Physics, Compact Modeling and TCAD of SiGe HBT for Wide Temperature Range
Operation
by
Lan Luo
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
December 12, 2011
Keywords: SiGe HBT, Cryogenic Temperature, Compact Modeling, TCAD, Carrier freezeout
Copyright 2011 by Lan Luo
Approved by
Guofu Niu, Chair, Alumni Professor of Electrical and Computer Engineering
Bogdan Wilamowski, Professor of Electrical and Computer Engineering
Fa Foster Dai, Professor of Electrical and Computer Engineering
Vishwani Agrawal, James J. Danaher Professor of Electrical and Computer Engineering
Abstract
One of the remarkable characteristics of SiGe HBT is the ability to operate over a wide tem-
perature range, from as low as sub 1K, to as high as over 400 K. The SiGe HBT investigated and
measured in this work is a first-generation, 0.5 mmSiGe HBT with fT=fmax of 50 GHz/65 GHz and
BVCEO=BVCBO of 3.3 V/10.5 V at 300 K. The base doping is below but close to the Mott-transition
(about 3 1018 cm 3 for boron in silicon). In this dissertation, some important SiGe HBTs physics
at cryogenic temperature are analyzed. New compact models equations for SiGe HBT are devel-
oped, which can function from 43 to 393K. Device TCAD simulations are used to help understand
the device physics at cryogenic temperatures.
First, the temperature dependence of semiconductors critical metrics are reviewed, including
bandgap energy Eg, effective conduction band density-of-states NC and valence band density-
of-states NV, intrinsic carrier concentration at low doping ni, bandgap narrowing DEg, carrier
mobility m, carrier saturation velocity nsat and carrier freezeout. The dc and ac low temperature
performance of SiGe HBT are analyzed, including collector current density, current gain, Early
effect, avalanche multiplication factor, transit time, cut-off frequency and maximum oscillation
frequency. This illustrates why SiGe HBT demonstrates excellent analog and RF performance at
cryogenic temperatures.
The current dependence of multiplication factor M-1 at low temperatures are investigated based
on a substrate current based avalanche multiplication technique. The M-1 at high current is con-
siderably lower than it at low current. Then, the temperature dependence of forced-IE pinch-in
maximum operation voltage limit, which is of interest for many space exploration application is
investigated. In particular, we discuss how the critical base current I B varies with temperature, and
introduce the concept of critical multiplication factor (M-1) , critical collector-base biasV CB where
M-1 reaches (M-1) . A decrease of the voltage limit is observed with cooling, and attributed to the
ii
increase of intrinsic base resistance due to freezeout as well as increase of avalanche multiplication
factor M-1. A practically high emitter current IE is shown to alleviate the decrease of V CB with
cooling, primarily due to the decrease of M-1 with increasing IE.
The existing commercial compact models are shown to fail below 110 K. In this work, new
temperature scaling equations are developed. As much physics basis are implemented as possible
to fit temperature dependence of SiGe HBTs dc and ac characteristics, such as ideality factor,
saturation current, series resistances and thermal resistance. In particular, carrier freezeout is
now modeled accounting for latest research on Mott transition, leading to successful modeling of
temperature dependences for all series resistances in SiGe HBT. These new temperature equations
give reasonably accurate fitting of the dc characteristics from 393 to 43 K, ac characteristics from
393 to 93 K.
Furthermore,theimpactofthenon-idealtemperaturedependenceofIC-VBE inSiGeHBTsonthe
outputofaBGRisexamined. Thesenon-idealitiesactuallyhelpmaketheBGRoutputvoltagevary
lessatcryogenictemperaturesthantraditionalShockleytheorywouldpredict. Successfulcryogenic
temperature modeling of both DVBE and VBE components of the BGR output is demonstrated for
the first time.
iii
Acknowledgments
Iwouldliketoexpressmygratitudetomysupervisor,Dr. GuofuNiu. Withouthisguidance,this
dissertationwouldnothavebeenfinishedsmoothly. Ithankhimforhispatienceandencouragement
that carried me on through difficult times, and for his insights and suggestions that helped to shape
my research skills. I appreciate his vast knowledge and skill in many areas, and his valuable
feedback that greatly contributed to my research and this dissertation. I would like to thank the
other members of my committee, Dr. Bogdan Wilamowski, Dr. Fa Foster Dai and Dr. Vishwani
Agrawal for the valuable assistance they provided.
I would like to extend my gratitude to former and current members in our SiGe HBT research
group. IwouldliketothankZiyanXuforourmemorablecollaborationsinSiGecompactmodeling,
Yan Cui, Kejun Xia and Tong Zhang for their fruitful discussions in semiconductor device physics
and TCAD. And I would like to thank Xiaoyun Wei for her patient guide in device DC/RF/Noise
characterization. I am also very grateful to the strong support from Professor John D. Cressler?s
SiGegroupatGeorgiaTech,especiallythecryogenictemperaturedeviceandcircuitmeasurements.
This work was supported by NASA, under grant NNL05AA37C and NNL06AA29C. I am
grateful for the support of NASA, JPL and the IBM SiGe development group, as well as the many
contributions of the SiGe ETDP team.
Finally, I am greatly indebted to my parents, Yuling Luo and Zhijiang Zhang, for the support
they provided me through my entire life. I would like to thank my sister, Qian Luo, who is always
there when I need help. In particular, I would like to thank my husband, Wei Jiang, who has been
my best friend since college. I would sincerely thank my little son, Richard Z. Jiang, whose arrival
is a precious gift from God.
iv
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 SiGe HBT physics over temperature . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 SiGe HBT compact modeling over temperature . . . . . . . . . . . . . . . . . . . 3
1.4 Dissertation contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Device Physics of SiGe HBT at Cryogenic Temperatures . . . . . . . . . . . . . . . . . 6
2.1 Semiconductor physics at cryogenic temperatures . . . . . . . . . . . . . . . . . . 6
2.1.1 Bandgap energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Density-of-states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.3 Intrinsic carrier concentration at low doping . . . . . . . . . . . . . . . . . . 11
2.1.4 Bandgap narrowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.5 Carrier freezeout and Mott-transition . . . . . . . . . . . . . . . . . . . . . 14
2.1.6 Low field carrier mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.7 Carrier saturation velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.8 Sheet resistance and resistivity . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 SiGe HBT characteristics at cryogenic temperature . . . . . . . . . . . . . . . . . 32
2.2.1 Limitations of Si BJT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.2 SiGe HBT fundamental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.3 Collection current density and current gain . . . . . . . . . . . . . . . . . . 33
v
2.2.4 Early effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.5 Avalanche multiplication and breakdown voltage . . . . . . . . . . . . . . . 35
2.2.6 AC characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Substrate Current Based M-1 Measurement at Cryogenic Temperatures . . . . . . . . . 44
3.1 Measurement theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Experimental results over temperatures and impact of current . . . . . . . . . . . . 47
3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Forced-IE Pinch-in Maximum Output Voltage Limit at Cryogenic Temperatures . . . . . 52
4.1 Physics of emitter current pinch-in and temperature dependence analysis . . . . . . 52
4.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.1 T-dependence of M-1 from forced-VBE measurements . . . . . . . . . . . . . 55
4.2.2 T-dependence of I B,V CB and (M-1) from forced-IE measurements . . . . . . 56
4.3 Circuit design implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5 Compact Modeling of SiGe HBT at Cryogenic Temperatures . . . . . . . . . . . . . . . 62
5.1 Introduction of compact modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Description of proposed wide temperature range compact model . . . . . . . . . . 66
5.3 Main current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.4 Depletion charges and capacitances . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.5 Early effects and Ge ramp effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.6 Diffusion charges and transit times . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.7 Non-quasi-static charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.8 Base current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.8.1 Ideal forward base current . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.8.2 Non-ideal forward base current . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.8.3 Trap-assisted tunneling current . . . . . . . . . . . . . . . . . . . . . . . . . 89
vi
5.8.4 Non-ideal reverse base current . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.8.5 Extrinsic base current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.9 Collector epilayer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.9.1 Epilayer current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.9.2 Epilayer diffusion charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.9.3 Intrinsic base-collector depletion charge . . . . . . . . . . . . . . . . . . . . 98
5.10 Self-heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.11 Temperature modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.11.1 Saturation current and ideality factor . . . . . . . . . . . . . . . . . . . . . . 101
5.11.2 Base tunneling current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.11.3 Series resistances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.11.4 Thermal resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.11.5 Hot carrier current IHC and epilayer space charge resistance SCRCV . . . . . 123
5.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6 Parameter Extraction and Compact Modeling Results . . . . . . . . . . . . . . . . . . . 126
6.1 Parameter extraction methodology in this work . . . . . . . . . . . . . . . . . . . 128
6.2 Saturation current and ideality factor . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.3 Base tunneling current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.4 Depletion capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.5 Avalanche and Early voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.6 Emitter resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.7 Base resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.8 Collector resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.9 Thermal resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.10 High current parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.11 Temperature parameter extraction . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.12 SiGe HBT compact modeling results . . . . . . . . . . . . . . . . . . . . . . . . . 176
vii
6.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7 Band Gap Reference Circuit Modeling Application . . . . . . . . . . . . . . . . . . . . 184
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.2 Technical approach and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.2.1 Slope of IC-VBE and impact on DVBE(T) and Iref(T) . . . . . . . . . . . . . . 185
7.2.2 IS(T) and impact onVBE(T) . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.2.3 Vref(T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
A Appendix A Full List of Temperature Scaling Equations . . . . . . . . . . . . . . . . . 203
A.1 Compact model parameter list . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
A.2 Saturation current and ideality factor . . . . . . . . . . . . . . . . . . . . . . . . . 208
A.3 Base tunneling current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
A.4 Non-ideal base current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
A.5 Early voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
A.6 pn junction diffusion voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
A.7 Depletion capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
A.8 Parasitic resistances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
A.9 IHC and SCRCV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
A.10 Knee current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
A.11 Transit times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
A.12 Thermal resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
A.13 Substrate Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
viii
List of Figures
1.1 Cross section of a first-generation SiGe HBT used in this work. . . . . . . . . . . . . 2
1.2 Vertical doping and Ge profile of a first-generation SiGe HBT used in this work. . . . 2
2.1 (a) Bandgap of silicon Eg;Si versus T. (b) Ge and temperature dependence of SiGe
bandgap Eg;SiGe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 (a) Ge and temperature dependence of NC;SiGe=NC;Si. (b) Ge and temperature depen-
denceofNV;SiGe=NV;Si. (c)Geandtemperaturedependenceof(NCNV)SiGe=(NCNV)SiGe.
10
2.3 Temperature dependence of intrinsic carrier concentration at low doping ni0. . . . . . 11
2.4 Doping dependence of bandgap narrowing DEg. . . . . . . . . . . . . . . . . . . . . 13
2.5 Concentration versus EF EI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Variation of Fermi level Ef Ei versus doping concentration over 40-500 K. . . . . . 16
2.7 (a) Doping dependence of IR at different temperatures. (b) Temperature dependence of
IR at different doping levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8 (a) Edop-Nd. (b) b-Nd. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.9 Doping dependence of IR at different temperatures using Altemmatt?s model. . . . . . 20
2.10 (a)Temperatureandconcentrationdependenceofholemobilitymh,w/ocarrierfreezeout
effect. (b) Temperature and concentration dependence of hole mobility mh, w/ carrier
freezeout effect. (c) Temperature dependence of hole mobility mh near Mott-transition. 24
ix
2.11 (a)Temperature and concentration dependence of electron mobility me, w/o carrier
freezeout effect. (b) Temperature and concentration dependence of electron mobil-
ity me, w/ carrier freezeout effect. (c) Temperature dependence of electron mobility me
near Mott-transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.12 (a) Calculated clustering function vs. impurity concentration. (b) Calculated concen-
tration dependence of electron mobilities me at 300 K. (c) Calculated concentration
dependence of electron mobilities me at 100 K. . . . . . . . . . . . . . . . . . . . . . 27
2.13 Saturation velocity as a function of temperature for Si and Ge. . . . . . . . . . . . . 28
2.14 calculate depletion layer thickness Wdep versus T from 30-300 K using Altermatt and
classic incomplete ionization models. . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.15 (a)Measuredsheetresistanceforcollector,intrinsicbase,extrinsicbaseandburiedlayer
from 30-300 K. (b) Measured substrate resistivity from 30-300 K. . . . . . . . . . . . 31
2.16 Energy band diagram for a Si BJT and graded-base SiGe HBT. . . . . . . . . . . . . 33
2.17 Measured current gain b-VBE for a graded-base SiGe HBT. . . . . . . . . . . . . . . 34
2.18 The avalanche multiplication process in a BJT. . . . . . . . . . . . . . . . . . . . . . 36
2.19 Measured M-1-T for a graded-base SiGe HBT over 43-393 K. . . . . . . . . . . . . . 37
2.20 Measured multiplication factor M-1 versus 1000=T at various VCB;s for a graded-base
SiGe HBT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.21 (a) Experimental setup. (b) Local emitter current density JE when IB < 0. . . . . . . . 38
2.22 (a) Calculated tb;SiGe-T and tb;Si-T for a graded-base SiGe HBT. (b) Calculated ratio
tSiGe=tSi-T for a graded-base SiGe HBT. . . . . . . . . . . . . . . . . . . . . . . . . 41
x
2.23 (a)Measured fT-JC foragraded-baseSiGeHBT.(b)Measured fmax-JC foragraded-base
SiGe HBT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1 Illustration of photo carrier generation in the SiGe HBTs used. . . . . . . . . . . . . . 44
3.2 Experimental setup of the substrate current based M-1 measurement technique. . . . . 45
3.3 Measured I0AVE, ISUB and ISUB=I0AVE ratio versus IE atVCB=4 V, 300 K. . . . . . . . . . 46
3.4 Extracted avalanche current IAVE versus IE atVCB=4 V, 300 K. . . . . . . . . . . . . . 47
3.5 Measured I0AVE, ISUB and ISUB=I0AVE ratio versus IE atVCB=4 V, 43 K. . . . . . . . . . 48
3.6 Extracted avalanche current IAVE versus IE atVCB=4 V, 43 K. . . . . . . . . . . . . . 48
3.7 Measured M-1 and fT vs. JE at: (a) 300 K; (b) 223 K; (c) 162 K; (d) 93 K; (e) 43 K. . 49
3.8 Measured M-1 versusVCB for IE=12.5 mA, 125 mA and 1.5 mA at 300 K. . . . . . . . 50
3.9 Measured fT and M-1 vs. IE over 162-300 K. . . . . . . . . . . . . . . . . . . . . . . 51
4.1 Measured and simulated intrinsic base sheet resistance RsBi-T. . . . . . . . . . . . . 54
4.2 Measured M-1 vs. VCB at forced-VBE over 43-300 K. . . . . . . . . . . . . . . . . . . 54
4.3 Measured IB vs. VCB at IE = 125m A over 43-300 K. . . . . . . . . . . . . . . . . . . 56
4.4 Measured IC vs. VCB at IE = 125m A over 43-300 K. . . . . . . . . . . . . . . . . . . 57
4.5 Measured ISUB vs. VCB at IE = 125m A over 43-300 K. . . . . . . . . . . . . . . . . . 57
4.6 Measured M-1 vs. VCB at IE = 125mA over 43-300 K. . . . . . . . . . . . . . . . . . 58
4.7 Measured b0 vs. VBE atVCB = 0 V over 43-300 K. . . . . . . . . . . . . . . . . . . . 58
4.8 Measured 1b , 1b0 vs. T at IE = 12:5mA, IE = 125mA and IE = 1mA. . . . . . . . . . 60
xi
4.9 Measured jI Bj,V CB vs. T at IE = 12:5mA, IE = 125mA and IE = 1mA. . . . . . . . . 60
4.10 Measured M-1 vs. VCB at IE = 12:5mA, IE = 125mA and IE = 1mA at 93 K. . . . . . 61
5.1 The Mextram equivalent circuit for the vertical NPN transistor. . . . . . . . . . . . . 64
5.2 Equivalentcircuitusedinthiswork,with1)addedforwardbasetunnelingcurrentIB;tun;
2) added RSUB andCSUB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3 (a) Doping profile for a NPN transistor. (b) Schematic illustrating the applied voltages
in normal operation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.4 A schematic cross-section of the base region for a NPN transistor, showing the Early
effect: EB and BC junction depletion thickness?s variation withVBE andVBC. . . . . 71
5.5 Measured IC-VBE for a graded-base SiGe HBT. . . . . . . . . . . . . . . . . . . . . . 72
5.6 The slope of IC-VBE from device simulation [1]. . . . . . . . . . . . . . . . . . . . . . 73
5.7 (a) Measured IC-VBE from 43 to 393 K. (b) Extracted NF at each temperature. (c)
Extracted IS at each temperature [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.8 (a) Simulated versus measured IC-VBE at high temperatures. (b) Simulated versus mea-
sured IC-VBE at low temperatures [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.9 Calculated Cj versus applied voltage V for an abrupt junction (p = 1=2) and a graded
junction (p = 1=3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.10 (a) Modeled Qte from 93-393 K. (b) Modeled ?Qte=?I from 93-393 K. . . . . . . . . 79
5.11 (a) Modeled Qtc from 93-393 K. (b) Modeled ?Qtc=?I from 93-393 K. . . . . . . . . 79
5.12 Schematic Ge and 1=n2ib profile of SiGe transistor with a gradient Ge content. . . . . 81
xii
5.13 Injected electron densities profile in the base region for a NPN transistor. . . . . . . . 83
5.14 (a) Modeled Qbe from 93-393 K. (b) Modeled ?Qbe=?I from 93-393 K. . . . . . . . 85
5.15 (a) Modeled Qbc from 93-393 K. (b) Modeled ?Qbc=?I from 93-393 K. . . . . . . . 86
5.16 (a) IB-VBE from forward Gummel measurement; (b) IB-VBC from reverse Gummel mea-
surement [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.17 Measured IB-VBE with illustration of TAT in forward-biased E-B junction [2]. . . . . 90
5.18 Electric field in the epilayer as a function of current. (a). Base-collector depletion
thickness decreases with current. (b). Base-collector depletion thickness increases
with current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.19 (a). Schematic of a bipolar transistor. (b). Doping, electron and hole densities in the
base-collector region. [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.20 Modeled xi=Wepi from 93-393 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.21 (a) Modeled Qepi from 93-393 K. (b) Modeled ?Qepi=?I from 93-393 K. . . . . . . . 98
5.22 Electric field in the epilayer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.23 Self-heating network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.24 IS extracted from measured IC-VBE vs. IS fitted by (5.67) and (5.69) for single 0.5 2.5
mm2 SiGe HBT from 43-300 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.25 (a)Measured sheet resistance for collector, intrinsic base, silicided extrinsic base and
buried layer from 30-300 K. (b) Measured substrate resistivity from 30-300 K. . . . . 109
5.26 (a) Measured R(T)R(T0), calculated m(T0)m(T) using PHUMOB and single power law approxi-
mation, for the substrate, from 40-320 K. (b) Calculated IR(T)IR(T0) using PHUMOB and a
single power law approximation, for substrate, from 40-320 K. . . . . . . . . . . . . 111
xiii
5.27 (a) Collector sheet resistance modeling using the classic model and doping dependent
activation energy from 30-300 K. (b) Substrate resistivity modeling using the classic
model and doping dependent activation energy from 40-320 K. . . . . . . . . . . . . 114
5.28 (a) Measured R(T)R(T0), calculated m(T0)m(T) using PHUMOB and a single power law approxi-
mation for intrinsic base, from 20-300 K. (b) Calculated IR(T)IR(T0) using PHUMOB and a
single power law approximation for the intrinsic base, from 20-300 K. . . . . . . . . 115
5.29 Intrinsic base sheet resistance modeling using the classic, the Altermatt, and the new
model, from 30-300 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.30 (a)CalculatedIRforintrinsicbaseusingtheclassical,theAltermatt,andthenewmodel,
from 30-300 K. (b) Calculated 1=IR for intrinsic base using the classic, the Altermatt,
and the new model, from 30-300 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.31 Measured R(T)R(T0) and calculated m(T0)m(T) using PHUMOB and a single power law approxi-
mation for the buried collector, from 30-300 K. . . . . . . . . . . . . . . . . . . . . 121
5.32 Silicidedextrinsicbaseandburiedcollectorsheetresistancemodelingusingdualpower
law approximation with IR = 1 and empirical approach, from 30-300 K. . . . . . . . 123
5.33 Extracted and modeled lnRTH-lnTamb. . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.1 Proposed parameter extraction methodology in this work. . . . . . . . . . . . . . . . 129
6.2 IS and NF extraction from the intercept and slope of IC-VBE. . . . . . . . . . . . . . . 130
6.3 (a) Comparison of the direct linear fitting of IB and fitting with iteration at 43 K. (b)
IB-VBE modeling results including TAT [2]. . . . . . . . . . . . . . . . . . . . . . . . 132
6.4 ExtractedCBE-VBE from 93-300 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.5 ExtractedCBC-VBC from 93-300 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
xiv
6.6 ExtractedCCS-VCS from 162-393 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.7 (a) ExtractedCSUB-VCS from 162-393 K; (b) Extracted and modeled RSUB-T. . . . . . 137
6.8 Extracted and modeled M-1-VCB from 93-393 K. . . . . . . . . . . . . . . . . . . . . 139
6.9 The measuredVCES versus -IE in the RE flyback measurement for the RE extraction. . 140
6.10 Derivation of the measuredVCES w.r.t the IE in the RE flyback measurement for the RE
extraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.11 ?(Z12) versus 1IE from 93-300 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.12 DVBE-VBE from 43-300 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.13 Extracted (RBC +RBV)-IC from 43-300 K. . . . . . . . . . . . . . . . . . . . . . . . 144
6.14 Extraction of re +rb at 93 K by fitting the high-frequency portion of 1=?(Y11)-w 2. . 145
6.15 Extracted re +rb-IC from 43-393 K by fitting the high-frequency portion of 1=?(Y11)-
w 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.16 Equivalent circuit used in extraction of rb using circuit impedance method. . . . . . . 146
6.17 Extracted of rb using the circle impedance method from 300-93 K, JC = 1mA=mm2. . 147
6.18 Extracted rb-JC using half-circle method from 300-93 K. . . . . . . . . . . . . . . . 147
6.19 Derivation of the measuredVCES w.r.t the IC in the RC flyback measurement for the RC
extraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.20 Measured junction temperature versus power dissipation over 93-300 K. The IE is fixed
near peak fT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.21 Impact of parameter IK on: (a) Force-IB output characteristics; (b) fT-IC; (c) b-VBE. . 152
xv
6.22 Impact of parameter RCV on: (a) Force-IB output characteristics; (b) fT-IC; (c) b-VBE. 153
6.23 Impact of parameterVDC on: (a) Force-IB output characteristics; (b) fT-IC; (c) b-VBE. 154
6.24 Impact of parameter IHC on: (a) Force-IB output characteristics; (b) fT-IC; (c) b-VBE. 155
6.25 Impact of parameter SCRCV on: (a) Force-IB output characteristics; (b) fT-IC; (c)b-VBE. 156
6.26 (a) Impact of parameter tb and te on fT-IC; (b) Impact of parameter tepi on fT-IC. . . 157
6.27 (a) Extracted and modeled IS for collector current as a function of temperature; (b)
Extracted and modeled NF for collector current as a function of temperature. . . . . . 159
6.28 (a) Extracted and modeled IBEI for base current as a function of temperature; (b) Ex-
tracted and modeled NEI for base current as a function of temperature. . . . . . . . . 160
6.29 Extracted and modeledCJE-T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.30 Extracted and modeledCJC-T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.31 Extracted and modeled Early voltageVER andVEF from 93-393 K. . . . . . . . . . . 162
6.32 Measured(symbols)andmodeled(lines)frequencydependenceofrealparty-parameters
at 300K. Four bias conditions are chosen near peak fT: (a) real partY11, RBC=28 W; (b)
real partY11, RBC=34 W; (c) real partY21, RBC=28 W; (d) real partY21, RBC=34 W. . . 164
6.33 Measured (symbols) and modeled (lines) frequency dependence of imaginary part y-
parameters at 300K. Four bias conditions are chosen near peak fT: (a) imaginary part
Y11, RBC=28 W; (b) imaginary part Y11, RBC=34 W; (c) imaginary part Y21, RBC=28 W;
(d) imaginary partY21, RBC=34 W. . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.34 Measured(symbols)andmodeled(lines)frequencydependenceofrealparty-parameters
at 300K. Four bias conditions are chosen near peak fT: (a) real part Y11, RBV=288 W;
(b) real partY11, RBV=348W; (c) real partY21, RBV=288W; (d) real partY21, RBV=348W. 166
xvi
6.35 Measured (symbols) and modeled (lines) frequency dependence of imaginary part y-
parameters at 300. Four bias conditions are chosen near peak fTK: (a) imaginary part
Y11, RBV=288 W; (b) imaginary partY11, RBV=348 W; (c) imaginary partY21, RBV=288
W; (d) imaginary partY21, RBV=348 W. . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.36 Measured(symbols)andmodeled(lines)frequencydependenceofrealparty-parameters
at 300K. Four bias conditions are chosen near peak fT: (a) real partY11, RE=12 W; (b)
real partY11, RE=15 W; (c) real partY21, RE=12 W; (d) real partY21, RE=15 W. . . . . 168
6.37 Measured (symbols) and modeled (lines) frequency dependence of imaginary part y-
parameters at 300K. Four bias conditions are chosen near peak fT: (a) imaginary part
Y11, RE=12 W; (b) imaginary part Y11, RE=15 W; (c) imaginary part Y21, RE=12 W; (d)
imaginary partY21, RE=15 W. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.38 Extracted and modeled RE-T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.39 Extracted and modeled RBc-T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.40 Extracted and modeled RBV from 93-300 K. . . . . . . . . . . . . . . . . . . . . . . 171
6.41 Extracted and modeled RCc-T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.42 Extracted and modeled thermal resistance RTH from 93-300 K. . . . . . . . . . . . . 172
6.43 Extracted and modeled IK from 93-300 K. . . . . . . . . . . . . . . . . . . . . . . . 172
6.44 Extracted and modeled RCV from 93-300 K. . . . . . . . . . . . . . . . . . . . . . . 173
6.45 Extracted and modeledVDC from 93-300 K. . . . . . . . . . . . . . . . . . . . . . . 173
6.46 Extracted and modeled IHC from 93-300 K. . . . . . . . . . . . . . . . . . . . . . . . 174
6.47 Extracted and modeled tB from 93-300 K. . . . . . . . . . . . . . . . . . . . . . . . 174
xvii
6.48 Extracted and modeled tepi from 93-300 K. . . . . . . . . . . . . . . . . . . . . . . . 175
6.49 (a)Measured and modeled IC-VBE from 136-393 K. (b)Measured and modeled IB-VBE
from 136-393 K. (c)Measured and modeled IC-VBE from 93-43 K. (d)Measured and
modeled IB-VBE from 93-43 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.50 Measured and modeled forced-IB output characteristics at low IB. (a) IC-VCE; (b) VBE-
VCE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.51 Measured and modeled forced-IB output characteristics at high IB. (a) IC-VCE; (b)VBE-
VCE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.52 Measured (symbol line) and modeled (solid line) fT-IC from 93-393 K. (a) VCB = 0V.
(b)VCB = 1V. (c)VCB = 2V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.53 (a) Measured (symbols) and modeled (curves) H21-JC at 5 GHz from 93-393 K. (b)
Measured (symbols) and modeled (curves) MUG-JC at 5 GHz from 93-393 K. . . . . 181
6.54 Measured (symbols) and modeled (curves) Y-parameters at 1,2,3 and 5 GHz for 162 K,
VCB = 0V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.1 Schematic of a first-order SiGe bandgap reference [2]. . . . . . . . . . . . . . . . . . 185
7.2 (a)MeasuredDVBE-T andcalculatedVT ln(8)-T from43to300K.(b)NF-T from43-300
K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.3 Measured and modeled IC-VBE for single 0.5 2.5 mm2 SiGe HBT at 43 K and 300 K. 189
7.4 Measured and modeledVBE from 43-300 K. . . . . . . . . . . . . . . . . . . . . . . 189
7.5 Measured and modeled K DVBE from 43-300 K. . . . . . . . . . . . . . . . . . . . 190
7.6 Measured and modeledVref from 43-300 K. . . . . . . . . . . . . . . . . . . . . . . 191
xviii
List of Tables
6.1 A typical grouping of parameters used in the extraction procedure in Mextram [4]. . . 127
7.1 Models examined in this work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
A.1 Compact model parameters overview. . . . . . . . . . . . . . . . . . . . . . . . . . . 204
xix
Chapter 1
Introduction
1.1 Motivation
SiGeheterojunctionbipolartransistor(HBT)technologyiscurrentlybeingusedtodevelopelec-
tronics for space applications due to the excellent analog and RF performance of SiGe HBTs over
an extremely wide range of temperatures, along with its built-in radiation hardness [5] [6]. SiGe
HBT technology has recently been used to develop electronic sub-systems for NASA?s envisioned
lunar missions. For instance, SiGe electronic components can operate reliably in the extreme am-
bient environmentfound on the lunar surface, where the extremely cold environmental temperature
drops to 180 C (93 K) during the lunar night and 230 C (43 K) in the shadowed polar craters.
To enable the design of circuits that can operate over such a wide temperature range, we need to
investigate the physics of SiGe HBTs at low temperatures, and importantly also develop robust
compact models that can work over a wide temperature range with good fidelity, particularly at
lower temperatures.
The SiGe HBT investigated and measured in this work is a first-generation, 0.5 mm SiGe HBT
with fT=fmax of 50 GHz/65 GHz and BVCEO=BVCBO of 3.3 V/10.5 V at 300 K, with base doping
below but close to the Mott-transition (about 3 1018 cm 3 for boron in silicon). The emitter area
is 0.5 2.5 mm2. The schematic cross section of this SiGe HBT is shown in Fig. 1.1. The vertical
doping and Ge profile from secondary ion mass spectrometry (SIMS) is shown in Fig. 1.2.
1
p-SiGe intrinsic base
n+ buried collector
n- collector n+
Emitter CollectorBase
poly-Si
Metal
p- substrate
Shallow Trench
Deep Trench
n+
Oxide
p+ extrinsic base p+ extrinsic base
Figure 1.1: Cross section of a first-generation SiGe HBT used in this work.
1016
1017
1018
1019
1020
1021
Dopant Concentration (cm
?3 )
0
0.02
0.04
0.06
0.08
0.1
0.12
Germanium
Depth (?m)
As poly
B
Ge?
P
As
n+ emitter p?SiGe base n? collector n+ buried collector
Figure 1.2: Vertical doping and Ge profile of a first-generation SiGe HBT used in this work.
2
1.2 SiGe HBT physics over temperature
Bandgap engineering has a positive impact on the low temperature characteristics of transistor.
Aswillbeshowninchapter2,theSiGeHBTisideallysuitedforcryogenicoperationasthebandgap
engineering induced improvements in current gain b, cutoff frequency fT, and Early voltageVA all
become more pronounced with cooling [7].
In this work, the impact of temperature on bandgap energy Eg, electron and holes density of
statesNC=NV,intrinsiccarrierconcentrationni,bandgapnarrowingBGN,carriermobilitym,carrier
saturation velocity nsat and carrier freezeout effect will be reviewed first. Following, the dc and ac
low temperature performance of SiGe HBT will be analyzed theoretically and experimentally.
1.3 SiGe HBT compact modeling over temperature
Today?s IC design heavily relies on circuit simulation and circuit simulation needs compact
device models. The industry standard bipolar transistor modeling is based on SPICE Gummel-
Poon (SGP) model [8]. Several advanced models, such as VBIC (vertical bipolar inter-company)
[9], HICUM (high current model) [10] and MEXTRAM (most exquisite transistor model) [3] have
been proposed.
The Mextram model was introduced by Philips Electronics in 1985 [11]. Appearing to be the
fifth existing bipolar transistor model (after the previous four described in Ref. [12]). The first
Mextram release was introduced as Level 501 in 1985. Mextram has appeared later in several
update releases: Level 502 in 1987 [13], Level 503 in 1994, and Level 504 in 2000 [3]. The latest
version of Mextram is 504.9.1 which was released in January, 2011.
In this work, the main purpose is to develop a temperature scalable SiGe HBT model that can
workoverthe desired cryogenic temperature range. Wechoose Most EXquisite TRAnsistor Model
(Mextram) [3] as our basis because of its excellent description of vertical bipolar transistors around
room temperature. We are particularly interested in its unique collector epilayer modeling [14],
which is very important for Kirk effect and quasi-saturation. Germanium induced effects in the
3
base are also taken into consideration. The new models developed in this work, however, can be
readily ported to other compact models, e.g. VBIC.
1.4 Dissertation contributions
Chapter 1 gives the motivation of this work, including an overview of topics related to low
temperature physics and compact modeling.
Chapter2reviewstheimpactoftemperatureonsemiconductorbandgapenergyEg, electronand
holes density of states NC=NV, intrinsic carrier concentration ni, bandgap narrowing BGN, carrier
mobility m, carrier saturation velocitynsat and carrierfreezeout effectfirstly. Following, thedcand
ac low temperature performance of SiGe HBT are analyzed.
Chapter 3 extends the substrate current based avalanche multiplication technique [15] down to
43 K and gives the current dependence of avalanche multiplication factor M-1 at low temperatures.
In chapter 4, the forced-IE pinch-in maximum output voltage limit in SiGe HBTs operating at
cryogenic temperatures is investigated. A decrease of the voltage limit is observed with cooling,
and attributed to the increase of intrinsic base resistance due to freezeout as well as increase of
avalanche multiplication factor M-1. A practically high IE is shown to alleviate the decrease ofV CB
with cooling, primarily due to the decrease of M-1 with increasing emitter current IE [16].
In chapter 5, some new temperature scaling models are presented [1][2] [17]. In particular,
the temperature characteristics of mobility and ionization rate are investigated. The classic ioniza-
tion model, together with a single power law mobility model, enables resistance vs. temperature
modeling of the substrate and the collector region, where doping levels are below Mott-transition.
Based on the Altermatt ionization model, a new incomplete ionization model that accounts for the
temperature dependence of the bound state fraction factor is developed. This new model enables
accurate temperature dependent modeling of the intrinsic base sheet resistance, which has a doping
level close to the Mott-transition. For the buried collector and the silicided extrinsic base, where
doping levels are well above the Mott-transition, two approaches are proposed and both give good
results. Inchapter6, anewparameterextractionstrategyisimplemented. Withtheextractedmodel
4
parameters, we obtain reasonably accurate fitting of the dc characteristics from 393 to 43 K. Good
ac fitting from 393 to 93 K have been achieved.
Chapter 7 examines the impact of the non-ideal temperature dependence of IC-VBE in SiGe
HBTs on the output of a BGR. These non-idealities are shown to actually help make the BGR
output voltage vary less at cryogenic temperatures than traditional Shockley theory would predict.
Successful cryogenic temperature modeling of both DVBE andVBE components of the BGR output
is demonstrated for the first time [18].
Chapter 8 concludes the work in this dissertation.
5
Chapter 2
Device Physics of SiGe HBT at Cryogenic Temperatures
In this chapter, the temperature characteristics of semiconductors critical metrics are reviewed
first, including bandgap energy Eg, effective conduction band density-of-states NC and valence
banddensity-of-statesNV, intrinsiccarrierconcentrationatlowdopingni, bandgapnarrowingDEg,
carrier mobility m, carrier saturation velocity and carrier freezeout. Further, the dc and ac low
temperature performance of SiGe HBT are analyzed. Bandgap engineering generally produces
positive influence on the low temperature operations of bipolar transistors [7]. As will be shown,
SiGe HBTs work very well in the cryogenic temperature.
