HIGH FREQUENCY NOISE MODELING AND MICROSCOPIC NOISE SIMULATION
FOR SIGE HBT AND RF CMOS
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee.
This dissertation does not include proprietary or classified information.
Yan Cui
Certificate of Approval:
Stuart M. Wentworth
Associate Professor
Electrical and Computer Engineering
Guofu Niu, Chair
Professor
Electrical and Computer Engineering
Foster Dai
Associate Professor
Electrical and Computer Engineering
Joe F. Pittman
Interim Dean
Graduate School
HIGH FREQUENCY NOISE MODELING AND MICROSCOPIC NOISE SIMULATION
FOR SIGE HBT AND RF CMOS
Yan Cui
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 15, 2005
HIGH FREQUENCY NOISE MODELING AND MICROSCOPIC NOISE SIMULATION
FOR SIGE HBT AND RF CMOS
Yan Cui
Permission is granted to Auburn University to make copies of this thesis at its discretion,
upon the request of individuals or institutions and at their expense.
The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
VITA
Yan Cui, daughter of Fengde Cui and Shuxian Song, spouse of Zhiming Feng, was born
on 4 September, 1975, in JiaMuSi, Heilongjiang Province, P. R. China. She received her BS
degree from Jilin University in 1995, majoring in Electronics Engineering. She received her MS
degree from Jilin University in 1998, majoring in Electronics Engineering. In Spring 2002, she
was accepted into the Electrical and Computer Engineering department of Auburn University,
Auburn, Alabama, where she has pursued her Ph.D degree.
iv
DISSERTATION ABSTRACT
HIGH FREQUENCY NOISE MODELING AND MICROSCOPIC NOISE SIMULATION
FOR SIGE HBT AND RF CMOS
Yan Cui
Doctor of Philosophy, December 15, 2005
(M.S., Jilin University, 1998)
(B.S., Jilin University, 1995)
351 Typed Pages
Directed by Guofu Niu
RF bipolar and CMOS are both important in RFIC applications. Modeling of noise pro-
vides critical information in the design of RF circuits. Unfortunately, available compact models
for both RF bipolar and CMOS, are typically not applicable for the GHz frequency range. In this
dissertation, a new technique of simulating the spatial distribution of microscopic noise contri-
bution to the input noise current, voltage, and their correlation is presented, and applied to both
RF SiGe HBT transistor and RF MOSFET transistor.
For RF SiGe HBT transistor, bipolar transistor noise modeling and noise physics are exam-
ined using microscopic noise simulation. Transistor terminal current and voltage noises result-
ing from velocity fluctuations of electrons and holes in the base, emitter, collector, and substrate
are simulated using the new technique proposed, and compared with modeling results. Major
physics noise sources in bipolar transistor are qualitatively identified. The relevant importance
as well as model-simulation discrepancy is analyzed for each physical noise source.
Moreover, the RF noise physics and SiGe profile optimization for low noise are explored
using microscopic noise simulation. A higher Ge gradient in a noise critical region near the EB
v
junction, together with an unconventional Ge retrograding in the base to keep total Ge content
below stability, when optimized, can lead to significant noise improvement without sacrificing
peak cuto frequency and without any significant high injection cuto frequency rollo degra-
dation.
For RF MOSFET transistor, RF noise of 50 nm Le CMOS is simulated using hydrody-
namic noise simulation. Intrinsic noise sources for the Y- and H- noise representations are ex-
amined and models of intrinsic noise sources are proposed. The relations between the Y- and
H- noise representations for MOSFETs are examined, and the importance of correlation for both
representations is quantified. The H- noise representation has the inherent advantage of a more
negligible correlation, which makes circuit design and simulation easier.
The extrinsic gate resistance is important as well as the intrinsic drain noise current for
noise modeling of scaled MOSFET. Accurately extract the gate resistance becomes an important
issue. The frequency and bias dependence of the e ective gate resistance are explained by
considering the e ect of gate-to-body capacitance, gate to source/drain overlap capacitances,
fringing capacitances, and Non-Quasi-Static (NQS) e ect. A new method of separating the
physical gate resistance and the NQS channel resistance is proposed.
Finally, drain current excess noise factors in CMOS transistors are examined as a function
of channel length and bias. The technology scaling are discussed for di erent processes. Using
standard linear noisy two-port theory, a simple derivation of noise parameters is presented. The
results are compared with the well known Fukui?s empirical FET noise equations. Experimental
data are used to evaluate the simple model equations. New figures-of-merit for minimum noise
figure is proposed.
vi
ACKNOWLEDGMENTS
I would like to express my gratitude to my supervisor, Dr. Guofu Niu. Without him, this
dissertation would not have been possible. I thank him for his patience and encouragement that
carried me on through di cult 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 this thesis. I would like to thank the other members of my
committee, Dr. Foster Dai, Dr. Stuart M. Wentworth, and Dr. John R. Williams for the assistance
they provided.
Several people deserve special recognition for their contributions to this work. I would
like to thank Yun Shi and Muthubalan Varadharajaperumal for their help with the DESSIS input
deck, and Qingqing Liang, Ying Li and Xiaoyun Wei for their help with device measurement. I
would like to thank Dr Susan Sweeney of IBM Microelectronics Communications R&D Center
for her great help with noise measurement data. I would like to thank Dr. Stewart S. Taylor of
Intel Corporation for helpful discussions. I would also like to thank Dr. J.D. Cressler of Georgia
Institute of Technology for his contributions.
Finally, I am forever indebted to my parents for the support they provided me through my
entire life and in particular, I must acknowledge my husband and best friend, Zhiming Feng,
without whose love, and encouragement, I would not have finished this dissertation.
In conclusion, I recognize that this research would not have been possible without the fi-
nancial assistance of the National Science Foundation under ECS-0119623 and ECS-0112923,
the Semiconductor Research Corporation under SRC #2001-NJ-937, and Intel Corporation.
vii
Style manual or journal used IEEE Transactions on Electron Devices (together with the
style known as ?aums?). Bibliography follows van Leunen?s A Handbook for Scholars.
Computer software used The document preparation package TEX (specifically LATEX)
together with the departmental style-file aums.sty. The plots were generated using VossPlot R?,
MATLAB R?, TecPlot R?and Microsoft Visio R?.
