE = Vb - V e . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 159 Using eq. (6.7) for C j e ( Y b e ), the total measured capacitance can be modeled by the following function: CjEth{VBE) = Cpe + ^ (7.6) Table 7.5. Parameters o f base-emitter junction capacitance model and Cpe (eq. (7.6)) after optimization C jeo *'e mE Cpe MAXIMUM error RMS error Parameters C jeo, 1.074 fF 1.576 V 2.081 18.14 fF 0.705% 0.315% $ £ and m # have been defined for eq. (6.7) and represent the zero- bias emitter-base junction capacitance, the effective built-in potential, and the base-emitter capacitance grading factor, respectively. The parameters o f eq. (7.6) can be determined using a computer optimization procedure, by fitting CjEth^YBE) to the measured CjEtot(YBE), as described in [12]. Capacitance Cpe (like Cpb o f eq. (7.3)) can thus be determined as an optimization parameter, despite the fact that it cannot be physically removed from the device under test. By configuring the optimization process to minimize the error function € c je given by: N ZCJE cje = £ I fo(CjEth(VBEn)) n=l ~ In(CjEtot(VBEn)) where Vbeti is the n-th dc base-emitter voltage and N cje (7.7) = 9 points in the range [ — 1.2 V, 0.4 V], the values o f table 7.5 have been obtained for the parameters o f the base-emitter Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 160 capacitance. Using the parameters o f table 7.5, CjEth is also shown in fig. 7.9 as a function o f Vb e ■The MNS input file for the circuit o f fig. 7.7 is given in table 7.6. Table 7.6. MNS input file for the circuit o f fig. 7.7 define hbt (C E B) Re = 1 Cjetot = 20f r :rrl E El r=Re r :rr2 B C r=50 c:ccl El C c=Cjetot end hbt 7.2 Determination o f Pad and Interconnection Parasitics The accurate determination o f the probe-pattem (pads and interconnections) parasitics is important for the high-frequency modeling o f the HBT, because these parasitics increase the complexity o f the device equivalent circuit and problems o f local minimae may occur for optimization processes in the larger parameter space [59]. The preliminary values o f the probe-pattem parasitics were determined in this work using a method similar to the one described in [57], Because measuring the 'open' and 'short' test structures as in [57] was infeasible, the S-parameters o f the HBT biased to cutoff were used in this work. The cutoff biasing o f the HBT ensures that the influence o f the intrinsic device is minimal, and that the frequency response is mainly determined by the pads and interconnections. A lumped-element model has been used for the pad and interconnection parasitics, instead o f the more accurate but impractical transmission-line model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 161 Fig. 7 .10(a) shows the equivalent circuit model o f the probe pattern o f the HBT under test, whose layout was shown in fig. 7.1. The circuit is similar to an accurate probe-pattem equivalent circuit used for high electron-mobility transistors (HEMT's) [60], Resistances R p b e , R p b c and R pce are very large and account for the dc leakage currents between the transistor terminals. intrinsic (Fig. 6.17) l- R PCE A /W r be Fig. 7.10. (a) Equivalent RF circuit model o f the probe pattern o f the microwave HBT. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 162 Fig. 7.10. Continued (b). C JCOth B<> (b) CutofF-mode small-signal equivalent circuit o f the intrinsic HBT for Vbc = 0 and VBE = V B E m in < 0. Fig. 7.10(b) shows the intrinsic HBT cutoff mode equivalent circuit, obtained from the full large-signal model o f fig. 6.17, in the case where Vbc = 0 and Vbe = VBEmin < 0; the complete set o f S-parameters corresponding to this dc bias situation was already available and has been used in the extraction o f the emitter capacitance o f the HBT (see previous section), VsEmm being the minimum base-emitter voltage used in that measurement. Corresponding to this bias situation, the following value has been used for capacitance EEmin C jei o f fig. 7 .10(b): (7.8) where the parameters have been taken from table 7.5. Capacitance Cjcatu o f fig. 7.10(b) has been defined in eq. (7.3) and its value has been taken from table 7.2. The series terminal resistances o f the HBT can be ignored in the model o f fig. 7 .10(b) due to the large reactances o f the capacitances. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 163 The preliminary values o f the elements o f the RF probe-pattem model o f fig. 7.10(a), with the circuit o f fig. 7.10(b) as the intrinsic HBT model, have been determined by optimization, following a procedure similar to the one described in [57], Due to the relative simplicity o f the probe-pattem equivalent circuit o f fig. 7 .10(a), the choice o f the initial values o f the parameters is not critical for the convergence o f the optimization process to a global minimum. The following initial values have been chosen for the elements o f the equivalent circuit: C b e = Cpe (table 7.5), C b c = Cp6 (table 7.2), C c e = R b e = R b c = R c e — 50 fi, L e = L b = L c = 50 pH; the above values o f the resistances and inductances are typical, in view o f the results reported in [60]. The optimization process was configured to minimize the error function espad given by: ^Spad ~ Z £ £ { I r e a l( S i j s ( f n ) ) ~ re a l( S i j m( f n )) 1 2 i= iy = ln = l L + I im a g (S y S( / n ) ) - im ag(Sijm{ f n)) \ 2 } where f n is the n-th measurement frequency, and (7.9) and Sijm denote the simulated and measured values, respectively, o f Sij, i, j = 1, 2. For the HBT under test, the preliminary values obtained for the elements o f the equivalent circuit o f the RF probe-pattem after optimization are listed in table 7.7, along with the corresponding final optimization errors. A number o f N f = 56 frequencies in the range [46 MHz, 25.87 GHz] have been used in the optimization. These preliminary values o f the pad and interconnection parasitics (table 7.7) will be adjusted in the extraction o f the forward transit time o f the transistor, to be described in the next section. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 164 Table 7.7. Optimized preliminary values o f the model parameters o f the RF probe-pattem o f the measured HBT C be C bc Cce R be R bc R ce L e Lb Lc MAXIMUM error RMS error 9.798 fF 26.15 fF 9.573 fF 84.74 Q. 22.49 Cl 0 80.61 pH 31.43 pH 0 1.563% 0.200% The MNS input file for the circuit o f fig. 7 .10(a), with the circuit o f fig. 7 .10(b) as the intrinsic HBT model, is given in table 7.8. 