2.1 Semiconductor physics at cryogenic temperatures
2.1.1 Bandgap energy
The bandgap Eg is the difference between the conduction band edge energy EC and the valence
band edge energy EV. The most popular nonlinear bandgap temperature relation of silicon is [19]:
Eg;Si(T) = Eg0;Si aT
2
T +b ; (2.1)
where Eg0;Si is the bandgap of silicon at 0 K, and a and b are material parameters with a =
4:45 10 4 V/K, b = 686 K.
FromopticalexperimentsofMacFarlane[20],aboveacertaintemperature(T >T0),thebandgap
can be approximated by a linear function of temperature Eg = Eg;0 aT. Such approximation is
widely used in bipolar transistor?s compact modeling. However, nonlinear behavior of Eg-T can be
observed below T0=250 K, as shown in Fig. 2.1.(a), and will impact the compact modeling at low
temperature.
6
TheresultantenergybandstructureobtainedinastrainedSiGealloyswithrespecttoitsoriginal
Si constituent is clearly key to its usefulness in transistor engineering. For the purpose of designing
a SiGe HBT, we desire a SiGe alloy which [7]:
Has a smaller bandgap than that of Si;
Has a band offset that is predominantly in the valence band;
Either improves or at least does not substantially degrade the carrier transport parameters
(motilities or lifetime) with respect to Si.
As will be seen below, strained SiGe fulfills all of these requisite conditions.
Because Ge has a significantly smaller bandgap than Si, the bandgap of SiGe will be smaller
than that of Si. The Ge-induced band offsets can be written as DEg DEV = 0:74xmol, where xmol
is the Ge mole fraction. Hence,
Eg;SiGe(T) = Eg;Si(T) 0:74xmol: (2.2)
Fig. 2.1.(b) illustrates the Ge and temperature dependence of bandgap Eg;SiGe.
2.1.2 Density-of-states
Itisgenerallyagreeduponthattheeffectiveconductionandvalencebanddensity-of-statesprod-
uct (NCNV) is reduced strongly due to strain-induced distortion of both the valence and conduction
band extreme, a consequence of which is the reduction in the electron and hole effective masses
[21].
The NC and NV ratio between SiGe and Si are given by [22] [23]:
NC;SiGe
NC;Si
4+2e DEcbkT
6 ;
NV;SiGe
NV;Si
1+e (DEhlkT ) +e (
DEso;SiGe
kT )
2+e (
DEso;Si
kT )
; (2.3)
7
0 50 100 150 200 250 300 350 4001.05
1.1
1.15
1.2
T (K)
E g (eV)
Si
Eg=Eg,0??T
Eg=Eg,0??T2/(T+?)
(a)
0 0.05 0.1 0.150.95
1
1.05
1.1
1.15
1.2
Xmol
E g,SiGe
(eV)
40 K100 K
200 K300 K
400 K500 KT?
(b)
Figure 2.1: (a) Bandgap of silicon Eg;Si versus T. (b) Ge and temperature dependence of SiGe
bandgap Eg;SiGe.
8
where DEhl is the splitting between the heavy and light hole valence bands in SiGe, and DEso;SiGe
and DEso;Si are the distances between the split-off band and valence band edge in SiGe and Si
respectively, and DEcb is the conduction band splitting due to the biaxial strain. The resulting
(NCNV)SiGe=(NCNV)Si is weakly dependent on temperature.
Fig. 2.2.(a) - (c) explain the Ge and temperature dependence of NC;SiGe=NC;Si, NV;SiGe=NV;Si and
(NCNV)SiGe=(NCNV)SiGe respectively. NC and NV decreases with increasing Ge mole fraction first,
and then ?saturates? when the band split exceeds a couple of kT. Such substantial reduction in
NCNV with increasing Ge mole fraction can be considered undesirable since it translate directly to
a reduction in collector current in the SiGe HBT, and hence reduce the current gain. Fortunately,
however, the same reduction in effective masses that produces the decrease in NCNV also increases
the carrier motilities, which partially offset the impact on the collector current.
9
0 0.05 0.1 0.15 0.20.65
0.7
0.75
0.8
0.85
0.9
0.95
1
xmol
N C,SiGe
/N C,Si
40 K100 K
200 K300 K
400 K500 K
T?
(a)
0 0.05 0.1 0.15 0.20.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
xmol
N V,SiGe
/N V,Si
40 K100 K
200 K300 K
400 K500 KT?
(b)
0 0.05 0.1 0.15 0.20.3
0.4
0.5
0.6
0.7
0.8
0.9
1
xmol
(N CN
V) SiGe
/(N C
N V) Si
40 K100 K
200 K300 K
400 K500 K
T?
(c)
Figure 2.2: (a) Ge and temperature dependence of NC;SiGe=NC;Si. (b) Ge and temperature depen-
dence of NV;SiGe=NV;Si. (c) Ge and temperature dependence of (NCNV)SiGe=(NCNV)SiGe.
10
2.1.3 Intrinsic carrier concentration at low doping
The band gap and band edge density-of-states are summarized in the intrinsic density ni(T) (for
undoped semiconductors):
ni0(T) =
p
NC(T)NV(T)exp
Eg(T)2kT
: (2.4)
Fig. 2.3 show the temperature dependence of intrinsic carrier concentration at low doping ni0
using (2.1), (2.2), (2.3) and (2.4). ni0 increases dramatically with increasing Ge content, especially
at low temperatures. At 30K, ni0 boosts by 12 orders of magnitude and 6 orders of magnitude for
xmol = 20% and xmol = 10%, respectively.
0 50 100 150 200 250 300
10?60
10?40
10?20
100
1020
Temperture (K)
n i0 (cm
?3 )
xmol=0%x
mol=5%x
mol=10%x
mol=15%x
mol=20%
xmol ?
Figure 2.3: Temperature dependence of intrinsic carrier concentration at low doping ni0.
2.1.4 Bandgap narrowing
It is well known that the bandgap narrows at heavy doping, which increases the pn products at
equilibrium. This is often referred to as heavy doping induced bandgap narrowing DEg.
11
The most widely used bandgap narrowing (BGN) model is the Slotboom bandgap narrowing
model [24] which is experimentally determined from the IC-VBE of NPN bipolar transistors. Slot-
boom bandgap narrowing model is derived based on some assumptions. Firstly, it assumes that
the minority carriers in the base obey the Boltzmann distribution law, which however is not valid
for heavily doped semiconductor or low temperature environment. Secondly, a linear temperature
dependence of bandgap energy Eg = Eg;0 aT is approximated in the equation of intrinsic carrier
concentrationni. Thirdly, thetotalnumberofholesQB inbaseregionisassumedasconstantacross
temperature which is not valid at low temperatures due to carrier freezeout effect.
Boltzmann statistics is no long accurate for heavily doping and Fermi-Dirac statistics is needed
instead. TheideaofSlotboomBGNmodelistoartificiallydecreasetheapparentelectricalbandgap
narrowingDEg;app byDEg;FD (correction to reduce the error introduced by Boltzmann statistics) so
that one can continue to apply Boltzmann statistics to describe the equilibrium pn product at heavy
doping. The apparent bandgap narrowing DEg;app can be given as [24].
DEg;app = Eref
ln
N
tot
Nref
+
s
ln
N
tot
Nref
2
+0:5
: (2.5)
Hence the true bandgap narrowing DEg;true can be written as:
DEg;true = DEg;app DEg;FD: (2.6)
and
DEg;FD
=
8
>>>>
<
>>>
>:
kT
h
ln
n
NC
F 11=2
n
NC
i
n-type doping;
kT
h
ln
p
NV
F 11=2
p
NV
i
p-type doping:
(2.7)
12
Fig. 2.4 shows the doping dependence of bandgap narrowing DEg;app, DEg;true and DEg;FD
respectively. OneshoulduseBoltzmannstatisticsindevicesimulationiftheapparentBGNDEg;app
parameters are used. For device simulation using Fermi-Dirac statistics, the true BGN DEg;true
parameters should be used.
1016 1017 1018 1019 10200
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Doping Concentration (cm?3)
?E g
(eV)
?Eg,app?E
g,true?E
g,FD
Figure 2.4: Doping dependence of bandgap narrowing DEg.
13
2.1.5 Carrier freezeout and Mott-transition
Complete ionization of dopants in Si and SiGe is typically assumed at room temperature. How-
ever, when the Fermi level EF is situated close to the dopant level, the dopant states are noticeably
occupied, leading to incomplete ionization even at room temperature. The free carrier density is
then noticeably smaller than the dopant density. This semiconductor physics is so called ?carrier
freezeout? and shows strong dependence on doping concentration and temperature.
The numerical solution of quasi-Fermi energy EF and ionization rage IR are illustrated here.
The analytical solution will be presented in 5.11.3 in detail.
p+N+D = n+N A ; (2.8)
n = NCexp( EC EFkT ); (2.9)
p = NV exp( EF EVkT ); (2.10)
N+D = ND1+g
Dexp(EF EDkT )
; (2.11)
N A = NA1+g
Aexp(EA EFkT )
; (2.12)
where EA is acceptor impurity energy, gA is acceptor degeneracy factor, ED is donor impurity
energy, gD is donor degeneracy factor, n is electron density, p is hole density, ND is active donor
concentration,NA isactiveacceptorconcentration,N+D isionizeddonorconcentration,N A isionized
acceptor concentration.
Substituting (2.10) and (2.11) into left-hand-side of (2.8) and substituting (2.9) and (2.12) into
right-hand-sideof(2.8)respectively,wecanplotoutthe p+N+D andn+N A asfunctionof(EF EI)
for different doping concentrations and different temperatures. By applying (2.8), Fermi energy
level EF can be solved numerically. Fig. 2.5 illustrates this graphical solution strategy for three
n-type doping levels at 300 K. Fig. 2.6 shows the variation of Fermi level Ef Ei versus doping
concentration over 40-500 K.
14
Because the EF locates near intrinsic Fermi level EI at low concentration and high temperature,
complete ionization assumption is valid for low doping and high temperature. However, when
concentration increases together with cooling, EF moves towards and even above dopant energy
level ED. Hence, incomplete ionization (carrier freezeout) occurs. Fig. 2.7.(a)-(b) show the solved
doping and temperature dependence of ionization rate IR for n-type doping. The IR drops towards
?0? at low temperature and high concentration. For example, at 300 K, for doping above 1 1018
cm 3, IR drops quickly, thus limits doping?s effectiveness in increasing carrier concentration.
?1 ?0.5 0 0.5 1 1.510?10
100
1010
1020
1030
1040
EF?EI (eV)
Concentration (cm
?3 )
Nd?
p+Nd+
n+Na?
300K n?type doping
Figure 2.5: Concentration versus EF EI.
However, experimentally, we use doping much higher than 1 1018 cm 3 in devices, such as
source/drain of CMOS transistors and emitter of BJT transistor. They continue to provide reduced
resistance. Hence this classic freezeout model fails at such high doping. In practice, the ionization
rate at 300 K is 100% at low doping, drops at higher doping near 1 1018 cm 3, but then increases
back to 100% when doping is around 1 1019 cm 3 or so.
15
1013 1014 1015 1016 1017 1018 1019?0.6
?0.4
?0.2
0
0.2
0.4
0.6
Nd (cm?3)
E F?E
I (eV)
40 K100 K
200 K300 K
400 K500 K
EC
EV
(EC+ED)/2
n?type doping
T?
Figure 2.6: Variation of Fermi level Ef Ei versus doping concentration over 40-500 K.
Numerous theories of how impurity concentration and temperature affect ionization rate have
been developed in the past several decades [25][26][27][28] [29][30][31][32]. At higher doping,
density-of-stateofdopantstatesbroadenintodopantbandanddopantbandtouchesEC=EV atMott-
transition ( 3 1018 cm 3). The ionization energy Edop drops towards ?0? hence causes dopants
completelyionized,despitecooling[33]. Toaccuratelymodeltheincompleteionizationforheavily
doping device, Mott-transition should be considered with freezeout model.
In the most recent Altemmatt?s incomplete ionization model [26][27], not all of the dopant
states are localized (or bound) states. At high concentration, once dopant band touches EC=EV,
they become free and do not contribute to the local states. The fraction of bound states is named as
?b?. In Altemmatt?s incomplete ionization model [26][27], both the bound state fraction ?b? and
dopantenergyEdop decreaseswithincreasingdoping. Fig.2.8.(a)-(b)showthedopingdependence
of b and Edop respectively. (2.11) and (2.12) then can be written as [17]:
N+D = (1 b)ND + bND1+g
Dexp(EF EDkT )
; (2.13)
16
1013 1015 1017 1019 10210
0.2
0.4
0.6
0.8
1
Nd (cm?3)
Ionization rate
40 K100 K
200 K300 K
400 K500 K
(a)
0 50 100 150 200 250 300 350 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T (K)
Ionization rate N
A=14 cm?3N
A=15 cm?3N
A=16 cm?3N
A=17 cm?3N
A=18 cm?3N
A=19 cm?3
NA ?
(b)
Figure 2.7: (a) Doping dependence of IR at different temperatures. (b) Temperature dependence of
IR at different doping levels.
17
N A = (1 b)NA + bNA1+g
Aexp(EA EFkT )
; (2.14)
Fig.2.9showthecalculateddopingdependenceofIRatdifferenttemperaturesusingAltemmatt?s
model. At 300 K, ionization rate is not complete around 1 1018 cm 3. Ionization rate decreases
with increasing doping concentration, but comes back at heavy doping levels towards complete
ionization (100%). At heavy doping, ionization remains complete even if temperature goes down-
this is where the ?b? factor comes into play, because at higher doping the ?b? factor drops to ?0?
and represents all states are free states, or ionized.
18
1015 1016 1017 1018 1019 1020 10210
0.01
0.02
0.03
0.04
0.05
0.06
Nd (cm?3)
Edop (eV)
Altermatt?s model
Arsenic
(a)
1015 1016 1017 1018 1019 1020 10210
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Nd (cm?3)
b
Altermatt?s model
Arsenic
(b)
Figure 2.8: (a) Edop-Nd. (b) b-Nd.
19
1013 1014 1015 1016 1017 1018 1019 1020 10210
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Nd (cm?3)
Ionization rate
40 K100 K
200 K300 K
400 K500 K
Arsenic
T?
Figure 2.9: Doping dependence of IR at different temperatures using Altemmatt?s model.
20
2.1.6 Low field carrier mobility
At low electric field, the relation between carrier velocity nsat and electric field ~E is given by
n = m~E, where m is the low field carrier mobility. At high electric field, the carrier velocity nsat
becomes saturate. The most popularly used mobility model is Philips unified mobility model [34]
[35], which unifies the description of majority and minority carrier bulk motilities. Besides lattice,
donorandacceptorscattering,electron-holescatteringisalsoincorporated. Screeningofimpurities
by charge carriers and the temperature dependence of both majority and minority carrier mobility
are included. Ultra-high concentration effects are also taken into account.
Based on Matthiesen?s rule to sum all the contributions to mobility,
1
me;h =
1
mL +
1
mDAeh; (2.15)
where mL represents the lattice scattering contribution, and the mDAeh accounts for all other bulk
scattering mechanisms due to free carriers and ionized donors and acceptors. The first term mL can
be written by the well-known power dependence on temperature [36][37]:
mL = mmax(300T )q; (2.16)
where q is determined in comparison with experimental data. The second term mDAeh is a compli-
catedfunctionofelectrondensityn,holedensity p,donorconcentrationND,acceptorconcentration
NA and the temperature T. The complete equations can be found in [34] [35].
In this section?s discussion, all of the calculations are using the equation and parameters of
Philips unified mobility model [34] [35].
21
Carrier freezeout effect
Based on the previous discussion of carrier freezeout effect, incomplete ionization occurs the
mostly when doping concentration is near Mott-transition ( 1018 cm 3) at low temperatures.
Hence in the following discussion, for better understanding the hole (majority carrier) and electron
(minoritycarrier)mobilitiesinthebaseregionofaNPNSiGeHBT,theholeandelectronmobilities
are calculated for p-type Boron from 50 K to 300 K, with doping concentration is from 1014
cm 3 to 1020 cm 3. In the Philips unified mobility model [34] [35], carrier freezeout effect is
taken in consideration from carrier scattering (electron-hole scattering) by introducing the strong
temperature dependence of hole density and electron density.
Fig. 2.10.(a)-(b) are the calculated hole mobilities with complete ionization assumption and
incomplete ionization assumption respectively. Altemmatt?s incomplete ionization model [26][27]
are used in these calculations. The first observation is, whatever the carrier freezeout is taken in
account or not, at low concentration and higher temperature, lattice scattering factor mL dominates.
At high concentration and low temperature, bulk scattering factor mDAeh mechanisms overwhelm
lattice scattering factor mL. Secondly, by comparing Fig. 2.10.(a) and Fig. 2.10.(b), it is found that
the most difference is near Mott transition ( 1018 cm 3), which is plotted in Fig. 2.10.(c). Since
the semiconductor sheet resistance is not only a function of majority density, but also a function
of majority mobility, it implies that not only the carrier freezeout effect should be considered into
majority density modeling but also into the majority mobility modeling, especially near Mott-
transition. The further discussion will be found in compact modeling chapter 5.11.3.
Fig. 2.11.(a)-(b) are the calculated electron mobilities with complete ionization assumption and
incomplete ionization assumption respectively. Similarly, whatever the carrier freezeout is taken in
account or not, at low concentration and higher temperature, lattice scattering factor mL dominates.
At high concentration and low temperature, bulk scattering factor mDAeh mechanisms overwhelm
lattice scattering factor mL. Secondly, by comparing Fig. 2.11.(a) and Fig. 2.11.(b), it is found that
the most difference is near Mott transition ( 1018 cm 3), which is plotted in Fig. 2.11.(c).
22
As the base transit time, which limits the frequency response of the SiGe HBT in most cases, is
determined by the minority electron mobility in the base region, the carrier freezeout effect should
be considered into the transit time calculation.
Ultra-high concentrations effect
The effects of ultra-high concentrations on the mobility can be accounted for by assuming that
above certain impurity concentration ( 1020 cm 3), the carriers are no longer scattered by impu-
rities possessing one electronic charge and a concentration N, but by impurities with Z electronic
charges and a ?cluster? concentration N . Therefore, the ultra-high concentration effects can be
modeled by replacing N by Z(N) N, where Z(N) is the ?clustering? function [34] [35].
Z = 1+ 1
c+(NrefN )2
: (2.17)
Fig. 2.12.(a) is the calculated clustering function vs. impurity concentration. The effects of
ultra-high concentrations on the mobility become noticeable when doping concentration is larger
than 1020 cm 3. Fig. 2.12.(b)-(c) are the calculated concentration dependence of electron majority
and minority mobility me at 300 K and 100 K respectively. The dotted lines represent the results
with ZA;D = 1. The Z factor introduces the second ?knee-like? shape transition for mobilities at
ultra-high concentration.
Inthepast, theminoritycarriermobilityhasbeenassumedequaltothemajoritycarriermobility
for simplicity in device modeling. However, results reported on carrier transport characteristics in
Si show higher minority carrier mobility than the majority carrier mobility. This discrepancy is
believed to originate from the difference in ionized impurity scattering [38]. In Fig. 2.12, the ratio
ofthetwomobilitiesis1.5nearMott-transition(1 1018cm 3)atroomtemperature. Andthisratio
increases to 3 near Mott-transition (1 1018cm 3) at 100 K.
However,becausebulkscatteringmechanismdominatesathighconcentrationandlowtempera-
ture, differencebetweenmajoritymobilityandminoritymobilityisobservedwhendopingisabove
23
50 100 150 200 250 300101
102
103
104
105
106
Temperature (K)
Hole Mobility (cm
2 /Vs)
?h?
h,L?h,DAehP?type NA=1014 cm?3
NA=1016 cm?3
NA=1018 cm?3
NA=1020 cm?3 complete ionization
(a)
50 100 150 200 250 300101
102
103
104
105
106
Temperature (K)
Hole Mobility (cm
2 /Vs)
?h?
h,L?h,DAehP?type NA=1014 cm?3
NA=1016 cm?3
NA=1018 cm?3
NA=1020 cm?3 incomplete ionization
(b)
50 100 150 200 250 300101
102
103
Temperature (K)
Hole Mobility (cm
2 /Vs)
?e?
e,DAehP?type
NA=1018 cm?3
solid: incomplete ionizationdash: complete ionization
(c)
Figure2.10: (a)Temperatureandconcentrationdependenceofholemobilitymh,w/ocarrierfreeze-
out effect. (b) Temperature and concentration dependence of hole mobility mh, w/ carrier freezeout
effect. (c) Temperature dependence of hole mobility mh near Mott-transition.
24
50 100 150 200 250 300101
102
103
104
105
106
Temperature (K)
Electron Mobility (cm
2 /Vs)
?e?
e,L?e,DAehP?type NA=1014 cm?3
NA=1016 cm?3
NA=1018 cm?3
NA=1020 cm?3 complete ionization
(a)
50 100 150 200 250 300101
102
103
104
105
106
Temperature (K)
Electron Mobility (cm
2 /Vs)
?e?
e,L?e,DAehP?typeN
A=1014 cm?3
NA=1016 cm?3
NA=1018 cm?3
NA=1020 cm?3
incomplete ionization
(b)
50 100 150 200 250 300102
103
Temperature (K)
Electron Mobility (cm
2 /Vs)
?e?
e,DAeh
P?type
NA=1018 cm?3
solid: incomplete ionization dash: complete ionization
(c)
Figure 2.11: (a)Temperature and concentration dependence of electron mobility me, w/o carrier
freezeout effect. (b) Temperature and concentration dependence of electron mobility me, w/ carrier
freezeout effect. (c) Temperature dependence of electron mobility me near Mott-transition.
25
1 1018cm 3 at 100 K, as shown in Fig. 2.12.(c). The concentration dependence of hole majority
and minority mobilities mh is similar to that of electron mobility me and is not shown here.
Ge effect
Many investigations have been taken on the germanium dependence of mobility [39][40][41].
Both the holes and electrons mobility enhancement have been demonstrated in strained Si devices,
includingdualstressliner(DSL),embeddedSiGesource/drainandstressmemorytechnique(SMT)
of CMOS technology. Uniaxial strain reduces carrier effective mass, increasing low field mobility
and velocity. NFET is enhanced by in-plane tension or vertical compression and PFET is enhanced
by compression along the channel or tension perpendicular to channel [39][40][41].
In[42][43], thetheoreticalanalysisreportedontheminorityelectronmobilityinp-typestrained
Si Ge alloys for low Ge compositions (x 0:3) and mainly focused on room temperature mobility.
In [43], the low-field minority and majority mobilities in strained SiGe perpendicular, mzz, and
parallel to the Si/SiGe interface, mxx, as well as the mobility m in unstrained SiGe, are displayed
as a function of Ge content for different doping concentrations. As a general trend, the mobility
is reduced with increasing Ge content at low doping concentrations by alloy scattering. At higher
doping, mzz in strained SiGe increases, mxx in strained SiGe and m in unstrained SiGe reduce more
or less equally strong.
In [44], temperature dependence of minority electron mobilities in p-type SiGe has been mea-
sured for the first time for Ge composition between 0:2 x 0:4. The measurements were made
on NPN SiGe HBT with a heavily doped base (7 1019 cm 3), from 5 K to 300 K. The mea-
sured minority electron mobilities show a sharp increase with decreasing temperature, and exhibit
saturation for temperature below 50 K.
26
1014 1016 1018 1020 10221
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Impurity Concentration (/cm3)
Clustering function Z
donors, ZDacceptors, Z
A
(a)
1014 1016 1018 1020 1022100
101
102
103
104
Impurity Concentration (/cm3)
Electron Mobility (cm
2 /Vs)
Majority mobilityMinority mobility300 K
ZA,D=1
(b)
1014 1016 1018 1020 1022101
102
103
104
105
Impurity Concentration (/cm3)
Electron Mobility (cm
2 /Vs)
Majority mobilityMinority mobility100 K
ZA,D=1
(c)
Figure 2.12: (a) Calculated clustering function vs. impurity concentration. (b) Calculated concen-
tration dependence of electron mobilities me at 300 K. (c) Calculated concentration dependence of
electron mobilities me at 100 K.
27
2.1.7 Carrier saturation velocity
The bulk carrier saturation velocity nsat used in high-field mobility is modeled as a function of
temperature with the following two-parameter model [45][46]:
nsat(T) = nsat;300(1 A
n)+An( T300)
; (2.18)
where nsat;300 represents the saturation velocity at room temperature and An is temperature co-
efficient. The model assumes that the nsat is independent of the doping concentration, which is
in good agreement with published data [46]. An reflects the temperature dependence of various
materials. For a material of A1 xBx, the saturation velocity can be interpolated according to the
material composition.
0 50 100 150 200 250 300 350 4000.5
1
1.5
2
2.5
3
3.5x 107
T (K)
v sat
(cm/s)
Si electronsSi holes
Ge electronsGe holes
Vassil?s model
Figure 2.13: Saturation velocity as a function of temperature for Si and Ge.
28
2.1.8 Sheet resistance and resistivity
Semiconductor?s sheet resistance or resistivity is closely related to majority carrier mobility,
majority carrier concentration and hence incomplete ionization rate.
Rsh =
0
@q
WZ
0
N dop(x)m(x)dx
1
A
1
; (2.19)
where W is neutral region width, N dop is ionized majority carrier density, m is majority carrier
mobility. Temperature dependence of W is from the variation of p-n junction depletion layer
thickness, which is much smaller than that of N dop and m and is thus neglected.
Forexample, intrinsicbaseneutralregionwidthWB isafunctionofthethicknessofemitter-base
and collector-base junction depletion layers on the base side. Here, we calculate depletion layer
thicknessWdep versus T from 30-300 K using Altermatt and classic incomplete ionization models
in Fig. 2.14. The chosen doping levels are similar to those of the SiGe HBT in this work. In the
calculations, we have included T-dependence of NV, NC and bandgap Eg. As shown in Fig. 2.14,
both of these two IR models give very weak T-dependence of Wdep, which is much smaller than
those of N dop and m. Hence T-dependence ofW is negligible compared to those of N dop and m.
Here, sheet resistance and substrate resistivity were measured on-wafer from 300 K to 30 K in
a first-generation 0.5 mm SiGe HBT technology featuring 50 GHz peak fT at 300 K, with base
doping below but close to the Mott-transition (about 3 1018 cm3 for boron in silicon). The detail
descriptionofteststructureandmeasurementtechniquewillbediscussedin5.11.3. Fig.2.15shows
the measured substrate resistivity, the collector sheet resistance, the intrinsic base sheet resistance,
the buried collector sheet resistance, and the silicided extrinsic base sheet resistance, from 30 to
300 K.
For the n+ buried collector, the resistance increases slightly with temperature at high tempera-
turesduetoadecreasesinthemajoritycarriermobility. Atlowtemperatures,theresistanceremains
approximately constant, indicating that freezeout of carriers is not occurring to any significant ex-
tent. This is expected since the doping is well above the Mott-transition and ionization is complete
29
0 50 100 150 200 250 3002
2.5
3
3.5
4
4.5
5
5.5
6
6.5x 10?6
T (K)
W Dep
(cm)
ND=5x1017 cm?3 ,NA=3x1018 cm?3
ND=1x1020 cm?3 ,NA=3x1018 cm?3
solid: Altermatt IR model dot: Classic IR model
base?collector junction depletion layer
base?emtter junction depletion layer
Figure 2.14: calculate depletion layer thicknessWdep versus T from 30-300 K using Altermatt and
classic incomplete ionization models.
at all temperatures. For the p+ silicided extrinsic base, it has similar temperature function as the
n+ buried collector but with smaller sheet resistance values. In contrast, for the p substrate, n
collector and the p-type intrinsic base, the resistance increases strongly at low temperatures, indi-
cating that freezeout of dopant is occurring. This is again expected because the p substrate and
n collector doping is well below the Mott-transition [33], while the doping of the p-type intrinsic
base is close to Mott-transition.
30
20 100 30010
0
102
104
106
108
T (K)
Sheet resistance (
?/sq)
40 100 30010
0
101
102
T (K)
Resistivity (
?.cm)
n? collector
p intrinsic base
n+ buried collector
p+ extrinsic base
p? substrate
(a) (b)
Figure 2.15: (a)Measured sheet resistance for collector, intrinsic base, extrinsic base and buried
layer from 30-300 K. (b) Measured substrate resistivity from 30-300 K.
31
2.2 SiGe HBT characteristics at cryogenic temperature
2.2.1 Limitations of Si BJT
Simultaneous increase of current gain b, cut-off frequency fT , and decrease of base resistance,
however, are conflicting goals when translated into device design. To increase fT , base width
WB needs to be reduced, which increases base resistance unfortunately. The total base dopants
NBWB must be kept at least to keep base resistance, requiring the increase of base doping NB.
The emitter structure, which determines the base current, is normally fixed with device scaling for
manufacturing bipolar technologies.
The emphasis is often to achieve a low but reproducible base current while minimizing the
emitter resistance. As a result, there is no b and base resistance improvement at all from such a
WB scaling even if realizable. The best result is an increase of fT and a smaller increase of fmax.
While a simultaneous increase of fT , increase of b, and reduction of base resistance is desired for
better transistor performance. Furthermore, a thinWB with high base doping NB is very difficult to
achieve in traditional implanted BJT technologies.
2.2.2 SiGe HBT fundamental
The essential operational differences between the SiGe HBT and the Si BJT are best illustrated
by considering a schematic energy band diagram. For simplicity, we consider an ideal, graded-
base SiGe HBT with constant doping in the emitter, base and collector regions. The Ge content is
linearlygradedfrom0% nearthemetallurgicalemitter-base(EB)junctiontosomemaximumvalue
of Ge content near the metallurgical collector-base (CB) junction, and then rapidly ramped back
down to 0%. The resultant overlaid energy band diagrams for both the SiGe HBT and the Si BJT,
biased identically in forward-active mode, are shown in Fig. 2.16. A Ge-induced reduction in base
bandgap occurs at the EB edge of the quasi-neutral base (DEg;Ge(x = 0)), and at the CB edge of
the quasi-neutral base (DEg;Ge(x =Wb)). This grading of the Ge across the neutral base induces a
32
built-inquasi-driftfield((DEg;Ge(x=Wb) DEg;Ge(x= 0))=Wb)intheneutralbasethatwillimpact
minority carrier transport [47][7].
EC
EV
n+Siemitter
p-SiGebase
n- Sicollector
e-
h+
?Eg,Ge(x=0)
?Eg,Ge(grade)=?Eg,Ge(x=Wb)-?Eg,Ge(x=0)
drift field !
p-Si
Ge
Figure 2.16: Energy band diagram for a Si BJT and graded-base SiGe HBT.
2.2.3 Collection current density and current gain
The theoretical consequences of the Ge-induced bandgap changes to collection current density
JC can be derived in closed-form for a constant base doping profile by considering the generalized
Moll-Ross collector current density relation, which holds for low-injection in the presence of both
non-uniform base doping and non-uniform base bandgap at fixedVBE and temperature (T) [47].
JC = q(e
qVBE=kT 1)
Wb
s
0
pb(x)dx
Dnb(x)n2ib(x)
; (2.20)
wherex=0andw=Wb aretheneutralbaseboundaryvaluesontheemitter-baseandcollector-base
sides of the base respectively. nib, Dnb and pb are intrinsic carrier concentration, electron diffusion
constant and hole density in the base respectively. Through the Ge-induced bandgap offset, nib and
Dnb are position-dependent.
33
The intrinsic carrier concentration in the SiGe HBT base can be written as eqn(2.4). In [7],
detailed derivations of the collector current in SiGe HBT are discussed and can be finally written
as:
JC = qDnbN
abWb
eqVBE=kT 1
n2ioeDEappgb =kT
ghDE
g;Ge(grade)=kTeDEg;Ge(0)=kT
1 e DEg;Ge(grade)=kT
;
DEg;Ge(grade) = DEg;Ge(Wb) DEg;Ge(0);h = Dnb;SiGeD
nb;Si
;g = (NCNV)SiGe(N
CNV)Si
: (2.21)
where Dnb and g is position-averaged quantities across the base region.
For a comparably structure SiGe HBT and Si BJT, the base current density JB should be com-
parable between the two devices, while JC at fixed VBE should be enhanced for the SiGe HBT.
10?3 10?2 10?1 100 1010
50
100
150
200
250
JC (mA/?m2)
?
43K60K
76K93K
110K136K
162K192K
223K262K
300K393K
T?
Figure 2.17: Measured current gain b-VBE for a graded-base SiGe HBT.
Fig.2.17showsthemeasuredcurrentgainb-VBE foragraded-baseSiGeHBT.Giventhenatureof
anexponentialdependence,itisobviousthatstrongenhancementinJC forfixedVBE canbeobtained
34
for small amounts of introduced Ge, and that the ability to engineer the device characteristics to
obtain a desired current gain is easily accomplished.
2.2.4 Early effects
The dynamic output conductance, ?JC?VCE at fixedVBE, of a transistor is a critical design parameter
for many analog circuits. As we increase collector-base voltage VCB, we deplete the neutral base
from the backside , thus moving the neutral base boundary (x =Wb) inward. SinceWb determines
the minority carrier density on the CB side of the neutral base, the slope of the minority electron
profile, and hence the collector current IC rises [48]. This mechanism is known as ?Early effect?.
For a linearly graded Ge profile, the ratio of VA between a comparably constructed SiGe HBT
and Si BJT to be [49][7] can be written as:
VA;SiGe
VA;Si
VBE
= eDEg;Ge(grade)=kT
1 e DEg;Ge(grade)=kT
DEg;Ge(grade)=kT
: (2.22)
The fundamental difference betweenVA in a SiGe HBT and Si BJT arises from the variation of
n2ib asafunctionofposition. Thebaseprofileiseffectively?weighted?bytheincreasingGecontent
on the collector side of the neutral base, making it harder to deplete the neutral base for a givenVCB
and it effectively increasing the Early voltage of the transistor.
2.2.5 Avalanche multiplication and breakdown voltage
Under reverse bias, the electric field in the space-charge region of the CB junction is large.
Electrons injected from the emitter drift to the collector through the CB space-charge region.
For a sufficiently high electric field, electrons can gain enough energy from the electric field to
create an electron-hole pair during the carrier generation process known as ?impact ionization?.
Electrons and holes generated by impact ionization can subsequently acquire energy from this
strong electric field, and create additional electron-hole pairs by further impact ionization. This
process of multiplicative impact ionization is known as ?avalanche multiplication?. The ratio of
35
the electron current leaving CB space-charge region to that entering the CB space-charge region
is known as the avalanche multiplication factor M. In practice, (M-1) instead of M is often used
because (M-1) better describes the yield of the resulting collector current increase.
Avalanche multiplication is an important effect that must be accurately measured and modeled.
The avalanche multiplication (M-1) determines the breakdown voltage as well as the base current
reversalvoltage,whichinturndeterminesthemaximumusefulVCB forstablecircuitoperation[50].
Figure 2.18: The avalanche multiplication process in a BJT.
Forced-IE (M-1)measurementiswidelyusedinsteadofforced-VBE (M-1)measurementtoavoid
self-heating effect at high JC or highVCB, because the total amount of current injected into the CB
space-charge region is always limited by emitter current IE [51].