viii
TABLE OF CONTENTS
LIST OF FIGURES xiii
LIST OF TABLES xxiii
1 INTRODUCTION 1
1.1 RF Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Noise Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Minimum Noise Figure NFmin . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Noise Resistance Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Optimum Source Admittance Yopt . . . . . . . . . . . . . . . . . . . . 6
1.3 RF Bipolar Transistor Compact Noise Modeling . . . . . . . . . . . . . . . . . 6
1.3.1 Lumped Base Resistance . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 SPICE Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.3 van Vliet Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.4 Time-delay and Phase-delay Model . . . . . . . . . . . . . . . . . . . 12
1.4 RF MOSFET Transistor Compact Noise Modeling . . . . . . . . . . . . . . . 17
1.4.1 Gate and Drain Noise currents Modeling . . . . . . . . . . . . . . . . 17
1.4.2 Gate Noise Voltage and Drain Noise Current Modeling . . . . . . . . . 29
1.4.3 Role of Gate Resistance . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.5 Dissertation Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 NOISE NETWORK ANALYSIS AND DE-EMBEDDING 36
2.1 Noise Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.1.1 Chain Noise Representation (ABCD- Noise Representation) . . . . . . 36
2.1.2 Y- Noise Representation . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.1.3 Z- Noise Representation . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.1.4 H- Noise Representation . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2 Transformation to Other Noise Representations . . . . . . . . . . . . . . . . . 49
2.3 Adding Noisy Passive Components to a Noisy Two-Port Network . . . . . . . 50
2.4 Open/Short De-embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.4.1 Open De-embedding of Y-parameters and Noise Parameters . . . . . . 57
2.4.2 Short De-embedding of Y-parameters and Noise Parameters . . . . . . 58
2.4.3 Problems Encountered in MATLAB Programming for Open-Short De-
embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5 Transistor Internal Noise De-embedding . . . . . . . . . . . . . . . . . . . . . 63
ix
2.5.1 MOSFET Transistor ig and id Noise De-embedding . . . . . . . . . . . 63
2.5.2 SiGe HBT Transistor ib and ic Noise De-embedding . . . . . . . . . . 69
2.6 Importance of Terminal Series Resistances to Noise parameters . . . . . . . . 76
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3 MICROSCOPIC NOISE CONTRIBUTIONS 81
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.2 Microscopic Noise Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3 New Technique: Microscopic Noise Contribution of Chain Noise Representation
Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.4 Spatial Distribution of Microscopic Noise Contributions in RF SiGe HBT Tran-
sistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4.1 Input Noise Voltage Sva;v?a . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4.2 Input Noise Current Sia;i?a . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.4.3 Input Noise Voltage and Current Correlation Sia;v?a . . . . . . . . . . . 90
3.5 Spatial Distribution of Microscopic Noise Contributions in RF MOSFET Transistor 95
3.5.1 Gate Noise Current Sig;i?g . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.5.2 Drain Noise Current Sid;i?d . . . . . . . . . . . . . . . . . . . . . . . . 96
3.5.3 Drain and Gate Noise Current Correlation Sig;i?d . . . . . . . . . . . . . 101
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4 BIPOLAR NOISE MODELING 102
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2 Technical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.2.1 Microscopic Input Noise Concentration . . . . . . . . . . . . . . . . . 104
4.2.2 Macroscopic Input Noise . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.2.3 Microscopic and Macroscopic Connections . . . . . . . . . . . . . . . 107
4.3 Chain Representation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.3.1 Sva;v?a , Sia;i?a and Sia;v?a . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.3.2 NFmin, Yopt and Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.4 Intrinsic Base and Collector Noise . . . . . . . . . . . . . . . . . . . . . . . . 114
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5 SIGE PROFILE OPTIMIZATION FOR LOW NOISE 123
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2 SiGe Profile Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.2.1 Distributive Transit Time Analysis . . . . . . . . . . . . . . . . . . . . 125
5.2.2 Input Noise Voltage and Current . . . . . . . . . . . . . . . . . . . . . 127
5.3 New Approach: Regional Electron and Hole Contributions . . . . . . . . . . . 128
5.3.1 Noise Critical Region and Ge Profile Impact . . . . . . . . . . . . . . 130
5.4 Optimization Under Constant Stability . . . . . . . . . . . . . . . . . . . . . . 133
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
x
6 MODELING OF INTRINSIC NOISE IN CMOS 140
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.2 Technical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.3.1 DC I  V Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.3.2 Noise Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.4 Intrinsic Noise Sources and Modeling . . . . . . . . . . . . . . . . . . . . . . 146
6.4.1 Y-representation Noise Sources . . . . . . . . . . . . . . . . . . . . . 146
6.4.2 H-representation Noise Sources . . . . . . . . . . . . . . . . . . . . . 150
6.5 Relations Between Y- and H- Noise Representations in MOSFETs . . . . . . . 156
6.5.1 Relations Between Y- and H- Noise Representation Coe cients . . . . 157
6.5.2 Noise Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.6 Importance of Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.7 Extraction and Modeling of H-Representation RF Noise Sources in CMOS . . 168
6.7.1 Experimental Extraction . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.7.2 Noise Source Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7 EFFECTIVE GATE RESISTANCE MODELING 182
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.2 h11 model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.3 Parameter Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.5 Length and Width E ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
8 EXCESS NOISE FACTORS AND NOISE PARAMETER EQUATIONS FOR RF CMOS 205
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
8.2 Excess Noise Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
8.3 Technology Discussion of Excess Noise Factor . . . . . . . . . . . . . . . . . 210
8.4 Vds Dependence of Excess Noise Factor . . . . . . . . . . . . . . . . . . . . . 212
8.4.1 0.24 ?m device, W = 4 ?m, Nf = 128. . . . . . . . . . . . . . . . . . 212
8.4.2 0.12 ?m Device, W = 5 ?m, Nf = 30. . . . . . . . . . . . . . . . . . 214
8.4.3 Simulation Results on 50 nm Le CMOS . . . . . . . . . . . . . . . . 218
8.5 Noise Parameter Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8.6 Comparison with Fukui?s Equations . . . . . . . . . . . . . . . . . . . . . . . 225
8.7 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
8.8 Figure-of-Merit for NFmin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
8.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
9 CONCLUSIONS 237
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BIBLIOGRAPHY 242
APPENDICES 247
A MATLAB PROGRAMMING FOR OPEN-SHORT DEEMBEDDING IN CHAPTER 2 249
B DESSIS INPUT DECK AND MATLAB PROGRAMMING FOR SIGE HBT NOISE SIM-
ULATION 255
B.1 5HP SiGe HBT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
B.1.1 Mesh files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
B.1.2 Noise Simulation CMD file . . . . . . . . . . . . . . . . . . . . . . . 263
B.1.3 Tecplot MCR file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
B.2 8HP SiGe HBT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
B.2.1 Mesh files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
B.2.2 Noise Simulation CMD file . . . . . . . . . . . . . . . . . . . . . . . 289
B.3 MATLAB Programming for Simulation Results . . . . . . . . . . . . . . . . . 300
B.3.1 Main file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
B.3.2 Z_from_Y.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
B.3.3 rb_from_h11.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
B.3.4 circle.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
B.3.5 myCostFunc.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
B.3.6 c_from_z_to_a.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
B.3.7 c_from_a_to_y.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
B.3.8 nf_from_ca.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
B.3.9 y_from_z.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
B.3.10 a_from_y.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
C DESSIS INPUT DECK AND MATLAB PROGRAMMING FOR 50 NM Le MOSFET
NOISE SIMULATION 308
C.1 Mesh files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
C.1.1 BND file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
C.1.2 CMD file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
C.2 Noise Simulation CMD file . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
C.3 MATLAB Programming for Simulation Results . . . . . . . . . . . . . . . . . 326
C.3.1 Main file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
C.3.2 c_from_y_to_h.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
xii
LIST OF FIGURES
1.1 Illustration of definition of noise figure for a noisy two-port. . . . . . . . . . . 5
1.2 RF Bipolar transistor noise modeling. . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Equivalent circuit proposed for the intrinsic transistor together with the resis-
tance of the pinched base [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 SPICE model for RF bipolar transistor. . . . . . . . . . . . . . . . . . . . . . . 9
1.5 The small-signal equivalent circuit for intrinsic bipolar device. . . . . . . . . . 11
1.6 Time-delay noise model in [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.7 Phase-delay noise model in [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.8 Thermal noise in MOSFETs [4]. . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.9 MOSFET noise model using gate noise current, drain noise currents, and their
correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.10 Illustration of drain noise current derivation. . . . . . . . . . . . . . . . . . . . 21