7.3 Determination o f Forward Transit Time The forward transit time and the remaining parameters o f the HBT can be determined using the measurement setup shown in fig. 7.11, where Ve = 0 and Vc > Vb > 0 (the HBT is biased in the forward active region). The reverse transit times r R and tqol o f eqs. (6.22) and (6.26), respectively, are not extracted in this work, because normally the base-collector junction o f the HBT (including the overlap diode) is not strongly forward biased, and the depletion components o f the basecollector capacitances o f the HBT ensure an accurate behavior o f the model in most applications [61]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 165 Table 7.8. MNS input file for the circuit o f fig. 7 .10(a), with the circuit o f fig. 7 .10(b) as the intrinsic HBT model ( Vbc = 0, Vb e = VBEmin < 0) define pads (C E B) LB = 5 Op LE = 5 Op LC = 5 Op CBC = 34,,01f CBE = 18,.14f CCE = 0 RBC = 50 RCE = 50 RBE = 50 Cjei = 0,.33f Cjci = 3..471f r :rrl B1 BC r r :rr2 BE El r r :rr3 CE El r 1:111 B B1 1=LB 1:112 E El 1=LE 1:113 C Cl 1=LC c:ccl BC Cl c=CBC c:cc2 B1 BE c=CBE c:cc3 Cl CE c=CCE c:cc4 B1 El c=Cjei c:cc5 B1 Cl c=Cjci end pads The measurement setup is similar to the setup used in the measurement o f the baseemitter junction capacitance (fig. 7.6), with the exception o f the bias voltages applied to the device. Because Vc > Vb , the base-collector diffusion capacitance and the diffusion capacitance o f the overlap diode are practically zero; also, the values o f the base-collector and overlap junction capacitances are relatively small. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 166 FORCE Vq > VB COLLECTOR DC 3A S D.U.T BAS EC BAS' RF PORT 1 RFPCRT 2 Fig. 7.11. Measurement configuration for the determination o f the forward transit time, showing the connection and placement o f the microwave probes. An HP4142 dc modular source/monitor controlled by IC-CAP via an HP-IB bus can be used to provide all voltages o f fig. 7.11, and to sweep Vg and Vc while holding Ve at ground potential. The dc collector and base currents are available as the measured currents through Vc and Vg, respectively. The S-parameters o f the device are measured at each dc bias voltage o f the emitter junction. For the device under test, the dc values o f the base and collector voltages, and the corresponding measured values o f the dc terminal currents (Ig m and Ic m > respectively) are given in table 7.9. The S-parameters o f the transistor have been measured in the frequency range [46 MHz, 25.87 GHz], using N / = 56 frequency points. The equivalent circuit o f the HBT under the bias conditions o f fig. 7.11 is shown in fig. 7.12, and has been obtained from fig. 6.17 and the RF probe-pattem parasitics o f fig. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 167 7.10(a); due to their high values, the shunt parasitic resistances o f the transistor have been omitted in fig. 7.12. In view o f the above considerations, the overlap diffusion capacitance, the base-collector diffusion capacitance, and the base-collector and overlap diode junctions o f fig. 6.17 do not appear in the equivalent circuit o f fig. 7.12. Table 7.9. Dc values o f the applied base and collector voltages (Vb and Vc, respectively), and o f the measured base and collector currents (IBm and I c m, respectively) o f the HBT under test, in the forward active region (VE = 0, Vc - VB = 1 V) VB (V ) 1.255 1.275 1.295 1.315 1.335 1.355 1.375 1.395 1.415 1.435 VC (V ) 2.255 2.275 2.295 2.315 2.335 2.355 2.375 2.395 2.415 2.435 iBm (M ) 0.276 0.424 0.656 1.035 1.620 2.611 4.246 7.064 11.94 20.79 Icm (M ) 3.691 7.518 15.13 30.23 59.62 115 221 418 773 1403 With the exception o f the probe-pattem parasitics, series resistances R b \ and R b 2 , the forward transit time t f of the HBT (eq. (6.19)) and thermal resistance R thcq o f the temperature dependent model (eqs. (4.98) and (4.101)), all the parameters o f the HBT model are assumed to be known at this point; the extraction o f the static model has been described in chapter 3, the extraction o f the temperature dependent model has been described in chapter 5, and the extraction o f the junction capacitances has been described in section 7.1 o f this chapter. Parameter R th c q has an effect in cases where the temperature o f the device increases due to self-heating, thereby affecting the temperature dependent base spreading resistance through temperature T jc o f the base-collector junction (eq. (4.66)). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 168 NETWCRK ANALYZER port 1 (a c + d c ) *-8 cc <—'W'v—J-VVV ^0 R ce VW NETWORK ANALYZER p o rt 2 (a c + d c)]1 Fig. 7.12. Equivalent circuit o f the HBT in the forward active region measurement, including the RF probe-pattem parasitics. It is important to note that the junction capacitances are known with precision only in the case where the corresponding junctions are in reverse bias. Thus, in the case o f the reverse-biased junction capacitances C jc and C jol o f fig. 7.12, the models defined by eqs. (6.13) and (6.25), with the extracted parameters listed in table 7.2, can be used. For the junction capacitance Cje (which in the case o f fig. 7.12 is the junction capacitance o f a forward-biased junction) the following semi-empirical model is used [62]: C j e (V b e ) = < C jE o [l + ’Tile (V b E - &E$' e )] » ^ a E$E (7.10) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 169 where m lE = ^ { l - c c E Y TTlE~l (7.11) The model o f eq. (7.10) is similar to the classical SPICE model for a junction capacitance in forward bias [12], in that the capacitance is a linear function o f voltage for Vbe > the capacitance-vs.-voltage function is also continuous in Q £$'£ . For consistency, the junction capacitances according to the model used for capacitance and C jol will be modeled in this work C j e - (l C j c (V b c ) = < C je ^ BC < a c ®'c C j c o \ X 4- r n i c ( V B c — a c $ c ) ] , V b c > & c $ ' c (7.12) C j o l {V b c ) = - i)C jc (V b c ) (7.13) In eqs. (7.12) and (7.13) the same voltage Vbc has been assumed for the basecollector junction and the overlap diode, due to the relatively small voltage drops across the series parasitic resistances of the transistor. In eq. (7.12), m ic = ^ ( 1 - a c ) - " * - 1 (7.14) Parameters cle, etc 6 (0,1) o f eqs. (7.10) and (7.12) are additional parameters o f the HBT model; because the base-collector junction and the overlap diode are assumed not to be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 170 in strong forward bias, a c = 0 . 