36
Fig.2.19showsthe(M-1)versuscollector-basevoltageVCB foratypicalgraded-baseSiGeHBT.
Fig. 2.20 shows the (M-1) as a function of 1000=T at various VCB;s. In contrast to the previous
observations of a strongly exponential increase with cooling in Si BJT [52], the increase with
cooling is much weaker in the SiGe BJT under study, particularly below 162 K. The difference in
temperature sensitivity is not due to any Ge effect, but instead is attributed to the higher collector
doping level in the device measured in this study than in the devices used in [52].
0 1 2 3 4 5
10?4
10?3
10?2
10?1
VCB (V)
M?1
43K93K
162K223K
300K393K
T?
Figure 2.19: Measured M-1-T for a graded-base SiGe HBT over 43-393 K.
The maximum operation voltage limit of a bipolar transistor is generally dictated by avalanche
multiplication. Two often used voltage limits are open-base BVCEO and short-base BVCBO, which
representtheworstandbestcasesforforced-IB andforced-VBE configurations,respectively. Another
frequently employed bias configuration is forced-IE, which has virtually the same collector-to-
emitter breakdown voltage as BVCBO, due to the fixed IE. However, at practically high currents, a
lateral current instability due to avalanche multiplication induced pinch-in effect [50] may occur
before the BVCBO limit is reached. This can occur for forced-VBE as well. Pinch-in occurs when
avalanche induced hole current dominates over the normal hole current injected into the emitter.
37
0 5 10 15 20 2510?3
10?2
10?1
100
1000/T (K?1)
M?1
VCB=2VV
CB=3VV
CB=4VV
CB=5V
Figure 2.20: Measured multiplication factor M-1 versus 1000=T at variousVCB;s for a graded-base
SiGe HBT.
VCB
B
JER
E
RBx RBi
C
E
(a) experimental setup
(b) local emitter current density JE when I
B<0
LE0
Figure 2.21: (a) Experimental setup. (b) Local emitter current density JE when IB < 0.
38
The net base current flows out of the base, creating a lateral VBE and emitter current density JE
variation that is highest at the emitter center and lowest at the edge, which is shown in Fig. 2.21.
At a critical base current I B, an abrupt pinch-in of the emitter current to a very small area of the
emitter center occurs, and sets an upper limit of stable operation.
In chapter 4, forced-IE pinch-in maximum output voltage limit in SiGe HBTs operating at cryo-
genictemperatureswillbeinvestigated,whichisofinterestformanyspaceexplorationapplications
[5].
39
2.2.6 AC characteristics
At low injection, the unity-gain cutoff frequency fT in a bipolar transistor can be written as:
fT = 12pt
ec
= 12p
V
T
IC (Cte +Ctc)+tb +te +
WCB
2vsat +rcCtc
1
; (2.23)
where gm is the intrinsic transconductance, Cte and Ctc are the emitter-base and base-collector
depletion capacitances,tb is the base transit time,te is the emitter transit time,WCB is the collector-
base space-charge region width, vsat is the saturation velocity, and rc is the dynamic collector
resistance, tec is the total emitter to collector transit time.
In most cases, the base transit time tb overwhelms other components and determines the peak
fT. The ratio of base transit time between SiGe HBTtB;SiGe and Si HBTtB;Si can be written as [7]:
tb;SiGe
tb;Si =
2
h
kT
DEg;Ge(grade)
1 kTDE
g;Ge(grade)
1 e DEg;Ge(grade)=kT
;
tb;Si = W
2
b
2Dnb: (2.24)
Fig. 2.22 are the calculated tb;SiGe, tb;Si and tSiGe=tSi-T ratio as functions of temperature for a
graded-base SiGe HBT. The base width is assumed as 50 nm with 1 1018 cm 3 uniform doping.
The total Ge-induced bandgap grading in base is assumed as 100 meV. Ge effect on low field
mobilityisnotincludedforsimplicity. ThePhilipsunifiedmobilitymodel[34][35]andAltemmatt?s
incomplete ionization model [26][27] are used in the calculation. The thermal energy kT residing
in the denominator of eqn(2.24) decreases tSiGe=tSi at low temperatures, which are demonstrated
in Fig. 2.22.(b). Therefore, whatever the carrier freezeout is taken in account or not, tb;SiGe is
smaller than tb;Si, especially at cryogenic temperatures. This demonstrates the advantage of SiGe
application at cryogenic temperatures.
Fig.2.23.(a)-(b)showthemeasured fT-JC and fmax-JC foragraded-baseSiGeHBTrespectively.
Asaresultofthedecreasingtb;SiGe withcooling,thepeak fT enhancesatlowtemperature. Another
40
0 50 100 150 200 250 300 350 4000
0.5
1
1.5
2
2.5
3
3.5
4
T (K)
Base transit time (ps)
?B,Si
?B,SiGe?Eg,Ge = 100 meV
WB = 50 nm
NA = 1? 1018 cm?3
solid: incomplete ionizationdash: complete ionization
(a)
0 50 100 150 200 250 300 350 4000.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
T (K)
? SiGe
/? Si
(b)
Figure 2.22: (a) Calculated tb;SiGe-T and tb;Si-T for a graded-base SiGe HBT. (b) Calculated ratio
tSiGe=tSi-T for a graded-base SiGe HBT.
41
observation from Fig. 2.23.(a) is the improved fT roll-off at low temperatures. Lower the temper-
ature, larger collector current density where base push out effect occurs. The critical collector
current density Jcr for the onset of base push-out is proportional to high field saturation velocity,
Jcr vsat. From the previous discussion in 2.1.7, high field saturation velocity vsat increases with
cooling, hence fT roll-off is improved due to the higher Jcr at low temperature.
The maximum oscillation frequency fmax is related to fT by:
fmax =
s
fT
8pCbcrb; (2.25)
whereCbc istheMillercapacitancebetweenbaseandcollector( Ctc),rb isdynamicbaseresistance.
Capacitance Cbc has much weaker temperature dependence than that of cut-off frequency fT and
dynamic base resistance rb. Carrier freezeout effect induced base resistance increasing at low
temperature lead to fmax degradation at 93 K, as shown in Fig. 2.23.(b).
2.3 Summary
In this chapter, the temperature characteristics of semiconductors critical metrics are studied,
including bandgap energy Eg, effective conduction band density-of-states NC and valence band
density-of-states NV, intrinsic carrier concentration at low doping ni, bandgap narrowing DEg, car-
rier mobility m, carrier saturation velocity and carrier freezeout. The dc and ac low temperature
performance of SiGe HBT are analyzed, including collector current density, current gain, Early
effect, avalanchemultiplicationfactor, transittime, cut-offfrequencyandmaximumoscillationfre-
quency. SiGe HBT demonstrates excellent analog and RF performance at cryogenic temperatures.
42
10?3 10?2 10?1 100 1010
10
20
30
40
50
60
70
JC (mA/um2)
f T (GHz)
93K162K
223K300K
393K
VCB=0V
(a)
10?3 10?2 10?1 100 1010
10
20
30
40
50
60
70
80
JC (mA/um2)
f max
(GHz)
93K162K
223K300K
393K
VCB=0V
T?
(b)
Figure 2.23: (a) Measured fT-JC for a graded-base SiGe HBT. (b) Measured fmax-JC for a graded-
base SiGe HBT.
43
Chapter 3
Substrate Current Based M-1 Measurement at Cryogenic Temperatures
3.1 Measurement theory
In [15], a new substrate current-based technique for measuring the avalanche multiplication
factor (M-1) in high-speed SiGe HBTs is proposed, which enables (M-1) measurement at high
operating current densities required for high-speed operation, where conventional techniques fail
because of self-heating. The proposed technique is based on photo carrier generation by hot carrier
induced light which produces electron-hole pairs in the collector-substrate junction, as shown in
Fig. 3.1. Collection of these electron-hole pairs leads to a substrate current (ISUB), which can be
used as a monitor for the occurrence of avalanche multiplication [53].
p-SiGe base
STISTI
n
n+
Avalanche
generation
light
n+ polyB
E
C Sub
~~ ~~
Isub
p-substrate
DT p
+
n+
-+ + -
Figure 3.1: Illustration of photo carrier generation in the SiGe HBTs used.
Fig. 3.2 shows a schematic of the measurement setup. The base is grounded. The collector
voltage VC is set to desired VCB. The substrate voltage is chosen such that the collector-substrate
44
Figure 3.2: Experimental setup of the substrate current based M-1 measurement technique.
biasVCS 0. An emitter current IE is forced, and the value of IE is swept. VBE, IB, IC and ISUB are
recorded during the IE sweep. In the absence of self-heating, the avalanche current can be obtained
as the difference in IB between highVCB andVCB = 0V, denoted as I0AVE:
I0AVE = IB(VBE;VCB = 0) IB(VBE;VCB); (3.1)
where IB(VBE;VCB = 0) is the IB at the VBE values recorded during the IE sweep for a 0 V VCB,
and can be determined using a separate measurement. IB(VBE;VCB = 0) represents the hole current
injected into the emitter. At high IE, self-heating becomes severe, and the junction temperature
increases withVCB significantly. Therefore IB(VBE;VCB = 0) gives the hole current injected into the
emitterlowerthanatthedesiredVCB. TheholecurrentinjectedintotheemitteratthedesiredVCB are
thus underestimated by IB(VBE;VCB = 0). Consequently, the avalanche current is underestimated
by I0AVE. Negative I0AVE can be obtained, which is clearly unphysical for avalanche current.
Fig.3.3showsthemeasuredI0AVE,ISUB andISUB=I0AVE ratioversusIE at300K,VCB=4V.I0AVE first
increases with IE, as expected, but becomes negative at an of 3 mA, because of self-heating. How-
ever, for medium IE (region B), ISUB increases proportionally with I0AVE, and a constant ISUB=I0AVE
45
10?6 10?5 10?4 10?3 10?210
?12
10?10
10?8
10?6
10?4
10?2
I SUB
and I?
ave
(A)
10?7
10?6
10?5
10?4
10?3
IE (A)
I SUB
/I? ave
300K
VCB=4V
I?ave
Isub
I?ave going
negative
?=4.7666?10?6
A B C
Figure 3.3: Measured I0AVE, ISUB and ISUB=I0AVE ratio versus IE atVCB=4 V, 300 K.
ratio can be identified. Intuitively, this ratio can be viewed as the efficiency of substrate current
generation due to avalanche, which we denote as h. Measurements show that h is independent of
VCB andVCS [53]. In region C, due to self-heating,h loses its accuracy. There, the ?true? avalanche
current IAVE in the high IE region can be extracted by ISUB=h. The overall avalanche current is
given by:
IAVE =
8
><
>:
I0AVE region A and B;
Isub
h region C:
(3.2)
Fig. 3.4 shows the extracted avalanche current IAVE by (3.2) as a function of IE at 300 K,VCB=4
V. The rapid increase of IAVE at very high IE is caused by the rapid increase of ISUB at very high IE,
due to the hole injection resulting from the forward biasing of the internal CB junction. This does
not present a problem as it occurs at IE well above the peak fT.
46
10?6 10?5 10?4 10?3 10?210
?7
10?6
10?5
10?4
IE (A)
I ave
(A)
300K
VCB=4V
Iave=ISUB/?
A B C
Iave=I?ave
Figure 3.4: Extracted avalanche current IAVE versus IE atVCB=4 V, 300 K.
3.2 Experimental results over temperatures and impact of current
This substrate current based avalanche multiplication technique has been investigated at 300K
before [15]. Here we extend this technique down to 43 K. Fig. 3.5 and Fig. 3.6 show the measured
I0AVE, ISUB and ISUB=I0AVE ratio versus IE and extracted avalanche current IAVE versus IE at 43 K for
example.
By definition, avalanche multiplication factor M-1 is obtained by:
M 1 = IAVEI
C IAVE
: (3.3)
Fig. 3.7 show the measured M-1 and fT versus JE from 300 K to 43 K. Note that the decrease
of M-1 starts much lower than the JE of peak fT. Physically this is reasonable because fT rolls off
when JE is high enough to cause base push out, while M-1 decreases as long as the JE is sufficient
to cause a decrease of the CB junction peak field.
47
10?6 10?5 10?4 10?3 10?210
?12
10?10
10?8
10?6
10?4
10?2
I SUB
and I?
ave
(A)
10?7
10?6
10?5
10?4
10?3
IE (A)
I SUB
/I? ave
43K
Isub
I?ave
?=3.83?10?6
I?ave going
negative
VCB=4V
A B C
Figure 3.5: Measured I0AVE, ISUB and ISUB=I0AVE ratio versus IE atVCB=4 V, 43 K.
10?6 10?5 10?4 10?3 10?210
?7
10?6
10?5
10?4
IE (A)
I ave
(A)
43K
Iave=ISUB/?
VCB=4V
A B C
Iave=I?ave
Figure 3.6: Extracted avalanche current IAVE versus IE atVCB=4 V, 43 K.
48
10?2 10?1 100 10110?3
10?2
10?1
JE (mA/?m2)
M?1
05
1015
2025
3035
4045
50
f T (GHz)
300K
(a) 10
?2 10?1 100 101
100
JE (mA/?m2)
M?1
0
f T (GHz)
223K
(b)
10?2 10?1 100 101
100
JE (mA/?m2)
M?1
0
f T (GHz)
162K
(c) 10?2 10?1 100 101
100
JE (mA/?m2)
M?1
0
f T (GHz)
93K
(d)
10?2 10?1 100 101JE (mA/?m2)
M?1
0
f T (GHz)
43K
(e)
Figure 3.7: Measured M-1 and fT vs. JE at: (a) 300 K; (b) 223 K; (c) 162 K; (d) 93 K; (e) 43 K.
49
For transistors used in radio frequency (RF) power amplifiers, the maximum voltage handling
capabilitydependsonthedetailsofM-1versusVCB characteristics. Theseapplicationsrequirehigh
IE biasing for high speed, and high power density. It is therefore important to understand the M-1
versusVCB characteristics at high biasing IE. The breakdown voltage at high IC is also an important
concern for operating with mismatched load. Fig. 3.8 shows the measured M-1 versus VCB for
IE=12.5 mA,125 mAand1.5mAat300K.IE=1mAiswherepeak fT located. TheM-1at1mAm
is much smaller than the M-1 at 12.5 mA and 125 mA. Fig. 3.9 is the measured fT and M-1 versus
IE over 162-300 K. It demonstrates that such current dependence of M-1 leads to a much higher
breakdown voltage at high IC, where fT is maximized, than BVCEO and BVCBO which are typically
measured at low IC.
1 1.5 2 2.5 3 3.5 4 4.5 510?4
10?3
10?2
10?1
VCB (V)
M?1
IE=12.5uA
IE=125uA
IE=1.5mA
300K
Figure 3.8: Measured M-1 versusVCB for IE=12.5 mA, 125 mA and 1.5 mA at 300 K.
3.3 Summary
In this chapter, the substrate current base avalanche multiplication technique has been extended
from 300 K to 43 K. The current dependence of avalanche multiplication factor M-1 has been
50
10?2 10?1 100 1010
20
40
60
f T (GHz)
10?2 10?1 100 10110
?3
10?2
10?1
IE (mA)
M?1
162K223K
300K
12.5?A 125?A
1mA
12.5?A
125?A
1mA
Figure 3.9: Measured fT and M-1 vs. IE over 162-300 K.
investigated. The M-1 at 1 mA m is much smaller than the M-1 at 12.5 mA and 125 mA. Such
current dependence of M-1 leads to a much higher breakdown voltage at high collector current,
where fT is maximized, than BVCEO and BVCBO which are typically measured at low IC.
51
Chapter 4
Forced-IE Pinch-in Maximum Output Voltage Limit at Cryogenic Temperatures
In chapter 3, we discussed the current dependence of avalanche multiplication factor M-1 at
cryogenic temperatures. In this chapter, we will investigate forced-IE pinch-in maximum output
voltagelimitinSiGeHBTsoperatingatcryogenictemperatures,whichisofinterestformanyspace
exploration applications [5]. In section 4.1, we first review the physics of emitter current pinch-in
and analyze its temperature dependence. In particular we discuss how the critical base current
I B varies with temperature, and introduce the concept of critical multiplication factor (M-1) . In
section 4.2, we present measurement results of I B, (M-1) , and V CB over temperature, from 43 to
300 K.V CB is the collector-base bias where M-1 reaches (M-1) . Implications to device design and
circuit application for cryogenic operation are discussed in section 4.3.
4.1 Physics of emitter current pinch-in and temperature dependence analysis
For long emitter stripe (lE wE), where lE is emitter length and wE is emitter width, the
differential equation for local current distribution can be analytically solved. The instabilities start
mainly at emitter length direction and can be modeled by a linear transistor chain. The critical base
current of instability I B for driving condition IE = const: can be modeled as [50]:
I B = VT +reIER
Bx +rbi
1+(wEl
E
)2
;
(4.1)
where VT = kT=q is thermal voltage, rbi is the small-signal value of the internal intrinsic base
resistance RBi, re is the small-signal value of emitter resistance RE, RBx is extrinsic base resistance.
52
From (4.1), it is clear that T-dependence of VT, re, RBx and rbi all affect I B. Due to the highly
doped emitter and extrinsic base, and the weak T-dependence of mobility at such doping, the T-
dependence of re and RBx are weak. We also neglect the difference between small-signal rbi/re and
large-signal RBi/RE respectively. T-dependence of I B comes from the VT and rbi terms. Intrinsic
base sheet resistance RsBi can be written as:
RsBi =
0
@q
WBZ
0
pp(x)mp(x)dx
1
A
1
; (4.2)
whereWB is base width, pp is base hole density, mp is hole mobility.
WB is almost constant over 43-300 K. mp is a function of temperature, and its temperature
dependence is also related to doping concentration [35]. As a SiGe HBT of around 2 1018 cm 3
peak base doping is used here, mp is a weak function of temperature over 70-300 K. pp decreases
withcoolingasaresultoffreezeoutbecausethebasepeakconcentrationisbelowtheMotttransition
( 3 1018 cm 3). RsBi will thus increase with cooling. Fig. 4.1 shows RsBi versus temperature
simulated using Sentaurus device [54]. Note that as a simple incomplete ionization model is
implemented in Sentaurus, without considering Mott transition effect, the simulated T-dependence
of rsBi is weaker than the measurement. Nevertheless, a sizable increase of RBi with cooling down
to 43 K is expected. Therefore, with cooling, jI Bj will decreases asVT decreases and RBi increases.
This suggests that instabilities can occur more easily at low temperatures.
Forced-IE avalanche multiplication factor M-1 can be written as [7]:
M 1 = IB0 IBI
E IB0
: (4.3)
IB0 = IB(VBE)jVCB=0 IEb
0 +1
;
b0 = b(VBE)jVCB=0; (4.4)
53
0 50 100 150 200 250 300 350 40010
4
105
106
107
T (K)
Base sheet resistance (
?/sheet)
measurementsimulation
Figure 4.1: Measured and simulated intrinsic base sheet resistance RsBi-T.
1 2 3 4 5 6
10?4
10?3
10?2
10?1
100
VCB (V)
M?1
43K93K
162K223K
300K
T?,M?1?
Figure 4.2: Measured M-1 vs. VCB at forced-VBE over 43-300 K.
54
where IB0 is found from the IB VBE curve obtained at VCB = 0V. Here we introduce critical
multiplication factor (M-1) which correspond to I B. V CB is the collector-base bias where M-1
reaches (M-1) .
(M 1) = IB0 I
B
IE IB0: (4.5)
Substituting (4.4) into (4.5),
(M 1) =
IE
b0+1 I
B
IE IEb0+1
1
b0
1
b ;
b = IEI
B
: (4.6)
b is negative when IB < 0. The T-dependence of (M-1) originates from T-dependence of
1
b0 and
1
b . The b0 of SiGe HBTs is large (around 100) and hence for all practical situations
1
b
dominates. For a given IE, the absolute value of I B decreases with cooling, causing a decrease of
(M-1) with cooling.
As IE increases, from (4.1), jI Bj increases accordingly. BecauseVT dominates the whole (VT +
reIE) term, 1b will decrease even though IE increases. This leads to a decreasing (M-1) with
increasing IE.
4.2 Experimental results
4.2.1 T-dependence of M-1 from forced-VBE measurements
Forced-VBE M-1 measurements are taken to analyze the temperature dependence of M-1. Small
VBE were chosen to avoid self-heating and instabilities which still occurred at 43 K. Fig. 4.2 is
the measured M-1 vs. VCB over 43-300 K. At 43 K, instability occurs above VCB = 3V due to the
relatively high VBE used to produce the same current for all temperatures. The oscillations seen
near VCB = 4V at 93 K and 162 K are believed to be measurement errors. At a given VCB, M-1
increases with cooling and this is consistent with earlier measurements [52].
55
4.2.2 T-dependence of I B,V CB and (M-1) from forced-IE measurements
1 2 3 4 5 6?4
?3.5
?3
?2.5
?2
?1.5
?1
?0.5
0
x 10?5
VCB (V)
I B (A)
43K93K
162K223K
300K
IE=125uA
IB?(VCB? )
T?
Figure 4.3: Measured IB vs. VCB at IE = 125m A over 43-300 K.
Forced-IE M-1 measurements are taken at IE = 12:5mA, 125mA and 1 mA which is near peak
fT. Fig. 4.3 shows the IB VCB at IE = 125mA. IB decreases first due to avalanche. Once pinch-in
occurs,IB becomesinstable. I B iswherethisabrupttransitionoccurs. Over43-300K,jI Bjingeneral
decreases with cooling, as shown in Fig. 4.9. It agrees with what we expect from (4.1). Fig. 4.4
shows the corresponding collector current IC-VCB. Sudden transitions occur at the same V CB as
shown in Fig. 4.3.
Fig. 4.6 shows the M-1-VCB obtained using forced-IE technique for IE = 125mA. Observe that
(M-1) decreases at low temperatures. The extracted 1=b0 and 1=b at three IE are plotted in
Fig. 4.8. It is clear that compared to 1=b , 1=b0 can be neglected safely in calculating (M-1)
from (4.6). With cooling, 1=b decreases. This can explain why (M-1) becomes smaller at low
temperatures. When instabilities occur above V CB, the forced-IE M-1 curves become ?saturated?
56
1 2 3 4 5 61.2
1.25
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
1.7x 10?4
VCB (V)
I C (A)
43K93K
162K223K
300K
IE=125uA
IC?(VCB? )
T?
Figure 4.4: Measured IC vs. VCB at IE = 125m A over 43-300 K.
1 2 3 4 5 60
0.5
1
1.5
2
x 10?10
VCB (V)
I SUB
(A)
43K93K
162K223K
300K
IE=125uA
ISUB? (VCB? ) 93K
162K
223K
300K
43K
Figure 4.5: Measured ISUB vs. VCB at IE = 125m A over 43-300 K.
57
1 2 3 4 5 610?4
10?3
10?2
10?1
VCB (V)
M?1
43K93K
162K223K
300K
IE=125uA
(M?1)?(VCB? )
T?
Figure 4.6: Measured M-1 vs. VCB at IE = 125mA over 43-300 K.
0.6 0.7 0.8 0.9 1 1.1 1.2
20
40
60
80
100
120
140
160
180
200
VBE (V)
?
43K93K
162K223K
300K
VCB=0 IE = 125?A
300K
223K
162K 93K
43K
Figure 4.7: Measured b0 vs. VBE atVCB = 0 V over 43-300 K.
58
at (M-1) or even drops down. The M-1 obtained after V CB is no longer the true M-1 because of
pinch-in effect.
At IE = 125mA, instabilities occur over 43-300 K in the VCB range used. At IE = 12:5mA,
instabilities occur only at 43 K and 93 K. At IE = 1mA, instabilities occur at 93 K and 162 K, and
43 K measurement was not successful. From Fig. 4.8, we can observe that 1=b decreases as IE
increases, which is consistent with what we expect from the analysis made in Section II.
Fig. 4.9 shows the extracted I B andV CB vs. T. As expected, jI Bj decreases with cooling. From
Fig. 4.2, the true M-1 monotonically increasing withVCB before instabilities occur. Now we know
that (M-1) decreases with cooling, even if M-1 versusVCB is independent of temperature, theV CB
will decrease with cooling. The increase of M-1 with cooling for a given VCB makes the decrease
ofV CB with cooling even worse.
AccordingtotheanalysisinSectionII,jI BjincreasesasIE increases. However,theV CB decrease
fromIE = 12:5mA toIE = 125mA, and then increase fromIE = 125mA toIE = 1mA. This can only
be understood by considering the current dependence of M-1 vs. VCB characteristics. Fig. 4.10
illustrates (M-1) andV CB at different IE at 93 K, for which M-1 vs. VCB varies. A higher IE leads
to a lower M-1. At low currents, such as IE = 12:5mA and IE = 125mA, the M-1 curves are very
close to each other. Hence in general (M-1) 12:5mA>(M-1) 125mA leads toV CB;12:5mA >V CB;125mA. At
a high current IE = 1mA, M-1 curve drops well below the low current curves. This makes possible
V CB;1mA > V CB;12:5mA even though (M-1) 1mA<(M-1) 12:5mA. Note that IE = 1mA is close to peak
fT, and of interest to practical circuits. The fact that the maximum operation voltage range does
not degrade as much with cooling at such high current density is certainly good news for circuit
applications.
4.3 Circuit design implications
BoththeT-dependenceofM-1-VCB andtheT-dependenceof(M-1) determinetheT-dependence
ofV CB. (M-1) in turn is related to I B, and henceVT = kT=q and rbi. The use of a high base doping
[55] is expected to decrease rbi. This is expected to increase the V CB. We still expect V CB at low
59
50 100 150 200 250 3000
0.05
0.1
0.15
0.2
0.25
T (K)
?1/?
? & 1/
? 0
1/?0
?1/??
IE=12.5uA
IE=125uA
IE=1mA
Figure 4.8: Measured 1b , 1b0 vs. T at IE = 12:5mA, IE = 125mA and IE = 1mA.
50 100 150 200 250 300
10?6
10?4
|I B? | (A)
50 100 150 200 250 3003
4
5
6
T (K)
V CB?
(V)
IE=12.5uA
IE=12.5uA
IE=125uA
IE=125uA
IE=1mA
IE=1mA
Figure 4.9: Measured jI Bj,V CB vs. T at IE = 12:5mA, IE = 125mA and IE = 1mA.
60
1 2 3 4 5 6
10?3
10?2
10?1
100
VCB (V)
M?1
IE=12.5uAI
E=125uAI
E=1mA
93K (M?1)?(VCB? )
IE?
Figure 4.10: Measured M-1 vs. VCB at IE = 12:5mA, IE = 125mA and IE = 1mA at 93 K.
temperatures to be lower than 300 K, given the increase of M-1 with cooling and the decrease ofVT
with cooling. Circuit designs for applications over a wide temperature range for space explorations
should take this reducedV CB into account to assure reliability of operations. Using a higher current
density helps to alleviate the decrease ofV CB with cooling, and may be considered in circuit design.
4.4 Summary
In this chapter, the forced-IE pinch-in maximum output voltage limit in SiGe HBTs operating
at cryogenic temperatures has been investigated . A decrease of the voltage limit is observed with
cooling,andattributedtotheincreaseofintrinsicbaseresistanceduetofreezeoutaswellasincrease
of avalanche multiplication factor M-1. A practically high IE is shown to alleviate the decrease of
V CB with cooling, primarily due to the decrease of M-1 with increasing IE.
61
Chapter 5
Compact Modeling of SiGe HBT at Cryogenic Temperatures
5.1 Introduction of compact modeling
A compact transistor model tries to describe the I-V characteristics of a transistor in a mathe-
matical way, such that the model equations can be implemented in a circuit simulator. The first
bipolar model is the Ebers-Moll model [56] developed in 1954, which is the predecessor of to-
day?s computer simulation models and contains only two back-to-back diode currents. The two
diodes represent the base-emitter and base-collector diodes. Depending on the base-emitter and
base collector junction biases four operation regions are modeled, including forward active region,
reverse active region, saturation region and cut-off region. The Gummel-Poon model invented in
1970?sisbasedontheintegralcharge-controlrelation[8]andhasservedasastandardformorethan
three decades. On the basis of a general integral relation, high injection effect and Early effect are
incorporated in Gummel-Poon model. When a device with light doped collector is operated at high
injection level, dc current gain and cut-off frequency fT roll off due to quasi-saturation effect [14].
Quasi-saturation effect is very necessary to be modeled accurately as high frequency circuits are
biased at high collector densities in order to obtain maximum operating speed. However, in both
Ebers-Moll model and Gummel-Poon compact model, the quasi-saturation effect is not addressed.
The VBIC model (Vertical Bipolar Inter-Company model) [9] was developed in 1995 as an
industry standard replacement for the SPICE Gummel-Poon (SGP) model, to improve deficiencies
of the SGP model that have become apparent over time because of the advances in BJT process
technology. TheMextram(MostEXquisiteTRAnsistorModel)modelreleasewasfirstlyintroduced
as Level 501 in 1985 [3] to improve on the standard Gummel-Poon model. HICUM (HIgh CUrrent
Model ) model was first described in [57][58] in 1987 to address modeling issue for high-speed
and high current density operations [10]. VBIC, Mextram and HICUM all include avalanche
62
breakdown, self-heating, quasi-saturation effect, non-quasi-static effect and temperature effect.
HICUM incorporates the epilayer charges in the total charge storage, and Mextram and VBIC do
modeltheepilayerexplicitly. HICUMmodelisusefulinhighcurrentdensityandhighspeedcircuits.
The Mextram and VBIC bipolar transistor models are comparable for low and medium collector
current densities and frequencies. The main deficiency of the VBIC model is the description of the
velocity saturated behavior of the current though the collector epilayer in combination with base
push-out. Therefore the VBIC model is not able to describe accurately the degradation of gain,
output conductance and cut-off frequency at high current densities [59].
Our basis is the MEXTRAM 504.6 [3]. Mextram model is a widely used vertical bipolar
transistor model. The first Mextram release was introduced as Level 501 in 1985 [60]. Later Level
502, 503 and 504 were respectively released in 1987 [13], 1994 and 2000 [3]. And development
wasneverstoppedfollowingtherequirementofupdatedtechnology. ThelatestversionofMextram
is 504.9.1 which was released in January, 2011.
Fig.5.1showstheequivalentcircuitoftheMextrammodelasitisspecifiedin[3]. Thebranches
representing model currents and charges are schematically associated with different physical re-
gions of a bipolar transistor separated by the base-emitter (BE), base-collector (BC), and substrate-
collector(SC)junctions. AllcurrentandchargebranchesinMextramaregivenasexplicitfunctions
of external and internal nodal potentials [60]. The main transfer current IN in Mextram, as in the
Gummel-Poon model [6], is evaluated in the quasi-neutral base (QNB). Moreover, the effects of
a graded Ge profile in QNB [61] are physically addressed in the transfer current description. A
distinguishing feature of the Mextram model is the description of the epilayer transfer current Iepi.
It is employed for intensive physical modeling of the quasi-saturation phenomena including the
base widening, Kirk effect [14] and hot-carrier behavior in the epilayer.
The diode-like injection currents IB1,ISB1 ,IB2, IB3, Iex, and XIex in the Mextram equivalent
circuit describe various diffusion and recombination currents in the quasi-neutral and depletion
transistor regions. The recombination in the modulated QNB, which is particularly important for
SiGe HBT applications, is also included. The effect of a distributed hole injection across BE
63
C2
IB1B2
CBEO
IN
QE
QtEQ
BE
IB2 IB1
QtES
QB1B2R
Bc
QBC
QtC Q
epi
RE
EB
B1
Isub
Qtex
QexIex+IB3
IC1C2
C1
XIsub
XQex
XQtex
XIex
RCc
CBCOC
QtS
S
n+buried layer
p base
n epilayer
p substrate
n+emitter
IB1s
E1
B2
Iavl
Isf
Parasitic PNP
Transfer Current
Avalanche Current
Epilayer Current
Thermal Network
CTHRTH
dT
Pdiss
Figure 5.1: The Mextram equivalent circuit for the vertical NPN transistor.
64
junction is described by an additional current branch IB1B2. Mextram provides also a sophisticated
model for the weak avalanche current in the branch Iavl. The contribution of the parasitic PNP
transistor transfer current to the substrate current, represented by the current sources Isub and XIsub,
is implemented using a simplified Gummel-Poon integral charge control relationship.
Below effects are contained in Mextram:
1. Bias-dependent Early effect
2. Low-level non-ideal base currents
3. High-injection effects
4. Ohmic resistance of the epilayer
5. Velocity saturation effects on the resistance of the epilayer
6. Hard and quasi-saturation (including Kirk effect)
7. Weak avalanche (optionally including snap-back behavior)
8. Charge storage effects
9. Split base-collector and base-emitter depletion capacitance
10. Substrate effects and parasitic PNP
11. Explicit modeling of inactive regions
12. Current crowding and conductivity modulation of the base resistance
13. First order approximation of distributed high frequency effects in the intrinsic base (high
frequency current crowding and excess phase-shift)
14. Recombination in the base (meant for SiGe transistors)
15. Early effect in the case of a graded bandgap (meant for SiGe transistor)
65
16. Temperature scaling
17. Self-heating
18. Thermal noise, shot noise and 1/f-noise
Some parts of the model are optional and can be switched on or off by setting flags. These are
the extended modeling of reverse behavior, the distributed high-frequency effects, and the increase
of the avalanche current when the current density in the epilayer exceeds the doping level.
5.2 Description of proposed wide temperature range compact model
In this work, various modifications and extensions are made to enable modeling of dc char-
acteristics from 43-393 K, and ac characteristics from 93-393 K. New modeling equations were
implemented using Verilog-A. Customized programs were written using the PEL (Parameter Ex-
traction Language) of ICCAP [62] for parameter extraction. Main improvements include:
? Adding forward bias trap-assisted tunneling current IB;tun[2].
? Adding substrate resistor RSUB and capacitorCSUB.
? developing new temperature-scaling equations of:
Saturation current and ideality factor for main current and base currents [1]
Base tunneling current IB;tun [2]
Series resistance [17]
Thermal resistance RTH
Epilayer current parameter IHC and SCRCV
Fig. 5.2 shows the equivalent circuit of the compact model. A new main current model IN is
developed to produce accurate collector current low and medium injection. Such new IN includes
66
new models of forward and reverse saturation current IS;F and IS;R, forward and reverse ideality
factor NF and NR. The T-dependence of NF is included in the T-dependence model of IS;F, which
is necessary to fit measured IC-VBE data at low temperatures[1]. Below 93 K, the trap-assisted
tunneling(TAT)[63]currentcanbeclearlyobservedinforwardbiasIB-VBE. IB;tun isaddedbetween
B2 and E1 to account for this forward bias TAT current. Substrate resistance RSUB and capacitance
CSUB are included as part of the model. New temperature scaling models of terminal resistance
[17], thermal resistance RTH, quasi-saturation parameters IHC and SCRCV are developed to model
high injection region.
C2
IB 1B 2
CB E O
IN
QE
QtEQ
B E
IB 2
IB 1
QtES
QB 1B 2
RB c
QB C
QtC
Qepi
RE
EB
B1
Isub
Qtex
QexIex+IB 3
IC 1C 2
C1
XIsub
XQex
XQtex
XIex
RC c
CB C OC
QtS
S
n+ buried layer
p base
n epilayer
p substrate
n+ emitter
IB 1s
E1
B2
Iavl
IB ,tun
S1
RSU B
CSU B
Isf
CT HRT H
Pdiss
dT
Figure5.2: Equivalentcircuitusedinthiswork,with1)addedforwardbasetunnelingcurrentIB;tun;
2) added RSUB andCSUB.