1.11 Schematic for BSIM4 channel thermal noise modeling [5]. . . . . . . . . . . . 26
1.12 Comparison of Sid;i?d for the data and BSIM holistic model for 0.18 ?m device.
W = 10 ?m, Nf = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.13 Comparison of noise parameters for the data and BSIM holistic model for 0.18
?m device. W = 10 ?m, Nf = 8. . . . . . . . . . . . . . . . . . . . . . . . . 28
1.14 MOSFET noise model: Pospieszalski model . . . . . . . . . . . . . . . . . . . 29
1.15 Role of gate resistance noise to gate noise current, drain noise current, and their
correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.16 Schematic layout of a single gate finger, showing the meaning of W , Wext, and
L in (1.105) [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
xiii
2.1 The chain noise representation of a linear noisy two-port network. . . . . . . . 37
2.2 Noisy linear two-port network. . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 The Y- noise representation of a linear noisy two-port network. . . . . . . . . 42
2.4 The Z- noise representation of a linear noisy two-port network. . . . . . . . . 45
2.5 The H- noise representation of a linear noisy two-port network. . . . . . . . . 47
2.6 Adding noisy passive components parallel to a linear noisy two-port network. . 51
2.7 Adding noisy passive components in series with a linear noisy two-port network. 54
2.8 Equivalent circuit diagram used for open-short de-embedding method, including
both the parallel parasitics Yp1, Yp2, Yp3, and the series parasitics ZL1, ZL2 and
ZL3 surrounding the transistor [7]. . . . . . . . . . . . . . . . . . . . . . . . 56
2.9 NFmin v.s. frequency. IDS = 148 ?A=?m. VDS = 1 V. . . . . . . . . . . . . . . 60
2.10 NFmin v.s. IDS normalized by size of device. f = 10 GHz. VDS = 1 V. . . . . . 61
2.11 Rn v.s. frequency. IDS = 148 ?A=?m. VDS = 1 V. . . . . . . . . . . . . . . . 62
2.12 Rn v.s. IDS normalized by size of device. f = 10 GHz. VDS = 1 V. . . . . . . . 63
2.13 Gopt v.s. frequency. IDS = 148 ?A=?m. VDS = 1 V. . . . . . . . . . . . . . . . 64
2.14 Gopt v.s. IDS normalized by size of device. f = 10 GHz. VDS = 1 V. . . . . . . 65
2.15 Bopt v.s. frequency. IDS = 148 ?A=?m. VDS = 1 V. . . . . . . . . . . . . . . . 66
2.16 Bopt v.s. IDS normalized by size of device. f = 10 GHz. VDS = 1 V. . . . . . . 67
2.17 The small signal equivalent circuit model used with Y-representation noise
sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.18 Y- noise representation input noise current for the whole and the intrinsic MOS-
FET transistor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.19 Y- noise representation output noise current for the whole and the intrinsic MOS-
FET transistor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
xiv
2.20 Y- noise representation correlation for the whole and the intrinsic MOSFET tran-
sistor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.21 Y- noise representation input and output noise currents for the whole and the
intrinsic SiGe HBT transistor. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.22 Y- noise representation correlation for the whole and the intrinsic SiGe HBT
transistor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.23 NFmin vs IDS with and without Rg, Rs and Rd at 5 GHz. . . . . . . . . . . . . 78
2.24 Rn vs IDSwith and without Rg, Rs and Rd at 5 GHz. . . . . . . . . . . . . . . 78
2.25 Gopt vs IDS with and without Rg, Rs and Rd at 5 GHz. . . . . . . . . . . . . . 79
2.26 Bopt vs IDS with and without Rg, Rs and Rd at 5 GHz. . . . . . . . . . . . . . 79
3.1 Impedance field method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2 2D distribution of the total noise concentration CSva;v?a at 2 GHz. JC=0.1 mA=?m2. 87
3.3 2D distribution of electron noise concentration CSva;v?a at 2 GHz. JC=0.1 mA=?m2. 88
3.4 2D distribution of hole noise concentration CSva;v?a at 2 GHz. JC=0.1 mA=?m2. 89
3.5 2D distribution of the total noise concentration CSva;v?a at 2 GHz. JC=0.5 mA=?m2. 90
3.6 2D distribution of noise concentration CSia;i?a at 2 GHz. JC=0.1 mA=?m2. . . . 91
3.7 2D distribution of electron contribution to noise concentration CSia;i?a at 2 GHz.
JC=0.1 mA=?m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.8 2D distribution of hole contribution to noise concentration CSia;i?a at 2 GHz.
JC=0.1 mA=?m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.9 2D distribution of the total noise concentration <(CSia;v?a ) at 2 GHz. JC=0.1
mA=?m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.10 2D distribution of the total noise concentration =(CSia;v?a ) at 2 GHz. JC=0.1
mA=?m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.11 2-D gate noise current concentration CSig;i?g at 5 GHz. Vds = 1 V. Vgs = 0.5 V. . 97
xv
3.12 2-D gate noise current concentration CSig;i?g at 5 GHz. Vds = 1 V. Vgs = 1 V. . . 97
3.13 2-D drain noise current concentration CSid;i?
d
at 5 GHz. Vds = 1 V. Vgs = 0.5 V. 98
3.14 2-D drain noise current concentration CSid;i?
d
at 5 GHz. Vds = 1 V. Vgs = 1 V. . 98
3.15 2-D real part of noise current correlation concentration <(CSig;i?
d
) at 5 GHz. Vds
= 1 V. Vgs = 0.5 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.16 2-D real part of noise current correlation concentration <(CSig;i?
d
) at 5 GHz. Vds
= 1 V. Vgs = 1 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.17 2-D imaginary part of noise current correlation concentration =(CSig;i?
d
) at 5
GHz. Vds = 1 V. Vgs = 0.5 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.18 2-D imaginary part of noise current correlation concentration =(CSig;i?
d
) at 5
GHz. Vds = 1 V. Vgs = 1 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.1 Chain noise parameter: measured vs compact model. JC=0.01 mA=?m2. . . . 103
4.2 Chain noise parameter: measured v.s. compact model, JC=0.63 mA=?m2. . . 104
4.3 Chain noise parameter: simulation v.s. compact model, JC=0.65 mA=?m2.. . 105
4.4 Sva;v?a , Sev
a;v?a
, and Shv
a;v?a
vs frequency at (a) JC=0.01 mA=?m2. (b) JC=0.65
mA/?m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.5 2D distribution of CeS
va;v?a
at 2 GHz, JC=0.65 mA/?m2. . . . . . . . . . . . . . 110
4.6 2D distribution of ChS
va;v?a
at 2 GHz, JC=0.65 mA/?m2. . . . . . . . . . . . . . 111
4.7 Regional contributions of Sev
a;v?a
(a) and Shv
a;v?a
(b) at JC=0.65 mA/?m2. . . . . 112
4.8 Sia;i?a, Sei
a;i?a
, and Shi
a;i?a
vs frequency at (a) JC=0.01 mA=?m2. (b) JC=0.65
mA/?m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.9 Regional contribution of Sei
a;i?a
(a) and Shi
a;i?a
(b) at JC=0.65 mA/?m2. . . . . . 114
4.10 (a) Rn, and (b) NFmin vs frequency. JC=0.65 mA/?m2. . . . . . . . . . . . . 115
4.11 (a) Gopt, and (b) Bopt vs frequency. JC=0.65 mA/?m2. . . . . . . . . . . . . . 116
xvi
4.12 Regional contributions of internal input noise current Sib;i?b (a) JC=0.01
mA/?m2. (b) JC=0.65 mA/?m2. . . . . . . . . . . . . . . . . . . . . . . . . 117
4.13 Regional contributions of internal output noise current Sic;i?c (a) JC=0.01
mA/?m2. (b) JC=0.65 mA/?m2. . . . . . . . . . . . . . . . . . . . . . . . . 119