9 will be imposed, and only a # will be an additional parameter o f the HBT model, which will be determined along with the other parameters. All the remaining parameters o f the HBT can be determined using the optimization feature o f IC-CAP and the forward active region S-parameter data. The optimization consists o f simultaneously fitting the MNS simulated S-parameters o f the model o f fig. 10(a) (whose temperature dependent parameters are modeled according to eqs. (4.41), (4.53), (4.54), (4.61), (4.63), (4.66), (4.97), (4.98), (4.99), (4.100), (4.101), and (4.102), whose junction capacitances are modeled according to eqs. (7.10), (7.12) and (7.13), and whose base-emitter diffusion capacitance is modeled according to eq. (6.23)) to the measured S-parameters using the configuration o f fig. 7.11, at different collector currents. The optimization consists o f minimizing the error function eps given by: £f s — real(5ijS(/n, VBmki Vcmk')') real(5jjm( /n, Vgm^, )I + |imag( S { j s ( f n , V sm ki imag( S ijm (f n 5 V sm k >V cm k ) ) | ^ (7.15) In eq. (7.15), f n is the n-th measurement frequency, Vsmk is the fc-th measured (applied) base voltage, Vcmjfc is the fc-th measured (applied) collector voltage, and Sij3 and Sijm denote the simulated and measured values, respectively, o f Sij, i , j = 1,2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 171 For the device under test, the preliminary values o f the probe-pattem parasitics listed in table 7.7 have been used as initial values in the minimization o f z f s defined above; the other initial values were: t f = 0, R b 2 = 10, R th c o = 3354 °CAV ( = R th e o , table 5.8), cie = 0.5. For the HBT model considered in this work it was assumed that R b \ = 0, in view o f the fact that this resistance is the contact resistance between the base terminal and the heavily doped base; resistance Rb-i is not negligible, because it accounts for the resistance of the base material between the base contact and the emitter edge o f the one-dimensional transistor. The results given by the optimizer for N f = 56 frequency points in the range [46 MHz, 25.87 GHz] and Ndc = 10 dc bias points according to table 7.9, are shown in table 7.10 along with the final optimization errors. The RMS error is sufficiently small, in view o f the large number o f points used in the optimization. The choice o f the dc bias voltages ensures that the dc collector o f the HBT takes on values ranging from very small (where a g has a dominant effect) to values which produce device self-heating (in the mA range, where t f , R b 2 and R th c o have a dominant effect). With the parameters o f table 7.10 and the ones extracted in chapters 3, 5, and section 7.1 for the device under test, the simulated S-parameters o f the HBT for Vb = 1.435 V, Vc = 2.435 V (last set o f dc voltages in table 7.9) are shown in fig. 7.13, along with the measured S-parameters. The agreement between the curves is very good, which indicates the correctness o f the model. For the other sets o f dc voltages o f table 7.9, the agreement between the S-parameters is also very good; the complete set o f simulated and measured dc characteristics and S-parameter curves for this device (referred to as the T8 device) will be given in chapter 8. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 172 Table 7.10. Parameters t f , a E, R m , R t h c o , and the probe-pattem parasitics o f the HBT after optimization tf R b2 R thco aE C be C bc C cE R be R bc R ce L e Lb Lc MAXIMUM error RMS error 2.393 ps 133.2 SI 4289°C/W 0.6495 2.540 fF 26.28 fF 9.465 fF 0 25.36 A 0 9.258 pH 51.27 pH 58.56 pH 11.66% 1.662% The MNS circuit used in the above optimization, which is the complete model for the HBT, is given in appendix B. The numerical values o f the model parameters in appendix B are the parameters for the T8 device. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 173 i o -i (a) S n 0 .3 3 (b) S' to Fig. 7.13. Measured (I) and simulated (2) S-parameters o f the HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.435 V, Vc = 2.435 V (jBm = 20.79//A, I Cm = 1.4 mA) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 174 Fig. 7.13. Continued (c and d). 1 .6 (C) S-2i t 0.5 0.2 [0.2 - 0.2 -0 .5 I (d) S-2-2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 8 RESULTS AND DISCUSSION This chapter presents the dc and high frequency simulated versus measured data for two HBT's o f different types, whose models have been extracted following the procedures described in the previous chapters. The two transistors are referred to as the T8 device and the 78 device, respectively; all the model parameters extracted in the previous chapters are for the T8 device. In the measurement and characterization o f the HBT's, the following equipment and computer software has been used: — an HP 9000 Series 300 computer and an HP Model 712 workstation, both running Hewlett-Packard's version 4.30HF IC-CAP software [13], with version B.06.20 o f the MNS simulator; — an HP 4142B dc modular source/monitor controlled by IC-CAP via an HP-IB bus, for providing and measuring all the transistor dc voltages and currents; — an HP 8510B network analyzer controlled by IC-CAP via an HP-IB bus, for measuring the high-frequency S-parameters o f the transistors at different dc biases; — a Cascade Microtech Model 12 microwave R&D probe fixture, and GGB Industries Model 40A microwave probes with 100 fim and 150 pm tip spacing, for on-wafer probing o f the transistors; — a refrigerated/heated circulating bath, for varying the temperature at which the transistors are measured. 175 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 176 The T8 and 78 transistors were two essentially different transistors, coming from different wafers and fabrication processes, and having different structures and probe-pattem layouts. These layouts are shown in figs. 8.1 and 8.2, respectively; the device information provided by the manufacturer is given in tables 8.1 and 8.2, respectively. Collector HBT v ii I Emitter Collector Fig. 8.1. Layout o f the T8 device probe pattern. Collector HBT 7 Base 7 Collector Emitter Fig. 8.2. Layout o f the 78 device probe pattern. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 177 Table 8.1. Device information for the T8 transistor (npn HBT, common collector, one emitter finger, two base fingers) Ae Ac Jm ax Pdm ax V icB m a x V B E m ax ^ B E re m iia j: estimated thermal resistance T-j m a x 8.25 p m 2 42.6 pm 2 6 x 104 A/cm2 0.03 W 3V 1.7 V 0.5 V 2000°C/W 150 °C Table 8.2. Device information for the 78 transistor (npn HBT, common collector, one emitter finger, two base fingers) A e Ac Jm ax V cE m ax estimated thermal resistance TJ -jm a x 9 pm 2 21 pm 2 6 x 104 A/cm2 8V 2400°C/W 150 °C In tables 8.1 and 8.2, J max is the maximum allowable current density through the device junctions and Tjr,iax is the maximum junction temperature; P<fmax in table 8.1 is the maximum dissipated power. Except for the junction areas A g and A c and the estimated thermal resistance, all the parameters in tables 8.1 and 8.2 are maximum values which have been used in determining the maximum dc voltages or currents applied to the device in the measurement process, as well as the corresponding compliance values o f the source/monitor units (SM Us) of the HP 4142B. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 178 Table 8.3. Model parameters for the T8 HBT at Tamt, = 23.5 °C Is IS E IS R O L Nf Nr Ne N rol Pf Pr Ka R btiiox o-r b I BO RpBE R pce R pbc Re Rb\ R b -i Rc R theo R thco R tholo K th c Ks Xpi 1.506 X 10"26 A 2.874 x 10-18 A 21.86 fA 1.044 1.027 1.948 1.977 109.3 1.451 4.164 11.01 Kfi 0.8686 1.76 64.42 Gf2 27.93 Gfi 311.6 Mft 7.422 Q 0 133.2 n 3.413 n 3354 °CAV 4289 °CAV 0 0.5114 11.16 x l O - ^ C '1 6.110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 179 Table 8.3. Continued X te X trol X jrr X fb Po X trbb C jco *c me occ C jeo mE <*E Lb Le Lc C bc C oe C be R bc R ce R be tf 7.218 4.242 - 1.993 0.1173 9.128 144.0 8.351 3.471 fF 2.513 V 3.712 0.9 1.074 fF 1.576 V 2.081 0.6495 51.27 pH 9.258 pH 58.56 pH 26.28 fF 9.465 fF 2.540 fF 25.36 ft 0 0 2.393 ps Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 180 Table 8.4. Model parameters for the 78 HBT at Tamb = 23.5 °C Is IsE IsR O L n f N r N e N rol Pf Pr Ka R -B m a x a RB I bq RpBE R pce R pbc R e R bi R b2 Rc R th eo R th co R th o lq K th c Ks X ti 1.447 X 10~25 A 1.386 x 1 0 "16 A 17.82 fA 1.049 1.040 2.238 1.968 225.0 5.154 1.333 2.010 Kft 1.742 7.987 /zA 889.1 M fl 46.50 Mf2 42.16 13.37 Q 0 102.7 n 5.037n 4190 °CAV 0 0 0.4811 0 2.529 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 181 Table 8.4. Continued X Te X trol X tbr X h A) X trbb C jco *'c me C jE Q mE <*E L b L e Lc C bc C cE C be R bc R ce R be tf 1.316 3.959 - 0.020 0.1322 18.76 299.1 0 6.450 fF 1.764 V 1.879 0.9 5.635 fF 2.252 V 0.8285 0.7617 55.40 pH 40.80 pH 0 43.19 fF 25.84 fF 17.50 fF 16.52 Q 0 71.10 a 2.822 ps The model parameters for the T8 transistor (including the probe-pattem parasitics) have been determined in the previous chapters o f this work, and are given in condensed form in table 8.3. The MNS circuit file for this transistor is given in appendix B, and has been used in section 7.3 to generate the simulated data shown in fig. 7.13. The model parameters for the 78 transistor have been determined using the same procedures that were employed for the T8 device. In the case o f the 78 transistor, however, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 182 because the base spreading resistance is relatively small, its temperature dependence could not be determined accurately. Therefore, X t r b b = 0 and R th c o = 0 (see eqs. (4.66) and (4.98)) have been imposed in the model for the 78 device, and R e has been assumed an optimization variable in the minimization o f eps (eq. (7.15)). Thus, a slight disagreement at large collector currents occurred in the case o f the forward Gummel configuration (V b c = 0) between the simulated and the measured data, which is inconsequential in view o f the excellent agreement in all the other configurations. The model parameters for the 78 transistor, including the probe-pattem parasitics, are given in table 8.4; the MNS circuit file for this device is given in appendix C. It is important to note that, in general, the parasitic series resistances o f the transistor model (R b i, R b 2 , R e , and R c o f fig. 6.17) are functions o f temperature, and may depend on the history o f the current passing through them at different temperatures [63]. In addition, at a given temperature, the series resistances o f a bipolar transistor may be current dependent; for example, the emitter resistance may approach surprisingly high values at low collector currents, as demonstrated in [64], Thus, especially in the self-heating regions o f the transistor characteristics, the current and temperature dependencies o f the series resistances and o f the other model parameters may interact in a complex manner. For reasonable simplicity, however, the series resistances o f the transistor have been assumed constant in this work (independent o f current and/or temperature). As a result, following the extraction procedures described in the previous chapters, some o f the other model parameters (such as X t r b b and R-thcq) may assume non-physical values due to the possible current/temperature dependence o f the series resistances. It is also important to notice that, at moderate dc collector currents in the forward active region o f the HBT, the base-emitter junction capacitance and the base-emitter diffusion Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 183 capacitance o f the transistor may have comparable values, and the standard model used in this work for the HBT junction capacitances (see chapter 7) is only approximate in forward bias, since the physical junction capacitance is actually a decreasing function for sufficiently large bias voltages [11]. Therefore, due to the fact that the optimization for determining the forward transit time o f the HBT (section 7.3) is performed using measured data at collector currents in a wide range (which includes large currents where the actual base-emitter junction capacitance is likely to be considerably different from the value predicted by the proposed model), the probe-pattem parasitics o f the device— which are also variables in the optimization— may converge to values which are substantially different from their preliminary values (see sections 7.2 and 7.3). Using the models o f appendix B and appendix C for the T8 and 78 HBT's, the simulated vs. measured data is shown in figs. 8.3-8.16 (T8), and figs. 8.17-8.30 (78), respectively. These sets o f figures show the dc simulated vs. measured data in the forward and reverse Gummel configurations (figs. 8.3, 8.4 for the T8 device, and 8.17, 8.18 for the 78 device, respectively), the dc simulated vs. measured data in the open-collector configuration (fig. 8.5 for T8, and fig. 8.19 for 78), the simulated vs. measured dc output characteristics (fig. 8.6 for T8, and fig. 8.20 for 78), and the simulated vs. measured S-parameters at 10 bias points in the forward active region (figs. 8.7-8.16 for T8, and figs. 8.21-8.30 for 78), in the frequency range [46 MHz, 25.87 GHz], For the T8 and 78 transistors, the S-parameters have been measured at the dc base and collector voltages and currents given in table 8.5 (table 7.9) and table 8.6, respectively. The agreement between the simulated and measured data is very good for both devices, and demonstrates the correctness of the models and o f the procedures used to extract the model parameters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 8.5. Applied dc base and collector voltages and measured dc base and collector currents, corresponding to the measured S-parameters o f the T8 device in the forward active region (VE = 0, Vc - VB = 1 V) Vb OO 1.255 1.275 1.295 1.315 1.335 1.355 1.375 1.395 1.415 1.435 VC (V) 2.255 2.275 2.295 2.315 2.335 2.355 2.375 2.395 2.415 2.435 h m (M ) 0.276 0.424 0.656 1.035 1.620 2.611 4.246 7.064 11.94 20.79 Icm ( M ) 3.691 7.518 15.13 30.23 59.62 115 221 418 773 1403 Table 8.6. Applied dc base and collector voltages and measured dc base and collector currents, corresponding to the measured S-parameters o f the 78 device in the forward active region (VE = 0, Vc - VB = 1 V) VB (V ) 1.181 1.201 1.221 1.241 1.261 1.281 1.301 1.321 1.341 1.361 Vfc(V) 2.181 2.201 2.221 2.241 2.261 2.281 2.301 2.321 2.341 2.361 h m (M ) 0.121 0.185 0.278 0.421 0.643 1.008 1.630 2.724 4.721 8.494 Ic,n ( M ) 1.974 4.121 8.560 17.79 36.55 74.51 149.2 295.6 578.2 1114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 185 10-2 2 1 0 - 3 aj 1 io -5 10“ 6 J iO ~ 7 *1 10“ 8 3 Civ) 10-9 10-10 1 0 - 11 0. 