67
5.3 Main current
Fig.5.3(a)showstheone-dimensionalrepresentationofaverticaln-p-ntransistor. Thetransistor
consists of an n+ emitter, p-type base and an n-type collector. The n-type collector consists of a
n epilayer and n+ buried layer. The starting substrate of a vertical n-p-n transistor is usually a
p silicon. Fig. 5.3(b) shows the bias condition for an n-p-n transistor in normal operation. The
emitter-base diode is forward biased byVBE, and the base-collector diode is reverse biased byVBC.
Electrons flow from emitter into the base and the holes flow from the base into the emitter. The
main current IN caused by those electrons not recombined in the base arriving at the collector gives
rise to collector current IC. The current IBE induced by the holes injected into the emitter gives
rise to the base current IB. Similarly, if the base-collector diode is forward biased, the current IBC
induced by the holes injected into the collector also contributes to the base current IB.
Recallingthat, the main currentdensityJN runningthrough abipolar transistor hasbeen derived
by Gummel?s integral charge control relation (ICCR) [8]:
JN = q(e
qVBE=kT eqVBC=kT)
Wb
s
0
pb(x)dx
Dnb(x)n2ib(x)
; (5.1)
wherex=0andw=Wb aretheneutralbaseboundaryvaluesontheemitter-baseandcollector-base
sides of the base respectively. nib, Dnb and pb are intrinsic carrier concentration, electron diffusion
constant and hole density in the base respectively. If we assume constant Dnb and constant nib
(indicating constant Eg, NC and NV), the IN becomes:
IN = q
2DnbA2emn2
ib
QB
exp
V
BE
VT
exp
V
BC
VT
;
QB = q
Wb
s
0
pb(x)dx; (5.2)
68
emitter base collector
VBE VBC
IBE
IN
IBC
CBE
=Cdep+Cdiff
CBC
=Cdep+Cdiff
(a)
(b)
n+ emitter
p base
n-epilayer collector
n+ buried layer
collector
0 WB Wepi0
IC
IB
IE
Figure 5.3: (a) Doping profile for a NPN transistor. (b) Schematic illustrating the applied voltages
in normal operation.
69
where Aem is effective emitter area, QB is the total base charge. If we assume complete ionization
and neutral region approximation, pb(x) can be approximated as:
pb(x) NA(x)+nBE(x)+nBC(x); (5.3)
whereNA(x)isacceptordopingconcentration,nBE(x)andnBC(x)aretheinjectedelectrondensities
due to appliedVBE andVBC respectively.
QB = q
Wb
s
0
pb(x)dx changes with appliedVBE andVBC in several ways:
1. As we increase VBE, the EB junction depletion thickness becomes smaller, this moves the
neutral base boundary (x = 0) towards the EB metallurgical junction location. This VBE
dependent base boundary ?0? causes extra depletion charges QtE, as shown in Fig. 5.4 (b).
This is so called reverse Early effect.
2. Similarly, as we increase VBC, the BC junction depletion thickness becomes smaller, this
moves the neutral base boundary (x =WB) towards the BC metallurgical junction location.
This VBC dependent base boundary ?WB? causes extra depletion charges QtC, as shown in
Fig. 5.4 (b). This is so called forward Early effect.
3. nBE(0) is proportional to exp(VBEVT ). Hence diffusion charge nBE(x) isVBE dependent.
4. nBC(WB) is proportional to exp(VBCVT ). Hence diffusion charge nBC(x) isVBC dependent.
QB(VBE;VBC) = QNA(VBE;VBC)+QBE(VBE)+QBC(VBC);
QNA(VBE;VBC) = q
Wb
s
0
NA(x)dx = QB0 +QJE +QJC;
QBE(VBE) = q
Wb
s
0
nBE(x)dx;
QBC(VBC) = q
Wb
s
0
nBC(x)dx; (5.4)
70
NA(x)
(a)
n+ emitter
p base
n-epilayer collector
0 WB 0
(b)
n+ emitter
p base
n-epilayer collector
0 WB 0
VBE=0 VBC=0
VBE>0 VBC>0
QJE
QJC
Figure 5.4: A schematic cross-section of the base region for a NPN transistor, showing the Early
effect: EB and BC junction depletion thickness?s variation withVBE andVBC.
71
whereQB0 = QBjVBE=0;VBC=0 = QNAjVBE=0;VBC=0 is the zero bias total base charge. QBE is the diffu-
sion charges due to VBE, QBC is the diffusion charges due to VBC. The normalized base charge is
then given by:
qB = QBQ
B0
= QB0 +QJE +QJC +QBE +QBCQ
B0
;
= 1+ QJEQ
B0
+ QJCQ
B0
+ QBEQ
B0
+ QBCQ
B0
: (5.5)
The main current (5.2) then can be reorganized as:
IN =
IS
exp
VBE
VT
exp
V
BCV
T
qB ;
IS = q
2DnbA2emn2
ib
Qp0 : (5.6)
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.210?12
10?10
10?8
10?6
10?4
10?2
VBE (V)
I C (A)
43K60K
76K93K
110K136K
162K192K
223K262K
300K393K
Figure 5.5: Measured IC-VBE for a graded-base SiGe HBT.
72
0 50 100 150 200 250 3000
50
100
150
200
250
300
1/ I C
*dI C/dV
BE (1/V)
temperature(K)
Ideal 1/VT
Drift?diffusionHydrodynamic
Measurement
Figure 5.6: The slope of IC-VBE from device simulation [1].
Fig. 5.5 shows the measured IC-VBE for a graded-base SiGe HBT. A logarithmic scale is used
for IC and it facilitates the examination of the slope of IC at fixed VBE at low temperatures. For
first-order approximation, the slope of IC-VBE is proportional to 1VT in semilog scale. Therefore,
lower the temperature, sharper the slope. However, the slope of measured IC-VBE significantly
deviates from the ideal 1VT approximation at low temperatures. To identify the physical reasons, we
performed both drift-diffusion and hydrodynamic device simulation of this graded-base SiGe HBT
using Sentaurus Device [54]. The simulated IC-VBE slope, however, shows a much less deviation
from ideal value than what we observed in measurement, as shown in Fig. 5.6. As all of the higher
order physics effects are naturally included in device simulation, such as Philips unified mobility
model,Ge-dependentbandgap,Ge-dependentdensity-of-states,incompleteionization,Earlyeffect
and Ge ramping effect, we conclude that such a deviation is due to unknown physics to the best of
our knowledge.
In Mextram, it was believed that qB is sufficient in modeling the slope of IC-VBE, and using
ideality factors like in SGP and VBIC could complicate the parameter extraction. However, at low
temperatures, we found that the deviation ofIC-VBE slope from ideal 1=VT is much larger than what
73
can be modeled with qB. Even though the underlying physics is not understood yet, one can model
this with a NF factor that increases with cooling, such that the slope 1=(NFVT) does not increase as
much as the ideal 1=VT. The T-dependence of NF is included in the T-dependence model of IS;F,
which is necessary to fit measured IC-VBE data [1].
Fig. 5.7 (b) and (c) show the NF and IS extracted versus temperature. Observe that NF is close
to 1 above 200 K, but increases rapidly below 200 K, to 1.35 at 43 K. Fig. 5.8 shows simulated
IC-VBE using default Mextram models from 43 K to 393 K. Above 110 K, default Mextram models
can produce reasonably good IC-VBE at moderate injection as shown in Fig. 5.8 (a). From 43-93 K,
however, default Mextram models fails, as shown in Fig. 5.8 (b). Clearly, NF is necessary to model
the IC slope correctly below 110 K.
0.2 0.4 0.6 0.8 1 1.210
?10
10?8
10?6
10?4
10?2
100
102
VBE (V)
I C (mA)
0 100 200 300 4001
1.1
1.2
1.3
1.4
temperature (K)
N F(T)
0 100 200 300 40010
?100
10?50
100
temperature (K)
I S(T)
(a)
(b)
(c)
temperature increase
162K
192K 300K
43K
393K
Figure 5.7: (a) Measured IC-VBE from 43 to 393 K. (b) Extracted NF at each temperature. (c)
Extracted IS at each temperature [1].
In our previous work [1], the main current IN is modeled as:
IN = IS;Fe
VB2E1
NFVT IS;Re
V?B2C2
NRVT
qB ; (5.7)
74
0.9 0.95 1 1.05 1.110
?10
10?5
100
I C (mA)
VBE (V)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.110
?10
10?5
100
I C (mA)
T=43K, 60K,76K, 93K
T=110K, 136K,162K, 192K, 223K, 262K, 300K, 393K
(a)
(b)
temperature increase
temperatureincrease
43K
93K
110K
393K Measurement Mextram default model
Figure 5.8: (a) Simulated versus measured IC-VBE at high temperatures. (b) Simulated versus
measured IC-VBE at low temperatures [1].
75
where all the symbols have their usual meanings in Mextram except for forward and reverse sat-
uration current IS;F and IS;R, forward and reverse ideality factor NF and NR. Ideality factor N is
extracted from medium injection region.
In section 2.1.8, we have clarified that the depletion thickness has weak temperature depen-
dence. In Mextram parameter AQB0 is mainly introduced to model carrier freezeout effect. As we
will discuss in chapter 5.11.3, the incomplete ionization rate IR(T) is a complicated function of
temperature. However, in Mextram, the carrier freezeout effect is only taken into account through
relation QB0;TQB0 =tAQB0N .
5.4 Depletion charges and capacitances
The early effect is the effect that the main current gets modulated due to a variation in effective
basewidthonbothbase-emittersideandbase-collectorside. Atlowinjection,thediffusioncharges
QBE and QBC can be neglected and the normalized base charge qB is usually denoted as q1:
q1 1+ QtEQ
B0
+ QtCQ
B0
: (5.8)
In compact modeling, we cannot determine depletion charges directly. The depletion charges
Qt are directly related to the well-known junction capacitance Cj. The basic model for depletion
capacitance is [64][65] [12],
Cj = Cj0(1 V=V
d)p
; (5.9)
where Cj0 is zero bias depletion capacitance, V is applied voltage, Vd is the diffusion voltage and
p is the grading coefficient. p has a theoretical value of 1=2 for an abrupt junction and 1=3 for a
graded junction. By taking the integration of Qt = Vs
0
CjdV, the depletion charge corresponding to
76
the ideal depletion capacitanceCj is:
Qt =Cj0 Vt;
Vt = Vd1 p
1 (1 V=Vd)1 p
; (5.10)
whereVt is a function of applied voltageV,Vd and p.
To avoid the singularity atV =Vd, an effective junction biasVj is employed in (5.10) instead of
V [12] [3]:
Vf =Vd
1 a
1/p
E
j
!
;
Vj =V 0:1Vd ln
n
1+exp
h V V
f
0:1Vd
io
;
Vt = Vd1 p
1
1 Vj Vd
1 p
+aj V Vj ;
Qt =Cj0 Vt: (5.11)
The quantity aj is a constant and is different for each of the depletion capacitances. Fig. 5.9 is
the calculated Cj versus applied voltage V for an abrupt junction (p = 1=2) and a graded junction
(p = 1=3). The effective junction biasVj help force the capacitance to asymptotically approach the
constant value ajCj0 forV >Vj.
The depletion charges of base-emitter junction is split into two parts, an intrinsic component
and the side-wall component [3].
QtE = (1 XCjE)CjEVtE;
QStE = XCjECjEVtE; (5.12)
whereCjE represents the zero bias base-emitter junction depletion capacitance,VtE is related to the
effective junction biasVjE of base-emitter depletion junction, XCjE is defined as the fraction of the
emitter-base depletion capacitance that belongs to the sidewall.
77
?0.2 0 0.2 0.4 0.6 0.8 1 1.20
10
20
30
40
V (V)
C j (fF)
p=1/2p=1/3
?jCj0
Cj0
Vj
Cj0/(1?V/Vd)p
abrupt junction
graded junction
Figure 5.9: Calculated Cj versus applied voltage V for an abrupt junction (p = 1=2) and a graded
junction (p = 1=3).
Similarly, for the intrinsic base-collector depletion capacitance [3]:
QtC = XCjCCjCVtC; (5.13)
where CjC represents the zero bias base-collector junction depletion capacitance, VtC is related to
the effective junction bias VjC of base-collector depletion junction. XCjC is the fraction of the
collector-base depletion capacitance under the emitter. For the extrinsic part of base-collector
depletion capacitance, it is partitioned between nodesC1 and B1 and nodesC1 and B respectively.
Fig. 5.10 shows the modeled Qte and ?Qte=?I from 93-393 K. Fig. 5.11 shows the modeled Qtc
and ?Qtc=?I from 93-393 K. The Qte shows strong temperature dependence however Qtc shows
very week temperature dependence. Meanwhile, the Qte shows weak current dependence however
Qtc shows very strong current dependence.
78
10?3 10?2 10?1 100 101
1
1.5
x 10?14
Q te
(C)
10?3 10?2 10?1 100 1010
100
200
300
400
JC (mA?m2)
?=?
Q te
/?I (ps)
93 K162 K
223 K300 K
393 K?=?Q
te/?I T?
(a)
(b)
Figure 5.10: (a) Modeled Qte from 93-393 K. (b) Modeled ?Qte=?I from 93-393 K.
10?3 10?2 10?1 100 10110
?19
10?18
10?17
10?16
10?15
10?14
Q tc
(C)
10?3 10?2 10?1 100 1010
50
100
150
JC (mA?m2)
?=?
Q tc
/?I (ps)
93 K162 K
223 K300 K
393 K?=?Q
tc/?I T?
(a)
(b)
Figure 5.11: (a) Modeled Qtc from 93-393 K. (b) Modeled ?Qtc=?I from 93-393 K.
79
5.5 Early effects and Ge ramp effect
ByintroducingthemodelparametersofreverseEarlyvoltageVer andforwardEarlyvoltageVef,
Ver = QB0(1 XC
jE)CjE
;Vef = QB0XC
jCCjC
: (5.14)
(5.8) can be written as:
q1 = 1+ (1 XCjE)CjEVtEQ
B0
+ XCjCCjCVtCQ
B0
= 1+VtEV
er
+VtCV
ef
: (5.15)
Model parameters Ver and Vef can be extracted from reverse-Early measurement and forward-
Early measurement respectively. Model parameters XCjE and XCjC can be estimated from C-V
curves. Although q1 is physically related to XCjE and XCjC through Early voltage parametersVer
andVef , there is no explicit couple between q1 and XCjE and XCjC directly.
With a close look of the derivation of Gummel?s charge control relation [8], the main current is
determined by the base Gummel number:
GB =
Z xC
xE
pb(x)
Dn(x)
n2i0
n2i (x)dx; (5.16)
where ni0 is intrinsic carrier concentration of un-doped Si. xC and xC are the boundary of base-
emitter junction and base-collector junction, which areVBE andVBC bias dependent due to reverse
and forward early effect. For a linearly graded base SiGe HBT, the intrinsic carrier concentration
including Ge gradient profile can be written as:
n2i = n2i0exp( xW
B
DEg
kT ); (5.17)
whereWB isbasewidth,DEg =DEg(WB) DEg(0) isthedifferenceinbandgapbetweentheneutral
edges of the base at zero bias. The major part of this bandgap narrowing is due to Ge, but also be a
result of bandgap narrowing due to heavy base doping.
80
emitter
base
epilayer
collector
xE xC
VBE=0
VBE>0
?Eg,Ge(x)
?Eg,Ge(0)
Ge
1/n2ib
Figure 5.12: Schematic Ge and 1=n2ib profile of SiGe transistor with a gradient Ge content.
Because ni is a much stronger function of position across base region, majority carrier con-
centration pb and electron diffusion constant Dn can be estimated as average values across base.
Assume complete ionization, pb(x) = NA(x), hence the Gummel number becomes:
GB = NAn2i0=Dn
Z xC
xE
1
n2i (x)dx
= NAn2i0=Dn
Z xC
xE
exp( xW
B
DEg
kT )dx
= NAn2i0=DnkTWBDE
g
exp( xEW
B
DEg
kT ) exp(
xC
WB
DEg
kT )
(5.18)
Due to reverse and forward early effect, the positions of xC and xC change withVBE andVBC. At
zero bias, xE = 0, xC =WB, the zero bias Gummel number can be written as:
GB0 = NAn2i0=DnkTWBDE
g
1 exp( DEgkT )
(5.19)
81
By constant base doping profile assumption,
QB0 = qAemNAWB;
QB = qAemNA(xC xE): (5.20)
Therefore the depletion charges QtE and QtC are given by:
QtE = qAemNA(0 xE);
QtC = qAemNA(xC WB); (5.21)
From the definitions of early voltage, we get:
VtE
Ver =
(0 xE)
WB ;
VtC
Vef =
(xC WB)
WB : (5.22)
Combining eqn(5.18), eqn(5.19) and eqn(5.22), we get the ratio of the Gummel number and
Gummel number at zero bias [66]:
GB
GB0 =
exp([VtEVer +1]DEgkT ) exp( VtCVef DEgkT )
exp(DEgkT ) 1
: (5.23)
If carrier freezeout effect is taken into account, pb(x) 6= NA(x). With assuming constant pb(x)
profileacrossbaseortakeanaverage pb(x)overthebaseregion,wecanreachthesame GBGB0 equation
because eqn(5.23) as the ionization rate appearing in the denominator and numerator cancels by
each other.
82
From the depletion charges modeling standpoint, we still use the base charge. However, from
the current modeling standpoint, we should use Gummel number [3].
qQ1 = QBQ
B0
= 1+VtEV
er
+VtCV
ef
;
qI1 = GBG
B0
=
exp([VtEVer +1]DEgkT ) exp( VtCVef DEgkT )
exp(DEgkT ) 1
: (5.24)
5.6 Diffusion charges and transit times
Foruniformlydopedbase, athighinjection,QBE andQBC ineqn(5.5)beginplayingarole. Both
nBE(x) and nBC(x) can be modeled as a linear function of position in a short base (nBE(Wb) = 0,
nBC(0) = 0), as shown in Fig. 5.13.
0 WB
nBE(x)
nBC(x)
total diffusion
charge nBE(x)+nBC(x)
nBE(WB)
nBC(WB)
nBC(0)
nBE(0)
Figure 5.13: Injected electron densities profile in the base region for a NPN transistor.
TheVBE induced diffusion charge QBE andVBC induced diffusion charge QBC can be expressed
as:
QBE(VBE) = q
WB
s
0
nBE(x)dx = 12qWBnBE(0) = 12QB0n0;
QBC(VBC) = q
WB
s
0
nBC(x)dx = 12qWBnBC(0) = 12QB0nB; (5.25)
83
where n0 = nBE(0)=NA and nB = nBC(WB)=NA are the normalized electron densities at the edges of
the neutral base region in terms of the zero bias base charge.
The base transit time tB is assumed constant at low injection of forward operation. Taking no
account of Early effect first, and using the charge control relations QBE = tBI;QB0 = tBIK (IK is
knee current), (5.7) becomes:
I =
ISexp
V
B2E1
NFVT
1+ tBIQB0 =
ISexp
V
B2E1
NFVT
1+ IIK : (5.26)
Current I can be solved as:
I =
2ISexp
V
B2E1
NFVT
1+
r
1+ 4ISIK exp
V
B2E1
NFVT
; (5.27)
From eqn(5.25),
n0 = 2QBEQ
B0
= 2II
K
; (5.28)
Hence
f1 = 4ISI
K
exp
V
B2E1
NFVT
;n0 = f11+p1+ f
1
: (5.29)
Similarly, for reverse operation,
f2 = 4ISRI
KR
exp
V
B2C2
NRVT
;nB = f21+p1+ f
2
; (5.30)
where we introduce knee current IKR for reverse operation.
Therefore the normalized base charge (neglecting the early effect) q2 is given by:
q2 =
1+ 12n0 + 12nB
84
The impact of Early effect on the diffusion charges is modeled by placing q1 into eqn(5.25).
QBE = 12q1QB0n0;
QBC = 12q1QB0nB; (5.31)
The Normalized base charge qB then can be written as qB = q1q2.
Near peak fT, diffusion charges dominate. Fig. 5.14 shows the modeled Qbe and ?Qbe=?I
from 93-393 K. Fig. 5.15 shows the modeled Qbc and ?Qbc=?I from 93-393 K. With cooling, the
diffusion charges decreases and hence the transit time decreases. This explains the higher peak fT
with cooling.
10?3 10?2 10?1 100 10110
?19
10?17
10?15
10?13
Q be
(C)
10?3 10?2 10?1 100 1010
2
4
6
8
JC (mA?m2)
?=?
Q be
/?I (ps)
93 K162 K
223 K300 K
393 K
?=?Qbe/?I
T?
(a)
(b)
AE = 0.5 ? 2.5 ?m2
T?
Figure 5.14: (a) Modeled Qbe from 93-393 K. (b) Modeled ?Qbe=?I from 93-393 K.
85
10?3 10?2 10?1 100 10110
?19
10?17
10?15
10?13
Q bc
(C)
10?3 10?2 10?1 100 1010
2
4
6
8
JC (mA?m2)
?=?
Q bc
/?I (ps)
93 K162 K
223 K300 K
393 K
?=?Qbc/?I
T?
(a)
(b)
AE = 0.5 ? 2.5 ?m2
T?
Figure 5.15: (a) Modeled Qbc from 93-393 K. (b) Modeled ?Qbc=?I from 93-393 K.
86
5.7 Non-quasi-static charges
Duetothedistributednatureofthetransistor,thequasi-staticapproximationisnolongervalidfor
high-speed application. AC current crowding is a consequence of the distributed RC components
in the lateral direction along the intrinsic base. At high frequencies this results in non-uniform
vertical ac currents, thus affecting the small-signal base and collector currents [67][68][69].
AC current crowding is modeled as a capacitance CBv between nodes B1 and B2, equal to the
capacitance of the base-emitter junction divided by 5, parallel to the variable base resistance RBv.
CBv = 15?QB2E1?V
B2E1
= 15?(QtE +QBE +QE)?V
B2E1
= 15
?Q
tE
?vB2E1 +
1
2QB0q
Q
1
?n0
?vB2E1 +
?QE
?vB2E1
; (5.32)
where
?n0
?VB2E1 = 2
IS
IK exp(
VB2E1
NFVT )
1
NFVT
1p
1+ f1 (5.33)
The non-quasi-static charge QB1B2 is:
QB1B2 =CBvVB1B2: (5.34)
5.8 Base current
For the ?ideal? BE and BC diode currents IB1, ISB1, Iex and XIex, we no longer use forward
current gain bF and reverse current gain bR to describe them, as at low temperature, current gain b
becomesmorecurrentdependent,makingbF andbR difficulttoextract. Instead,eachdiodecurrent
is modeled the same way as main current IN, with its own saturation current IS and N factor. At the
same temperature, the N factor for IB is found to be larger than for IC [1].
87
Fig. 5.16 show the measured base currents from typical forward and reverse Gummel measure-
ments. Clearly, a non-ideal base current component in the forward mode IB is observed below
100 K, however, is not observed in reverse mode. Note that this non-ideal base current cannot be
modeled as the space-charge recombination current, typically with an ideality factor of ?2? and
a slope of q=2kT, as the slope of TAT is nearly temperature independent. In [2], we contribute
this non-ideal base current as the trap-assisted tunneling (TAT) current caused by the heavy doping
nature of the base-emitter junction. The non-ideal forward base current IB2 (space-charge recombi-
nationcurrent)isnolongersufficienttomodelsuchbehaviorwithaconstantidealityfactorMLF=2.
This TAT current is modeled by IB;tun between B2 and E1 in Fig. 5.2.
0.60.8 1 1.2
10?10
10?5
VBE (V)
I B (A)
300K
43K
TAT
(a)
0.6 0.8 1 1.2
10?10
10?5
I B (A)
VBC (V)
300K
43K
(b) V
CB = 0 V VEB = 0 V
Figure 5.16: (a) IB-VBE from forward Gummel measurement; (b) IB-VBC from reverse Gummel
measurement [1].
88
5.8.1 Ideal forward base current
The ideal forward base current is separated into a bulk part and a side-wall part, which has a
fraction factor XIB1. Both depend on separate voltages.
IB1 = (1 XIB1)IBEI
exp( VB2E1N
BEIVT
) 1
;
ISB1 = XIB1IBEI
exp( VB1E1N
BEIVT
) 1
; (5.35)
where IBEI is the saturation current, NBEI is the ideality factor.
5.8.2 Non-ideal forward base current
The non-ideal forward base current is given by:
ISB2 = IBf
exp( vB2E1m
LfVT
) 1
; (5.36)
where mLf is a non-ideality factor with a value close to ?2?.
5.8.3 Trap-assisted tunneling current
In [2], a physics based trap-assisted tunneling (TAT) current expression was then parameterized
as follows:
IB;tun = IBT(T)exp
V
B2E1
VTUN
IBT(T) = IBTptN exp(KTN (Eg;TN Eg;T))
IBT isthenominaltemperaturesaturationcurrent, orinterceptofIB;tun-VBE, andVTUN representsthe
temperature independent slope of IB;tun-VBE. VTUN, IBT, KTN are three model parameters specific
to TAT. Eg;TN and Eg;T are the band gap at Tnom and T.
89
n p
q( bi V)Ec
Efn
EvEfp
Figure 5.17: Measured IB-VBE with illustration of TAT in forward-biased E-B junction [2].
5.8.4 Non-ideal reverse base current
The non-ideal reverse base current is given by [3]:
IB3 = IBr exp(
vB1C1
VT ) 1
exp(vB1C12VT )+exp( vLr2VT ) (5.37)
This expression is basically an approximation to the Shockley-Read-Hall recombination.
5.8.5 Extrinsic base current
Extrinsic base current Iex is expressed in terms of the electron density, in this case nBex, at the
end of the extrinsic base:
Iex = IBCI
2exp VB1C1
NCIVT
1+
q
1+ 4IS;RIK;EX exp(VB1C1=NCIVT)
1
; (5.38)
where we introduce knee current IK;EX for extrinsic base current.
90
5.9 Collector epilayer model
5.9.1 Epilayer current
The epilayer collector of a bipolar transistor is the most difficult part to model. At low current
densities, the current Iepi flowing across epilayer is determined mainly by the main current IN. At
high current densities, the epilayer is flooded by holes and electrons. Since then, the main current
IN and the epilayer current Iepi depend on each other and their equations become coupled.
When a device with a lightly doped collector is operated at high injection levels, the minority
carriers are injected into epilayer region, widening the electrical base of the device, which then
degrades the current gain, increases transit time and decreases cut-off frequency. There are two
competing effects that make this width either increases or decreases. First, the internal base-
collector junction potential decreases with current when the ohmic potential drop over the epilayer
increases. Hence the base-collector depletion width decreases with current. At certain current
level, the base-collection depletion layer thickness vanishes and the whole electric filed is applied
on the ohmic field region, as shown in Fig. 5.18 (a). At higher current, the internal base-collector
metallurgical junction becomes forward biased while the external base-collector terminal remains
reverse biased. The holes get injected into the epilayer and this effect is commonly known as
quasi-saturation effect.
Secondly, the slope of the electric filed decreases due to the decreasing of the net charges with
increasing current. Consequently, the depletion width continues to increase with increasing current
until it reaches the highly doped buried collector. At some current level, the net charge and hence
theelectricfielddroptozeroandthenchangethesign, theelectricfieldshiftstothen /n+ junction,
as shown in Fig. 5.18 (b). From this point, the high injection effects start to play a role. This is
again quasi-saturation regime. When quasi-saturation is caused by the potential drop as a result of
91
the reversal of the slope of the electric field, this effect is better known as the Kirk effect.
dE
dx =
qNepi
e (1
Iepi
Ihc );
Ihc = qNepiAemvsat (5.39)
0 Wepi
I=0
IIqs
(a) (b)
0 Wepi
I=0
IIqs
I=Iqs
Figure 5.18: Electric field in the epilayer as a function of current. (a). Base-collector depletion
thickness decreases with current. (b). Base-collector depletion thickness increases with current.
The quasi-saturation can be either due to voltage drop dominated by an ohmic resistance
(Fig. 5.18 (a)), or due to a space-charge limited resistance (Fig. 5.18 (b)). As be shown in Fig. 5.19
(a), the potential of the buried layer, at the interface with the epilayer, is given by the node VC1.
The potential of the internal base is given byVB2. The potential of intrinsic collector node is given
by VC2. VC2 is very important for the description of collector epilayer model, both for low current
where it determines the depletion capacitance and for high currents where it determines the quasi-
saturation effect. In Mextram 504 implementation, it extends the Kull model [14] by including
velocity saturation. The intrinsic collector node potential has a double function, one is for low
current and the other is for high injection. When the quasi-saturation occurs, holes from the base
will be injected in the epilayer. Charge neutrality is maintained in this injection region layer, so
also the electron density increase consequently, as shown in Fig. 5.19 (b). The effective width is
92
fromthebase-emitterjunctiontothen /n+ junctionandhenceitdegradestransistor?sperformance
considerably.
OuranalysisstartswiththeKullmodel[14],butincludingtheidealityfactorintotheI-V relation.
The Kull model gives a good description of the currents and charges in the epilayer as long as the
whole epilayer is quasi-neutral (n = p+1). We can express the hole densities at both ends of the
epilayer p0 and pw in terms of the node voltages:
p0(p0 +1) = exp nB2C2 VdC =NRVT (5.40a)
pW(pW +1) = exp nB2C1 VdC =NRVT (5.40b)
Following Kull model [14], we introduce
K0 =
q
1+4exp nB2C2 VdC =NRVT ; (5.41a)
Kw =
q
1+4exp nB2C1 VdC =NRVT : (5.41b)
Following the derivation of Kull model [14], we get modified Kull result (without velocity
saturation):
IC1C2 = Iepi = EC +nC1C2R
Cv
; (5.42a)
EC = NRVT
K0 KW ln K0 +1K
W +1
; (5.42b)
where nC1C2 = nB2C2 nB2C2, the epilayer resistance RCv = Wepiqmn0NepiAem.
For low injection, K0 and Kw are very close to ?1? and EC is close to ?0?. The epilayer current
Iepi shows ohmic behavior by nC1C2RCv . For high injection, in the case of quasi-saturation, the internal
base-collector junction bias is in forward although the external base-collector bias is reverse. The
epilayerconsistsoftwoparts. Thefirstpartistheinjectionpartwheretheholedensityiscomparable
to the electron density, which is between x = 0 and xi. The second part is the ohmic region where
the hole density is negligible, which is between x = xi andWepi. The voltage of the ohmic region is
93
almostequaltothetotalvoltagedropsincethevoltagedropacrosstheinjectionregionisnegligible.
nC1C2 = nB2C2 nB2C1 ndC nB2C1 = IepiRCv(1 xi=Wepi): (5.43)
ndC is used instead of nB2C2 because both are almost the same as long as the injection occurs
(xi>0). Therefore, the injection region thickness xi can be given as:
xi
Wepi = 1
VdC nB2C1
IepiRCv : (5.44)
By defining the onset current of quasi-saturation Iqs (when xi=0), we have:
xi
Wepi = 1
Iqs
?Iepi; (5.45)
where
Iqs = VdC nB2C1R
Cv
;
?Iepi = Iqs1+axiln
1+exp I
epi=Iqs 1
=a
xi
1+axilnf1+exp[ 1=axi]g ; (5.46)
where axi is smooth parameter. To assure a non-negative xi, ?Iepi is always larger than Iqs, unless
Iepi = 0 when ?Iepi = Iqs.
By combining eqn.(5.42b) and eqn.(5.43), we get:
xi
WepiIepiRCv = EC =VT
2p 0 2pW ln
1+ p
0
1+ pW
(5.47)
Following the derivation in [3], we can solve the internal base-collector bias V B2C2 via the hole
density p 0. In order to calculate p 0 directly, eqn.(5.47) is approximated as.
xi
WepiIepiRCv = 2NRVT (p
0 pW) p 0 + pW +1
p 0 + pW +2 (5.48)
94
V B2C2 =VdC +NRVT ln[p 0(p 0 +1)] (5.49)
Hence p 0 can be solved from the second-order equation from pW, Iepi and xiWepi.
g = IepiRCV2N
RVT
xi
Wepi; (5.50a)
p 0 = g 12 +
r
(g 12 )2 +2g+ pW(pW +g+1); (5.50b)
The internal base-collector biasV B2C2 can be expressed by:
V B2C2 =VdC +VT ln[p 0(p 0 +1)] (5.51)
In the original Kull model [14], velocity saturation is included under the assumption of the
quasi-neutral, which no longer holds when Iepi IHC. In [3], eqn. (5.43) becomes:
VdC nB2C1 = IhcRCv
1 xiW
epi
+(Iepi Ihc)SCRCv
1 xiW
epi
2
; (5.52)
whereIHC =qNepiAemvsat isthehot-carriercurrent,SCRCv = W
2epi
2evsatAem isthespace-chargeresistance
of the epilayer.
Further, eqn. (5.46) becomes:
Iqs = VdC nB2C1SCR
Cv
VdC nB2C1 +IhcSCRCv
VdC nB2C1 +IhcRCv ;
?Iepi = VdC nB2C1
SCRCv
1 xiWepi
2
VdC nB2C1 +IhcSCRCv
1 xiWepi
VdC nB2C1 +IhcRCv : (5.53)
VB2C2 is used to calculate the current IC1C2 through the epilayer, using the Kull model .
V?B2C2 is the most physical one and it takes quasi-saturation effect into account. V?B2C2 will be
used in IN, QBC and Qepi modeling. First the current IC1C2 is calculated using the external base-
collector bias VB2C1 and VB2C2 based on Kull model [14]. Then the injection layer thickness xi is
95
calculated using IC1C2 and VB2C1. Finally the hole density at collector side p?0 and V?B2C2 can be
solved by using IC1C2, xi andVB2C1.
Vjunc is the bias which is used to calculate the intrinsic base-collector depletion capacitance.
Vjunc is calculated using IC1C2 andVB2C1.
emitterbase n-epilayer collector n+ buried layercollector
0 Wepi
E1
B2 C2
C1
(a)
(b)
n+ emitter p base n-epilayer
collector n+ buried layercollector
0 Wepi
Electron densityHole densityDoping density
xi
Figure 5.19: (a). Schematic of a bipolar transistor. (b). Doping, electron and hole densities in the
base-collector region. [3]
5.9.2 Epilayer diffusion charge
The epilayer diffusion charge Qepi is the charge of the holes in epilayer.
Qepi = 12Qepi0 xiW
epi
(p?0 + pW +2); (5.54)
96
where p?0 and pW is the normalized hole density at the both sides of epilayer. And p?0 and pW are
related toV?B2C2 andVB2C1 respectively. Physically, Qepi0 = qNepiAemWepi is the background charge
of epilayer. In Mextram504, however, it introduce an extra transit time tepi with typically value of
W2epi
4Dn ,
Qepi0 = 4tepiVTR
CV
: (5.55)
Fig. 5.20 is the modeled xi=Wepi from 93-393 K. With cooling, the normalized thickness of
the injection region xi=Wepi starts to rise slowly, which indicates the larger onset current of quasi-
saturation at low temperatures. Hence at high current, both epilayer diffusion charge Qepi and
?Qepi=?I are weakly dependent on current with cooling, as shown in Fig. 5.21.
10?1 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
JC (mA/?m2)
x i/W
epi
93 K162 K
223 K300 K
393 K
Figure 5.20: Modeled xi=Wepi from 93-393 K.