4.14 Output noise current of whole transistor Si2;i?2 at 2 GHz. . . . . . . . . . . . . 120
4.15 Regional contributions of internal noise current correlation Sic;i?b. JC=0.01
mA/?m2. (a) <Sic;i?b. (b) =Sic;i?b. . . . . . . . . . . . . . . . . . . . . . . . . 121
4.16 Regional contributions of internal noise current correlation Sic;i?b. JC=0.65
mA/?m2. (a) <Sic;i?b. (b) =Sic;i?b. . . . . . . . . . . . . . . . . . . . . . . . . 122
5.1 (a) Ge profile I and II. (b) Gummel curves for profile I and II. . . . . . . . . . 125
5.2 fT vs JC for Ge profile I and II. . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.3 Noise parameters vs JC for profile I and II at 40 GHz. . . . . . . . . . . . . . 127
5.4 1-D center cut of ?di for profile I and II. JC = 2 mA=?m2. . . . . . . . . . . 128
5.5 Sia;i?a, Sva;v?a and their correlation vs JC. f = 40 GHz. . . . . . . . . . . . . . . 129
5.6 (Fmin  1) contributions. f = 40 GHz. . . . . . . . . . . . . . . . . . . . . . . 130
5.7 Comparison of Regional contributions of Sva;v?a . f=40 GHz. . . . . . . . . . . 131
5.8 Comparison of Regional contributions of Sia;i?a. f=40 GHz. . . . . . . . . . . 132
5.9 1-D center cut of CnS
ia;i?a
and CpS
ia;i?a
for Ge profile I and II. JC = 2 mA=?m2. f =
40 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.10 1-D center cut of CnSi and jGn;iaj2 for profile I and II. JC = 2 mA=?m2. f = 40
GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.11 (a) Constant stability Ge profiles: profile I, II and III. (b) Gummel curves for
profile I, II and III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.12 fT vs JC for profile I, II and III. . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.13 1-D center cut of ?di for profile I, II and III. JC = 2 mA=?m2. . . . . . . . . 137
xvii
5.14 1-D center cut of CnS
ia;i?a
and CpS
ia;i?a
for profile I, II and III. f = 40 GHz. JC = 2
mA=?m2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.15 NFmin vs JC at 10 and 60 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.16 NFmin vs frequency at JC = 2 and 10 mA=?m2. . . . . . . . . . . . . . . . . . 138
6.1 (a) IDS, and (b) gm vs Vgs at Vds = 1 V. . . . . . . . . . . . . . . . . . . . . . 143
6.2 Output Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.3 Simulation vs data reported in [8]: NFmin at 5 GHz vs IDS. . . . . . . . . . . . 144
6.4 fT vs IDS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.5 Gopt vs IDS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.6 Bopt vs IDS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.7 Rn vs IDS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.8 Sid;i?d=[4kTgm] vs frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.9 Sig;i?g=[4kT<(Y11)] vs frequency. . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.10 Normalized correlation c vs frequency. . . . . . . . . . . . . . . . . . . . . . . 149
6.11 NFmin and Rn vs frequency, with and without correlation. . . . . . . . . . . . . 150
6.12 Yopt vs frequency, with and without correlation. . . . . . . . . . . . . . . . . . 151
6.13 Svh;v?h vs frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.14 Svh;v?h=[4kT<(h11)] vs frequency. . . . . . . . . . . . . . . . . . . . . . . . . 153
6.15 Svh;v?h=[4kT<(h11)] vs IDS. . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.16 Sih;i?h vs frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.17 Correlation of H-representation noise sources vs frequency. . . . . . . . . . . . 154
6.18 NFmin, Rn and Yopt with and without correlation between vh and ih. . . . . . . . 155
6.19 Small signal equivalent circuit for intrinsic MOSFET. . . . . . . . . . . . . . 158
xviii
6.20 Importance of H- and Y- noise representation correlations:  NFmin vs frequency
for 50nm NMOS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.21 Importance of H- and Y- noise representation correlations:  Rn=Rn vs IDS for
50nm NMOS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.22 Importance of H- and Y- noise representation correlations:  Gopt=Gopt vs IDS
for 50nm NMOS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.23 Importance of H- and Y- noise representation correlations:  Bopt=Bopt vs IDS
for 50nm NMOS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.24 The small signal equivalent circuit model used with H-representation noise
sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.25 Data-model comparison of Y-parameter vs frequency at VGS = 1.2 V. VDS = 1.2
V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.26 Data-model comparison of Y-parameter at f = 10 GHz. VDS = 1.2 V. . . . . . 173
6.27 SIIv
h;v?h
=(4kTRgs) and SIIi
h;i?h
=(4kTgm) (symbols) vs frequency. VGS = 1.2 V.
VDS = 1:2 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.28 CIIv
h;i?h
vs frequency, VGS = 1.2 V. VDS = 1.2 V. . . . . . . . . . . . . . . . . . 175
6.29 (a) NFmin and Rn vs frequency; (b) real and imaginary parts of Yopt vs frequency.
SIIv
h;i?h
= 0. VGS = 1.2 V. ? = 0.6, ?ih = 1.75. VDS = 1.2 V. . . . . . . . . . . . 176
6.30 ? = SIIv
h;v?h
=(4kTRgs) and ?ih = SIIi
h;i?h
=(4kTgm) vs VGS. Symbols are extracted
values, lines are model results. . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.31 (a) NFmin and Rn vs IDS; (b): real and imaginary parts of Yopt vs IDS. f = 10
GHz. VDS = 1.2 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.32 (a) NFmin and Rn vs IDS; (b) real and imaginary parts of Yopt vs IDS. f = 10
GHz. VDS = 0.2 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.1 MOSFET small signal equivalent circuit model. . . . . . . . . . . . . . . . . 183
7.2 <(h11) vs frequency for 0.18 ?m CMOS device, W = 10?m, Nf = 8. Vds=1 V. 184
7.3 <(h11) vs IDS for 0.18 ?m CMOS device, W = 10?m, Nf = 8. Vds=1 V. . . . 185
xix
7.4 CMOS small signal model in [9]. . . . . . . . . . . . . . . . . . . . . . . . . 186
7.5 Data-model comparison of <(h11) vs IDS for 0.18?m CMOS device, W =
10?m, Nf = 8, using the small signal model in Fig. 7.4. Vds=1 V. . . . . . . . 187
7.6 A more complete MOSFET small signal model. . . . . . . . . . . . . . . . . 188
7.7 h11 derivation illustration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.8 Frequency dependence of (<(h11)  Rg) in logarithm scale. . . . . . . . . . . 191
7.9 Frequency dependence of (<(h11)  Rg) in linear scale. . . . . . . . . . . . . 192
7.10 Influence of !1 on the frequency dependence of (<(h11)  Rg). . . . . . . . . 193
7.11 Extraction of p1, p2, q1 and q2 at Vgs = 0.5 V for 0.18 ?m device, W=10 ?m,
Nf = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
7.12 Extracted capacitances Cp, Cperi;s, Cperi;d, and Cgs, and extracted Rnqs vs Vgs for
0.18 ?m device, W=10 ?m, Nf = 8. . . . . . . . . . . . . . . . . . . . . . . 196
7.13 !1, !2 and R1 vs Vgs for 0.18 ?m device. W = 10 ?m, Nf = 8. . . . . . . . . 196
7.14 <(h11) vs frequency. Symbols are measurement data. Lines are modeling re-
sults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.15 Modeled (<(h11)  Rg) vs frequency for 0.18 ?m device. W = 10 ?m, Nf = 8.