8 m 1 1. 2 VBm - VEm 14 1. 6 00 Fig. 8.3. T8 HBT dc measured collector and base currents Icm (0 and /em(ii), and corresponding simulated currents I c 3 (iii) and Ib 3 (iv), in the forward Gummel configuration. -2 —4 (4 K| •"i -6 (4 (iv) .«* E K| •v E 0.2 0.4 0.6 0.8 VBm ~ VCm 1 00 Fig. 8.4. T8 HBT dc measured base and emitter currents ig m (i) and I Em (ii), and corresponding simulated currents I b 3 (iii) and I e* (iv), in the reverse Gummel configuration. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 186 6 <P) 2 £*. VJ 0.8 J•k E 0 .4 8? 0 8 r 6 -5 4 3 -2 *Bm <A > Fig. 8.5. T8 HBT dc measured base and collector voltages V#m (i) and Vcm (ii), and corresponding simulated voltages Vb 3 (iii) and V c3 (iv), in the open collector configuration (Vg = 0, I q = 0). + k 1 *1-44 j- M - H I H H 4 + | I I I + + + 4 j I I H I I I I I j I I H V C m ~ VE m 00 Fig. 8.6. T8 HBT dc measured collector current I c m (symbols), and simulated collector current I c a (lines), as functions o f the collector-emitter voltage (Vcm — VEm)- The base current step is 6 fjA. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 187 I 0 .5 0. 2 , . 0.2 - Th5 0 .2 -0 .5 t (a )S „ 0.2?I0.2l40.l630.l0dk0.0j (b) S 12 Fig. 8.7. Measured (1) and simulated (2) S-parameters o f the T8 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.255 V, Vc = 2.255 V ( I Bm = 0.276 /^A, I Cm = 3.691 M ) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 8.7. Continued (c and d). 10.2. (C)52l I 0.5 0 .2 0.2 - l0 .5 0.2 -0 .5 -I (d) 5 o2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 189 t 0 .5 0.2 , ,0.2 - ,0.5 0.2 -2 - 0 .5 I (a )5 u (b) S 12 Fig. 8.8. Measured (1) and simulated (2) S-parameters o f the T8 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for Vg = 1.275 V, Vc = 2.275 V ( I Bm = 0.424 M , I Cm = 7.518 /xA) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 190 Fig. 8.8. Continued (c and d). (c) S -21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 191 I 0 .5 0.2 0 .2 - .0.5 0.2 -0 .5 -I (a )S n 0 .2 9 0 .2 3 2 0 .1 7 4 0 .1 1 4 0 .0 ' (b) 5 12 Fig. 8.9. Measured (1) and simulated (2) S-parameters o f the T8 HBT including theRF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for Vb = 1.295 V, Vc = 2.295 V (I f a = 0.656/izA, I Cm = 15.13 /zA) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 192 Fig. 8.9. Continued (c and d). (c) So i I 0.5 0.2 0 .2 - ,0.5 0.2 -2 -0 .5 -I (d) S-2-2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 193 I (a) S i i 0 .3 0 .2 i 10.18 10.12 \0.0l (b )S I2 Fig. 8.10. Measured (1) and simulated (2) S-parameters o f the T8 HBT including theRF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.315 V, Vc = 2.315 V {IBm = 1.035 fiA, I Cm = 30.23 fiA) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 194 Fig. 8.10. Continued (c and d). (c) So, I 0.5 0.2 0 .2 - 0.5 0.2 -2 -0 .5 -I (d) So-, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 195 (a ) S 11 0.31 (b) 5,2 Fig. 8.11. Measured (I) and simulated (2) S-parameters o f the T8 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.335 V, Vc = 2.335 V ( I Bm = 1.620/xA, I Cm = 59.62 fiA ) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 8.11. Continued (c and d). (c) So 1 L 0.5 0.2 iQ .2 - 0 .5 0.2 -0 .5 I (d) Soo Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (b) 5 ,2 Fig. 8.12. Measured (1) and simulated (2) S-parameters o f the T8 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for V# = 1.355 V, Vc = 2.355 V ( I Bm = 2.611 ^A , I Cm = 115 n A) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 198 Fig. 8.12. Continued (c and d). (C) Sol (d) 5*2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 199 i (a) S u 0.35 0 .2 8 10.21 \Q .M \0.0, G>) S 12 Fig. 8.13. Measured (1) and simulated (2) S-parameters o f the T8 HBT including theRF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.375 V, Vc = 2.375 V ( I Bm = 4.246/iA, I Cm = 221 fiA ) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 200 Fig. 8.13. Continued (c and d). (c) So 1 (d) Soo Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 201 i ( a )S „ 0 .3 6 10 .2 0 8 1 0 .2 1 ^ 0 .1 4 ^ 0 .0 . (b)5ia Fig. 8.14. Measured (1) and simulated (2) S-parameters o f the T8 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for Vjg = 1.395 V, Vc = 2.395 V (IBm = 7.064/iA, I Cm = 418 pA) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 202 Fig. 8.14. Continued (c and d). (c) 5o[ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 203 (a)5„ 0.38 0.30110.22610.15210.0, (b) 5,2 Fig. 8.15. Measured (1) and simulated (2) S-parameters o f the T8 HBT including theR F probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.415 V, Vc = 2.415 V (IBm = 11.94/aA, I Cm = 773 /xA) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 8.15. Continued (c and d). (C) S o l I 0.5 0.2 ,0.5 0.2 * -2 -0 .5 I (d) Soo Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 205 (a ) S U 0.39 [0.3L2lO .23M 0.L5^0.q; (b) 5,0 Fig. 8.16. Measured (1) and simulated (2) S-parameters o f the T8 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for Vg = 1.435 V, Vc = 2.435 V (.I Bm = 20.79/xA, I Cm = 1.4 mA) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 206 Fig. 8.16. Continued (c and d). 1.6 (C)S2| 0 .5 0.2 0.2 - 0.2 -0 .5 “I (d) S-22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -4 m -6 <m -9 0.8 00 VBm ~ VE m Fig. 8.17. 78 HBT dc measured collector and base currents I c m (i) and Ism (ii), and corresponding simulated currents I d (iii) and Is* (iv), in the forward Gummel configuration. 10~2 K? 1 0 “ 5 I cq K'« 10“6 i.-r (Iv) „ 10“ 8 10~9 0.2 0 4 0.6 0.8 VB m - l VCm 1.2 1.4 00 Fig. 8.18. 78 HBT dc measured base and emitter currents Ib m (0 and I Em and corresponding simulated currents Ib* (iii) and Ies (iv), in the reverse Gummel configuration. 00, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 208 (HI) t* i - Z £ s?o.* I i 04 !? (* ) (iv) “8 -4 10 -2 Fig. 8.19. 