97
10?3 10?2 10?1 100 10110
?20
10?19
10?17
10?15
10?13
Q epi
(C)
10?3 10?2 10?1 100 1010
2
4
6
8
JC (mA?m2)
?=?
Q epi
/?I (ps)
93 K162 K
223 K300 K
393 K
?=?Qepi/?I T?
(a)
(b)
AE = 0.5 ? 2.5 ?m2
T?
Figure 5.21: (a) Modeled Qepi from 93-393 K. (b) Modeled ?Qepi=?I from 93-393 K.
5.9.3 Intrinsic base-collector depletion charge
Theintrinsicbase-collectordepletionchargeQtC iscurrentdependent. Firstofall, thedepletion
layer thickness xd changes due to the charges of the electrons moving at the saturated velocity in
the epilayer. Secondly, the voltage drop across the epilayer, IepiRCV, need to be taken into account
when calculating junction voltage, Vjunc, of the depletion charges, where the voltage drop is also
current dependent. Therefore, Vjunc =VB2C1 +IepiRCV for low current. To calculate the intrinsic
depletion thickness, from
dE
dx =
qNepi
e (1
Iepi
Ihc ); (5.56)
we get the solution of the electric field:
E(x) = E0 + qNepie (1 IepiI
hc
)x: (5.57)
98
The depletion region is between 0 and xd. The electric field is constant when x > xd. By
assuming the ohmic region between xd to Wepi at low current, voltage drop over the epilayer is
IepiRCV, then xd can be solved by:
qNepi
2e (1
Iepi
Ihc )x
2
d =Vdc VB2C1 IepiRCV: (5.58)
The modeling of intrinsic base-collector depletion charge QtC is given by:
VCV = VdcT1 p
C
[1 fI(1 VjC=VdcT)1 pC]+ fIbjC(Vjunc VjC);
VtC = (1 XpT)VCV +XpTVB2C1;
QtC = XCjCCjCTVtC; (5.59)
where fI is introduced to model the current dependence of the capacitance,
fI = IhcIepiI
hc +Iepi
: (5.60)
In the description of VjC, a current dependent Vch =VdcT(0:1+ 2IepiIepi+Iqs) is introduced to avoid
sudden changes in the capacitance and hence transit time when the transistor runs into hard satura-
tion.
VFC =VdcT(1 b 1=pCjc );
VjC =Vjunc Vchlnf1+exp[(Vjunc VFC)=Vch]g: (5.61)
The electric field in the epilayer is directly related to the base-collector depletion capacitance.
99
0 xd
EW
E0
Wepi x
E
Figure 5.22: Electric field in the epilayer.
5.10 Self-heating
Since the junction temperature is determined by the thermal resistance for a given power dis-
sipation, accurate modeling of the thermal resistance RTH is critical for the modeling of junction
temperature, and therefore the temperature characteristics of device.
In dc case, a linear relation DT = RTHPdiss is used between the dissipated power Pdiss and tem-
perature changeDT. In non-stationary case, driven by a temperature gradient, the dissipated power
generates an energy flow from the transistor to some heat sink far away. Larger the temperature
gradient, largertheenergyflowis. ThisincreasedtemperatureDT isrelatedtotheincreasedenergy
density through a heat capacitanceCTH [70].
Pdiss = DTR
TH
+CTH dDTdT : (5.62)
5.11 Temperature modeling
As we discussed previously in chapter 2, SiGe HBTs operate very well in the cryogenic en-
vironment [7][5]. In this chapter we will discuss the temperature modeling of SiGe HBT. The
100
CTHRTH
Pdiss
dT
Figure 5.23: Self-heating network.
main purpose is to develop a temperature scalable SiGe HBT model that can work over the desired
cryogenic temperature range over 93-300 K. In this chapter, some new temperature scaling models
are developed to help obtain good dc and ac modeling. The same temperature scaling model as
Mextram will not be duplicated here.
The actual device temperature is expressed as:
T = TEMP+DTA+273:15+VdT; (5.63)
whereTEMPistheambienttemperatureindegreecentigrade,DTAspecifiesaconstanttemperature
shift to ambient temperature. VdT is the increase in temperature DT due to self-heating. The
difference in thermal voltage is given by:
1
VDT =
1
VT
1
VTnom =
q
k
1
T
1
Tnom
:
tN = TT
nom
(5.64)
5.11.1 Saturation current and ideality factor
New temperature scaling models of saturation current and ideality factor were proposed in our
previous work [1]. For the ideality factor of main current?s forward and reverse parts, NF and NR,
101
the temperature dependence is modeled by [1]:
NF(T) = NF
"
1 T TnomT
nom
ANF TnomT
XNF#
;
NR(T) = NR
"
1 T TnomT
nom
ANRTnomT
XNR#
; (5.65)
where Tnom is the nominal temperature, NF and NR are the forward and reverse ideality factor at
nominal temperature respectively. ANF, XNF, ANR and XNR are fitting parameters.
Similarly, for the ideality factor of ideal base current?s forward and reverse parts, NEI and NCI,
the temperature dependence is modeled by [1]:
NEI(T) = NEI
"
1 T TnomT
nom
ANE TnomT
XNE#
;
NCI(T) = NCI
"
1 T TnomT
nom
ANCTnomT
XNC#
; (5.66)
where NEI and NCI are the forward and reverse ideality factor at nominal temperature respectively.
ANE, XNE, ANC and XNC are fitting parameters.
ThemostcompleteIS(T)expressionderivedusingidealShockleytransistortheorywasgivenby
Tsividis[71]. Usingthepopularnonlinearbandgap-temperaturerelationEg;t =Eg;0 aT2=(T +b)
[72],theresultcanberewritteninaformthatresemblestheIS(T)equationsfoundincompactmodels
[1]:
IS(T) = IS;nom
T
Tnom
XIS
exp
0
@ Ea;t
1 TTnom
VT
1
A; (5.67)
Ea;t = Ea;nom abTnom
2
(Tnom +b)2 +
abTTnom
(T +b)(Tnom +b); (5.68)
102
where Ea;t appears in the place of bandgap activation energy VGB in Mextram (other compact
models use different symbols), but is now temperature dependent due to the nonlinear temperature
dependence of Eg;t. Ea;nom is the extrapolated 0 K Eg at nominal temperature, usually 300 K, and
IS;nom is IS at nominal temperature. XIS includes the temperature coefficient of mobility and density
of states, a=4:45 10 4 V/K, b=686 K, and Ea;nom, IS;nom, and XIS are model parameters.
0 50 100 150 200 250 30010?150
10?100
10?50
100
T (K)
I S (A)
fitted by (5.67) fitted by (5.69)
extracted from measurement
Figure 5.24: IS extracted from measuredIC-VBE vs. IS fitted by (5.67) and (5.69) for single 0.5 2.5
mm2 SiGe HBT from 43-300 K.
We observe that the measured IS(T) is much higher than prediction of (5.67), by orders of
magnitudes at lower temperatures, as shown in Fig. 5.24. Even though the underlying physics is
not yet understood, the measured IS(T) -T can be well modeled by [1]:
IS(T) = IS;nom
T
Tnom
XIS
NF(T) exp
0
@ Ea;t
1 TTnom
NF(T)VT
1
A: (5.69)
The effectiveness of (5.69) in modeling IS(T) can be seen in Fig. 5.24.
103
For main current IN?s forward and reverse component, IS;F and IS;R, they share the same Ea;t,
IS;nom and XIS. However, NF(T) and NR(T) are used in eqn(5.69) for IS;F and IS;R expression
respectively.
The temperature scaling of forward base saturation current IBEI and reverse base saturation
current IBCI are very similar. IBEI;nom, XIBEI, Ea;BEI;nom, IBCI;nom, XIBCI, Ea;BCI;nom are model
parameters.
IBEI(T) = IBEI;nom
T
Tnom
XIBEI
NEI(T) exp
0
@ Ea;BEI;t
1 TTnom
NEI(T)VT
1
A:
Ea;BEI;t = Ea;BEI;nom abTnom
2
(Tnom +b)2 +
abTTnom
(T +b)(Tnom +b); (5.70)
IBCI(T) = IBCI;nom
T
Tnom
XIBCI
NCI(T) exp
0
@ Ea;BCI;t
1 TTnom
NCI(T)VT
1
A:
Ea;BCI;t = Ea;BCI;nom abTnom
2
(Tnom +b)2 +
abTTnom
(T +b)(Tnom +b): (5.71)
5.11.2 Base tunneling current
In our previous work [2], The physics based TAT current expression is then parameterized as
follows:
IB;tun = IBT(T)exp
V
B2E1
VTUN
IBT(T) = IBTptN exp(KTN (Eg;TN Eg;T))
IBT isthenominaltemperaturesaturationcurrent, orinterceptofIB;tun-VBE, andVTUN representsthe
temperature independent slope of IB;tun-VBE. VTUN, IBT, KTN are three model parameters specific
to TAT. Eg;TN and Eg;T are the band gap at Tnom and T.
104
5.11.3 Series resistances
The resistances of a bipolar transistor strongly influence device large-signal and small-signal
performance. Transistor I-V curves at practical operational biases are affected by the voltage
drops across terminal resistances and hence modeling of resistances as a function of temperature
is an important part of compact model development. The intrinsic and extrinsic base resistance
fundamentallydegradenoisefigure, astheyproducethermalnoiseatthetransistorinput. Theyalso
consumeinputpoweranddegradepowergainandhencemaximumoscillationfrequency fmax. The
intrinsic base sheet resistance is also related to the Gummel number (or effective Gummel number
for HBTs) and hence the collector current. Furthermore, a decrease of the voltage limit can be
observed with cooling, due to the increase of intrinsic base resistance as a result of freezeout, as
well as the increase in the avalanche multiplication factor (M-1) [16].
This subsection investigates the physics and modeling of the temperature dependence of the
resistances in various regions of SiGe HBTs, including the p substrate, the n collector, the p-
type intrinsic base, the n+ buried collector and the p+ silicided extrinsic base. Sheet resistance
and substrate resistivity were measured on-wafer from 300 to 30 K . The sheet resistance data of
the p+ non-silicided extrinsic base are not available in this work. Although the results presented
are for first-generation SiGe HBTs, we expect the same resistance models to work for second- and
third-generation SiGe HBTs given the similarities in resistor structures.
Forthep substrateandn collector,thedopingiswellbelowtheMott-transition[33],whilefor
the p-type intrinsic base, the doping is close to Mott-transition. The ionization rates are estimated
frommeasuredresistancevs. temperaturedatausingthePhilipsunifiedmobilitymodel(PHUMOB)
[35], and the single power law mobility approximation. For doping well below the Mott-transition,
the classic ionization model [25], together with single power law mobility approximation, enables
accurate modeling of resistance vs. temperature down to 30 K. For doping close to the Mott-
transition, however, the classic ionization model significantly underestimates the ionization rate,
while the recent Altermatt model [26][27] significantly overestimates the ionization rate below 100
K. Analysis of experimental data shows that bound state fraction factor ?b? increases towards ?1?
105
with cooling below certain threshold temperature, while ?b? is fixed at ?1? in Altermatt et al.?s
model. Anempiricalequationfortemperature-dependent?b?isproposedanditenablesresistances
modeling from 300-30 K for doping close to the Mott-transition.
For the n+ buried collector, the doping is well above the Mott-transition, and ionization is
complete at all temperatures. In this case, the temperature dependence of the resistance is solely
determinedbythetemperaturedependenceofthemajoritycarriermobility. However,asinglepower
lawmobilityapproximationcannolongerbeused, andadoublepowerlawmobilityapproximation
utilizing the Mathiessen?s rule of combination of the lattice and impurity scattering components
is proposed. An alternative approach is to continue using a single power law approximation for
mobilityandthenmodelpartofthemobilitytemperaturedependenceasaneffectivedecreaseofthe
ionizationratewithtemperature,which,whilepurelyempirical,alsoworkswell. Thisapproachhas
the advantage of allowing the use of the same model equation for all resistances. The calculations
also show that the PHUMOB is not very accurate for doping well above the Mott-transition at
the cryogenic temperatures. This inaccuracy of PHUMOB, however, does not affect its use in
estimating the ionization rate for doping below and close to the Mott-transition, as the ionization
rate variation with temperature is larger than the inaccuracy of the mobility model. For the p+
extrinsic silicided base, it is not doping related. However, from modeling standpoint, we need to
model the silicide resistance too. The same equation for the n+ buried collector works for silicided
p+ extrinsic base very well [17].
Resistance measurements
The SiGe HBT under investigation employs a heavily-doped polysilicon emitter, an ultra-high-
vacuum/chemical vapor deposition (UHV/CVD) grown SiGe base layer, a lightly-doped n-type
collector epilayer, a heavily-doped buried collector on a p substrate and a heavily-doped silicided
extrinsic base. The peak base doping is below but close to Mott-transition.
All resistance measurements used an Agilent 4156 Semiconductor Parameter Analyzer to per-
formKelvinmeasurementsonavarietyofspeciallydesignedteststructures. Temperaturedependent
106
measurementsofpackagedteststructureswerecarriedoutusingaclosed-cycleliquid-heliumcryo-
genic test system capable of DC to 100 MHz operation from 10 K to 400 K. The measurements
of p-type resistances reported here include the resistivity of the lightly-doped substrate, the sheet
resistance of intrinsic base region, and the sheet resistance of the p+ silicided extrinsic base. Mea-
surementsofn-typeresistancesincludethelightly-dopedepilayerinwhichtheSiGeHBTcollector
is defined, and the heavily-doped n+ buried collector. The substrate resistivity was measured using
the standard four-point-probe technique. A custom test structure was designed with four collinear
1.6 mm substrate contacts which were equally spaced by 150 mm. The base sheet resistance was
measuredusingaconventionalring-dotstructure, whichconsistsofanemitterringboundedbytwo
inner and two outer base contacts. For comparison of measurement and calculation using PHU-
MOB mobility model and Altermatt incomplete ionization model, the base sheet resistance shown
belowis from aSi teststructure, asboth modelsare developedwithexperimentaldata onSi. Under
identical process conditions, Si and SiGe structures show similar base sheet resistance [73]. The
slight difference is mainly due to suppression of boron out diffusion by Ge. The Ge in base slightly
improvesholemobility,buttheimprovementissmallfortherelativelylowaverageGemolefraction
found in the SiGe HBTs used. For compact modeling, the model equations presented below are
applicable to both Si and SiGe bases, only the parameters are slightly different. The remaining
sheet resistances were measured using rectangular Kelvin structures of varying geometry.
Experimental results and analysis
Considerthesheetresistanceofasemiconductorlayer(Rsh). Rsh =
qmN dopW
1
, whereN dop
is the ionized carrier concentration, m is majority carrier mobility, and W is neutral region width.
For example, W is the neutral base width for intrinsic base resistance. Temperature dependence
of W is from the variation of PN junction depletion layer thickness, which is much smaller than
that of N dop and m, and is thus neglected. In current compact models intended for applications
near room temperature, only the mobility variation with temperature is considered through a single
power law relation of m = mT0(T=T0) AR, where T0 is the nominal temperature, mT0 is the nominal
107
mobility, and AR is a doping dependent parameter. This leads to resistance temperature scaling
model R(T) = RT0(T=T0)AR, with RT0 being the nominal temperature resistance.
We first examine the measured temperature dependence of the various resistances and the appli-
cabilityofcurrentresistancetemperaturescalingmodel,andthenanalyzethevariationofionization
rate with temperature for doping below and close to Mott-transition using PHUMOB. For the ex-
trinsic base and buried collector, both doped above the Mott-transition, ionization is complete, and
thus the temperature dependence of the resistances directly reflects the temperature dependence of
the majority carrier mobility.
Fig. 5.25 shows the measured substrate resistivity, the collector sheet resistance, the intrinsic
base sheet resistance, the buried collector sheet resistance, and the silicided extrinsic base sheet
resistance, from 30 to 300 K. A logarithmic scale is used for both resistance and temperature to
facilitate examination of the validity of the current resistance scaling model R = RT0(T=T0)AR. A
straight line or linear relation of lnR-lnT would indicate good applicability of the conventional
model. Observe that such a linear relation is seen only at higher temperatures.
For the lightly-doped substrate and collector, lnR - lnT is linear above 100 K. For the intrinsic
base, however, lnR-lnT is much more complicated and it is only possible to identify a linear
relationship over a smaller temperature range. This more complex temperature dependence, we
believe, is due to the higher doping level, which is close to Mott-transition (the transition occurs
at 3 1018 cm 3 for boron in silicon). For the heavily-doped buried collector and silicided
extrinsicbase, lnR-lnT islinearabove150Kandbecomes?flattened?below100K.Thedeviation
oflnR-lnT fromlinearityhastwopossibleexplanations: 1)themobilityvariationwithtemperature
can no longer be described using a single power relation, which is not surprising given the wide
temperaturerangeinvolved; and2)thecarrierconcentrationvarieswithtemperatureviaincomplete
ionization. For doping below and close to the Mott-transition, both variations are important, and
henceweneedtoestimatetheirindividualcontributionstofacilitateaccuratemodeling. Fordoping
wellabovetheMott-transition, theresistancetemperaturedependenceisdetermineddirectlybythe
mobility temperature dependence due to its complete ionization at all temperatures.
108
20 100 30010
0
102
104
106
108
T (K)
Sheet resistance (
?/sq)
40 100 30010
0
101
102
T (K)
Resistivity (
?.cm)
n? collector
p intrinsic base
n+ buried collector
p+ extrinsic base
p? substrate
(a) (b)
Figure 5.25: (a)Measured sheet resistance for collector, intrinsic base, silicided extrinsic base and
buried layer from 30-300 K. (b) Measured substrate resistivity from 30-300 K.
To estimate the temperature dependence of the ionization rate IR(T) for doping below and close
totheMott-transition,weneedtoobtainthetemperaturedependenceofmobility. Here,wecalculate
mobility using PHUMOB, since it is widely used in device simulators. PHUMOB includes lattice
scattering, impurity scattering, and carrier-carrier scattering, as well as their temperature depen-
dences. We implemented this model in a Matlab program, and verified our implementation using
Sentaurus Device [54]. Complete ionization is assumed in mobility calculation using PHUMOB.
Formally, IR is written as:
IR = N
dop
Ndop; (5.72)
where Ndop is the active doping concentration. Using N dop = Ndop IR and R =
qmN dopW
1
,
where R represents sheet resistance, we obtain:
R(T)
R(T0) =
m(T0)
m(T)
IR(T0)
IR(T) : (5.73)
109
The same relationship holds for resistivity. IR(T)IR(T0) can thus be calculated from measured R(T)R(T0) as:
IR(T)
IR(T0) =
m(T0)
m(T) =
R(T)
R(T0); (5.74)
Wewilldiscussthemeasured R(T)R(T0) andcalculated m(T0)m(T) fordopingbelow,closetoandwellabove
theMott-transitioninSectionIV,VandVI,respectively. FordopingwellabovetheMotttransition,
IR = 1. The comparison of measured R(T)R(T0) with calculated m(T0)m(T) should give us information on the
accuracy of PHUMOB. For doping below and close to Mott-transition, IR varies so much with
temperature that inaccuracy of PHUMOB should not be an issue for the purpose of estimating and
modeling IR(T).
PHUMOB, however, is too complicated for practical (efficient) compact modeling. So we will
also examine the single power law mobility approximation, and show that it is useful for doping
levels well below the Mott-transition. This results in a IR(T) that can be successfully modeled
using a classic incomplete ionization model. Fordoping close to Mott-transition, a newincomplete
ionizationmodelisdevelopedbasedontheAltermattincompleteionizationmodel. Fordopingwell
above Mott-transition, we will present two approaches, one using mobility modeling and complete
ionization, and the other using single power law mobility approximation, but with an artificial
incomplete ionization.
Doping below Mott-transition
DopingconcentrationsinthesubstrateandthecollectorarebelowtheMott-transition[33]. Here
we discuss the substrate resistivity. Fig. 5.26(a) shows measured R(T)R(T0) and m(T0)m(T) calculated using
PHUMOB and a single power law mobility model, for the substrate, over the range of 40-320 K.
The parameters AR for a single power law model were determined from the slope of m-T from 100
K to 300 K using PHUMOB. Above 100 K, the calculated m(T0)m(T) using these two mobility models
both are very close to measured R(T)R(T0) data. This indicates that the impurities in the substrate are
completely ionized above 100 K and PHUMOB can be simplified to a single power law.
110
Fig.5.26(b)shows IR(T)IR(T0) calculatedusing(5.74). IR(T)IR(T0) beginstodecreasefromunitybelow100
K. Although a single power law mobility approximation leads to lower IR than that of PHUMOB,
the shape of IR-T is similar to that obtained from PHUMOB. Therefore, we opted to use a single
power law mobility approximation in this case. A consequence is that the temperature dependence
of the mobility is partially attributed to incomplete ionization below 100 K. This is acceptable for
compact modeling, however, as long as IR(T) can still be successfully modeled, which is the case
here, as shown below.
40 100 30010?2
10?1
100
101
R(T)/R(T
0) &
?(T
0)/?
(T)
40 100 30010?3
10?2
10?1
100
T (K)
IR(T)/IR(T
0)
measured R(T)/R(T0) calculated ?(T
0)/?(T) using PHUMB calculated ?(T
0)/?(T) using single power law approx.
calculated using PHUMB calculated using single power law approx.
p? substrate R(T)/R(T0)
?(T0)/?(T)
(a)
(b)
Figure 5.26: (a) Measured R(T)R(T0), calculated m(T0)m(T) using PHUMOB and single power law approxi-
mation, for the substrate, from 40-320 K. (b) Calculated IR(T)IR(T0) using PHUMOB and a single power
law approximation, for substrate, from 40-320 K.
Using m = mT0(T=T0) AR, R(T), the resistance at temperature T can be related to that at T0,
R(T0), by:
R(T) = RT0
T
T0
AR IR(T
0)
IR(T) ; (5.75)
111
(5.75)involvesthecalculationofIRatbothT andT0. Thiscanbeavoidedbydefiningafictitious
nominal temperature, complete ionization resistance RCI;T0:
RCI;T0 = RT0 IR(T0): (5.76)
Note that RCI;T0 is not the measured resistance at T0, which always includes the impact of
incomplete ionization. (5.75) can be rewritten as:
R(T) = RCI;T0
T
T0
AR 1
IR(T): (5.77)
Numerous theories of how impurity concentration and temperature affect ionization rate have
been developed in the past several decades [25] [26] [27] [28] [29] [30] [31] [32]. Here we will
examine the popular model of using doping dependent activation energy [28][30][31][32][74] in
the classic ionization model [25]. An explicit expression of this classic model IR can be found in
[25]. We briefly review its derivation below, since it subsequently used in the work to derive IR
for the most recent Altemmatt?s incomplete ionization model [26][27], which will then be applied
to the intrinsic base resistance. We will use a p-type dopant for the derivation, but all results can
obviously be applied to n-type dopants as well.
The classic model assumes a Fermi-Dirac-like impurity distribution function. That is, the prob-
ability of finding an electron at an acceptor impurity energy EA is given by [25]:
f(EA) = 1
1+gAexp (EA EF)kT
; (5.78)
where EA = Edop+EV, EV is the valence band edge energy, Edop is the impurity activation energy,
EF is the Fermi level and kT is the thermal energy, and gA is degeneracy factor (gA = 4 for boron).
N dop is then given by:
N dop = Ndop f(EA) = Ndop
1+gAexp (EA EF)kT
; (5.79)
112
Substituting EA = Edop +EV into (5.79) leads to
N dop = Ndop
1+gAexp
E
dop
kT
exp
E
V EFkT
: (5.80)
The EF-EV term can be related to p and hence N dop because:
p = NV exp
E
V EF
kT
; (5.81)
where NV is the valence band effective density-of-states (DOS); NV = 3:14 1019 T300 1:5cm 3
[75]isusedhere. Usingexp
E
V EFkT
= pNV ,N dop =Ndop IRand p=N dop,oneobtainsaquadratic
equation of IR from (5.80),
G 1IR2(T)+IR(T) 1 = 0;
G = g 1A NVN
dop
exp
EdopkT
: (5.82)
Then IR(T) is solved as[25]:
IR(T) = G+
pG2 +4G
2 : (5.83)
(5.83), together with (5.77), are implemented in Agilent ICCAP [62] to fit measured resistance
vs. temperature data. Four model parameters are involved: the impurity activation energy Edop,
the impurity concentration Ndop, the fictitious nominal temperature complete ionization resistance
RCI;T0, and the mobility temperature coefficient AR. Edop, RCI;T0 and AR are dependent on doping
level Ndop, dopant species and device geometry, and thus vary from technology to technology.
Given process and layout information, their values can also be estimated from PHUMOB mobility
model and Altermatt incomplete ionization model. As shown in Fig. 5.27, using this approach we
obtain good modeling accuracy for the collector sheet resistance and the substrate resistivity, from
30-300 K. This, however, does not work for the other resistances as expected.
113
30 100 30010
3
104
105
106
107
108
T (K)
Sheet resistance (
?/sq)
40 100 30010
0
101
102
103
T (K)
Resistivity (
?.cm)n? collector
p+
p? substrate
solid: classic modelsymbol: measured
(a) (b)
Figure 5.27: (a) Collector sheet resistance modeling using the classic model and doping dependent
activation energy from 30-300 K. (b) Substrate resistivity modeling using the classic model and
doping dependent activation energy from 40-320 K.
114
Doping close to Mott-transition
Fig. 5.28(a) shows the measured R(T)R(T0) and m(T0)m(T) calculated using PHUMOB and a single power
law mobility model for the intrinsic base, whose peak doping is close to the Mott-transition. The
parameter AR for the single power law model is determined from the average slope of m-T from
30-300KusingPHUMOB.Thesetwomobilitymodelsgivealmostthesame m(T0)m(T) . Thismeansthat
PHUMOBcan indeedbesimplified assinglepowerlaw, and the IR(T)IR(T0) obtainedshouldbeaccurate.
Due to incomplete ionization, the resulting m(T0)m(T) for both models are much smaller than measured
R(T)
R(T0) below 250 K, as shown by the
IR(T)
IR(T0) curves in Fig. 5.28(b). The decrease of
IR(T)
IR(T0) from unity
starts at 250 K, much higher than the 100 K in the substrate case, because the doping is close to the
Mott-transition.
20 100 30010
?1
100
101
102
103
R(T)/R(T
0) &
?(T
0)/?
(T)
20 100 30010
?3
10?2
10?1
100
T (K)
IR(T)/IR(T
0)
measured R(T)/R(T0) calculated ?(T
0)/?(T) using PHUMB calculated ?(T
0)/?(T) using single power law approx.
calculated using PHUMB calculated using single power law approx.
p intrinsic base
R(T)/R(T0)
?(T0)/?(T)
(a)
(b)
Figure 5.28: (a) Measured R(T)R(T0), calculated m(T0)m(T) using PHUMOB and a single power law approxi-
mation for intrinsic base, from 20-300 K. (b) Calculated IR(T)IR(T0) using PHUMOB and a single power
law approximation for the intrinsic base, from 20-300 K.
The classic incomplete ionization model is applied first to fit the IR(T) obtained. However,
it overestimates resistance below 100 K. The more recent Altermatt incomplete ionization model
115
[26][27]wasthenapplied. However,itunderestimatesresistanceatlowertemperatures,particularly
below 50 K. A new incomplete ionization model is therefore proposed by introducing temperature
dependence into the bound state fraction factor in the Altermatt model. This new model enables
modeling of the intrinsic base sheet resistance from 30-300 K. Fig. 5.29 shows the intrinsic base
sheetresistancemodelingresultsusingtheclassic,theAltermatt,andthenewlyderivedmodel,from
30-300 K. Fig. 5.30 plots the IR and 1=IR using the classic, the Altermatt, and the new ionization
models.
20 100 300104
105
106
107
T (K)
Sheet resistance (
?/sq)
Classic modelAltermatt model
New model Meas.
p intrinsic base
Figure 5.29: Intrinsic base sheet resistance modeling using the classic, the Altermatt, and the new
model, from 30-300 K.
The overestimation of resistance by the classic model is caused by underestimation of the
ionization rate. From (5.82), G reduces towards ?0? when temperature decreases. Hence, from
(5.83), IR decreases exponentially towards ?0? and thus leads to an infinitely large resistance at low
temperatures, as can be seen from the dot-dash line in Fig. 5.30. However, the measured resistance
increase does not suffer from carrier freezeout as much as one would have naively expected from
116
20 100 30010?6
10?5
10?4
10?3
10?2
10?1
100
T (K)
IR
20 100 300100
101
102
103
104
105
106
T (K)
1/IR
classic model Altermatt?s model
new model
(a) (b)
p intrinsic base
p intrinsic base
Figure 5.30: (a) Calculated IR for intrinsic base using the classical, the Altermatt, and the new
model, from 30-300 K. (b) Calculated 1=IR for intrinsic base using the classic, the Altermatt, and
the new model, from 30-300 K.
117
the classic ionization model, even if the activation energy is artificially made to decrease with
increasing doping.
Altermattincompleteionizationmodel Intherecent Altermattmodel[26][27], at highimpurity
concentration, some of the impurity states are no longer localized, and become free states. These
impuritystatesarealwayscompletelyionized,independentoftemperatureandFermilevelposition.
Both the activation energy and the fraction of bound impurity states are functions of the doping
concentration. Detailed physics considerations underlying this model can be found in [26] and
[27]. We will describe here the essence of the model and then derive an explicit expression of IR
for this model, since the original model was developed for numerical simulations and not compact
modeling.
Again, consider p-type dopants with a concentration of Ndop. Denoting the fraction of bound
impurity states as b, we have (1 b)Ndop free impurity states that are always completely ionized.
Out of the bNdop bound states, bNdop f(EA) are ionized. The total ionized concentration N dop is
given by:
N dop = (1 b)Ndop +bNdop f(EA); (5.84)
where the (1 b)Ndop term represents the free states? contribution and the bNdop f(EA) term
represents the bound states? contribution. The ionization rate IR becomes:
IR(T) = (1 b)+bf(EA): (5.85)
Substituting (5.78) into (5.85) leads to:
IR(T) = (1 b)+ b
1+gAexp EdopkT exp (EV EF)kT
: (5.86)
118
We then eliminate EF-EV using (5.81) to obtain IR(T):
IR(T) = (1 b)+ b1+G 1IR(T); (5.87)
(5.87) can be rewritten as a quadratic equation:
G 1IR2(T)+ 1 (1 b)G 1 IR(T) 1 = 0: (5.88)
An explicit solution of IR(T) is thus obtained:
IR(T) = G+(1 b)+
q
[G (1 b)]2 +4G
2 : (5.89)
Besides Edop, Ndop, RT0 and AR, an additional model parameter ?b? is used. When b reaches its
upper limit of ?1?, (5.89) reduces to the classical model. We apply (5.89) and (5.77) in IC-CAP to
fit the intrinsic base sheet resistance vs. temperature data. During the fitting, we only adjusted the
?b? value and kept Edop, Ndop, RCI;T0 and AR the same between the classic and Altermatt models.
The results are shown in Fig. 5.29.
The intrinsic base resistance is underestimated below 50 K using the Altermatt model. This is
expected, as IR(T) has a minimum of (1 b) within the model (e.g., (5.85) and (5.86)). When G
decreasestowards?0?withcooling,IRbecomes?1 b?. InFig.5.30,theAltermattmodelgivesusa
largerionizationrateatlowtemperatures. Below50K,theionizationrateisfixedat1 b=0:05%,
which is why the intrinsic base resistance is underestimated below 50 K. We note that in [26] and
[27], ?b? is temperature independent from 300 K to 30 K. Such a temperature independence of ?b?
inevitably leads to overestimation of the IR value at low temperatures.
New incomplete ionization model When the temperature is sufficiently low, ionization rate IR
decreases exponentially towards ?0? in the classic model and becomes ?1 b? in the Altermatt
model. To fit the measured intrinsic base sheet resistance below 50 K, we need to produce a IR(T)
119
thatisinbetweentheclassicmodelandtheAltermattmodel,wheretheexperimentalvalueofIR(T)
lies. SuchIR(T) can only be obtained when ?b? increases towards unity with cooling below certain
threshold temperature. These results strongly suggest that ?b? becomes temperature dependent at
lower temperatures and further investigation into its physical interpretation is needed. For compact
modeling, we propose the following empirical expression:
b = 1 b1+ T
T0
a : (5.90)
Two model parameters, b and a, are used. As shown by the solid line in Fig. 5.29, this new
model enables accurate intrinsic base sheet resistance vs. temperature modeling from 30-300 K.
If b is set to ?0?, ?b? reduces to ?1?, and this new model reduces to the classic model. The same
model equation can therefore be used for doping below or close to the Mott-transition.
For n-type dopants, the derivation is similar with obvious changes. The IR equation is given by:
IR(T) = G+(1 b)+
q
[G (1 b)]2 +4G
2 ;
G = g 1D NCN
dop
exp
EdopkT
; (5.91)
whereNC =2:8 1019 T300 1:5cm 3 [75]istheeffectivedensity-of-statesintheconductionband,
and gD = 2 for both arsenic and phosphorus.
Heavy doping above Mott-transition
For the heavily-doped buried collector, doping levels are well above Mott transition. Impurities
are thus always completely ionized, even at cryogenic temperatures. Device simulation using
the Altermatt model [26][27] through the physical model interface (PMI) in Sentaurus Device
[54] indeed gives us ionization rate that is very close to unity from 300 K to 30 K. Therefore
the temperature dependence of the mobility directly determines the temperature dependence of
resistance. The same modeling methodology will be applied to the p+ silicided extrinsic base
120
although it is not doping related. Here we use the buried collector for illustration. Fig. 5.31 shows
the measured R(T)R(T0), calculated m(T0)m(T) using PHUMOB and a single power law mobility model for
buried collector from 30-300 K. The parameter AR for the single power law model is determined
from the slope of m-T above 150 K using PHUMOB.
30 100 3000.3
0.5
1
T (K)
R(T)/R(T
0) &
?(T
0)/?
(T)
measured calculated using PHUMB
calculated using single power law approx.
R(T)/R(T0)
?(T0)/?(T)
n+ buried collector
Figure 5.31: Measured R(T)R(T0) and calculated m(T0)m(T) using PHUMOB and a single power law approx-
imation for the buried collector, from 30-300 K.
Over the whole temperature range, the PHUMOB gives a mobility temperature dependence that
canbewell-approximatedwithasinglepowerlawrelation. However,alargedifferenceisobserved
betweencalculated m(T0)m(T) andmeasured R(T)R(T0) below150K.Thisindicatesthateventhecomplicated
PHUMOB model is not sufficiently accurate at low temperatures for the heavily doped buried
collector. From Fig. 5.31, the ln( R(T)R(T0)) - lnT is linear above 150 K and then becomes ?flattened?
with further cooling. For compact modeling, we have two options. The first option is to directly
model the mobility temperature dependence. The second option is to continue to use the single
power law mobility model, but model the deviation from the true mobility as a variation of IR with
temperature. Both approaches work well, as we discuss below.
121
Dual power law mobility approximation with IR = 1 As IR(T) = 1, R(T)R(T0) is essentially m(T0)m(T) .