Vgs = 0.4, 0.6, and 0.9 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7.16 <(h11) vs IDS. Symbols are measurement data. Lines are modeling results. . . 200
7.17 Cgs=Cp ratio vs Vgs for 0.18 ?m device. W = 10 ?m, Nf = 8. . . . . . . . . . 201
7.18 The real and imaginary parts of Y11 and Y12 vs Vgs for 0.18?m CMOS device.
W = 10 ?m, Nf = 8. Symbols are measurement data. Lines are modeling results. 202
7.19 The real and imaginary parts of Y21 and Y22 vs Vgs for 0.18?m CMOS device.
W = 10 ?m, Nf = 8. Symbols are measurement data. Lines are modeling results. 203
7.20 <(h11) vs Vgs for 0.5 ?m and 1 ?m CMOS device. W = 10 ?m, Nf = 1. . . . 203
8.1 Measured ratio of gd0=gm vs Vgs for di erent channel lengths from a 0.13 ?m
CMOS process. Vds = 1.5 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
xx
8.2 Measured ?gd0 and ?gm for a 0.18 ?m CMOS process. Vds = 1 V. . . . . . . . . 209
8.3 IDS vs Vgs in saturation region for gate length of 0.24 ?m, 0.18 ?m, and 0.12
?m devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.4 fT vs (a) IDS, and (b) Vgs for gate length of 0.24 ?m, 0.18 ?m, and 0.12 ?m
devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
8.5 (a) Sid;i?d, and (b) gm normalized by (W ? Nf) vs IDS for gate length of 0.24
?m, 0.18 ?m, and 0.12 ?m devices. . . . . . . . . . . . . . . . . . . . . . . . 211
8.6 ?gm and ?gd0 vs IDS for gate length of 0.24 ?m, 0.18 ?m, and 0.12 ?m devices. 212
8.7 ?id and ?ih (a) vs IDS, and (b) vs Vgs at Vds = 0.2 V and 1.2 V for 0.24 ?m
device. W = 4 ?m, Nf = 128. . . . . . . . . . . . . . . . . . . . . . . . . . 213
8.8 IDS vs Vgs at Vds = 1 V and 1.5 V for 0.12 ?m device. W = 5 ?m, Nf = 30. . 214
8.9 IDS vs Vds at Vgs = 0.7, 1.0 and 1.5 V for gate length of 0.12 ?m device. . . . 214
8.10 (a) Sid;i?d, and (b)?gm and ?gd0 vs Vgs at Vds = 1 V and 1.5 V for 0.12 ?m device.
W = 5 ?m, Nf = 30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8.11 (a) Sid;i?d, and (b)?gm and ?gd0 vs IDS at Vds = 1 V and 1.5 V for 0.12 ?m device.
W = 5 ?m, Nf = 30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8.12 (a) Sid;i?d, (b) ?gd0, and (c) ?gm vs Vds at Vgs = 0.7, 1.0 and 1.5 V for gate length
of 0.12 ?m device. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
8.13 (a) Sid;i?d, (b) ?gd0, and (c) ?gm vs Vgs at Vds = 0.1 V to 1.0 V with step of 0.1 V
for 50 nm Le CMOS simulation. . . . . . . . . . . . . . . . . . . . . . . . . . 218
8.14 (a) Sid;i?d, (b) ?gd0, and (c) ?gm vs IDS at Vds = 0.1 V to 1.0 V with step of 0.1 V
for 50 nm Le CMOS simulation. . . . . . . . . . . . . . . . . . . . . . . . . 219
8.15 ?gm (a) vs IDS, and (b) vs Vgs at Vds = 0.1 V to 1.0 V with step of 0.1 V for 50
nm Le CMOS simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
8.16 ?id;0, ?id;1 and ?id;2 vs Vds for 50 nm Le CMOS simulation. . . . . . . . . . . . 221
8.17 MOSFET equivalent circuit with drain current noise and gate resistance noise. . 222
8.18 Model-data comparison of noise parameters vs frequency. Vgs = 0.7 V, Vds = 1 V. 228
xxi
8.19 Model-data comparison of noise parameters vs Vgs, f = 5 GHz, Vds = 1 V. . . . 229
8.20
q
Sid;i?d=(W ? Nf), fT , and KNF vs (a) log scale IDS, and (b) linear scale IDS
for the 0.18 ?m process, with Sid;i?d in unit of A2/Hz, W in unit of ?m, fT in unit
of GHz, and KNF in unit of A=
p
?mHz3. . . . . . . . . . . . . . . . . . . . . 232
8.21
q
Sid;i?d=(W ? Nf), fT , and KNF vs IDS for (a) the 0.25 ?m process, (b) the
0.18 ?m process, and (c) the 0.12 ?m process, with Sid;i?d in unit of A2/Hz, W
in unit of ?m, fT in unit of GHz, and KNF in unit of A=
p
?mHz3. . . . . . . . 233
8.22 (a)
q
Sid;i?d=(W ? Nf), (b) fT , and (c) KNF vs IDS comparison between a 0.25
?m process, a 0.18 ?m process, and a 0.12 ?m process. . . . . . . . . . . . . 234
8.23 KNF;Rg vs IDS comparison between a 0.18 ?m process device with W = 10 ?m,
a 0.25 ?m process device with W = 4 ?m, and a 0.12 ?m process device with
W = 5 ?m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
xxii
LIST OF TABLES
2.1 Transformation matrices to calculate other noise representations . . . . . . . . 49
2.2 Conversions between two-port network parameters. . . . . . . . . . . . . . . . 50
4.1 Connections between noise modeling and simulation for Sva;v?a , Sia;i?a and Sia;v?a . 107
4.2 SPICE model and van Vliet model for intrinsic transistor Sib;i?b, Sic;i?c and Sic;i?b. 115
8.1 Comparison of Fukui empirical constants with our derivation. . . . . . . . . . 227
xxiii
CHAPTER 1
INTRODUCTION
Wireless communications have been thrived in the last decade, due to tremendously increas-
ing demand for information need for connectivity. The rapid development of personal communi-
cation systems, such as cellular phones, cordless phones, GPS (global positioning system), and
WLAN (wireless local area network), have aroused considerable interests in high frequency de-
vices. The RFICs (radio frequency integrated-circuits) designs are among the most demanding
design tasks. The recent bipolar and CMOS technologies provide relatively high cuto frequen-
cies (fT ). The RFICs have been the primary domain for modern bipolar and CMOS applications
in GHz frequency range.
RF bipolar and CMOS are both important in RFIC applications. RF bipolar device is known
for its low noise, low power consumption, high reliability and better thermal management, hence
it can be used as the first stage LNA (low noise amplifier) of RF transceiver design. RF CMOS
device is known for its high speed and high level of integration, therefore it is perfect for large-
scale digital applications in RF transceiver design. Accurate models are critical in order to reduce
design cycles and to achieve first-time success in implementation. Unfortunately, available com-
pact models for both RF bipolar and CMOS, are typically not applicable for the GHz frequency
range.
Modeling of noise provides critical information in the design of RF circuits. Lack of un-
derstanding of RF device noise presents a substantial barrier to the design of RF circuits. It is
extremely inevitable procedure to understand the physical origin of broadband noise and incor-
porate in the noise modeling of RF devices.
1
1.1 RF Noise
Noise can be defined as ?everything except desired signal?. Noise sources that can be
reduced or eliminated using good shield system is called Artificial noise. For example, the noise
sources that interfering with broadcasting signal for radio and TV. On the other hand, noise
sources that is inherent in the system or devices itself and is irreducible is called Fundamental
noise. For example, the snow pictures in analog TV sets. Fundamental noise is random yet
can be statistically characterized. There are several types of fundamental noise sources: thermal
noise, shot noise, flicker noise, and generation-recombination noise (G-R noise).
In RF bipolar and MOSFET transistors, thermal noise and shot noise are the main noise
sources. Flicker noise is negligible in RF noise modeling, since its 1=f characteristic. G-R
noise is much smaller than flicker noise, and can be generally neglected. Therefore we will not
discuss flicker noise and G-R noise in this work. We will focus on thermal noise and shot noise
in RF bipolar and MOSFET transistors.
1.1.1 Thermal Noise
A thermally-excited carrier in a conductor undergoes a random walk Brownian motion via
collisions with the lattice of the conductor. As a result it produces fluctuations in the terminal
characteristics. In 1927, Johnson discovered that the noise power spectrum of a conductor is
independent of its material and the measurement frequency. He also found that noise power
spectrum is determined only by the temperature T and electrical resistance R under thermal
2
equilibrium:
< v2n > = Svn f; (1.1)
Svn = 4kTR; (1.2)
< i2n > = Sin f; (1.3)
Sin = 4kTR ; (1.4)
where Svn and Sin are the power spectral density of vn and in, respectively. k is the Boltzmann
constant. Thermal noise is also called Johnson noise or Nyquist noise.