78 HBT dc measured base and collector voltages Vsm (i) and Vcr (ii), and corresponding simulated voltages VBs (iii) and V c3 (iv), in the open collector configuration (V e = 0, I c = 0). 1 11 M l i m i I IM L L IL L I 111 ni i ii ii ii'nii wj v up u t I 1 2 ~ 00 Fig. 8.20. 78 HBT dc measured collector current I c m (symbols), and simulated collector current I c s (lines), as functions of the collector-emitter voltage (Vcm — VEm ). The base current step is 4 n A. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 209 I (a) 5 „ 0 .3 5 10.28 10.21 10.11 \0.0; (b) 5,2 Fig. 8.21. Measured (I) and simulated (2) S-parameters o f the 78 HBT including theRF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.181 V, Vc = 2.181 V ( I Bm = 0.121 a*A, I Cm = 1.974 p A ) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 210 Fig. 8.21. Continued (c and d). (c)Soi I 0.5 0.2 0.2 - iO.S 0.2 -0 .5 -I (d) 522 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 211 (a)5n 0 .3 5 0.28 0.21 \0.M \0.0. (b) 5 I2 Fig. 8.22. Measured (1) and simulated (2) S-parameters o f the 78 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.201 V, Vc = 2.201 V ( I Bm = 0.185 imA, I Cm = 4.121 /*A) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 212 Fig. 8.22. Continued (c and d). (c) S2l i 0.5 0.2 ,0.2 - ,0.5 0.2 -2 -0 .5 -I (d) S2., Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (b) 5 ia Fig. 8.23. Measured (1) and simulated (2) S-parameters o f the 78 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for Vq = 1.221 V, Vc = 2.221 V (I Bm = 0.278//A, I Cm = 8.560 (j,A) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 214 Fig. 8.23. Continued (c and d). (c) 5-21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 215 (a )S n 0 .3 7 10.29* 0 .2 2 2 0 . MAO.Oj (b) S 12 Fig. 8.24. Measured (1) and simulated (2) S-parameters o f the 78 HBT including theRF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for Vq = 1.241 V, Vc = 2.241 V (.l Bm = 0.421/zA, I Cm = 17.79 p A ) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 216 Fig. 8.24. Continued (c and d). (c) S , i L 0.5 0. 2 , 0 .2 - 0.2 •0 .5 •I (d) So-2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (b) S v2 Fig. 8.25. Measured (1) and simulated (2) S-parameters o f the 78 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for V# = 1.261 V, Vc = 2.261 V (IBm = 0.643 y K I Cm = 36.55 y,A) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 218 Fig. 8.25. Continued (c and d). (c) 5 2t I 0.5 0.2 ;0 .2 - 0.5 0.2 -2 - 0 .5 I (d) S 2o Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 219 (a )S H CM 10.32 0.24 10.16 \0.0I (b) S l2 Fig. 8.26. Measured (1) and simulated (2) S-parameters o f the 78 HBT including theRF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.281 V, Vc = 2.281 V ( I Bm = 1.008 M , Icm = 74.51 iiA ) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 8.26. Continued (c and d). (c)So, L 0.5 0.2 0.2 - K5 0.2 -2 -0 .5 I (d) So2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 221 (a )S „ [0 .1 2 0.33dO.252SO.L6ao.OI (b) S\o Fig. 8.27. Measured ( l) and simulated (2) S-parameters o f the 78 HBT including theRF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for Vg = 1.301 V, Vc = 2.301 V (IBm = 1.630/xA, I Cm = 149.2 fiA) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 8.27. Continued (c and d). (c) S 21 1 0.5 0.2 0.2 - 0.5 1 0.2 -2 -0 .5 I (d) S-2-2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 223 i 0.5 0 .2 ^2 - [0.5 0.2 -0 .5 -I (a)5„ 0.352 0 .2 6 4 0 .1 7 4 0 .0 1 (b) 5 ,2 Fig. 8.28. Measured (1) and simulated (2) S-parameters o f the 78 HBT including theRF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.321 V, Vc = 2.321 V (IBm = 2.724, n K Icm = 295.6 ft A) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 224 Fig. 8.28. Continued (c and d). (c)S221 (d) s>2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 225 (a) S n 0 .4 6 10.36810.27610.1840.0; (b) S a Fig. 8.29. Measured (1) and simulated (2) S-parameters o f the 78 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for VB = 1.341 V, Vc = 2.341 V ( I Bm = 4.721 /xA, I Cm = 578.2 /xA) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 226 Fig. 8.29. Continued (c and d). . 0.8 (c) S-n Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission. 227 (a) S n 0.4 7 IO.376tO.282lO.l8AO.OJ (b) 5 ia Fig. 8.30. Measured (1) and simulated (2) S-parameters o f the 78 HBT including the RF probe-pattem parasitics, in the frequency range [46 MHz, 25.87 GHz], for Vb = 1.361 V, Vc = 2.361 V (JSm = 8.494 /uA, l Cm = 1.11 mA) (a and b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 228 Fig. 8.30. Continued (c and d). 1.6 (c) S -n i 0 .5 0.2 , - 0 .2 10.5' 0.2 -0 .5 I (d) 522 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 9 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH An accurate physics-based model for the heterojunction bipolar transistor (HBT) has been developed in this work. The model is large-signal and temperature dependent, and is intended for use in RF and microwave applications. Unlike the existing models for HBT's, the present model represents planar singleheterojunction structures with the collector junction area larger than the emitter junction area, by including an additional base-collector overlap diode in the model o f a one-dimensional transistor structure; all devices modeled in this work have been planar devices o f the abovementioned type. An accurate, temperature dependent model for the base spreading resistance o f the HBT has been developed; in the case o f HBT's with small base spreading resistances, the temperature dependence o f this resistance model can be relaxed, as demonstrated by the results on one type o f test devices. The proposed temperature dependence o f the base spreading resistance assumes that the temperature in the base region is constant and equal to the average o f the emitter and collector junction temperatures; by considering the temperature distribution in the base, a more accurate expression for this dependence can be found. The temperature characterization o f the transistor model parameters is implemented by using the explicit temperature dependencies o f the energy gaps o f the emitter, base, and collector regions o f the transistor. Physics-based mathematical expressions for the temperature 229 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 230 dependencies o f the transistor parameters are obtained from the boundary conditions corresponding to an arbitrary bias, and by taking into account the recombination/generation currents in the space-charge regions. The temperature dependent model o f the HBT uses an accurate thermal circuit, which includes the temperature responses o f the em itter and collector junctions and o f the overlap diode. The temperature dependencies