An inspection of the measured R(T)R(T0) vs. temperature shows that we can combine two power law
mobility temperature dependencies using Mathiessen?s rule to produce the desired R(T)R(T0). Such a
dual power law mobility model can be written as:
1
m =
1
m1 +
1
m2;
m1 = g1Ta;m2 = g2Tb; (5.92)
where m1 and m2 represent two scattering mechanisms, g1 and g2 are two model parameters, and a
and b are their temperature coefficients. By defining g = g2g1 , R(T) can be related to m by:
R(T) = RT0
1
g1Ta +
1
g2Tb
1
g1Ta0 +
1
g2Tb0
= RT0g
1
Ta +
1
Tb
g 1Ta
0
+ 1Tb
0
; (5.93)
(5.93)isimplementedusingIC-CAPtofitmeasuredsheetresistancevs. temperaturedata. Here,
g, a, b and RT0 are used as model parameters. The modeling results are accurate from 30-300 K,
as shown by solid lines in Fig. 5.32.
Empirical approach Instead of modeling the true mobility variation with a physical IR = 1, we
can also continue to use the single power law mobility approximation (for numerical efficiency),
and model the deviation from true mobility as a variation of IR. We implemented this approach
using both the classic incomplete ionization model and the Altermatt model for the buried collector
andtheextrinsicbase. WefindthattheAltermattmodelworks, whereastheclassicmodeldoesnot.
As shown in Fig. 5.32, where the dotted lines are completely overlaid by solid lines, this approach
enables accurate resistance vs. temperature modeling from 30-300 K. As we discussed in Section
III, this approach is empirical, and gives us IR smaller than unity because part of the temperature
dependence of mobility has been modeled by carrier freezeout. This approach, though empirical,
122
20 100 300100
101
T (K)
Sheet resistance (
?/sq)
p+ extrinsic base
n+ buried collector
solid: dual power law model with IR=1 dot: empirical approach
symbol: measured
Figure 5.32: Silicided extrinsic base and buried collector sheet resistance modeling using dual
power law approximation with IR = 1 and empirical approach, from 30-300 K.
does have the advantage of allowing the use of the same model equation for all resistances, which
can be attractive for compact modeling efficiency.
5.11.4 Thermal resistance
Fig. 5.33 shows measured and modeled RTH-Tamb on a log-log scale. Current RTH T-scaling
equation can only model the linear portion, so a 3rd order polynomial is used. The increase of RTH
with cooling below 150 K is consistent with thermal resistivity T-dependence reported in [76].
Rth;Tamb = aT3 +bT2 +cT +Rth;nom
5.11.5 Hot carrier current IHC and epilayer space charge resistance SCRCV
Hot-carrier current IHC and epilayer space charge resistance SCRCV strongly affect fT roll-off.
IHC vsat,andSCRCV 1=IHC. BothIHC andSCRCV areregardedasT-independentinMEXTRAM.
123
100 200 300
3500
4000
4500
Tamb (K)
R TH
(K/W)
modeledextracted
Figure 5.33: Extracted and modeled lnRTH-lnTamb.
Here we find it necessary to include T-dependence of vsat into IHC and SCRCV as follows:
IHC(T) = IHC;nomtN l1(T Tnom) l2;
SCRCV(T) = SCRCV;nomtNl1(T Tnom)+l2; (5.94)
where IHC;nom and SCRCV;nom are nominal temperature values, and l1 and l2 are fitting parameters.
124
5.12 Summary
In our previous work [1], we pointed out ideality factor is necessary to accurately model the
slope of I-V characteristics over wide temperature range. A new temperature scaling model of
saturationcurrentincludingtheimpactofidealityfactorwasdeveloped. Also, inourpreviouswork
[2], a physics based trap-assisted tunneling current equation is presented to model the base current
at low bias over wide temperature range.
In this chapter, new temperature scaling models for series resistance, thermal resistance RTH,
epilayer current parameter IHC and SCRCV are presented. Particularly, temperature characteristics
ofmobilityandionizationrate,whicharethetwomainfactorsaffectingthetemperaturedependence
of various device resistances, are investigated. The classic ionization model, together with a single
power law mobility model, enables resistance vs. temperature modeling of the substrate and the
collector region, where doping levels are below Mott-transition. Based on the Altermatt ionization
model, a new incomplete ionization model that accounts for the temperature dependence of the
bound state fraction factor is developed. This new model enables accurate temperature dependent
modelingoftheintrinsicbasesheetresistance,whichhasadopinglevelclosetotheMott-transition.
For the buried collector and the silicided extrinsic base, where doping levels are well above the
Mott-transition, two approaches are proposed and both give good results. The first approach uses a
dual power law mobility model and complete ionization. The second approach uses the Altermatt
incompleteionizationmodelandasinglepowerlawmobilityapproximation. Thesecondapproach
allows one to use a single model equation for compact modeling of all resistances in SiGe HBTs.
125
Chapter 6
Parameter Extraction and Compact Modeling Results
A reliable, robust and unambiguous parameter extraction method is very important. The use
of a very accurate compact model with poorly extracted parameters will produce bad prediction
of device and circuit performance. The electrical parameter extraction includes low-current pa-
rameters extraction and high-current parameters extraction. Low-current parameters extraction is
straightforward. However,high-currentparametersextractionismuchmoredifficultbecauseinthat
regime many physics effects play a role. Table. 6.1 is the typical grouping of parameters extraction
at reference temperature in Mextram [4]. In Mextram, most of the parameters can be extracted
directly from measured data, including depletion capacitance C-V, dc Gummel plots, dc output
characteristics, dc Early voltage measurement, and ac S-parameter measurement. Some special
measurements are taken to extract terminal resistance, such as RE-flyback and RCc-active methods.
There are different methods to extract temperature parameters. One method is to optimize the
temperature parameters of all data over temperatures. The disadvantage of this method is that we
do not extract the individual parameters at each parameter. If there are unexpected differences
between the model simulation and hardware data, it is difficult to know whether it is the weakness
of the electrical model or temperature scaling model. The second method is to extract the electrical
parameters at all temperatures isothermally. The main advantage is that one can check the cor-
rectness of existing temperature scaling equations by comparing extracted and simulated electrical
parameters [4].
126
Table 6.1: A typical grouping of parameters used in the extraction procedure in Mextram [4].
Base-emitter depletion capacitance CjE, pE, (VdE)
Base-collector depletion capacitance CjC, pC, Xp
Collector-substrate depletion capacitance CjS, pS, (VdS)
Forward-Early Wavl,Vavl
Reverse-Early Ver
Forward-Early Vef
Forward-Gummel Is
Forward-Gummel bf, IBf, mLf
RE-flyback RE
RCc-active RCc
Reverse-Gummel ISs, (Iks)
Reverse-Gummel bri, IBr,VLr
Output-characteristic Rth, Ik
Forward-Gummel RCv, (VdC)
Cut-off frequency SCRCv, Ihc, tE, tepi, (tB;aXi)
Reverse-Gummel Xext
127
6.1 Parameter extraction methodology in this work
A wide temperature range (43-393 K) is covered in this work, hence isothermal parameter
extraction is mainly used to verify the validity of existing temperature models. For example, as we
discussed in chapter 5, Mextram?s temperature models of saturation current and ideality factor no
longer work below 110 K. In this work, complete DC and AC measurements are made at 393, 300,
223, 162, and 93 K Additional Gummel measurements are made at more temperatures to allow
sufficient modeling of temperature dependence.
Self-heating has a large impact at high current parameters because of the strong temperature
dependence of currents, making it difficult to accurately determine isothermal values of high injec-
tion parameters, especially at low temperatures. A 80 K junction temperature rise is much more
significant for an ambient temperature of 43 K than for 300 K. To minimize the impact of the
correlation between self-heating and high current parameters, in our work, we present following
parameter extraction methodology, as shown in Fig. 6.1.
Firstly, low injection parameters (saturation current IS and ideality factor NF, junction capaci-
tance parameters, early voltage, avalanche parameters) are extracted isothermally and temperature
mapped.
Then parasitic resistances are extracted isothermally and temperature mapped;
Then thermal resistances RTH are extracted isothermally and temperature mapped;
Then, high current parameters are extracted isothermally and temperature mapped, with the
implementation of temperature mapped low injection parameters, parasitic resistances and RTH.
During this step, self-heating is turned on.
Finally, once the electrical parameters are extracted and all the temperature scaling equations
are updated, temperature parameters will be fine-tuned again within the whole temperature range,
with self-heating turned on. Usually, several iterations between the last two steps are necessary.
In chapter 5, we present new temperature scaling models for saturation current, ideality factor,
base tunneling current, series resistance, thermal resistance, hot carrier current and epilayer space
charge resistance. In this chapter, we have not developed corresponding isothermal parameter
128
extraction functions based on our new temperature scaling models. Saturation current, ideality
factor and base tunneling current parameters are extracted from the intercept and slope of I-V
through customized transforms coded in Matlab. Junction capacitance parameters are extracted
by using Mextram default functions MXT_cbe and MXT_cbc. Forward and reverse voltages are
extracted by using Mextram default functions MXT_VEF and MXT_VER. Parasitic resistance and
thermal resistance are extracted through customized transforms coded in Matlab. High current
parameters are extracted through Verilog-A simulator together with global optimizer.
isothermal extraction of RTH
and T-mapping
isothermal extraction of saturation
current and ideality factor and T-
mapping
isothermal extraction of parasitic
resistances and T-mapping
isothermal extraction of
avalanche and Early voltage
parameters and T-mapping
Temperature parameters fine
tuning in whole T-range
Forward Gummel
Reverse Gummel
WAVL,VAVL,VER,VEF
RE,RCC,RBC,RBV
RTH
IK,VDC,RCV,SCRCV,IHC,?
B, (?E),?EPI,AXI,MC,M?
IS,IBEI,IBCI,NF,NR,NEI,NCI,IBT
T-mapped low current
parameters, RTH, parasitic
resistance are applied
Forward/reverse Early
VBE-T/VBE-Pdiss
Sheet resistance/resistivity
RE flyback, RCC active, etc
Output characteristic
Current gain
fT-IC
isothermal extraction of
depletion capacitance parameters
and T-mapping
C-V
CJE,PE,VDE,CJC,PC,(VDC),XP
CJS,PS,VDS
TREF=Tnom,TEMP=393,...43K,
self-heating on
isothermal extraction of high
current parameters and T-mapping
self-heating on
Figure 6.1: Proposed parameter extraction methodology in this work.
129
6.2 Saturation current and ideality factor
Let us take the collector current IC-VBE for example. The IS and NF are extracted from low bias
region where the Early effects are negligible. Hence we can extract the saturation current IS and
ideality factor NF from the linear region of semi-log I-V curve where the collector current IC can
be approximated as:
IC IS(T)[exp( VBEN
F(T)VT
) 1] (6.1)
IS canbeextractedfromtheinterceptionofcurrentatzerovoltage,andNF canbeextractedfrom
the slope of this linear region. Fig. 6.2 show the extraction range of IS and NF from IC-VBE.
0.5 0.6 0.7 0.8 0.9 1 1.110?10
10?8
10?6
10?4
10?2
VBE (V)
I C (A)
(VBE,1, IC,1)
(VBE,2, IC,2)
Figure 6.2: IS and NF extraction from the intercept and slope of IC-VBE.
130
6.3 Base tunneling current
At low temperatures, due to the large TAT current (IS;TATeVBE=VTUN) and the sharp slope of
base diffusion current (IS;BEeVBE=VDiff ), it is difficult to directly identify such extreme VBE ranges
where either TAT or diffusion current overwhelmingly dominates. In [2], an iterative procedure is
presented to separate the TAT current from the ?ideal? base current. We achieve the fitting using
the iterative procedure below:
(1)SelectanidealhighVBE region,performlinearfitting,determineIS;BE andVDiff initialvalues.
(2) Subtract IS;BEeVBE=VDiff from total IB, fit the resulting current in a relatively lowerVBE region
to determine IS;TAT andVTUN.
(3) Subtract IS;TATeVBE=VTUN from total IB. Perform linear fitting and update IS;BE andVDiff.
(5) Repeat the Step 2, 3 and 4 until the difference of slope extracted by two successive times are
smaller than a set limit.
Fig.6.3.(a)showsthedifferencebetweenlinearfittingsofbasecurrentwithandwithoutiteration
at 43 K. Without iteration means that main current is directly fitted only by Step 1 from the ?ideal?
base region without excluding the TAT current. It shows that with iteration the summation of two
linear fitting can better cover the measurement. Thus, the TAT current is recognized as the total
base current subtracting the ?ideal? IB after iterated fitting. Fig. 6.3.(b) shows good overall fitting
results of IB-VBE including TAT.
131
0.95 1 1.05 1.1 1.15 1.210?10
10?8
10?6
10?4
VBE (V)
I B (A)
MeasurementWith Iteration
Without Iteration
43 K
(a)
0.9 1 1.110?10
10?9
10?8
10?7
10?6
10?5
10?4
10?3
VBE(V)
I B (A)
0.9 0.95 1
10?10
10?9
VBE (V)
I TAT
(A)
MeasurementModeling
110 K
43 K
(a) (b)
110 K
43 K
(b)
Figure 6.3: (a) Comparison of the direct linear fitting of IB and fitting with iteration at 43 K. (b)
IB-VBE modeling results including TAT [2].
132
6.4 Depletion capacitance
C-V data are obtained from performing S-parameter measurements on transistors biased in the
?cold? operation (low current). Measurement is made from 100 MHz to 1 GHz with a lower
frequency VNA for C-V purpose, so that the parasitic resistances have negligible effect. RSUB and
CSUB are extracted from substrate Y-parametersY22+Y12 [77].
Base-emitter depletion capacitance
The measured base-emitter capacitanceCBE consists of a depletion capacitance, an overlap ca-
pacitanceandadiffusioncapacitance. Thedepletioncapacitanceandoverlapcapacitancedominate
so long asVBE is not so high.
The base-emitter capacitance can be written as [4]:
CBE = sECjE(1 V
jE=VdE)pE
+ (1 sE)CjE(1 V
FE=VdE)pE
+CBEO: (6.2)
This describes a transition from a normal depletion capacitance (if sE = 1) to a constant capac-
itance (sE = 0). The formula of VjE and VFE have been given in (5.11). In practical parameter
extraction, the constant capacitance does not appear as it usually occurs at higher bias, where the
diffusion capacitance dominates. CBEO represents any overlay (peripheral) capacitance between
base and emitter. Actually, the sum of CjE and CBEO give the overall zero bias capacitance and
cannot be separated clearly.
The zero-biasCjE, grading coefficient pE and built-in potentialVdE can be extracted directly by
applying (6.2) in IC-CAP [62] to fit measuredCBE-VBE isothermally. Fig. 6.4 shows the extracted
CBE-VBE from `(1=(Y11 +Y12)) from 393 K to 93 K. Theoretically, the depletion capacitance will
generally decrease with cooling due to the increase in junction built-in voltage. Additionally, our
TCAD simulatedCBE-VBE decreases with cooling monotonically whatever the carrier freezeout is
taken into account or not. Also, the temperature dependence of base-emitter depletion capacitance
CBE is weaker than base-collector depletion capacitanceCBC due to higher doping in base than that
133
of collector. However, theCBE-VBE extracted from measurement do not give monotonic trend with
temperatures, as shown in Fig. 6.4. Hence in our work, we model a weak temperature dependent
CBE by fitting theCBE-T averagely.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.96
8
10
12
14
16
18
VBE (V)
C BE
(fF)
93K162K
223K 300K
393K
Figure 6.4: ExtractedCBE-VBE from 93-300 K.
Base-collector depletion capacitance
The extraction of the base-collector depletion capacitanceCBC is similar to that of base-emitter
capacitance. Parameter Xp is introduced to describe the finite thickness of collector epilayer. The
base-collector depletion capacitance can be written as:
CBC = sC(1 Xp)CjC(1 V
jC=VdC)pC
+ (1 sC)(1 Xp)CjC(1 V
FC=VdC)pC
+XpCjC +CBCO; (6.3)
where XpCjC models constant part when the base-collector depletion region reaches the collector
buried layer.
134
Fig. 6.5 shows the extracted CBC-VBC from `( Y12) from 393 K to 93 K. The zero-bias CjC,
grading coefficient pC and Xp can be extracted directly by applying (6.3) in IC-CAP [62] to fit
measured CBC-VBC isothermally. Help parameter VdC will be re-extracted with other high current
parameter later, because it has strong impact on the current gain roll-off, cut-off frequency roll-off
and output characteristics quasi-saturation region.
?2.5 ?2 ?1.5 ?1 ?0.5 0 0.53
4
5
6
7
8
9
10
VBC (V)
C BC
(fF)
93K162K
223K 300K
393K
Figure 6.5: ExtractedCBC-VBC from 93-300 K.
Collector-substrate depletion capacitance
To better model the device characteristics in high frequency range, we include a pair of sub-
strate capacitanceCSUB and substrate resistance RSUB in parallel with collector-substrate depletion
capacitance CCS, consisting substrate network, as shown in Fig. 5.2. The CCS, CSUB and RSUB are
extractedbyfittingtherealpartandimaginarypartofZsub (Zsub = (Y22+Y12) 1)atdifferentVCS,as
shown in Fig. 6.6 and Fig. 6.7. The extractedCSUB is constant across temperatures. The extracted
temperature dependence of RSUB agrees with what we obtain from measured substrate resistivity,
as shown in Fig. 6.7.(b).
135
The collector-substrate depletion capacitance can be written as:
CSC = sSCjS(1 V
jS=VdS)pS
+ (1 sS)CjS(1 V
FS=VdS)pS
: (6.4)
0 0.5 1 1.5 2
5
6
7
8
9
x 10?15
VCS(V)
C CS
(F)
393K300K
223K162K
Figure 6.6: ExtractedCCS-VCS from 162-393 K.
136
0 0.5 1 1.5 24
4.5
5
5.5
6
6.5
7x 10?15
VCS(V)
C SUB
(F)
393K300K
223K162K
(a)
50 100 150 200 250 300 350 4000
500
1000
1500
2000
2500
Temperature (K)
R S(
?)
Extraction from small?signal circuitR
S Model
(b)
Figure 6.7: (a) ExtractedCSUB-VCS from 162-393 K; (b) Extracted and modeled RSUB-T.
137
6.5 Avalanche and Early voltage
Avalanche
Effective width of the epilayer for avalanche currentWAVL and voltage describing the curvature
of the avalanche currentVAVL are extracted from the avalanche currentIavl and multiplication factor
M 1 in the Forward-Early measurement. The avalanche current Iavl is the difference between the
base current atVCB = 0 and at higherVCB.
Iavl = IB0 IB; (6.5)
where IB0 is a help parameter. Iavl can be expressed by [3]:
EM = Vdc +vCB +2VavlW
avl
r V
dc +vCB
Vdc +vCB +Vavl;lD =
W2avl
2VavlEM;
GEM = AnB
n
EMlD
exp
BnE
M
exp
BnE
M
1+Wavl
lD
;
Iavl = ICGEM; (6.6)
where EM is the maximum electric field in the depletion region, lD is the extrapolated depletion
thickness where the electric field is zero, GM is the generation coefficient. An and Bn are material
constants, which are avalanche coefficient and critical electric field respectively.
Therefore:
M 1 = IAVEI
C IAVE
= GEM1 G
EM
: (6.7)
Fig. 6.8 shows the extracted and modeled M-1-VCB from 93-393 K. Forced-IE measurement
technique at a low current (IE = 12:5mA) is used here to avid self-heating effect. In chapter 3, the
currentdependenceofavalanchemultiplicationfactorM-1isinvestigated. Suchcurrentdependence
of M-1 leads to a much higher breakdown voltage at high collector current. However, this current
dependence of M-1 has not been modeled in compact modeling.
138
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510
?6
10?5
10?4
10?3
10?2
10?1
100
VBE (V)
M?1
93162
223300
393
symbol: measurementsolid: model
IE=12.5 ?A
Figure 6.8: Extracted and modeled M-1-VCB from 93-393 K.
Early voltage
TheextractionofforwardEarlyvoltageVEF andreverseEarlyvoltageVER dependonthebandgap
difference in base DEg = Eg(0) Eg(WB). VEF andVER have a value that corresponds to a pure Si
transistor with the same doping as that of the SiGe transistor.
In this work, DEg is estimated directly by using process knowledge and is fixed as a constant
value over temperatures. And the effective Early voltage:
VForw_Early;eff = IC( ?IC?V
CB
) 1 =VEF e
DEg=kT 1
DEg=kT ;
VRev_Early;eff = IE( ?IE?V
EB
) 1 =VER1 e
DEg=kT
DEg=kT : (6.8)
VEF andVER are extracted isothermally from IC-VCB and IE-VEB respectively.
139
6.6 Emitter resistance
RE flyback method
OneofthesimplestwaytoextracttheemitterresistanceisfromtheGiacolettomethod[78]. The
collector current is kept zero and theVBE is increased . The collector-emitter saturation voltage can
be estimated asVCES IERE. Then the emitter resistance can be obtained by taking the derivative
ofVCES with regard to IE:
RE = ?VCES?I
E
: (6.9)
Fig. 6.9-Fig. 6.10 illustrate the measured VCES and the derivation of the measured VCES with
regard to the IE in the RE flyback measurement. Physically, the series emitter resistance of the
heavily doped emitter region is expected to be current independent. However, the derivation of
measured VCES with regard to the IE decreases as function of IE. If it has reached a plateau, the
value there will be the RE. If the plateau is not reached one will over-estimate the RE.
0 0.002 0.004 0.006 0.008 0.010
0.01
0.02
0.03
0.04
0.05
0.06
0.07
?IE (A)
V CE
(V)
162 K223 K
300 K393 K
DC RE flyback measurement
Figure 6.9: The measuredVCES versus -IE in the RE flyback measurement for the RE extraction.
140
0 0.0010.0020.0030.0040.0050.0060.0070.0080.0090.015
10
15
20
?IE (A)
dV CE
/dI E
(?)
162K223K
300K393K
DC RE flyback measurement
Figure 6.10: Derivation of the measured VCES w.r.t the IE in the RE flyback measurement for the
RE extraction.
Extracted from Z12
In our work, the ac S-parameter measurement is also taken to extract RE for comparison. In the
high frequency range, the real part of Z12 is the sum of dynamic input resistance re of the emitter
base junction and emitter resistance RE.
?(Z12) = re +RE: (6.10)
The dynamic input resistance is defined by:
re = ?VBE?I
E
?VBE?I
C
= 1
ISe
VBE
NFVT 1N
FVT
NFVTI
E
; (6.11)
141
Hence,
?(Z12) = NFVTI
E
+RE;
: (6.12)
RE can be extracted by extrapolating ?(Z12) versus 1IE from the intercept.
0 1 2 3 4 50
50
100
150
1/IE (mA?1)
?(Z
12)
300K223K
162K93K
Figure 6.11: ?(Z12) versus 1IE from 93-300 K.
142
6.7 Base resistance
dc method
ThedeviationbetweenidealandmeasuredbasecurrentIB athighcurrentlevelsisduetovoltage
drops DVBE across base resistance RB and emitter resistance RE [79].
DVBE =VBE VB2E1 = RE IE +RB(IB) IB;
= RE IE +(RBC +RBV(IB)) IB: (6.13)
(6.14)
Once emitter resistance RE is determined, the total base resistance RBC +RBV(IB) as function of
current can be extracted from Gummel curves.
0.8 0.9 1 1.1 1.2 1.30
0.05
0.1
0.15
0.2
0.25
0.3
VBE (V)
?V BE
(V)
43K60K
76K93K
110K136K
162K192K
223K262K
300K
Figure 6.12: DVBE-VBE from 43-300 K.
143
10?3 10?20
100
200
300
400
500
600
700
800
IC (A)
R BC
+R BV
(?)
43K60K
76K93K
110K136K
162K192K
223K262K
300K
Figure 6.13: Extracted (RBC +RBV)-IC from 43-300 K.
ac method
Two ac methods are applied here to extract base resistances by using s-parameter data.
At very high frequency, 1=?(Y11)-w 2 curves can be fitted by a straight line with an intercept
of re +rb [80], where re and rb are ac emitter and ac base resistances respectively.
144
0 1 2 3 4 5x 10?220
500
1000
1500
2000
2500
3000
3500
4000
4500
1/?2 (1/rad2)
1/Re.(Y11)
93K
Figure 6.14: Extraction of re +rb at 93 K by fitting the high-frequency portion of 1=?(Y11)-w 2.
10?2 10?1 100 10150
100
150
200
250
300
350
400
450
500
IC (mA)
r e+r
b (?
)
93K162K
223K300K
393K
extracted from 1/?Y11
Figure 6.15: Extractedre+rb-IC from 43-393 K by fitting the high-frequency portion of 1=?(Y11)-
w 2.
145
Anotherpopulartechniquetoextractsmallsignalbaseresistancerb istousetheinputimpedance
with a shorted output, which by definition is equal to h11.
h11 = re + 11
rb + jwCbv
+ 1g
be + jwCp
;
=
re +rb + gbeg2
be +w2C2p
jw
C
p
g2be +w2C2p
;Cp =Cbe +Cbc: (6.15)
Cbv
re rbc rbv
gbeCbe
Cbc
gmvb
B C
EE
vb
Figure 6.16: Equivalent circuit used in extraction of rb using circuit impedance method.
The (?(h11);`(h11)) ordered pairs at different frequencies form a semicircle on the complex
impedance plane. The (?(h11);`(h11)) impedance point moves clockwise with increasing fre-
quency. the center of this circle is x0 = rb+re+ 12gbe, the radius is r = 12gbe. This circle is shown as
the dotted circle in Fig. 6.17.
146
0 500 1000 1500 2000 2500
?1200
?800
?400
0
?(h
11)
?(h11)
JC=1mA?m2
VCB=0V
AE=0.5 ? 2.5 ?m2
rb
T?
Figure 6.17: Extracted of rb using the circle impedance method from 300-93 K, JC = 1mA=mm2.
10?2 10?1 100 1010
50
100
150
200
250
300
350
400
450
500
JC (mA/?m2)
r b (
?)
93K162K
223K300K
393K
AE=0.5 ? 2.5 ?m2
Figure 6.18: Extracted rb-JC using half-circle method from 300-93 K.
147
6.8 Collector resistance
Similarly to RE-flyback measurement, if the emitter and collector are interchanged to operate in
a common-collector configuration with zero emitter current, the extrinsic collector resistance RCc
can be obtained by taking the derivative ofVCES with regard to IC:
RCc = ?VCES?I
C
: (6.16)
Fig. 6.19 illustrates the derivation of the measuredVCES with regard to the IC in the RC flyback
measurement. The increasing of RC with increasing IC at 93 K and 43 K may due to self-heating
effect. Therefore, the accuracy of the extracted RC at low temperature are in doubt.
0 1 2 3 4 5 6 7 8 9
x 10?3
?10
?5
0
5
10
15
20
25
30
IC (A)
R Cc
(?)
43K93K
162K223K
300K393K
DC RC?flyback measurement
Figure 6.19: Derivation of the measuredVCES w.r.t the IC in the RC flyback measurement for the RC
extraction.
148
6.9 Thermal resistance
Accurate information of device junction temperature is of importance in predicting the device
performance. For Mextram?s default extraction method, RTH is extracted from VBE-VCE relation
of output characteristics. However, in this way, RTH?s extraction relies on the accuracy of base
resistance and emitter resistance, which are also challenges at low temperatures.
Thermal resistance can be extracted from the relation between the power dissipation Pdiss and
the junction temperature Tjunc, for which a temperature-sensitive electrical parameter (TSEP) is
utilized in order to link the two parameters experimentally [81]. Here, base-emitter voltageVBE is
usedasTSEP.ThefirststepisbiasingdevicewithfixedemittercurrentIE andcollector-basevoltage
VCB = 0 V, and thenVBE is measured for different substrate temperatures TS swept around a center
temperature (300 K, 223 K, 162 K and 93 K) with 10 K temperature step. Secondly, the device is
biased with the same IE with fixed substrate temperatures TS (300 K, 223 K, 162 K and 93 K) and
VBE is measured for different power dissipation (Pdiss = ICVCE +IBVVBE) with sweepingVCB. The
relation between TS and Pdiss can be extracted by eliminating VBE from these two measurements.
In our measurement, emitter current IE is chose near peak fT to represents the actual power range
in practical operation of device. More details about this measurement setup can be found in [81].
Fig. 6.20 shows the junction temperature as a function of dissipated power over ambient tem-
perature 93-300 K. The thermal resistance RTH can be extracted by Tjunc = Tamb +RTHPdiss.
149
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10?3
50
100
150
200
250
300
350
Pdiss (W)
T junc
(K)
93 K162 K
223 K300 K
Figure 6.20: Measured junction temperature versus power dissipation over 93-300 K. The IE is
fixed near peak fT
150
6.10 High current parameters
The initial values of high current parameters can be estimated from process information and
layout geometries [4]. However, due to strong correlation between each other, the extraction of the
high-currentparametersisnotasstraightforwardasthatofthelowcurrentparameters. InFig.6.21-
Fig. 6.25, we simulate the influences of IK, RCV, VDC, IHC and SCRCV on output characteristic, fT
and current gain, respectively. In each simulation, we change only one high current parameter and
fix others. Ambient temperature is 300 K andVCB=0 V for current gain and fT simulations. It can
be seen that IK affects both the output characteristic at high VCB, fT roll off and current gain roll
off. The quasi-saturation parameters RCV, VDC, IHC and SCRCV affect output characteristic at low
VCB, fT roll off and current gain roll off.
Basically,duringparameterextraction,IK canbeextractedfromforward-Gummelmeasurement
(current gain measurement) or can be extracted from output characteristic at highVCE alternatively,
where quasi-saturation is of minor importance. RCV and VDC are strongly correlated. They can
be extracted from current gain roll off region. The base transit time tb and emitter transit time te
can be extracted near peak fT, as shown in Fig. 6.26.(a). In modern SiGe HBT technology, base
transit time tb is dominant over emitter transit time te in determining fT because te is reciprocally
proportional to the ac current gain. This is particularly true at decreased temperatures since the
current gain of SiGe HBT is enhanced with cooling. In this work, we fix emitter transit time te=0
and hence it is far easier to tune tb only to fit peak fT. Epilayer parameter tepi can be extracted
from fT roll off, as shown in Fig. 6.26.(b).
Several iterations between these high current parameters are inevitable. For example, knee
current IK needs to be repeated after RCV and VDC extraction because the collector current IC has
been changed.
151
0 0.5 1 1.5 2 2.5 3 3.5 40
1
2
3
4
5
6x 10?3
VCE (V)
I C (A)
IK?
(a)
10?2 10?1 100 1010
5
10
15
20
25
30
35
40
45
50
IC (mA)
f T (GHz)
IK?
(b)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20
20
40
60
80
100
120
140
160
VBE (V)
?
IK?
(c)
Figure 6.21: Impact of parameter IK on: (a) Force-IB output characteristics; (b) fT-IC; (c) b-VBE.
152
0 0.5 1 1.5 2 2.5 3 3.5 40
1
2
3
4
5
6x 10?3
VCE (V)
I C (A)
RCV?
(a)
10?2 10?1 100 1010
5
10
15
20
25
30
35
40
45
50
IC (mA)
f T (GHz) R
CV?
(b)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20
20
40
60
80
100
120
140
160
VBE (V)
?
RCV?
(c)
Figure 6.22: Impact of parameter RCV on: (a) Force-IB output characteristics; (b) fT-IC; (c) b-VBE.
153
0 0.5 1 1.5 2 2.5 3 3.5 40
1
2
3
4
5
6x 10?3
VCE (V)
I C (A)
VDC?
(a)
10?2 10?1 100 1010
5
10
15
20
25
30
35
40
45
50
IC (mA)
f T (GHz) V
DC?
(b)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20
20
40
60
80
100
120
140
160
VBE (V)
?
VDC?
(c)
Figure 6.23: Impact of parameterVDC on: (a) Force-IB output characteristics; (b) fT-IC; (c) b-VBE.
154
0 0.5 1 1.5 2 2.5 3 3.5 40
1
2
3
4
5
6x 10?3
VCE (V)
I C (A)
IHC?
(a)
10?2 10?1 100 1010
5
10
15
20
25
30
35
40
45
50
IC (mA)
f T (GHz) I
HC?
(b)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20
20
40
60
80
100
120
140
160
VBE (V)
?
IHC?
(c)
Figure 6.24: Impact of parameter IHC on: (a) Force-IB output characteristics; (b) fT-IC; (c) b-VBE.
155
0 0.5 1 1.5 2 2.5 3 3.5 40
1
2
3
4
5
6x 10?3
VCE (V)
I C (A)
SCRCV?
(a)
10?2 10?1 100 1010
5
10
15
20
25
30
35
40
45
50
IC (mA)
f T (GHz)
SCRCV?
(b)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20
20
40
60
80
100
120
140
160
VBE (V)
?
SCRCV?
(c)
Figure 6.25: Impact of parameter SCRCV on: (a) Force-IB output characteristics; (b) fT-IC; (c)
b-VBE.
156
10?2 10?1 100 1010
5
10
15
20
25
30
35
40
45
50
IC (mA)
f T (GHz)
?b ? or ?e ?
(a)
10?2 10?1 100 1010
5
10
15
20
25
30
35
40
45
50
IC (mA)
f T (GHz)
?epi?
(b)
Figure 6.26: (a) Impact of parameter tb and te on fT-IC; (b) Impact of parameter tepi on fT-IC.
157
6.11 Temperature parameter extraction
The initial parameters can be obtained by using the parameter extraction methods discussed in
previous sections.
Fig. 6.27 shows the extracted and modeled IS and NF of collector current as functions of tem-
perature. Fig. 6.28 shows the extracted and modeled IBEI and NEI of ?ideal? forward base current
IB-VBE as functions of temperature. Fig. 6.29 is the extracted zero bias CjE-T from Fig. 6.4. The
measurement data of base-emitter depletion capacitance CBE are not very reasonable and a weak
temperature-dependentfittedlineisusedinourmodeling. Fig.6.30istheextractedzerobiasCjC-T
from Fig. 6.5. Fig. 6.31 is the extracted forward and reverse Early voltages.
158
0 50 100 150 200 250 30010?120
10?100
10?80
10?60
10?40
10?20
T (K)
I S (A)
extractedmodeled
(a)
0 50 100 150 200 250 3001
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
T (K)
N F
extractedmodeled
(b)
Figure 6.27: (a) Extracted and modeled IS for collector current as a function of temperature; (b)
Extracted and modeled NF for collector current as a function of temperature.
159
0 50 100 150 200 250 30010?100
10?90
10?80
10?70
10?60
10?50
10?40
10?30
10?20
10?10
T (K)
I BEI
(A)
extractedmodeled
(a)
0 50 100 150 200 250 3001
1.1
1.2
1.3
1.4
1.5
1.6
1.7
T (K)
N EI
extractedmodeled
(b)
Figure 6.28: (a) Extracted and modeled IBEI for base current as a function of temperature; (b)
Extracted and modeled NEI for base current as a function of temperature.
160
50 100 150 200 250 300 350 4004
5
6
7
8
9
10
11
12
13
14
15
T (K)
C JE
(fF)
extractedmodeled
Figure 6.29: Extracted and modeledCJE-T.
50 100 150 200 250 300 350 4004
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8
6
T (K)
C JC
(fF)
extractedmodeled
Figure 6.30: Extracted and modeledCJC-T.
161
0 100 200 300 4005
6
7
8
9
10
11
12
13
14
15
T (K)
V ER
(V)
0 100 200 300 40015
20
25
30
35
T (K)
V EF
(V)
extractedmodeled
Figure 6.31: Extracted and modeled Early voltageVER andVEF from 93-393 K.
162
Although the initial values of parasitic resistances can be obtained from specific dc and ac
measurements we discussed, due to their coupling with high current parameters near and above
peak fT, in order to achieve good fitting of frequency dependence, the parasitic resistances need to
be tuned in the last step together with high current parameters, as shown in Fig. 6.1.
