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# 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
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Copyright © by Alexandru Aurelian Ciubotaru 1996
All Rights Reserved
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To my parents
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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
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UMI Number: 9718536
Copyright 1996 by
Ciubotaru, Alexandra Aurelian
All rights reserved.
UMI Microform 9718536
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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.
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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
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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).
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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.
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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).
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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.
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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
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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
.
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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)
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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
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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.
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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:
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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)
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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)
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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)
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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.
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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).
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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.
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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).
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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)
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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
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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.
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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.
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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
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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 .
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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
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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):
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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.
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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.
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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
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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
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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-
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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)
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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:
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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).
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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.
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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)
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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.
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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)
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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 .
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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
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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.
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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
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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.
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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.
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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.
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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:
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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)
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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],
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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)
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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
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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
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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
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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
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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.
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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.
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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],
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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.
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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.
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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.
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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:
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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
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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.
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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).
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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)
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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 .
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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.
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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.
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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).
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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.
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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
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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
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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).
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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
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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
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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.
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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).
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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
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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
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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).
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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
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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.
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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
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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.
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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.
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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.
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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).
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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
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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.
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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.
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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
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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
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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
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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,
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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
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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.
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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
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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.
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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.
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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).
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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
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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).
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190
Fig. 8.8. Continued (c and d).
(c) S -21
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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).
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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
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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).
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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-,
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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).
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198
Fig. 8.12. Continued (c and d).
(C) Sol
(d) 5*2
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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).
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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. Further reproduction prohibited without permission.
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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