o f the model parameters have been extracted from the dc characteristics o f the transistors measured at different temperatures. Since the measurements o f the HBT's in this work have been only dc and smallsignal, no effort has been put into determining the thermal capacitances o f the HBT thermal circuit. The extraction o f these capacitances, however, is desirable for a more complete characterization o f the devices in transient regimes. Compared to the existing high-frequency models for the HBT, the model proposed in this work is an improvement in that a more appropriate placement o f the junction and diffusion capacitances is used. The proposed high-frequency model allows an accurate characterization o f the HBT in a wide range o f collector currents and frequencies. In order to keep the proposed HBT model reasonably simple, the spreading effects in the overlap diode have not been modeled here. One has to be aware, however, that the distributed nature o f the overlap diode must be accounted for in cases where this diode is in forward bias and the frequency range is sufficiently large. An equivalent circuit similar to a circuit previously used for HEMT's has been used in this work to model the probe-pattem parasitics o f the HBT. The advantage to using this circuit is that the RF portion has no influence on the dc characteristics o f the device, and ensures an accurate high-frequency characterization. In this way, the high-frequency model extraction does not affect the previously-extracted static model o f the transistor. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 231 The extraction procedures for all the HBT model parameters have been developed and described in detail. The procedures rely on the IC-CAP software, which is a modern system that allows a variety o f operations on the data, including mathematical transformation and optimization. The HBT models for the measured devices have been implemented in the MNS simulator, also controlled by IC-CAP. The MNS simulator has been preferred over other simulators due to the possibility o f implementing high-complexity mathematical functions using symbolically-defined devices. The validity o f the proposed HBT model and o f the model extraction procedures has been demonstrated by the excellent agreement between the measured dc characteristics and Sparameters o f the transistor, and the corresponding curves obtained by simulating the extracted model; the agreement between the simulated and measured data has been verified for two essentially different HBT's, which demonstrates the versatility o f the model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX A MNS CIRCUIT FILE USED IN THE DETERMINATION OF THE TEMPERATURE DEPENDENDENT MODEL 232 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 233 define hbt (C E B) ISROL = 2 . 186e-14 ISE = 2 . 875e-18 IS = 1.506e-26 NROL = 1.977 NE = 1.948 NF = 1.044 NR = 1.027 BF = 109.3 BR = 1.451 VT = 0.02556 Ka = 5.479 RBmax = 1.101e4 aRB = 0.8686 IBO = 1.76e-6 RPBE = 6.442el0 RPCE = 2 .793el0 RPBC = 3.116e8 RE = 1 RC = 1 RTHEO = 2000 RTHOLO = 100 KTHC = 0 . 5 KS = 0.002 TEMP = 2 3 . 5 XTI = 6.11 XTE = 7.218 XTROL = 4.242 XTBR = -1.993 x_Al = 0.1173 fb = 9.128 betaO = 144 XTRBB = 1 r:rrl r:rr2 r :rr3 r:rr4 r:rr5 r :rr6 r :rr7 E El r=RE C Cl r=RC B E r=RPBE B C r=RPBC C E r=RPCE NTE 0 r=RTHE0 NTOL 0 r=RTHOLO diode ( w l , isat, n, w 2 ) = \ isat* (exp_soft ( w l / ( n * w 2 ) ) - 1) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 234 maxexp = le27 small = le-50 max_arg = ln(max_exp) exp_soft(x) = if (x<max_arg) (x+l-max_arg) *max_exp endif then exp(x) else \ Tn = TEMP + 273.15 deltatE = _V6 deltatOL = _V7 TjE = Tn + deltatE TjC = Tn TOL = Tn + deltatOL V Tn = T n * 8 .61738e-5 VTjE = T j E * 8 .61738e-5 VTjC = T n * 8 .61738e-5 VTOL = TOL* 8 . 61738e-5 KRE = RTHE0*KS KROL = RTHOLO*KS EgB(T) = 1.519 - (5.405e-4)* (TA2)/ (T + 204) EgGAlAs(T) = 3.114 - (5.41e-4)* (TA2) / (T + 204) EgE(T) = (1 - x_Al )* E g B (T ) + x_Al*EgGAlAs(T) + \ 0.37*x _ A l *(1 - x_Al) deltaEg (T) = EgE(T) - EgB(T) tISl = (TjE/Tn)"XTI 151 = IS*tISl*exp_soft(-EgB(TjE)/VTjE + EgB(Tn)/VTn) tIS2 = (TjC/Tn)~XTI 152 = IS*tIS2*exp_soft(-EgB(TjC)/VTjC + EgB(Tn)/VTn) tIS3 = (TOL/Tn)~XTI 153 = IS*tIS3*exp_soft(-EgB(TOL)/VTOL + EgB(Tn)/VTn) tISE = (TjE/Tn)~XTE ISE1 = ISE*tISE*exp_soft(-EgE(TjE)/ (2*VTjE) EgE(Tn)/ (2*VTn)) + \ tISROL = (TOL/Tn)AXTROL ISROL1 = ISROL*tISROL*exp_soft(-EgB(TOL)/ (2*VTOL) EgB(Tn)/ (2*VTn)) + \ tBRl = (TjC/Tn)~XTBR BR1 = BR*tBRl tBR2 = (TOL/Tn)"XTBR BR2 = BR*tBR2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. numBF = fb*exp_soft(-deltaEg(Tn)/VTn) + 1/betaO denBF = fb*exp_soft(-deltaEg(TjE)/VTjE) + 1/betaO BF1 = BF*numBF/denBF vbel = __V3 - _V4 vbcl = _V1 - _V2 vbc2 = _V3 - _V2 ice = diode(vbel, IS1, NF, VTjE) iec = diode(vbc2, IS2, NR, VTjC) ict = icc - iec ibl = diode(vbcl, IS3*Ka*(BR2+1)/BR2, NR, VTOL) diode(vbcl, ISR0L1, NROL, VTOL) ib2 = diode(vbel, IS1/BF1, NF, VTjE) + \ diode(vbel, ISE1, NE, VTjE) ib3 = diode(vbc2, IS2/BR1, VTjC) pde = pdc = pdol = pdel = + \ vbel*(ib2 + ict) vbc2*(ib3 - ict) vbcl*ibl pde + KTHC*pdc rbase = (((TjE + TjC) / (2*Tn) )''XTRBB) *\ RBmax/ (1+ ( ( (_i5+small) /IBO) ''aRB) ) sdd:sddhbt B 0 Cl 0 B1 0 El 0 B B1 NTE 0 NTOL 0 \ i [1,0]=ibl i [2,0]=-ibl+ict-ib3 \ i [3,0]=ib2+ib3 i [4,0]=-ib2-ict f [5,0]=_v5-_i5*rbase \ i [6, 0]=-pdel/(l-KRE*pdel) \ i [7,0]= - p d o l / (l-KROL*pdol) end hbt Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX B COMPLETE MNS CIRCUIT FILE FOR THE T8 DEVICE MODEL 236 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. define hbt (C E B) ISROL = 21.86f ISE = 2 . 8745E-18 IS = 1. 506E-26 NR O L = 1.977 NE = 1.94812 N F = 1.044 N R = 1.027 B F = 109.3 BR = 1.451 VT = 0.0255635 Ka = 4.164 RBmax = 11.OIK aRB = 0.8686 IBO = 1.760U CJCO = 3 .471f PHIC = 2.513 me = 3.712 alphac = 0.9 CJE0 = 1. 074f PHIE = 1.576 me = 2.081 alphae = 0.6495 LB = 51.27p LE = 9.258p LC = 58.56p CBC = 26.28f CCE = 9.4 65f CBE = 2 . 540f RBC = 25.36 RCE = 1.000 RBE = 1.000 RPBE = 64.42G RPCE = 27.93G RPBC = 311.6MEG tauf = 2.393p RE = 7.422 RB2 = 133.2 RC = 3.413 RTHEO = 3354 RTHCO = 4289 RTHOLO = 0 . 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. KTH C = 0.5114 KS = 1 1 . 16e-3 TEMP = 2 3 . 