1.1.2 Shot Noise
Shot noise refers to the fluctuations associated with the dc current IDC flow across a poten-
tial barrier. Shot noise is white noise, and is described as
< i2n >= 2qIDC f: (1.5)
Two conditions are required for shot noise to occur: a flow of direct current and a potential
barrier over which the carriers are extracted. In RF bipolar devices, base current shot noise and
collector current shot noise are considered for the intrinsic device. In RF MOSFET transistors,
shot noise dominates the noise characteristics only when the device is in the subthreshold region
owing to the carrier transport in this region.
3
1.2 Noise Parameters
Signal-to-noise ratio describes the ratio of useful signal power and the unwanted noise
power. When a combination of signal and noise go through a noisy two-port network, as shown
in Fig. 1.1, both the signal and unwanted noise will be amplified at the same factor. In addition,
the two-port network adds its own noise. Therefore, the signal-to-noise ratio becomes smaller
after a noisy two-port network. Noise factor F is defined as the signal-to-noise ratio at the input
divided by the signal-to-noise ratio at the output.
F = Si=NiS
o=No
; (1.6)
it defines noise figure NF according to
NF = 10log10(F): (1.7)
It is a useful measure of the amount of noise added by the noisy two-port network. [10]
The noise figure of a two-port network is determined by the source admittance Ys = Gs +
jBs, and the noise parameters of the circuit, including the minimum noise figure NFmin, the noise
resistance Rn, and the optimum source admittance Yopt = Gopt +jBopt, through [11]
F = Fmin + RnG
s
??Y
s  Yopt
??2 ; (1.8)
NFmin = 10log10(Fmin): (1.9)
4
Noisy
two port
min
,s  opt
n
NF
Y
R
s
Y
L
Y
i
i
S
N
o
o
S
N
gain
input output
Figure 1.1: Illustration of definition of noise figure for a noisy two-port.
1.2.1 Minimum Noise Figure NFmin
The minimum noise figure NFmin is a very important parameter for noise. As self-explained
in its name, NFmin determines the minimum noise figure for a noisy two-port network. NF
reaches its minimum NFmin when Ys = Yopt. It indicates the attribute of the noisy two-port. The
lowest possible NFmin is accordingly desired. For RF bipolar transistor and MOSFET transistor,
NFmin is dependent on both bias and frequency.
5
1.2.2 Noise Resistance Rn
The noise resistance Rn determines the sensitivity of noise figure to deviations from Yopt. A
small Rn is desired to alleviate the deviations. For RF bipolar transistor and MOSFET transistor,
Rn is frequency independent. Rn is only dependent on bias.
1.2.3 Optimum Source Admittance Yopt
The optimum source admittance Yopt determines the source admittance where NF reaches
its minimum. The value of Yopt indicates the ?noise matching? source admittance for minimum
noise figure, which normally di ers from the ?gain matching? source admittance for maximum
power transfer. Yopt has a real part of Gopt and an imaginary part of Bopt. For RF bipolar transistor
and MOSFET transistor, Gopt and Bopt are dependent on both bias and frequency.
1.3 RF Bipolar Transistor Compact Noise Modeling
The noise of an RF bipolar device can be considered as a lumped base resistance with
thermal noise voltage Svb;v?b , connected to an intrinsic transistor with an input noise current Sib;i?b
and an output noise current Sic;i?c , as shown in Fig. 1.2. At low injection, the noise of the lumped
base resistance can be modeled as 4kTrb [1].
1.3.1 Lumped Base Resistance
It is possible to separate current crowding e ects from all the e ects that play a role in the
intrinsic transistor [1]. This means the intrinsic transistor noise model is independent of base
resistance and current crowding. All the current crowding e ects are taken care of by a branch
that contains the base resistance as shown in Fig. 1.3 [1]. The resulting noise current associated
6
S
Vb
S
ib
Noiseless
Intrinsic
BJT
Y
int
V
1
I
1
V
2
I
2
S
ic
r
b
+
_
+
_
Y
Figure 1.2: RF Bipolar transistor noise modeling.
with the lumped base resistance is no longer 4kTrb, instead [1] showed,
<(yR) = 1=rb:SiR;i?R = 4kTr
b
+ 103 qIB: (1.10)
At low injection, where IB contribution can be neglected, 4kTrb can still be used to describe the
noise of the lumped base resistance. At high injection, the noise of the lumped base resistance is
dominated by 103 qIB [1].
The intrinsic transistor noise modeling is separated from the lumped base resistance branch.
Accurate noise modeling for the intrinsic transistor is needed. Di erent noise models have dif-
ferent expressions for the input noise current, the output noise current, and their correlation for
the intrinsic transistor, as will be detailed below.
7
Figure 1.3: Equivalent circuit proposed for the intrinsic transistor together with the resistance of
the pinched base [1].
1.3.2 SPICE Model
The SPICE model as shown in Fig. 1.4, is the essence of noise modeling in major CAD
tools. The noise physics accounted for include: base resistance thermal noise Svb;v?b , and base
current shot noise 2qIB, and collector current shot noise 2qIC for the intrinsic transistor.
8
4kTr
b
2qI
B
Noiseless
Intrinsic
BJT
Y
int
V
1
I
1
V
2
I
2
2qI
C
r
b
+
_
+
_
Y
Figure 1.4: SPICE model for RF bipolar transistor.
In SPICE model, the noise of the intrinsic transistor is described by
Sib;i?b = 2qIB; (1.11)
Sic;i?c = 2qIC; (1.12)
Sic;i?b = 0: (1.13)
Since the input noise current and the output noise current are both shot noise, they are only
bias dependent, and do not depend on frequency. Moreover, the input and output noise currents
are not correlated to each other in this model. This approach is used by SPICE Gummel-Poon,
VBIC, Mextram, and Hicum models. The accuracy of such compact noise modeling, however,
becomes worse at higher current densities required for high speed [3]. At high frequency or
high current densities, the base and collector current noises are no longer shot like, and their
correlation can becomes appreciable [12], as will detailed in chapter 4.
9
1.3.3 van Vliet Model
About 30 years ago, van Vliet proposed a general noise model in three-dimensional junction
device of arbitrary geometry using transport noise theory for low injection [13]. The structure
of the model is the same as the intrinsic transistor shown in Fig. 1.2. The van Vliet model is
derived from rigorous microscopic noise theory of minority carrier transportation in the base
region. Di erent from the SPICE model, the input noise current of van Vliet model is frequency
dependent, which comes from the intrinsic Y parameter Yint11 . Moreover, the input noise current
and the output noise current are correlated to each other. The correlation term is related to the
intrinsic Y parameters Yint12 and Yint21 , hence both bias and frequency dependent. In van Vliet
model, the noise of the intrinsic transistor at low injection is described by
Sib;i?b = 4kT<(Yint11 )  2qIB; (1.14)
Sic;i?c = 2qIC + 4kT<(Yint22 ); (1.15)
Sic;i?b = 2kT(Yint21 +Yint?12 )  2qIC: (1.16)
The noise of the intrinsic transistor is obtained from dc currents and ac Y-parameters, and no
additional parameter is required.
For a simple small signal model of the intrinsic bipolar transistor as shown in Fig. 1.5,
Yint11 = gbe +j!(Cbe +Ccb); (1.17)
Yint12 =  j!Ccb; (1.18)
Yint21 = gme j!?  j!Ccb; (1.19)
Yint22 = j!Ccb; (1.20)
10
where gbe is the input conductance, gm is the transconductance, Cbe is the EB capacitance, Ccb is
the CB capacitance, and go is the output conductance. ? is the second-order time delay owing to
the transcapacitance. Since
IB ? gbekT=q; (1.21)
IC ? gmkT=q: (1.22)
(1.16) can be further derived to
Sic;i?b = 2kT(gme j!?)  2qIC; (1.23)
= 2qIC  e j!?  1?: (1.24)
Although the van Vliet model does not consider the CB space-charge-region (SCR) e ect in its
derivation, the correlation equation has included the carrier transport delay term as will discussed
in the section 1.3.4.