Fig. 6.32 and Fig. 6.33 are the measured and modeled frequency dependence of Y11 and Y21
at 300K. Four bias conditions are chosen near peak fT. Here, by changing the RBC by 25%,
the difference of simulated Y11 and Y21 can be found at high frequencies. Similar comparison of
changing RBV and RE by 25% can be found in Fig. 6.34-Fig. 6.37.
Fig. 6.38-Fig. 6.41 are the extracted parasitic resistances as function of temperature.
Fig.6.42showsmeasuredandmodeledthermalresistanceRTH versusambienttemperatureTamb
onalog-logscale. CurrentRTH T-scalingequationcanonlymodelthelinearportion, soa3rdorder
polynomial is used.
Fig. 6.43-Fig. 6.48 show the extracted high current parameters, IK, RCV,VDC, IHC, tB and tepi.
163
(a) (b)
(c) (d)
Figure 6.32: Measured (symbols) and modeled (lines) frequency dependence of real part y-
parameters at 300K. Four bias conditions are chosen near peak fT: (a) real part Y11, RBC=28
W; (b) real partY11, RBC=34 W; (c) real partY21, RBC=28 W; (d) real partY21, RBC=34 W.
164
(a) (b)
(c) (d)
Figure 6.33: Measured (symbols) and modeled (lines) frequency dependence of imaginary part y-
parameters at 300K. Four bias conditions are chosen near peak fT: (a) imaginary partY11, RBC=28
W; (b) imaginary part Y11, RBC=34 W; (c) imaginary part Y21, RBC=28 W; (d) imaginary part Y21,
RBC=34 W.
165
(a) (b)
(c) (d)
Figure 6.34: Measured (symbols) and modeled (lines) frequency dependence of real part y-
parameters at 300K. Four bias conditions are chosen near peak fT: (a) real part Y11, RBV=288
W; (b) real partY11, RBV=348 W; (c) real partY21, RBV=288 W; (d) real partY21, RBV=348 W.
166
(a) (b)
(c) (d)
Figure 6.35: Measured (symbols) and modeled (lines) frequency dependence of imaginary part y-
parametersat300. Fourbiasconditionsarechosennearpeak fTK:(a)imaginarypartY11,RBV=288
W; (b) imaginary partY11, RBV=348 W; (c) imaginary partY21, RBV=288 W; (d) imaginary partY21,
RBV=348 W.
167
(a) (b)
(c) (d)
Figure 6.36: Measured (symbols) and modeled (lines) frequency dependence of real part y-
parameters at 300K. Four bias conditions are chosen near peak fT: (a) real part Y11, RE=12 W;
(b) real partY11, RE=15 W; (c) real partY21, RE=12 W; (d) real partY21, RE=15 W.
168
(a) (b)
(c) (d)
Figure 6.37: Measured (symbols) and modeled (lines) frequency dependence of imaginary part
y-parametersat300K.Fourbiasconditionsarechosennearpeak fT: (a)imaginarypartY11, RE=12
W; (b) imaginary part Y11, RE=15 W; (c) imaginary part Y21, RE=12 W; (d) imaginary part Y21,
RE=15 W.
169
50 100 150 200 250 300 350 4004
6
8
10
12
14
16
18
T (K)
R E
(?
)
extracted from ac measurementextracted from dc R
E flyback measurement
extracted from fitting isothermally
modeled
Figure 6.38: Extracted and modeled RE-T.
0 50 100 150 200 250 300 350 400
10
20
30
40
50
60
T (K)
R Bc
(?)
extracted from dc measurementextracted from fitting isothermally
modeled
Figure 6.39: Extracted and modeled RBc-T.
170
50 100 150 200 250 300 350 4000
1000
2000
3000
T (K)
R BV
(?)
extracted isothermallymodeled
Figure 6.40: Extracted and modeled RBV from 93-300 K.
50 100 150 200 250 300 350 4000
10
20
30
40
50
60
70
80
90
100
T (K)
R Cc
(?)
extracted from dc RC flyback measurement
extracted from fitting isothermallymodeled
Figure 6.41: Extracted and modeled RCc-T.
171
100 200 300
3500
4000
4500
Tamb (K)
R TH
(K/W)
modeledextracted
Figure 6.42: Extracted and modeled thermal resistance RTH from 93-300 K.
50 100 150 200 250 300 350 4002
4
6
8
10
12
14x 10?3
T (K)
I K (A)
extracted isothermallymodeled
Figure 6.43: Extracted and modeled IK from 93-300 K.
172
50 100 150 200 250 300 350 40020
40
60
80
100
120
140
160
T (K)
R CV
(?)
extracted isothermallymodeled
Figure 6.44: Extracted and modeled RCV from 93-300 K.
50 100 150 200 250 300 350 400
0.8
1
1.2
T (K)
V DC
(V)
extracted isothermallymodeled
Figure 6.45: Extracted and modeledVDC from 93-300 K.
173
50 100 150 200 250 300 350 4000.004
0.006
0.008
0.01
0.012
0.014
0.016
T (K)
I HC (A)
extracted isothermallymodeled
Figure 6.46: Extracted and modeled IHC from 93-300 K.
50 100 150 200 250 300 350 4000
0.5
1
1.5
2
T (K)
? B (ps)
extracted isothermally
modeled
Figure 6.47: Extracted and modeled tB from 93-300 K.
174
50 100 150 200 250 300 350 4000
200
400
600
800
1000
1200
1400
1600
T (K)
? epi
(ps)
extracted isothermally
modeled
Figure 6.48: Extracted and modeled tepi from 93-300 K.
175
6.12 SiGe HBT compact modeling results
Fig. 6.49 (a)-(d) show the IC-VBE and IB-VBE modeling results from 43-393 K. For both IC and
IB, the new saturation current and ideality factor scaling equations are essential for the good fitting
in the medium current range, particularly below 100 K. Trap-assisted tunneling current is well
modeled, as can be seen from Fig. 6.49 (d).
Fig. 6.50.(a) and Fig. 6.50.(b) show the measured and modeled IC-VCE andVBE-VCE for forced
low IB bias over 93-393 K respectively. Fig. 6.51.(a) and Fig. 6.51.(b) show the measured and
modeled IC-VCE andVBE-VCE for forced high IB bias over 93-393 K respectively.
Fig. 6.52 shows fT-IC from 93-393 K at VCB = 0;1;2V. Fig. 6.53 shows the measured and
modeled H21-JC and MUG-JC at 5GHz from 93-393 K. Fig. 6.54 shows measured and modeled
Y-parameters at 1,2,3 and 5 GHz for 162 K.
176
0.2 0.6 1.210
?12
10?6
10?1
VBE (V)
I C (A)
0.9 1 1.1 1.210
?12
10?7
10?2
VBE (V)
I C (A)
0.9 1 1.1 1.210
?12
10?7
10?3
VBE (V)
I B (A)
0.2 0.6 1.210
?12
10?7
10?2
VBE (V)
I B (A)
Measured data
Simulated data
393K
136K
(a) (b)
93K
43K Forward trap?assisted
tunneling current
(c) (d)
kink
Figure 6.49: (a)Measured and modeled IC-VBE from 136-393 K. (b)Measured and modeled IB-VBE
from136-393K.(c)MeasuredandmodeledIC-VBE from93-43K.(d)MeasuredandmodeledIB-VBE
from 93-43 K.
177
0 1 2 3 410
?6
10?5
10?4
10?3
VCE (V)
I C (A)
0 1 2 3 410
?6
10?5
10?4
10?3
VCE (V)
I C (A)
0 1 2 3 410
?6
10?5
10?4
10?3
VCE (V)
I C (A)
0 1 2 3 410
?6
10?5
10?4
10?3
VCE (V)
I C (A)
Measured dataSimulated data
300K 223K
162K 93K
IB = 480.7nA
IB = 185.2nA
IB = 85.74nA
IB = 39.54nA
IB = 1.010uA
IB = 383.7nA
IB = 140.2nA
IB = 50.38nA
IB = 339.5nA
IB = 125.5nA
IB = 45.23nA
IB = 16.17nA
IB = 182.4nA
IB = 59.89nA
IB = 19.80nA
IB = 6.830nA
(a)
0 1 2 3 4
0.75
0.8
0.85
VCE (V)
V BE
(V)
0 1 2 3 4
0.86
0.88
0.9
0.92
0.94
0.96
VCE (V)
V BE
(V)
0 1 2 3 4
0.92
0.96
1
VCE (V)
V BE
(V)
0 1 2 3 4
1
1.02
1.04
1.06
VCE (V)
V BE
(V)
Measured dataSimulated data
300K 223K
162K
93K
(b)
Figure 6.50: Measured and modeled forced-IB output characteristics at low IB. (a) IC-VCE; (b)
VBE-VCE.
178
0 1 2 3 40
2
4
6x 10
?3
VCE (V)
I C (A)
0 1 2 3 40
2
4
6x 10
?3
VCE (V)
I C (A)
0 1 2 3 40
1
2
3
4
5x 10
?3
VCE (V)
I C (A)
0 1 2 3 40
1
2
3
4
5x 10
?3
VCE (V)
I C (A)
Measured dataSimulated data
300K 223K
162K
93K
(a)
0 1 2 3 40.8
0.85
0.9
0.95
VCE (V)
V BE
(V)
0 1 2 3 4
0.95
1
1.05
VCE (V)
V BE
(V)
0 1 2 3 40.98
1
1.02
1.04
1.06
1.08
VCE (V)
V BE
(V)
0 1 2 3 41
1.05
1.1
1.15
VCE (V)
V BE
(V)
Measured dataSimulated data
300K 223K
162K 93K
(b)
Figure 6.51: Measured and modeled forced-IB output characteristics at high IB. (a) IC-VCE; (b)
VBE-VCE.
179
10?6 10?5 10?4 10?3 10?20
10
20
30
40
50
60
IC (A)
f T (GHz)
93 K162 K
223 K300 K
393 K
VCB=0V
symbol: measuredline: modeled
(a)
10?6 10?5 10?4 10?3 10?20
10
20
30
40
50
60
IC (A)
f T (GHz)
93 K162 K
223 K300 K
393 K
VCB=1V
symbol: measuredline: modeled
(b)
10?6 10?5 10?4 10?3 10?20
10
20
30
40
50
60
70
IC (A)
f T (GHz)
93 K162 K
223 K300 K
393 K
VCB=2V
symbol: measured line: modeled
(c)
Figure 6.52: Measured (symbol line) and modeled (solid line) fT-IC from 93-393 K. (a)VCB = 0V.
(b)VCB = 1V. (c)VCB = 2V.
180
10?3 10?2 10?1 100 101?20
?15
?10
?5
0
5
10
15
20
25
30
JC (mA/?m2)
H 21
(dB)
93162
223300
393
AE=0.5 ? 2.5 ?m2
5 GHz
SiGe HBT 5AM
(a)
10?3 10?2 10?1 100 101?40
?30
?20
?10
0
10
20
30
JC (mA/?m2)
MUG (dB)
93162
223300
393
AE=0.5 ? 2.5 ?m2
5 GHz
SiGe HBT 5AM
(b)
Figure 6.53: (a) Measured (symbols) and modeled (curves) H21-JC at 5 GHz from 93-393 K. (b)
Measured (symbols) and modeled (curves) MUG-JC at 5 GHz from 93-393 K.
181
10?6 10?4 10?210
?6
10?4
10?2
IC (A)
?Y
11 (A/V)
10?6 10?4 10?210
?4
10?3
10?2
IC (A)
?Y 11
(A/V)
10?6 10?4 10?210
?7
10?5
10?3
IC (A)
?Y
12 (A/V)
10?6 10?4 10?210
?5
10?4
10?3
IC (A)
?Y 12
(A/V)
10?6 10?4 10?210
?6
10?4
10?1
IC (A)
?Y
21 (A/V)
10?6 10?4 10?2?10
0
?10?5
IC (A)
?Y 21
(A/V)
10?6 10?4 10?210
?6
10?4
10?2
IC (A)
?Y
22 (A/V)
10?6 10?4 10?2
10?4
10?3
IC (A)
?Y 22
(A/V)
@1,2,3,5 GHz, VCB=0V
162K
Figure 6.54: Measured (symbols) and modeled (curves) Y-parameters at 1,2,3 and 5 GHz for 162
K,VCB = 0V.
182
6.13 Summary
In this chapter, a wide temperature range parameter extraction strategy is presented.
Firstly, low injection parameters (saturation current IS and ideality factor NF, junction capaci-
tance parameters, early voltage, avalanche parameters) are extracted isothermally and temperature
mapped.
Then parasitic resistances are extracted isothermally and temperature mapped;
Then thermal resistances RTH are extracted isothermally and temperature mapped;
Then, high current parameters are extracted isothermally and temperature mapped, with the
implementation of temperature mapped low injection parameters, parasitic resistances and RTH.
During this step, self-heating is turned on.
Finally, once the electrical parameters are extracted and all the temperature scaling equations
are updated, temperature parameters will be fine-tuned again within the whole temperature range,
with self-heating turned on. Usually, several iterations between the last two steps are necessary.
With the extracted model parameters, we obtain reasonably accurate fitting of the dc character-
istics from 393 to 43 K for a first-generation SiGe HBT. Good ac fitting from 393 to 93 K have
been achieved.
183
Chapter 7
Band Gap Reference Circuit Modeling Application
7.1 Introduction
Precision bandgap references (BGRs) are extensively used in a wide variety of circuits required
for such missions, and have been demonstrated to work well at cryogenic temperatures [82] [83].
To further optimize BGR performance at cryogenic temperatures, it is necessary to understand and
model the non-idealities found in existing designs, which we address in this work for the first time.
Typical BGR design assumes an ideal temperature dependence of the IC-VBE characteristics
predictedfromShockley?stransistortheory,withvariousdegreesofassumptionsonthetemperature
dependence of the bandgap. At cryogenic temperatures, however, the slope of measured IC-VBE
significantlydeviatesfromideal1=VT [84][1]. Aswediscussedpreviously,atemperaturedependent
non-ideality factor NF(T) is necessary to describe the slope of IC-VBE. The measured intercept of
IC-VBE,knownasthesaturationcurrentIS,alsoshowsatemperaturedependencedrasticallydifferent
from traditional theory below 200 K [84] [1]. It is therefore logical to examine how these device-
leveldeviationsfrom traditional theoriesaffectBGR output atcryogenic temperatures. In addition,
a key question centers of whether we can successfully model BGR performance over extremely-
wide temperature ranges using the transistor IC VBE model of [1]. The first-order BGR design
from [82] is used here, which was fabricated using a first-generation SiGe BiCMOS technology
with 50 GHz cut-off frequency at room temperature.
7.2 Technical approach and results
Fig.7.1 shows the schematic of the BGR analyzed [82]. M4-M7 and Q1-Q2, along with resistor
R1, generate the PTAT bias current IPTAT, which is set byDVBE. Q1 and Q3 consists of four parallel
184
M1
M3
M2
M6
IPTAT
ICQ1 Q
1
M4 M
5
M8
M7
IPTAT
Q2
R1
R2
Q3
Iref
Vref
VCC
Figure 7.1: Schematic of a first-order SiGe bandgap reference [2].
copies of 0.5 2.5 mm2 SiGe HBTs. The emitter area of Q2 is eight times of that of Q1 and Q3.
VBE of Q3 is controlled by IPTAT, and increases with cooling. Vref is the sum ofVBE of Q3 and the
voltage across R2, PTAT voltage proportional to DVBE through Iref(T).
7.2.1 Slope of IC-VBE and impact on DVBE(T) and Iref(T)
The ?PTAT? DVBE(T) is generated from theVBE difference of Q1 and Q2, two transistors oper-
ating at different current densities. Shockley theory predicts a IC VBE slope of 1=VT, and a DVBE
ofVT ln
A
Q2
AQ1
, which isVT ln(8). However, the measured DVBE of the BGR clearly deviates from
VT ln(8) below 200 K, which is shown in Fig. 7.2(a). This deviation from strict PTAT behavior
originates from deviation of IC VBE slope from 1=VT, and can be modeled using a non-ideality
factor NF(T) [1]. DVBE(T) is then given by:
DVBE(T) = NF(T)VT ln(8): (7.1)
185
NF(T) can be modeled using [1]:
NF(T) = NF;nom
1 T TnomT
nom
ANF TnomT
XNF!
; (7.2)
where Tnom is nominal temperature, NF;nom is non-ideality factor at nominal temperature, which is
close to unity, and ANF and XNF are technology dependent fitting parameters. Using (7.2), NF(T)
and consequently DVBE(T) can be well-modeled, as shown in Fig. 7.2(b).
0 50 100 150 200 250 3000
0.02
0.04
0.06
T (K)
?V BE
(V)
0 50 100 150 200 250 3001
1.1
1.2
1.3
1.4
T (K)
N F
calcualted VTln(8)measured ?V
BE
extracted from measurement by ?VBE/VTln(8)
fitted by (2)
(a)
(b)
Figure 7.2: (a) Measured DVBE-T and calculated VT ln(8)-T from 43 to 300 K. (b) NF-T from
43-300 K.
Transistors M4-M5 and M6-M7 are identically-sized pairs. IPTAT is then amplified through
transistor M8, with an amplification factor k = (W=L)M8 =(W=L)M5.
Iref(T) = kIPTAT(T) = kNF(T)VT ln(8)R
1
: (7.3)
186
Table 7.1: Models examined in this work.
Name Model 1 Model 2 Model 3
NF(T) 1.025 (7.2) (7.2)
IS(T) (5.67) (5.67) (5.69)
We mention in passing that the ?PTAT? designation is no longer strictly accurate below 200 K
because NF(T) > 1. The slope of IC-VBE, or 1=NF(T)VT, directly affects the BGR?s DVBE(T) and
hence Iref(T), which then determinesVBE of Q3, as detailed below.
7.2.2 IS(T) and impact onVBE(T)
VBE of Q3,VBE;3 is given by:
VBE;3 = NF(T)VT ln Iref(T)I
S;3(T)
= NF(T)VT ln k
DVBE(T)
R1
IS;3(T) ; (7.4)
where IS;3 is the IS of Q3.
We have seen that the slope of IC VBE affectsVBE;3 through the NF(T)VT term. The intercept
of IC-VBE, IS(T), further affectsVBE;3 through the IS;3(T) term in (7.4).
We now examine the impact of IS(T) onVBE;3. To examine the roles of the IC VBE slope and
intercept, that is, NF(T) and IS(T), we chose 3 model combinations, as shown in Table 1. Model 1
produces the classic slope and intercept models, both of which are ?wrong? at lower temperatures.
Model 2 produces the correct slope but the ?wrong? intercept. Model 3 produces the correct slope
and the correct intercept of IC-VBE.
Fig. 7.3 shows IC-VBE for a 0.5 2.5 mm2 SiGe HBT at 300 K and 43 K. The symbols represent
the (VBE;3;Iref=4) pair using (7.3) and (7.4). The factor of ?4? represents the multiplicity number
of Q3. At 300 K, the nominal temperature, all of the three models have the same Iref value, and
give the sameVBE;3, because they all have the same NF and IS at 300 K, as expected.
187
At 43 K, with NF(T), the slope of IC VBE is correctly modeled by both model 2 and model 3.
Model 2 and 3 thus have the same and correct Iref. However, for model 2, IS is underestimated by
traditional theory, leading to a much higherVBE;3 than model 3.
For model 1, the situation is more complex. The slope of IC VBE is overestimated as NF is
fixed at its 300 K value. The intercept of IC VBE (IS) is underestimated. The final IC VBE from
a model that gives a wrong slope and a wrong intercept, however, is surprisingly not too far off
from measured data in the current range of interest for producingVBE;3. As a result, theVBE;3 from
model 1, the traditional model, is much closer to model 3 than model 2, which actually has the
correct slope.
The above process can be repeated for all temperatures to yield VBE;3-T, which is shown in
Fig. 7.4. This is completely equivalent to a calculation using (7.4), but the graphical illustration
yields a much better intuitive understanding of the model differences. Model 3 successfully repro-
duces the measuredVBE;3-T characteristics. TheVBE;3-T prediction from traditional theory, model
1, is not terribly inaccurate, because of the cancellation between underestimated intercept, IS(T),
and the overestimated slope, 1=NF(T)VT. This cancellation is circuit design dependent, however,
and is to a large extent a coincidence for the BGR design analyzed.
7.2.3 Vref(T)
The BGR outputVref is given by:
Vref(T) =VBE;3(T)+Iref(T)R2 =VBE;3(T)+KDVBE(T); (7.5)
where K = kR2=R1. The temperature dependence of K can be negligibly small by choosing high
precision, low temperature coefficient resistors and careful layout design to minimize resistor mis-
match. The resistor used in the BGR exhibits a temperature coefficient of 17.8 ppm/ C over the
temperature range of 43-300 K. In practice, K is often chosen to make the positive temperature
188
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.410?8
10?7
10?6
10?5
10?4
10?3
VBE (V)
I C (A)
Model 1Model 2
Model 3Measurement
(VBE,3,Iref/4)
@43K
(VBE,3,Iref/4)
@300K
Figure 7.3: Measured and modeled IC-VBE for single 0.5 2.5 mm2 SiGe HBT at 43 K and 300 K.
0 50 100 150 200 250 3000.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
T (K)
V BE
(V)
Model 1Model 2
Model 3Measurement
Figure 7.4: Measured and modeledVBE from 43-300 K.
189
coefficient of DVBE(T) cancel the negative temperature coefficient of VBE(T), to produce a zero
temperature coefficientVref(T) at a nominal temperature, usually 300 K [71] [85].
0 50 100 150 200 250 3000.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
T (K)
K??
V BE
(V)
Model 1Model 2
Model 3Measurement
Figure 7.5: Measured and modeled K DVBE from 43-300 K.
Fig. 7.5 shows the measured and modeled K DVBE versus temperature. Due to the use of a
constant NF, model 1 deviates from measurements below 200 K. Model 2 and 3 accurately capture
K DVBE.
The sum of VBE;3(T) and K DVBE(T) gives Vref, which is shown in Fig. 7.6. Additional
calculation shows that the VBE;3(T) difference between the models dominates over the K DVBE
differences. Model 2 has the largestVBE;3 and henceVref deviation from measurement. Traditional
theory, model 1, is not terrible in predicting Vref, because of coincidental cancellation between
underestimated intercept, IS(T), and overestimated slope, 1=NF(T)VT.
On the other hand, we observethat the overallVref variation with temperature for model 3 is less
thanthatformodel1. Thedifferenceisthatmodel3accountsforthedeviationsofboththeslopeand
intercept of IC VBE from traditional Shockley transistor theory, model 1. This suggests that such
deviations actually help make the BGR output vary less with temperature than traditional theories
190
50 100 150 200 250 3001.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
T (K)
Vref (V)
Model 1Model 2
Model 3Measurement
Figure 7.6: Measured and modeledVref from 43-300 K.
would predict, and explains why BGRs experimentally perform much better than predictions using
traditional BGR design equations at lower temperatures.
7.3 Summary
Inthischapter,weexaminetheimpactofthenon-idealtemperaturedependenceofIC-VBE inSiGe
HBTsacrosstemperatureontheoutputofafirst-orderSiGebandgapreference. Thesenon-idealities
are shown to actually help make the BGR output voltage vary less at cryogenic temperatures
than traditional Shockley theory would predict. For the particular BGR design examined, the
overallVref(T) prediction from Shockleytheoryis not too bad, because of the cancellation between
underestimated intercept and overestimated slope. Successful cryogenic temperature modeling of
bothDVBE andVBE componentsoftheBGRoutputisdemonstratedforthefirsttime. Themodeling
methodpresentedprovidesbasisforfurtheroptimizationofSiGeBGRsoperatingacrossextremely
widetemperatureranges,anddowntodeepcryogenictemperatures,whereexistingdesignequations
fail.
191
Chapter 8
Conclusions
In this dissertation, physics and compact Modeling of SiGe HBT for wide temperature range
operation were investigated.
Chapter1gaveanintroductionofthemotivation. SiGeHBTtechnologiesarebeingincreasingly
deployed in ?extreme environments?, including operation to very low temperature (e.g., to 77 K
or even 4.2 K). Thorough understanding of SiGe HBT physic at cryogenic temperature is very
necessary. Meanwhile, to enable circuit design, robust and accurate compact models that can work
overawidetemperaturerangearetrulyimportanttobedeveloped. TheSiGeHBTinvestigatedand
measured in this work is a first-generation, 0.5 mmSiGe HBT with fT=fmax of 50 GHz/65 GHz and
BVCEO=BVCBO of 3.3 V/10.5 V at 300 K, with base doping below but close to the Mott-transition
(about 3 1018 cm 3 for boron in silicon).
Chapter 2 analyzed several import physics, including bandgap energy Eg, electron and holes
densityofstatesNC=NV,intrinsiccarrierconcentrationni,bandgapnarrowingBGN,carriermobility
m,carriersaturationvelocitynsat andcarrierfreezeouteffect. Following,thedcandacperformance
of SiGe HBT were discussed. During the analysis, not only the influence of temperature, but also
the impact of germanium wasillustrated with details. The bandgap engineering generally produces
positive influence on the low temperature operations of bipolar transistors, including higher current
gain b, larger Early voltageVA, lower base transit time tB and hence higher cut-off frequency fT.
Chapter3extendedthesubstratecurrentbasedavalanchemultiplicationtechniquedownto43K.
This measurement technique gave the current dependence of avalanche multiplication factor M-1
at low temperatures. However conventional techniques fail because of self-heating at high current
densities. Lower M-1 was observed at higher current densities where cut-off frequency is high (of
interest to practical circuits). In chapter 4, the forced-IE pinch-in maximum output voltage limit
192
in SiGe HBTs operating at cryogenic temperatures was investigated. A decrease of the voltage
limit was observed with cooling. We attributed it to the increase of intrinsic base resistance due
to freezeout as well as increase of avalanche multiplication factor M-1. A practically high IE was
shown to alleviate the decrease of V CB with cooling, primarily due to the decrease of M-1 with
increasing emitter current IE. The fact that the maximum operation voltage range does not degrade
as much with cooling at such high current density is certainly good news for circuit applications.
In chapter 5, wide temperature range compact models of SiGe HBTs were presented based on
Mextram and some new temperature scaling models were presented. In particular, the temperature
characteristics of mobility and ionization rate were investigated. The classic ionization model,
together with a single power law mobility model, enables resistance vs. temperature modeling of
the substrate and the collector region, where doping levels are below Mott-transition. Based on the
Altermatt ionization model, a new incomplete ionization model that accounts for the temperature
dependence of the bound state fraction factor was developed. This new model enables accurate
temperaturedependentmodelingoftheintrinsicbasesheetresistance,whichhasadopinglevelclose
to the Mott-transition. For the buried collector and the silicided extrinsic base, where doping levels
are well above the Mott-transition, two approaches were proposed and both give good results. In
chapter 6, a new parameter extraction strategy was implemented. Isothermal electrical parameters
extraction and temperature parameters extraction were illustrated in detail. With the extracted
model parameters, reasonably accurate fitting of the dc characteristics from 393 to 43 K, and good
ac fitting from 393 to 93 K have been achieved.
Typical BGR design assumes an ideal temperature dependence of the IC-VBE characteristics
predictedfromShockley?stransistortheory,withvariousdegreesofassumptionsonthetemperature
dependence of the bandgap. At cryogenic temperatures, however, the slope of measured IC-VBE
significantly deviates from ideal 1=VT. A temperature dependent non-ideality factor NF(T) is
necessarytodescribetheslopeofIC-VBE. ThemeasuredinterceptofIC-VBE,knownasthesaturation
current IS, also shows a temperature dependence drastically different from traditional theory below
200 K. Chapter 7 examined the impact of the non-ideal temperature dependence of IC-VBE in SiGe
193
HBTs on the output of a BGR. These non-idealities were shown to actually help make the BGR
output voltage vary less at cryogenic temperatures than traditional Shockley theory would predict.
Successful cryogenic temperature modeling of both DVBE andVBE components of the BGR output
was demonstrated for the first time.
194
Bibliography
[1] Z. Xu, X. Wei, G. Niu, L. Luo, D. Thomas, and J. D. Cressler, ?Modeling of temperature
dependentIC-VBE characteristicsofSiGeHBTsfrom43-400k,?inDig.ofIEEEBCTM,pp.81?
84, oct 2008.
[2] Z. Xu, G. Niu, L. Luo, and J. D. Cressler, ?A physics-based trap-assisted tunneling current
model for cryogenic temperature compact modeling of SiGe HBTs,? in ECS Trans., vol. 33,
pp. 301?310, 2010.
[3] J. C. J. Paasschens, W. J. Kloosterman, and R. v. d. Toorn, Model derivation of Mextram 504:
the physics behind the model. Nat Lab Unclassified Report, NL-UR 2002/806, Koninklijke
Philips Electronics, Oct 2004.
[4] J. C. J. Paasschens, W.J. Kloosterman, and R. J. Havens, Parameter extraction for the bipolar
transistor model Mextram. Nat Lab Unclassified Report, NL-UR 2001/801, Koninklijke
Philips Electronics, May 2001.
[5] J. D. Cressler, ?On the potential of SiGe HBTs for extreme environment electronics,? in
Proceedings of the IEEE, vol. 93, pp. 1559?1582, Sep 2005.
[6] Z. Xu, G. Niu, L. Luo, J. D. Cressler, M. L. Alles, R. Reed, H. A. Mantooth, J. Holmes,
and P. W. Marshall, ?Charge collection and SEU in SiGe HBT current mode logic operating
at cryogenic temperatures,? IEEE Transactions on Nuclear Science, vol. 57, pp. 3206?3211,
December 2010.
[7] J.D.CresslerandG.F.Niu,Silicon-GermaniumHeterojunctionJunctionBipolarTransistors.
Artech House, 2003.
[8] H. K. Gummel and H. C. Poon, ?An integral charge control model of bipolar transistors,? The
Bell System Technique Journal, pp. 827?852, May 1970.
[9] C. C. McAndrew, J. A. Seitchik, D. F. Bowers, M. Dunn, M. Foisy, I. Getreu, M. McSwain,
S.Moinian, J.Parker,D.J.Roulston, M.Schroter,P.vanWijnen,andL.F.Wagner,?VBIC95,
the vertical bipolar inter-company model,? IEEE Journal of Solid-State Circuits, vol. 31,
no. 10, pp. 1476?1483, 1996.
[10] A. Koldehoff, M. Schroter, and H. M. Rein, ?A compact bipolar transistor model for very-
high-frequency applications with special regard to narrow emitter stripes and high current
densities,? Solid-State Electronics, vol. 36, pp. 1035?1048, July 1993.
195
[11] H. C. Graaff and W. J. Kloosterman, ?New formulation of the current and charge relations
in bipolar transistor modeling for CACD purposes,? IEEE Transactions on Electron Devices,
vol. 32, pp. 2415?2419, 1985.
[12] I. E. Getreu, Modeling the Bipolar Transistor. Amsterdam: Elsevier, 1978.
[13] H.C.deGraaffandF.M.Klaassen,CompactTransistorModelingforCircuitDesign. Vienna,
Austria: Springer-Verlag, 1990.
[14] G. Kull, L. Nagel, S.-W. Lee, P. Lloyd, E. Prendergast, and H. Dirks, ?A unified circuit model
for bipolar transistors including quasi-saturation effects,? IEEE Transactions on Electron
Devices, vol. 32, pp. 1103?1113, Jun 1985.
[15] J. Pan, G. Niu, J. Tang, Y. Shi, A. J. Joseph, and D. L. Harame, ?Substrate current based
avalanche multiplication measurement in 120 GHz SiGe HBTs,? IEEE Electron Devices Let-
ters, vol. 24, pp. 736?738, December 2003.
[16] L. Luo, G. Niu, D. Thomas, J. Yuan, and J. D. Cressler, ?Forced-IE pinch-in maximum output
voltage limit in SiGe HBTs operating at cryogenic temperatures,? in Dig. of IEEE BCTM,
pp. 41?44, oct 2008.
[17] L. Luo, G. Niu, K. A. Moen, and J. D. Cressler, ?Compact modeling of the temperature
dependence of parasitic resistances in SiGe HBTs down to 30 K,? IEEE Transactions on
Electron Devices, vol. 56, pp. 2169?2177, Oct 2009.
[18] L. Luo, G. Niu, L. Najafizadeh, and J. D. Cressler, ?Impact of the non-ideal temperature
dependenceofIC-VBE onultra-widetemperaturerangeSiGeHBTbandgapreferencecircuits,?
in Dig. of IEEE BCTM, pp. 220?223, Oct 2010.
[19] W. Bludau, A.Onton, and W.Heinke, ?Temperature dependence of the band gap in silicon,?
Journal of Applied Physics, vol. 45, no. 4, pp. 1846?1848, 1974.
[20] G. MacFarlane, J. P. McLean, J. E. Quarrington, and V. Roberts, ?Fine structure in the
absorption-edge spectrum of Si,? Physical Review B, vol. 111, pp. 1245?1254, May 1958.
[21] T. Fromherz and G. Bauer, Properties of strained and relaxed Silicon Germanium. INSPEC,
London, 1995.
[22] Z. Matutinovic-Krstelj, V. Venkataraman, E. J. Prinz, J. C. Sturm, and C. W. Magee, ?Base
resistance and effective bandgap reduction in npn Si/Si1-xGex/Si HBT?s with heavy base
doping,? IEEE Transactions on Electron Devices, vol. 43, pp. 457?466, March 1996.
[23] R. People, ?Indirect band gap of coherently strained gexsi1 x bulk alloys on ???001???
silicon substra,? Physical Review B, vol. 32, pp. 1405?1408, July 1985.
[24] J. W. Slotboom and H. C. D. Graaff, ?Measurements of bandgap narrowing in si bipolar
transistors,? Solid-State Electronics, vol. 19, pp. 857?862, 1976.
[25] J. Lindmayer and C. Y. Wrigley, Fundamentals of Semiconductor Devices. Van Nostrand
Reinhold Company, 1965.
196
[26] P. P. Altermatt, A. Schenk, and G. Heiser, ?A simulation model for the density of states and
for incomplete ionization in crystalline silicon. I. establishing the model in Si:P,? Journal of
Applied Physics, vol. 100, pp. 113715?1137157, 2006.
[27] P. P. Altermatt, A. Schenk, B. Schmithen, and G. Heiser, ?A simulation model for the density
ofstatesandforincompleteionizationincrystallinesilicon.II.investigationofSi:AsandSi:B
and usage in devicesimulation,? Journal of Applied Physics, vol.100, pp. 113714?11371410,
2006.
[28] G.L.PearsonandJ.Bardeen,?Electricalpropertiesofpuresiliconandsiliconalloyscontaining
boron and phosphorus,? Physical Review, vol. 75, pp. 865?883, Mar 1949.
[29] F. J. Morin and J. P. Maita, ?Electrical properties of silicon containing arsenic and boron,?
Physical Review, vol. 96, pp. 28?35, Oct 1954.
[30] T. F. Lee and T. C. McGill, ?Variation of impurity-to-band activation energies with impurity
density,? Journal of Applied Physics, vol. 46, pp. 373?380, Jan 1975.
[31] G. D. Mahan, ?Energy gap in si and ge: Impurity dependence,? Journal of Applied Physics,
vol. 51, pp. 2634?2646, May 1980.
[32] S. R. Dhariwal, V. N. Ojha, and G. P. Srivastava, ?On the shifting and broadening of impurity
bandsandtheircontributiontotheeffectiveelectricalbandgapnarrowinginmoderatelydoped
semiconductors,? IEEE Transactions on Electron Devices, vol. 32, pp. 44?48, Jan 1985.