5 XTI = 6.110 XTE = 7.218 XTROL = 4.242 XTBR = -1.993 x_Al = 117.3m fb = 9.128 betaO = 144.0 XTRBB = 8.351 r:rrl B1 BC r=RBC r:rr2 El BE r=RBE r :rr3 El CE r=RCE r:rr4 B1 B2 r=RB2 r:rr5 El E2 r=RE r :rr6 Cl C2 r=RC r:rr7 B1 El r=RPBE r :rr8 B1 Cl r=RPBC r:rr9 Cl El r=RPCE r :rrlO NTE 0 r=RTHEO r :rrll NTC 0 r=RTHCO r :rrl2 NTOL 0 r=RTHOL0 1:111 B B1 1=LB 1:112 E El 1=LE 1:113 C Cl 1=LC c:ccl BC Cl c=CBC c:cc2 Cl CE c=CCE c:cc3 B1 BE c=CBE diode ( w l , isat, n, w 2 ) = \ isat * (exp_soft ( w l / ( n * w 2 ) ) - 1) max_exp = le27 small = le-50 max_arg = In(max_exp) exp_soft(x) = if (x<max_arg) (x+l-max_arg)*max_exp endif then exp(x) else \ charge(cO, v, m, vO, alpha) = if (v < alpha*v0) then \ chargel(cO, v, m, vO, alpha) else \ charge2(c0, v, m, vO, alpha) endif chargel(cO, v, m, vO, alpha) = \ - cO*vO*((l - v/ vO)A (l - m))/(l - m) charge2(c0, v, m, vO, alpha) = cO*(v - alpha*vO + \ 0.5*m*(v - alpha*vO)^ 2 / (vO*(1 - alpha)))/(! - alpha)Am + Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 239 chargel(cO, alpha*vO, m, vO, alpha) Tn = TEMP + 273.15 deltatE = _V7 deltatC = _V8 deltatOL = _V9 TjE = Tn + deltatE TjC = Tn + deltatC TjOL = Tn + deltatOL V T n = T n * 8 .61738e-5 VTjE = T j E * 8 .61738e-5 VTjC = TjC*8.61738e-5 VTjOL = TjO L*8 . 61738e-5 KRE = RTHE0*KS KRC = RTHC0*KS KROL = RTHOLO*KS EgB(T) = 1.519 - (5.405e-4)* (TA2)/ (T + 204) EgGAlAs (T) = 3.114 - (5 .4 le-4) * (TA2 )/ (T + 204) EgE(T) = (1 - x_Al)*EgB(T) + x_Al*EgGAlAs(T) + \ 0.3 7 * x _ A l * (1 - x_Al) deltaEg (T) = EgE(T) - EgB (T) tISl = (TjE/Tn)AXTI 151 = IS*tISl*exp_soft(-EgB(TjE)/VTjE + EgB(Tn)/VTn) tIS2 = (TjC/Tn)AXTI 152 = IS*tIS2*exp_soft(-EgB(TjC)/VTjC + EgB(Tn)/VTn) tIS3 = (TjOL/Tn)AXTI 153 = IS*tIS3*exp_soft(-EgB(TjOL)/VTjOL + EgB(Tn)/VTn) tISE = (TjE/Tn)AXTE ISE1 = ISE*tISE*exp_soft(-EgE(TjE)/ (2*VTjE) EgE(Tn)/ (2*VTn)) + \ tISROL = (TjOL/Tn)AXTROL ISROL1 = ISROL*tISROL*exp_soft(-EgB(TjOL)/ (2*VTjOL) EgB(Tn)/ (2*VTn)) + \ tBRl = (TjC/Tn)AXTBR BR1 = BR*tBRl tBR2 = (TjOL/Tn)AXTBR BR2 = BR*tBR2 numBF = fb*exp_soft(-deltaEg(Tn)/VTn) + 1/betaO denBF = fb*exp_soft(-deltaEg(TjE)/VTjE) + 1/betaO Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 240 BF1 = BF*numBF/denBF vbel vbe2 vbcl vbc2 vbc3 = = = = = _V3 _V1 _V6 _V3 _V1 - V4 V4 V2 V2 V2 icc = diode(vbel, IS1, NF, VTjE) iec = diode(vbc2, IS2, NR, VTjC) ict = icc - iec ibl = diode(vbcl, IS3*Ka*(BR2+1)/BR2, NR, VTjOL) diode(vbcl, ISROL1, NROL, VTjOL) ib2 = diode(vbel, IS1/BF1, NF, VTjE) + \ diode(vbel, ISE1, NE, VTjE) ib3 = diode (vbc2, IS2/BR1, NR, VTjC) pde pde pdol pdel = vbel*(ib2 + ict) = vbc2*(ib3 - ict) = vbcl*ibl = pde + KTHC*pdc qbcl qbc2 qbel qdel = = = = + \ c h a r g e ((Ka/(Ka + 1))*CJC0, vbcl, me, PHIC, alphac) c h a r g e ((1/(Ka + 1))*CJC0, vbc3, me, PHIC, alphac) charge(CJEO, vbe2, me, PHIE, alphae) tauf*icc rbase = (((TjE + TjC)/ (2*Tn))AXTRBB)*RBmax/\ (1+(((_i5+small)/IBO)"aRB)) sdd:sddhbt B2 0 C2 0 B3 0 E2 0 B2 B3 B1 0 NTE 0 NTC 0 \ NTOL 0 \ i[6,0]=ibl i [2,0]=-ibl+ict-ib3 \ i [3,0]=ib2+ib3 i [4,0]=-ib2-ict f [5,0]=_v5-_i5*rbase \ i[7, 0]=-pdel/(l-KRE*pdel) i [8,0]=-pdc/(l-KRC*pdc) \ i [9, 0]=-pdol/(l-KROL*pdol) \ i [1,1]=qbc2+qbel+qdel \ i [2, 1]=-qbcl-qbc2 i [4,1]=-qbel-qdel i[6,l]=qbcl end hbt Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX C COMPLETE MNS CIRCUIT FILE FOR THE 78 DEVICE MODEL 241 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 242 define hbt (C E B) ISROL = 17.82f ISE = 1.38682E-16 IS = 1.447E-25 NROL = 1.968 NE = 2.23811 N F = 1.049 N R = 1.040 BF = 225.0 BR = 5.154 VT = 0.0255635 Ka = 1.333 RBmax = 2.01OK aRB = 1.742 IBO = 7.987u CJCO = 6.450f PHIC = 1.764 me = 1.879 alphac = 0.9 CJEO = 5 . 635f PHIE = 2.252 me = 0.8285 alphae = 7 61.7m LB = 55.40p LE = 40.80p LC = O.lp CBC = 43.19f CCE = 2 5 .84f CBE = 17.50f RBC = 16.52 RCE = 0 . 1 RBE = 71.10 RPBE = 889.1MEG RPCE = 4 6.50MEG RPBC = 42.16MEG tauf = 2 . 822e-12 RE = 13.37 RB2 = 102.7 RC = 5.037 RTHEO = 4190 RTHCO = 0 . 1 RTHOLO = 0 . 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. KTHC = 0.4811 KS = lOu TEMP = 2 3 . 5 XTI = 2.529 XTE = 1.316 XTROL = 3.959 XTBR = -20.42m x_Al = 132.2m fb = 18.76 betaO = 299.1 XTRBB = 0 r:rrl B1 BC r=RBC r :rr2 El BE r=RBE r :rr3 El CE r=RCE r :rr4 B1 B2 r=RB2 r:rr5 El E2 r=RE r:rr6 Cl C2 r=RC r :rr7 B1 El r=RPBE r:rr8 B1 Cl r=RPBC r :rr9 Cl El r=RPCE r :rrlO NTE 0 r=RTHEO r :rrll NTC 0 r=RTHCO r :rrl2 NTOL 0 r=RTHOLO 1:111 B B1 1=LB 1:112 E El 1=LE 1:113 C Cl 1=LC c:ccl BC Cl c=CBC c:cc2 Cl CE c=CCE c:cc3 B1 BE c=CBE diode ( w l , isat, n, vv2) = \ isat * (exp_soft(vvl/(n*vv2)) - 1) max_exp = le27 small = le-50 max_arg = ln(max_exp) exp_soft (x) = if (x<max_arg) (x+l-max_arg)*max_exp endif then exp(x) else \ charge(cO, v, m, vO, alpha) = if (v < alpha*vO) then \ chargel(cO, v, m, vO, alpha) else \ charge2(cO, v, m, vO, alpha) endif chargel(cO, v, m, vO, alpha) = \ - cO*vO*((l -v/vO)A (l - m))/(l - m) charge2(cO, v, m, vO, alpha) = cO*(v - alpha*vO + \ 0.5*m*(v - alpha*vO)A2 / ( v O * (1 - alpha)))/(! - a l p h a ) Am + Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 244 chargel(cO, alpha*vO, m, vO, alpha) Tn = TEMP + 273.15 deltatE = _V7 deltatC = _V8 deltatOL = _V9 TjE = Tn + deltatE TjC = Tn + deltatC TjOL = Tn + deltatOL VTn = T n * 8 .61738e-5 VTjE = T j E * 8 .61738e-5 VTjC = T j C * 8 .61738e-5 VTjOL = TjOL*8.61738e-5 KRE = RTHE0*KS KRC = RTHC0*KS KROL = RTHOLO*KS EgB (T) = 1.519 - (5.405e-4) * (T/'2) / (T + 204) EgGAlAs(T) = 3.114 - (5.41e-4)*(TA2)/ (T + 204) EgE(T) = (1 - x_Al)*EgB(T) + x_Al*EgGAlAs(T) + \ 0.3 7 * x _ A l * (1 - x_Al) deltaEg(T) = EgE(T) - EgB(T) tISl = (TjE/Tn)"XTI 151 = IS*tISl*exp_soft(-EgB(TjE)/VTjE + EgB(Tn)/VTn) tIS2 = (TjC/Tn) ''XTI 152 = IS*tIS2*exp_soft(-EgB(TjC)/VTjC + EgB(Tn)/VTn) tIS3 = (TjOL/Tn)AXTI 153 = IS*tIS3*exp_soft(-EgB(TjOL)/VTjOL + EgB(Tn)/VTn) tISE = (TjE/Tn)AXTE ISE1 = ISE*tISE*exp_soft(-EgE(TjE)/ (2*VTjE) EgE(Tn)/ (2*VTn)) + \ tISROL = (TjOL/Tn)"XTROL ISROL1 = ISROL*tISROL*exp_soft(-EgB(TjOL)/ (2*VTjOL) EgB(Tn)/ (2*VTn)) + \ tBRl = (TjC/Tn)AXTBR BR1 = BR*tBRl tBR2 = (TjOL/Tn)"XTBR BR2 = BR*tBR2 numBF = fb*exp_soft(-deltaEg(Tn)/VTn) + 1/betaO denBF = fb*exp_soft(-deltaEg(TjE)/VTjE) + 1/betaO Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. BF1 = BF*numBF/denBF vbel vbe2 vbcl vbc2 vbc3 = = = = = V3 _V1 _V6 _V3 _V1 - V4 V4 V2 V2 V2 icc = diode(vbel, IS1, NF, VTjE) iec = diode(vbc2, IS2, NR, VTjC) ict = icc - iec ibl = diode(vbcl, IS3*Ka*(BR2+1)/BR2, NR, VTjOL) diode(vbcl, ISR0L1, NROL, VTjOL) ib2 = diode(vbel, IS1/BF1, NF, VTjE) + \ diode(vbel, ISE1, NE, VTjE) ib3 = diode(vbc2, IS2/BR1, NR, VTjC) + \ pde = pde = pdol = pdel = vbel*(ib2 + ict) vbc2*(ib3 - ict) vbcl*ibl pde + KTHC*pdc qbcl qbc2 qbel qdel c h a r g e ( (Ka/(Ka + 1))*CJC0, vbcl, me, PHIC, alphac) c h a r g e ((1/ (Ka + 1))*CJC0, vbc3, me, PHIC, alphac) charge(CJEO, vbe2, me, PHIE, alphae) tauf*icc = = = = rbase = (( (TjE + TjC)/\ (2*Tn))''XTRBB)*RBmax/(1+(((_i5+small)/IBO)"aRB)) sdd:sddhbt B2 0 C2 0 B3 0 E2 0 B2 B3 B1 0 NTE 0 NTC 0 \ NTOL 0 \ i [6,0]=ibl i [2,0]=-ibl+ict-ib3 \ i [3,0]=ib2+ib3 i [4,0]=-ib2-ict f [5,0]=_v5-_i5*rbase \ i [7, 0]=-pdel/(l-KRE*pdel) i [8,0]=-pdc/(l-KRC*pdc) \ i [9,0]=-pdol/(l-KR0L*pdol) \ i [1,1]=qbc2+qbel+qdel \ i [2,1]=-qbcl-qbc2 i [4,1]=-qbel-qdel i[6,l]=qbcl end hbt Reproduced with permission of the copyright owner. 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