 
 
+ 
-
+ 
- 
C
be
v
be
g
beV
1
I
1
I
2
V
2
C
cb
-j??
  
m               be
g  e     v
Figure 1.5: The small-signal equivalent circuit for intrinsic bipolar device.
11
At low frequency where ? ? 0, (1.14), (1.15) and (1.16) reduce to their low frequency
expressions:
Sib;i?b = 2qIB; (1.25)
Sic;i?c = 2qIC; (1.26)
Sic;i?b = 0; (1.27)
which are the same as the SPICE model expressions.
As will be discussed in details in chapter 4, the van Vliet model describes RF bipolar tran-
sistor noise well in low injection. For high current density, however, the van Vliet model for
the low injection cannot accurately model the noise in the transistor. In [13], extra modification
parameters are introduced for high current density based on low injection results. For example,
Sib;i?b = A(4kT<(Yint11 )  2qIB); (1.28)
where A is a modification factor. This provides us a way leading to a new noise model for bipolar
transistor as discussed in chapter 4.
1.3.4 Time-delay and Phase-delay Model
Time-delay noise model is proposed by M. Rudolph in 1999 using common-emitter con-
figuration, as shown in Fig. 1.6 [2]. The noise contributions of the input and output current
sources i0c and i"c related to the collector current IC are caused by the same electrons. The elec-
tron noise sources injected from the emitter into the base, cross the CB junction, and then reach
12
the collector. Therefore the correlation of these sources is given by a time delay e j!?, i.e.,
i"c = i0ee j!?; (1.29)
i0c = i0e  i"c; (1.30)
= i"c ej!?  1?: (1.31)
Therefore Si?c;i0?c , Si"c;i"?c , and their correlation are
Si"c;i"?c = 2qIC; (1.32)
Si0c;i0?c = Si"c;i"?c ??ej!?  1??2 = 2qIC ??ej!?  1??2 ; (1.33)
Si0c;i"?c = Si"c;i"?c  ej!?  1?= 2qIC  ej!?  1?: (1.34)
The noise current source related to the base current Ib is assumed not to correlated with the
others [2]. Therefore the input noise current Sib;i?b, the output noise current Sic;i?c , and their
correlation Sic;i?b for time-delay model are
Sib;i?b = 2qIB + 2qIC ??1  ej!???2 ; (1.35)
Sic;i?c = 2qIC; (1.36)
Sic;i?b = 2qIC  e j!?  1?: (1.37)
The phase-delay noise model is proposed by G.F. Niu in 2001 using common-base con-
figuration [3]. The essence of the phase-delay noise model is shown in Fig. 1.7. The collector
current shows shot noise only because the electron current being injected into the collector-base
13
Figure 1.6: Time-delay noise model in [2].
junction from the emitter already has shot noise. The emitter current short noise consists of
two parts, Sine;i?ne = 2qIC, due to the electron injection into the base, and Sipe;i?pe = 2qIB, due
to the hole injection into the emitter. The electron injection process and the hole injection pro-
cess are independent of each other and hence not correlated. The transition of electrons across
the collector-base junction, which is usually reverse biased, is a drift process, causing a delay
version of the emitter electron injection induced shot noise,
inc = inee j!?n; (1.38)
where ?n is the transit time associated with the transport of emitter-injected electron shot noise
current, which includes both the transit time in the base and the transit time in the CB junction.
14
Figure 1.7: Phase-delay noise model in [3].
In common-base configuration model, the noise sources associated with the collector and
emitter currents, ic and ie, are used,
Sic;i?c = Sinc;i?nc = 2qIC; (1.39)
Sie;i?e = Sine;i?ne +Sipe;i?pe = 2qIC + 2qIB; (1.40)
Sie;i?c = 2qICej!?n: (1.41)
Common-base noise sources ic and ie can be easily converted to common-emitter noise sources
ib and ic by equivalent circuit analysis
Sib;i?b = Sie;i?e +Sic;i?c  2<(Sic;i?e ); (1.42)
Sic;i?c = Sic;i?c; (1.43)
Sic;i?b = Sic;i?e  Sic;i?c: (1.44)
15
Therefore (1.39) ? (1.41) can be converted to the common-emitter version using (1.42) ? (1.44)
Sib;i?b = 2qIE + 2qIC  4qIC< ej!?n?; (1.45)
Sic;i?c = 2qIC; (1.46)
Sic;i?b = 2qIC  e j!?n  1?: (1.47)
(1.45) can be further simplified to
Sib;i?b = 2qIB + 4qIC  4qIC< ej!?n?; (1.48)
= 2qIB + 2qIC ?2  2< ej!?n??; (1.49)
= 2qIB + 2qIC ??1  ej!?n??2 ; (1.50)
Note that if ? = ?n, (1.50), (1.46), and (1.47) are the same as (1.35), (1.36), and (1.37). Al-
though derived from di erent angle, the time-delay model and phase-delay model ultimately
give the same noise model expressions. At low frequency, the time-delay model and phase-delay
model can be further simplified to,
Sib;i?b = 2qIB; (1.51)
Sic;i?c = 2qIC; (1.52)
Sic;i?b = 0; (1.53)
which are the same as the SPICE model expressions.
16
1.4 RF MOSFET Transistor Compact Noise Modeling
1.4.1 Gate and Drain Noise currents Modeling
The thermal noise of a MOSFET originates from the thermal noise sources in the channel
as illustrated in Fig. 1.8, leading to drain thermal noise current Sid;i?d and induced gate thermal
noise current Sig;i?g through capacitive coupling to the gate. Since both Sid;i?d and Sig;i?g are ag-
itated by the thermal noise sources in the channel, they are correlated, and the correlation are
imaginary due to the capacitive nature. This noise representation with gate noise current, drain
noise current, and their correlation, as shown in Fig. 1.9, is called Y- noise representation as will
further introduced in chapter 2.
Figure 1.8: Thermal noise in MOSFETs [4].
17
i
d
i
g
Figure 1.9: MOSFET noise model using gate noise current, drain noise currents, and their cor-
relation.
1.4.1.1 van der Ziel Model
Based on the fact that the MOSFET is a modulated resistor, capacitively coupled to the
gate, van der Ziel has proposed a thermal noise model for MOSFETs using impedance field
method [14] [15]. This well-known van der Ziel model are widely used in MOSFET noise
modeling. The drain noise current, induced gate noise current, and their correlation are modeled
as [15],
Sid;i?d = ?gd0 ? 4kTgd0; (1.54)
Sig;i?g = ?4kTgg; (1.55)
gg = ?!
2C2gs
gd0 ; (1.56)
c = Sig;i
?
dp
Sid;i?dSig;i?g = jx: (1.57)
18
Here gd0 is the zero Vds output conductance, gg is the input conductance, and Cgs is the gate-to-
source capacitance. ?gd0, ?, ? and x are model parameters. ?gd0 = 23 , ? = 43 , ? = 15 and x = 0:395
for long channel device in saturation region [15]. For short channel device, however, these model
parameters deviate from their long channel value, and become bias dependent, as will discussed
in chapter 6.