[33] N. F. Mott, ?Metal-insulator transition,? Reviews of Modern Physics, vol. 40, pp. 677?683,
Oct 1968.
[34] D. B. M. Klaassen, ?A unified mobility model for device simulation-i. model equations and
concentration dependence,? Solid-State Electronics, vol. 35, no. 7, pp. 953?959, 1992.
[35] D.B.M.Klaassen,?Aunifiedmobilitymodelfordevicesimulation-ii.temperaturedependence
of carrier mobility and lifetime,? Solid-State Electronics, vol. 35, no. 7, pp. 961?967, 1992.
[36] F. J. Morin and J. P. Maita, ?Electrical properties of Silicon containing arsenic and boron,?
Physical Review, vol. 96, pp. 28?35, October 1954.
[37] G.W.LudwigandR.L.Watters,?DriftandconductivitymobilityinSilicon,?PhysicalReview,
vol. 101, pp. 1699?1701, March 1956.
[38] A. B. Sproul, M. A. Green, and A. W. Stephens, ?Accurate determination of minority carrier-
and lattice scattering-mobility in silicon from photoconductance decay,? Journal of Applied
Physics, vol. 72, pp. 4161?4171, November 1992.
[39] M. V. Fischetti and S. Laux, ?Band structure, deformation potentials, and carrier mobility in
strained Si, Ge and SiGe alloys,? Journal of Applied Physics, vol. 80, no. 4, pp. 2234?2252,
1996.
197
[40] G. Hock and E. K. et al., ?High hole mobility in Si0:17Ge0:83 channel metal-oxide-
semiconductorfield-effecttransistorsgrownbyplasma-enhancedchemicalvapordeposition,?
Applied Physical Letters, vol. 76, pp. 3920?3922, June 2000.
[41] T. Krishnamohan, Physics and Technology of High Mobility, Strained Germanium Channel,
Heterostructure MOSFETs. PhD Dissertation, Stanford University, 2006.
[42] L. E. Kay and T. W. Tang, ?Monte Carlo calculation of strained and unstrained electron
mobilitiesinSi1 xGex usinganimprovedionizedimpuritymodel,?JournalofAppliedPhysics,
vol. 70, pp. 1483?1488, August 1991.
[43] F. M. B. et al., ?Low- and high-field electron-transport parameters for unstrained and strained
Si1 xGex,? IEEE Electron Device Letters, vol. 18, pp. 264?266, June 1997.
[44] J.-S.Rieh,P.K.Bhattacharya,andE.T.Croke,?Temperaturedependentminorityelectronmo-
bilities in strained Si1 xGex (0:2 x 0:4) layers,? IEEE Transactions on Electron Devices,
vol. 47, pp. 883?890, April 2000.
[45] R. Quay, C. Moglestue, V. Palankovski, and S. Selberherr, ?A temperature dependent model
for the saturation velocity in semiconductor materials,? Materials Science in Semiconductor
Processing, vol. 3, pp. 149?155, March 2000.
[46] V. Palankovski and R. Quay, Analysis and simulation of heterostructure devices. Springer-
Verlag Wien NewYork, 2004.
[47] H.Kroemer,?Towintegralrelationspertainingtoelectrontransportthroughabipolartransistor
with a nonuniform energy gap in the base region,? Solid-State Electronics, vol. 28, pp. 1101?
1103, 1985.
[48] J. M. Early, ?Effects of space-charge layer widening in junction transistors,? in Proceedings
of Institute of Radio Engineers, vol. 40, pp. 1401?1406, November 1952.
[49] A. J. Joseph, J. D. Cressler, and D. M. Richey, ?optimization of early voltage for cooled SiGe
HBT precision current sources,? Journal De PhysiqueQ IV, vol. 6, pp. 125?129, 1996.
[50] M. Rickelt, H. M. Rein, and E. Rose, ?Influence of impact-ionization-induced instabilities on
the maximum usable output voltage of Si-bipolar transistors,? IEEE Transactions on Electron
Devices, vol. 48, pp. 774?783, Apr 2001.
[51] G. Niu, J. D. Cressler, S. Zhang, U. Gogineni, and D. C. Ahlgren, ?Measurement of collector-
base junction avalanche multiplication effects in advanced UHV/CVD SiGe HBT?s,? IEEE
Transactions on Electron Devices, vol. 46, pp. 1007?1015, May 1999.
[52] H.M.R.M.RickeltandE.Rose, ?Low-temperatureavalanchemultiplicationinthecollector-
base junction of advanced n-p-n transistors,? IEEE Transactions on Electron Devices, vol. 37,
pp. 762?767, Mar 1990.
[53] J. H. Klootwijk, J. W. Slotboom, M. S. Peter, V. Zieren, and D. B. de Mooy, ?Photo carrier
generation in bipolar transistors,? IEEE Transactions on Electron Devices, vol. 49, pp. 1628?
1631, September 2002.
198
[54] S. Device, ?2-d device simulator, version 10.0, synopsys,? Synopsys, 2006.
[55] J. Yuan, J. D. Cressler, Y. Cui, G. Niu, S. Finn, and A. Joseph, ?On the profile design of sige
hbts for rf lunar applications down to 43 k,? in Dig. of IEEE BCTM, pp. 25?28, oct 2008.
[56] J. J. Ebers and J. L. Moll, ?Large-signal behavior of junction transistor,? in Proceedings of
Institute of Radio Engineers, vol. 42, pp. 1761?1772, December 1954.
[57] H. Stubing and H. M. Rein, ?A compact physical large-signal model for high-speed bipolar
transistors at high current densities-part i: One-dimensional model,? IEEE Transactions on
Electron Devices, vol. 34, pp. 1741?1751, August 1987.
[58] H. Stubing and H. M. Rein, ?A compact physical lar e-signal model for high-speed bipolar
transistorsathighcurrentdensities-partii: Two-dimensionalmodelandexperimentalresults,?
IEEE Transactions on Electron Devices, vol. 34, pp. 1752?1761, August 1987.
[59] W. J. Kloosterman, Comparison of Mextram and the Vbic95 bipolar transistor model. Nat
Lab Unclassified Report, NL-UR 034/96, Philips Electronics, 1996.
[60] J. D. Cressler, ed., Silicon Heterostructure Handbook: Materials, Fabrication, Devices, Cir-
cuits, and Applications of SiGe and Si Strained-Layer Epitaxy. CRC press, 2006.
[61] S. Salmon, J. D. Cressler, R. C. Jaeger, and D. L. Harame, ?The influence of Ge grading on
the bias and temperature characteristics of SiGe HBTs for precision analog circuits,? IEEE
Transactions on Electron Devices, vol. 47, pp. 292?298, Feb 2000.
[62] Agilent 85190A IC-CAP 2006 user?s guide. Agilent Technologies, 2006.
[63] J. A. D. Alamo and R. M. Swanson, ?Forward-bias tunneling: a limitation to bipolar device
scalings,? IEEE Electron Devices Letters, vol. 7, pp. 629?631, Nov 1986.
[64] S. M. Sze, Physics of semiconductor devices. Wiley, New York, 1981.
[65] R. S. Muller and T. I. Kamins, Device electronics for integrated circuits. Wiley, New York,
1986.
[66] J. Paasschens, W. Kloosterman, and R. Havens, ?Modelling two SiGe HBT specific features
for circuit simulation,? in Dig. of IEEE BCTM, pp. 38?41, oct 2001.
[67] R. L. Pritchard, ?Two-dimensional current flow in junction transistors at high frequencies,?
Proceedings of Institute of Radio Engineers, vol. 46, pp. 1152?1160, June 1958.
[68] H.N.Ghosh,?Adistributedmodelofthejunctiontransistoranditsapplicationintheprediction
of the emitter-base diode characteristic, base impedance, and pulse response of the device,?
IEEE Transactions on Electron Devices, vol. 12, pp. 513?531, oct 1965.
[69] M.P.J.G.Versleijen,?Distributedhighfrequencyeffectsinbipolartransistors,?inProceedings
of Bipolar Circuits and Technology Meeting, pp. 85?88, Sep 1991.
[70] C. Kittel and H. Kroemer, Thermal Physics. Freeman & Co., 1980.
199
[71] Y. P. Tsividis, ?Accurate analysis of temperature effects in IC-VBE characteristics with appli-
cationtobandgapreferencesources,? IEEE Journal of Solid-State Circuits, vol.15, pp.1076?
1084, Dec 1980.
[72] C. D. Thurmond, ?The standard thermodynamic functions for the formation of electrons
and holes in Ge, Si, GaAs, and GaP,? Journal of Electrochemical Society, vol. 122, no. 8,
pp. 1133?1141, 1975.
[73] G. Niu, S. Zhang, J. D. Cressler, A. J. Joseph, J. S. Fairbanks, L. E. Larson, C. S. Webster,
W. E. Ansley, and D. L. Harame, ?Noise modeling and SiGe profile design tradeoffs and RF
applications,? IEEE Transactions on Electron Devices, vol. 47, pp. 2037?2044, Nov 2000.
[74] M. R. Shaheed and C. M. Maziar, ?A physically based model for carrier freeze-out in si- and
sige- base bipolar transistors suitable for implementation in device simulators,? in Dig. of
IEEE BCTM, pp. 191?194, 1994.
[75] M. A. Green, ?Intrinsic concentration, effective densities of states, and effective mass in
silicon,? Journal of Applied Physics, vol. 67, pp. 2944?2954, Mar 1990.
[76] C. J. Glassbrenner and G. A. Slack, ?Thermal conductivity of silicon and germanium from 3
K to the melting point,? Physical Review, vol. 134, pp. 1058?1069, May 1964.
[77] Z.Xu,G.Niu,L.Luo,P.S.Chakraborty,P.Cheng,D.Thomas,andJ.D.Cressler,?Cryogenic
RFsmall-signalmodelingandparameterextractionofSiGeHBTs,?inDigest.ofIEEETopical
Meeting on Silicon Monolithic Integrated Circuits in RF Systems, pp. 1?4, 2009.
[78] L.J.Giacoletto,?Measurementofemitterandcollectorseriesresistances,?IEEETransactions
on Electron Devices, vol. 19, pp. 692?693, May 1972.
[79] T.NingandD.Tang,?Methodfordeterminingtheemitterandbaseseriesresistancesofbipolar
transistors,? IEEE Transactions on Electron Devices, vol. 31, pp. 409?412, April 1984.
[80] W. Z. Cai and S. Shastri, ?Extraction of base and emitter series resistances from the net
transadmittance parameter,? IEEE Transactions on Electron Devices, vol. 52, pp. 622?626,
April 2005.
[81] J. S. Rieh, D. Greenberg, B. Jagannathan, G. Freeman, and S. Subbanna, ?Measurement
and modeling of thermal resistance of high speed SiGe heterojunction bipolar transistors,?
Digest. of IEEE Topical Meeting on Silicon Monolithic Integrated Circuits in RF Systems,
pp. 110?113, Sep 2001.
[82] L. Najafizadeh, A. K. Sutton, R. M. Diestelhorst, M. Bellini, B. Jun, J. D. Cressler, P. W.
Marshall, and C. J. Marshall, ?A comparison of the effects of X-Ray and proton irradiation
on the performance of SiGe precision voltage references,? IEEE Transactions on Nuclear
Science, vol. 54, pp. 2238?2244, Dec 2007.
[83] L. Najafizadeh, J. S. Adams, S. D. Phillips, K. A. Moen, J. D. Cressler, T. R. Stevenson, and
R. M. Meloy, ?Sub-1-K operation of SiGe transistors and circuits,? IEEE Electron Device
Letters, vol. 30, pp. 508?510, May 2009.
200
[84] Z. Feng, G. Niu, C. Zhu, L. Najafizadeh, and J. D. Cressler, ?Temperature scalable modeling
of SiGe HBT DC current down to 43 K,? in ECS Trans., vol. 3, pp. 927?936, 2006.
[85] P. R. Gray and R. G. Meyer, eds., Analysis and Design of Analog Integrated Circuits. New
York: Wiley, 2001.
201
Appendices
202
Appendix A
Appendix A Full List of Temperature Scaling Equations
The actual device temperature is expressed as:
T = TEMP+DTA+273:15+VdT; (A.1)
whereTEMPistheambienttemperatureindegreecentigrade,DTAspecifiesaconstanttemperature
shift to ambient temperature. VdT is the increase in temperature DT due to self-heating. The
difference in thermal voltage is given by:
1
VDT =
1
VT
1
VTnom =
q
k
1
T
1
Tnom
:
tN = TT
nom
(A.2)
A.1 Compact model parameter list
203
Table A.1: Compact model parameters overview.
# Symbol Name Description
1 LEVEL LEVEL Model Level
2 EXMOD EXMOD Flag for extended modeling of the external regions
3 EXPHI EXPHI Flag for extended modeling of distributed HF effects in transients
4 EXAVL EXAVL Flag for extended modeling of avalanche current
5 MULT MULT Number of parallel transistors modeled together
6 TREF TREF Reference temperature
7 DTA DTA Difference between the local and global ambient temperatures
8 IS IS Collector-emitter saturation current at Tnom
9 XTI XTI Temperature coefficient of IS
10 Ea;nom Ea_nom Temperature coefficient of IS
11 IBEI IBEI Forward base saturation current at Tnom
12 XTE XTE Temperature coefficient of IBEI
13 Ea;BEI;nom EaBEI_nom Temperature coefficient of IBEI
14 IBCI IBCI Reverse Base saturation current at Tnom
15 XTC XTC Temperature coefficient of IBCI
16 Ea;BCI;nom EaBCI_nom Temperature coefficient of IBCI
17 NF NF Forward collector-emitter current ideality factor at Tnom
18 XNF XNF Temperature coefficient of the NF
19 ANF ANF Temperature coefficient of the NF
20 NR NR Reverse collector-emitter current ideality factor at Tnom
21 XNR XNR Temperature coefficient of the NR
22 ANR ANR Temperature coefficient of the NR
23 NEI NEI Forward base current ideality factor at Tnom
24 XNE XNE Temperature coefficient of the NEI
25 ANE ANE Temperature coefficient of the NEI
26 NCI NCI Reverse base current ideality factor Tnom
27 XNC XNC Temperature coefficient of the NCI
28 ANC ANC Temperature coefficient of the NCI
29 IBT IBT Base tunneling current at Tnom
30 KTN KTN Temperature coefficient of IBT
31 VTUN VTUN Temperature coefficient of base tunneling current
204
# Symbol Name Description
32 Ndop;RE Ndop_RE Temperature coefficient of the RE
33 Edop;RE Edop_RE Temperature coefficient of the RE
34 aRE alpha_RE Temperature coefficient of the RE
35 bRE beta_RE Temperature coefficient of the RE
36 ARE A_RE Temperature coefficient of the RE
37 RE R_E Emitter resistance at Tnom
38 Ndop;RBC Ndop_RBC Temperature coefficient of the RBC
39 Edop;RBC Edop_RBC Temperature coefficient of the RBC
40 aRBC alpha_RBC Temperature coefficient of the RBC
41 bRBC beta_RBC Temperature coefficient of the RBC
42 ARBC A_RBC Temperature coefficient of the RBC
43 RBC R_BC Constant part of the base resistance at Tnom
44 Ndop;RBV Ndop_RBV Temperature coefficient of the RBV
45 Edop;RBV Edop_RBV Temperature coefficient of the RBV
46 aRBV alpha_RBV Temperature coefficient of the RBV
47 bRBV beta_RBV Temperature coefficient of the RBV
48 ARBV A_RBV Temperature coefficient of the RBV
49 RBV R_BV Variable part of the base resistance at zero-bias and at Tnom
50 Ndop;RCV Ndop_RCV Temperature coefficient of the RCV
51 Edop;RCV Edop_RCV Temperature coefficient of the RCV
52 aRCV alpha_RCV Temperature coefficient of the RCV
53 bRCV beta_RCV Temperature coefficient of the RCV
54 ARCV A_RCV Temperature coefficient of the RCV
55 RCV R_CV Resistance of the un-modulated epilayer at Tnom
56 Ndop;RCC Ndop_RCC Temperature coefficient of the RCC
57 Edop;RCC Edop_RCC Temperature coefficient of the RCC
58 aRCC alpha_RCC Temperature coefficient of the RCC
59 bRCC beta_RCC Temperature coefficient of the RCC
60 ARCC A_RCC Temperature coefficient of the RCC
61 RCC R_CC Constant part of the collector resistance at Tnom
62 Ndop;RSUB Ndop_RSUB Temperature coefficient of the RSUB
63 Edop;RSUB Edop_RSUB Temperature coefficient of the RSUB
64 aRSUB alpha_RSUB Temperature coefficient of the RSUB
65 bRSUB beta_RSUB Temperature coefficient of the RSUB
66 ARSUB A_RSUB Temperature coefficient of the RSUB
67 RSUB R_SUB Substrate resistance at Tnom
205
# Symbol Name Description
68 CJE CJE Zero-bias emitter-base depletion capacitance
69 VDE VDE Emitter-base diffusion voltage
70 PE PE Emitter-base grading coefficient
71 CBEO CBEO Emitter-base overlap capacitance
72 XCjE XCJE Sidewall fraction of the emitter-base depletion capacitance
73 VGB VGB Band-gap voltage of the base
74 CJC CJC Zero-bias collector-base depletion capacitance
75 VDC VDC Collector-base diffusion voltage
76 PC PC Collector-base grading coefficient
77 XP XP Constant part ofCjc
78 MC MC Coefficient for current modulation of CB depletion capacitance
79 CBCO CBCO Collector-base overlap capacitance
80 XCjC XCJC Fraction of CB depletion capacitance under the emitter
81 VGC VGC Band-gap voltage of the collector
82 CJS CJS Zero-bias collector-substrate depletion capacitance
83 VDS VDS collector-substrate diffusion voltage
84 PS PS collector-substrate grading coefficient
85 VGS VGS band-gap voltage of the substrate
86 CCS CCS Substrate capacitance
87 CDT CDT Substrate distributive capacitance
88 Mt MTAU Non-ideality factor of the emitter stored charge
89 tE TAUE Minimum transit time of stored emitter charge
90 tB TAUB Transit time of stored base charge
91 tepi TEPI Transit time of stored epilayer charge
92 tR TAUR Transit time of reverse extrinsic stored base charge
93 AtE AB_TAUE Temperature coefficient of the resistivity of the tE
94 dVgtE DVGTE Band-gap voltage difference of emitter stored charge
95 AtB ATAUB Temperature coefficient of the resistivity of the tB
96 AtEPI AEPI_TEPI Temperature coefficient of the resistivity of the tepi
97 IHC IHC Critical current for velocity saturation in the epilayer
98 SCRCV SCRCV Space charge resistance of the epilayer
99 AIHC AIHC Temperature coefficient of IHC
100 BIHC BIHC Temperature coefficient of IHC
101 VAVL VAVL Voltage determining curvature of avalanche current
102 WAVL WAVL Epilayer thickness used in weak-avalanche model
103 SFH SFH Current spreading factor of avalanche model when EXAVL=1
104 AXI AXI Smoothness parameter for the onset of quasi-saturation
206
# Symbol Name Description
105 AS AS Temperature coefficient of the ISS and IKS
106 ISS ISS Base-substrate saturation current
107 IKS IKS Base-substrate high injection knee current
108 IK IK Forward collector-emitter high injection knee current
109 ABIK ABIK Temperature coefficient of IK
110 IKR IKR Reverse collector-emitter high injection knee current
111 AIKR AIKR Temperature coefficient of IKR
112 XIKR XIKR Temperature coefficient of IKR
113 IKEX IKEX Collector-emitter high injection knee current for Iex
114 AIKEX AIKEX Temperature coefficient of IKEX
115 XIKEX XIKEX Temperature coefficient of IKEX
116 VER VER Reverse Early voltage
117 VEF VEF Forward Early voltage
118 DEg DEG Bandgap difference over the base
119 XREC XREC Pre-factor of the recombination part of Ib1
120 AQBO;VER AQBO_VER Temperature coefficient of theVER
121 AQBO;VEF AQBO_VEF Temperature coefficient of theVEF
122 AQBO;DEG AQBO_DEG Temperature coefficient of the DEg
123 IBF IBF Saturation current of the non-ideal forward base current
124 MLF MLF Non-ideality factor of the non-ideal forward base current
125 VGJ VGJ Band-gap voltage recombination emitter-base junction
126 IBR IBR Saturation current of the non-ideal reverse base current
127 VLR VLR Cross-over voltage of the non-ideal reverse base current
128 XIB1 XIBI Part of ideal base current that belongs to the sidewall
129 XEXT XEXT Part of currents and charges that belong to extrinsic region
# Symbol Name Description
130 AF AF Exponent of the Flicker-noise
131 KF KF Flicker-noise coefficient of the ideal base current
132 KFN KFN Flicker-noise coefficient of the non-ideal base current
133 KAVL KAVL Switch for white noise contribution due to avalanche
134 RTH RTH Thermal resistance
135 CTH CTH Thermal capacitance
136 ATH ATH Temperature coefficient of the thermal resistance
137 C1;RTH C1_RTH Temperature coefficient of RTH
138 C2;RTH C2_RTH Temperature coefficient of RTH
139 C3;RTH C3_RTH Temperature coefficient of RTH
207
A.2 Saturation current and ideality factor
Ideality factor and saturation current of forward IC-VBE
NF(T) = NF
"
1 T TnomT
nom
ANF TnomT
XNF#
; (A.3)
where nominal temperature ideality factor NF and temperature coefficient ANF, XNF are model
fitting parameters.
Ea;t = Ea;nom abTnom
2
(Tnom +b)2 +
abTTnom
(T +b)(Tnom +b);
IS(T) = IS;nom
T
Tnom
XIS
NF(T) exp
0
@ Ea;t
1 TTnom
NF(T)VT
1
A; (A.4)
where Ea;nom is the extrapolated 0 K Eg at nominal temperature and IS;nom is IS at nominal tem-
perature. XIS includes the temperature coefficient of mobility and density of states, a=4:45 10 4
V/K, b=686 K. Ea;nom, IS;nom, and XIS are model fitting parameters.
# Parameter Unit Values
1 NF - 1.004
2 ANF - 0.006115
3 XNF - 0.944
4 Ea;nom eV 1.089
5 IS;nom A 2.723E-18
6 XIS - 4.195
208
Ideality factor and saturation current of reverse IE-VBC
NR(T) = NR
"
1 T TnomT
nom
ANRTnomT
XNR#
; (A.5)
wherenominaltemperatureidealityfactorNR andtemperaturecoefficientANR,XNR aremodelfitting
parameters.
Ea;t = Ea;nom abTnom
2
(Tnom +b)2 +
abTTnom
(T +b)(Tnom +b);
ISR(T) = IS;nom
T
Tnom
XIS
NR(T) exp
0
@ Ea;t
1 TTnom
NR(T)VT
1
A; (A.6)
# Parameter Unit Values
1 NR - 1
2 ANR - 0.08383
3 XNR - 2
Ideality factor and saturation current of forward IB-VBE
NEI(T) = NEI
"
1 T TnomT
nom
ANE TnomT
XNE#
; (A.7)
where nominal temperature ideality factor NEI and temperature coefficient ANE, XNE are model
fitting parameters.
209
Ea;BEI;t = Ea;BEI;nom abTnom
2
(Tnom +b)2 +
abTTnom
(T +b)(Tnom +b);
IBEI(T) = IBEI;nom
T
Tnom
XIBEI
NEI(T) exp
0
@ Ea;BEI;t
1 TTnom
NEI(T)VT
1
A; (A.8)
where IBEI;nom, XIBEI and Ea;BEI;nom are model fitting parameters.
# Parameter Unit Values
1 NEI - 1.02
2 ANE - 0.09063
3 XNE - 2.986
4 Ea;BEI;nom eV 1.091
5 IBEI;nom A 2.498E-20
6 XIBEI - 5.323
Ideality factor and saturation current of reverse IB-VBC
NCI(T) = NCI
"
1 T TnomT
nom
ANCTnomT
XNC#
; (A.9)
where nominal temperature ideality factor NCI and temperature coefficient ANC, XNC are model
fitting parameters.
Ea;BCI;t = Ea;BCI;nom abTnom
2
(Tnom +b)2 +
abTTnom
(T +b)(Tnom +b);
IBCI(T) = IBCI;nom
T
Tnom
XIBCI
NCI(T) exp
0
@ Ea;BCI;t
1 TTnom
NCI(T)VT
1
A; (A.10)
where IBCI;nom, XIBCI and Ea;BCI;nom are model fitting parameters.
210
# Parameter Unit Values
1 NCI - 0.9997
2 ANC - 0.1194
3 XNC - 2.798
4 Ea;BCI;nom eV 0.9948
5 IBCI;nom A 1.343E-19
6 XIBCI - 8.38
A.3 Base tunneling current
IB;tun = IBT(T)exp
V
B2E1
VTUN
IBT(T) = IBTptN exp(KTN (Eg;TN Eg;T))
IBT is the nominal temperature saturation current andVTUN represents the temperature independent
slope of IB;tun-VBE. VTUN, IBT, KTN are three model parameters specific to TAT. Eg;TN and Eg;T are
the band gap at Tnom and T.
# Parameter Unit Values
1 VTUN V 0.018
2 IBT A 60.85E-31
3 KTN - 192.11
4 Eg;TN eV 1.02451923
A.4 Non-ideal base current
IBF(T) = IBFt(6 2MLF)N exp
V
GJ
MLFVDT
;
IBR(T) = IBRt2N exp
V
GC
2VDT
: (A.11)
211
# Parameter Unit Values
1 IBF A 1.002E-16
2 VGJ V 1.18
3 MLF - 2.2
4 IBR A 4.441E-31
5 VGC V 1.222
A.5 Early voltage
We introduce individual temperature parameter AQB0;VEF, AQB0;VER and AQB0;DEg for VEF, VER
and DEg respectively.
VEF(T) =VEFtAQB0;VEFN
"
(1 XP)
V
dC
VdCT
PC
+XP
# 1
;
VER(T) =VERtAQB0;VERN
V
dE
VdET
pE
;
DEg(T) = DEgtAQB0;DEgN : (A.12)
# Parameter Unit Values
1 VEF V 23.55
2 AQBO;VEF - -0.138
3 VER V 9.379
4 AQBO;VER - -0.138
5 AQB0;DEg - 0
A.6 pn junction diffusion voltage
Base-emitter junction diffusion voltageVdE
VdE;T = 3VT lntN +tNVdE +(1 tN)VGB; (A.13)
212
where VdE is the nominal temperature base-emitter junction diffusion voltage, VGB is the bandgap
voltage of the base. VdE andVGB are model fitting parameters.
# Parameter Unit Values
1 VdE V 1.028
2 VGB V 1.07
Base-collector junction diffusion voltageVdC
VdC;T = 3VT lntN +tNVdC +(1 tN)VGC; (A.14)
whereVdC is the nominal temperature base-collector junction diffusion voltage,VGC is the bandgap
voltage of the collector. VdC andVGC are model fitting parameters.
# Parameter Unit Values
1 VdC V 0.8697
2 VGC V 1.222
Collector-substrate junction diffusion voltageVdS
VdS;T = 3VT lntN +tNVdS +(1 tN)VGS; (A.15)
where VdS is the nominal temperature collector-substrate junction diffusion voltage, VGS is the
bandgap voltage of the substrate. VdS andVGS are model fitting parameters.
213
# Parameter Unit Values
1 VdS V 1.102
2 VGS V 1.102
A.7 Depletion capacitance
CjE;T =CjE
V
dE
VdET
PE
;
CjC;T =CjC
"
(1 XP)
V
dC
VdCT
PC
+XP
# 1
;XP;T = XP CJCC
JCT
;
CjST =CjS
V
dS
VdST
PS
; (A.16)
# Parameter Unit Values
1 CjE fF 12.42
2 PE 0.2562
3 CjC fF 5.897
4 PC 0.6789
5 XP 0.515
6 CjS fF 8.8
7 PS 0.160
A.8 Parasitic resistances
NC = 2:8 1019
T
300
1:5
;
NV = 3:14 1019
T
300
1:5
: (A.17)
214
Emitter resistance RE
G = g 1D NCN
dop;RE
exp
Edop;REkT
;gD = 2;
b = 1 bRE1+ T
T0
aRE ;;
IR(T) = G+(1 b)+
q
[G (1 b)]2 +4G
2 ;
RE(T) = RE
T
Tnom
ARE 1
IR(T); (A.18)
where the impurity concentration Ndop;RE, the impurity activation energy Edop;RE, the fraction
of bound impurity states coefficients bRE and aRE, the fictitious nominal temperature complete
ionization resistance RE and the mobility temperature coefficient ARE are model fitting parameters.
# Parameter Unit Values
1 Ndop;RE cm 3 4.177E18
2 Edop;RE mV 5.366
3 aRE - -0.4506
4 bRE - 1
5 ARE - -0.2409
6 RE W 11.84
215
Constant part of the base resistance RBC
G = g 1A NVN
dop;RBC
exp
Edop;RBCkT
;gA = 4;
b = 1 bRBC1+ T
T0
aRBC ;;
IR(T) = G+(1 b)+
q
[G (1 b)]2 +4G
2 ;
RBC(T) = RBC
T
Tnom
ARBC 1
IR(T); (A.19)
where the impurity concentration Ndop;RBC, the impurity activation energy Edop;RBC, the fraction
of bound impurity states coefficients bRBC and aRBC, the fictitious nominal temperature complete
ionizationresistanceRBC andthemobilitytemperaturecoefficientARBC aremodelfittingparameters.
# Parameter Unit Values
1 Ndop;RBC cm 3 1.564E18
2 Edop;RBC mV 6.641
3 aRBC - -0.7477
4 bRBC - 0.9897
5 ARBC - 1.108
6 RBC W 28.02
216
Constant part of the collector resistance RCC
G = g 1D NCN
dop;RCC
exp
Edop;RCCkT
;gD = 2;
b = 1 bRCC1+ T
T0
aRCC ;;
IR(T) = G+(1 b)+
q
[G (1 b)]2 +4G
2 ;
RCC(T) = RCC
T
Tnom
ARCC 1
IR(T); (A.20)
where the impurity concentration Ndop;RCC, the impurity activation energy Edop;RCC, the fraction
of bound impurity states coefficients bRCC and aRCC, the fictitious nominal temperature complete
ionizationresistanceRCC andthemobilitytemperaturecoefficientARCC aremodelfittingparameters.
# Parameter Unit Values
1 Ndop;RCC cm 3 7.438E18
2 Edop;RCC mV 41.54
3 aRCC - -0.4506
4 bRCC - 1
5 ARCC - 0.6656
6 RCC W 34.89
217
Resistance of the un-modulated epilayer RCV
G = g 1D NCN
dop;RCV
exp
Edop;RCVkT
;gD = 2;
b = 1 bRCV1+ T
T0
aRCV ;;
IR(T) = G+(1 b)+
q
[G (1 b)]2 +4G
2 ;
RCV(T) = RCV
T
Tnom
ARCV 1
IR(T); (A.21)
where the impurity concentration Ndop;RCV, the impurity activation energy Edop;RCV, the fraction of
boundimpuritystatescoefficientsbRCV andaRCV, thefictitiousnominaltemperaturecompleteion-
ization resistance RCV and the mobility temperature coefficient ARCV are model fitting parameters.
# Parameter Unit Values
1 Ndop;RCV cm 3 8.118E16
2 Edop;RCV mV 22.83
3 aRCV - -4.142
4 bRCV - 1
5 ARCV - -1.34
6 RCV W 82.5
218
Variable part of the base resistance at zero-bias RBV
G = g 1A NVN
dop;RBV
exp
Edop;RBVkT
;gA = 4;
b = 1 bRBV1+ T
T0
aRBV ;;
IR(T) = G+(1 b)+
q
[G (1 b)]2 +4G
2 ;
RBV(T) = RBV
T
Tnom
ARBV 1
IR(T); (A.22)
where the impurity concentration Ndop;RBV, the impurity activation energy Edop;RBV, the fraction
of bound impurity states coefficients bRBV and aRBV, the fictitious nominal temperature complete
ionizationresistanceRBV andthemobilitytemperaturecoefficientARBV aremodelfittingparameters.
# Parameter Unit Values
1 Ndop;RBV cm 3 5.314E16
2 Edop;RBV mV 54.05
3 aRBV - -1.36
4 bRBV - 0.2958
5 ARBV - -0.1434
6 RBV W 288.6
219
Substrate resistance RSUB
G = g 1A NVN
dop;RSUB
exp
Edop;RSUBkT
;gA = 4;
b = 1 bRSUB1+ T
T0
aRSUB ;;
IR(T) = G+(1 b)+
q
[G (1 b)]2 +4G
2 ;
RSUB(T) = RSUB
T
Tnom
ARSUB 1
IR(T); (A.23)
where the impurity concentration Ndop;RSUB, the impurity activation energy Edop;RSUB, the fraction
of bound impurity states coefficients bRSUB and aRSUB, the fictitious nominal temperature com-
plete ionization resistance RSUB and the mobility temperature coefficient ARSUB are model fitting
parameters.
# Parameter Unit Values
1 Ndop;RSUB cm 3 5.011E14
2 Edop;RSUB mV 59.4
3 aRSUB - -4.309
4 bRSUB - 0
5 ARSUB - 1.676
6 RSUB W 1500
A.9 IHC and SCRCV
IHC;T = IHCtN AIHC(T Tnom) BIHC;
SCRCV;T = SCRCVtNAIHC(T Tnom)+BIHC; (A.24)
220
# Parameter Unit Values
1 AIHC - 0.0008409
2 BIHC - 0.6874
A.10 Knee current
We introduce individual temperature parameters for IK, IKR and IKEX.
IK(T) = IKt1 ABIKN ;
IKR(T) = IKRtAIKRN exp
X
IKR(1 tN)
VT
;
IKEX(T) = IKEXtAIKEXN exp
X
IKEX(1 tN)
VT
;
(A.25)
# Parameter Unit Values
1 IK A 0.0077
2 ABIK - 0.3254
3 IKR A 0.015
4 AIKR - 0.02
5 XIKR - 0.09
6 IKEX A 0.01058
7 AIKEX - 0.07
8 XIKEX - 0.005
221
A.11 Transit times
WeintroduceindividualtemperatureparameterAtE,AtB andAtepi fortE,tB andtepi respectively.
tE(T) = tEt(AtE 2)N exp( dVgtEV
DT
);
tB(T) = tBt(AtB 1)N ;
tepi(T) = tepit(Atepi 1)N ;
tR(T) = tRtB(T)+tepi(T)t
B +tepi
: (A.26)
# Parameter Unit Values
1 tE s 0
2 AtE - 1
3 dVgtE - 0.1099
4 tB s 1.179E-12
5 AtB - 1.9
6 tepi s 9.74E-11
7 Atepi - -1.34
8 tR - 1.3E-9
A.12 Thermal resistance
RTH;Tamb =C1;RTHTamb3 +C2;RTHTamb2 +C3;RTHTamb +RTH: (A.27)
222
# Parameter Unit Values
1 RTH C/W 4235
2 C1;RTH - -0.0001154
3 C2;RTH - 0.09645
4 C3;RTH s -17.64
A.13 Substrate Currents
ISS(T) = ISSt(4 AS)N exp( VGSV
DT
);
IKS(T) = IKSt(1 AS)N (IS(T)I
S
)( ISSI
SS(T)
): (A.28)
# Parameter Unit Values
1 ISS A 3.661E-21
2 IKS A 0.5
3 AS - 0.5
223