1.4.1.1 Klaassen-Prins Equation
Klaassen and Prins [16] have derived an equation to calculate the noise of a device using the
local channel conductivities of the device. The so called Klaassen-Prins equation is extensively
used to calculate the noise for long channel MOSFETs [17] [18] [6] [19]. The quasi-static dc
di erential equation for current Id of a device is [16] [20],
Id = g(V (x))dV (x)dx ; (1.58)
where g(V (x)) is the local channel conductivity and V (x) is the di erence in electron quasi-
Fermi potential in the inversion layer and the hole quasi-Fermi potential in the bulk at position
x. For a very simple MOSFET,
g(V (x)) = ?CoxW (Vgs  Vth  V (x)); (1.59)
= W?Q0I(x); (1.60)
where Vgs is the gate-source voltage, Vth is the threshold voltage, W is the width of the device,
? is the mobility, and Cox is the oxide capacitance per unit area. Q0I(x) is the local inversion
19
charge, whose integration over area gives the total inversion charge QI,
QI =
ZL
0
WQ0I(x)dx: (1.61)
(1.59) shows that g(V (x)) is the highest near the source, and the lowest dear the drain.
The derivation of drain noise current can be best illustrated in Fig. 1.10. For the noise
segment from x to x+ x, a small voltage contribution vn(x) is added on top of V (x). The noise
voltage also leads to a change in the dc current through the device, with boundary condition
vn(x)jx=0;L = 0 for input and output ac short ended condition [16] [20].
Id + id = g[V (x) +vn(x)] ddx(V (x) +vn(x)) +in(x); (1.62)
=
?
g(V (x)) + dg(V (x))dV (x) vn(x)
 ?dV (x)
dx +
dvn(x)
dx
?
+in(x); (1.63)
= g(V (x))dV (x)dx +g(V (x))dvn(x)dx + dg(V (x))dx vn(x) + dg(V )dV vn(x)dvn(x)dx +in(x):
(1.64)
Here
g[V (x) +vn(x)] =
?
g(V (x)) + dg(V (x))dV (x) vn(x)
 
(1.65)
is used. Substituting (1.58) in (1.64), and
g(V (x))dvn(x)dx + dg(V (x))dx vn(x) = ddx(g(V (x))vn(x)); (1.66)
20
 id, the fluctuation in Id, is
 id = ddx(g(V (x))vn(x)) +in(x); (1.67)
0 L
channel
X00X00
X00X00
noisy
section
S DI
d
g(V(x))
I
d
+? i
d
V(x)+v
n
(x) V(x+? x)+v
n
(x+? x)
x x+? x
i
n
(x)
x x+? x
? x
Figure 1.10: Illustration of drain noise current derivation.
Integrating both sides of (1.67), we have [16] [20],
 idL =
ZL
0
d
dx(vn(x))g(V (x))dx+
ZL
0
in(x) ?dx; (1.68)
=
ZL
0
in(x) ?dx; (1.69)
21
since
ZL
0
d
dx(vn(x))g(V (x))dx = vn(x)g(V (x))j
L
0 = g(V (L))vn(L)  g(V (0))vn(0) = 0; (1.70)
Therefore the noise fluctuation in Id is,
 id = 1L
ZL
0
in(x) ?dx: (1.71)
 id has a zero average  id = 0, and the noise spectral density is [16] [20],
Sid;i?d =  id; i
?
d
 f =
1
L2
ZL
0
ZL
0
in(x);i?n(x0) ?dxdx0: (1.72)
For in(x), we have [16] [17],
in(x);i?n(x0) = 4kTg(V (x)) ffi(x x0); (1.73)
where fi is the Dirac delta function. The drain thermal noise current is then found by substituting
(1.73) into (1.72),
Sid;i?d = 4kTL2
ZL
0
g(V (x)) ?dx: (1.74)
From (1.58), we have
dx = g(V (x))I
d
dV (x): (1.75)
22
Substituting into (1.74), we have [16] [17] [18] [20] [21] [22] [23],
Sid;i?d = 4kTL2I
d
ZVd
0
g2(V ) ?dV: (1.76)
(1.76) is known as Klaassen-Prins equations for thermal noise of a long channel MOSFET.
(1.74) can be also expanded using (1.60),
Sid;i?d = 4kTL2
ZL
0
W?Q0I(x)dx; (1.77)
= 4kTL2 ?QI: (1.78)
(1.78) is used in models like BSIM.
For long channel device, the drain current Id in saturation region is
Id = ?WCoxL ? 12V 2gt; (1.79)
where Vgt = Vgs  Vth. Substituting (1.59) and (1.79) into (1.76),
Sid;i?d = 4kTL2I
d
ZVgt
0
?2W 2C2ox(Vgt  V )2 ?dV; (1.80)
= 4kTL2 2L?WC
oxV 2gt
??2W 2C2ox ? 13 (Vgt  V )3
??
??
Vgt
0
; (1.81)
= 4kT ? 23 ? ?WCoxL Vgt; (1.82)
= 4kT ? 23gd0: (1.83)
23
with
gd0 = ?WCoxL Vgt: (1.84)
(1.83) is the same as (1.54) in van der Ziel model for long channel device operating in saturation
region.
1.4.1.3 Velocity Saturation in Short Channel Devices
In case of velocity saturation e ects play a role, the general expression of noise source
in(x), (1.73), becomes [21] [24] [25] [20],
Sin(x);in(x)? = 4kTg0Dn(E)D
n(0)
; (1.85)
where g0(x) = q?0n(x)WL is the zero-field channel conductivity, n(x) is the electron con-
centration at position x, Dn(0) = kT?0=q is the di usion coe cient at zero electric field at
the ambient temperature T. The velocity saturation make e ects via the scalar noise di usion
coe cient Dn(E),
Dn(E) = ?n(E)kTq : (1.86)
[20] and [6] argue that it is incorrect to take explicit carrier heating into account by using
a temperature Te > T in (1.85) and (1.86) since Dn(E) has already taken into account all
nonequilibrium e ects.
24
Moreover, consider the velocity saturation e ects for short channel device, (1.60) becomes
[20],
g(V (x)) = W?0Q0I(x) 1
1 + ?0vsat dV (x)dx
; (1.87)
= g0(V (x))
1 + 1Esat dV (x)dx
; (1.88)
where Esat = vsat?0 is the saturation electric field, and g0(V (x)) = W?0Q0I(x). Therefore dc
current for velocity saturation becomes,
Id = g0(V (x))
1 + 1Esat dV (x)dx
dV (x)
dx : (1.89)
Integration on both sides gives,
Id = 1
1 + VdsEsatL
? 1L
ZVds
0
g0(V )dV; (1.90)
= 1
1 + VdsEsatL
Id0; (1.91)
where Id0 is the Id without velocity saturation e ect. Similar derivation are performed, and the
resulting drain noise current for velocity saturation is [20],
Sid;i?d = 4kTI
dL2
1
 
1 + VdsEsatL
?2
ZVds
0
g20 (V )dV; (1.92)
[4] showed that this improved Klaassen-Prins equation has properly accounted for the velocity
saturation e ects.
25
In the case of channel length modulation, the conductivity g(V (x)) in the pinch-o region
is low compared to that in the channel, which is shown in (1.59). From the improved Klaassen-
Prins equation (1.92), the contribution of the pinch-o region can be neglected [26] [4]. How-
ever, the e ective gate length Le should be used instead of L in (1.92) [26] [4].
1.4.1.4 BSIM4 Channel Thermal Noise Model
There are two channel thermal noise models in BSIM4, as shown in Fig. 1.11 [5]. One is
charge-based model by selecting tnoiMod=0. The drain noise current is given by
Sid;i?d = 4kT?effL2
eff
jQinvj?NTNOI; (1.93)
which is essentially the same as (1.78). Here the parameter NTNOI is introduced for more
accurate fitting of short-channel devices.
Figure 1.11: Schematic for BSIM4 channel thermal noise modeling [5].
The other is the holistic model by selecting tnoiMod=1. In this thermal noise model, all
the short-channel e ects and velocity saturation e ect are automatically included. In addition,
a source thermal noise voltage vd is used to contribute to the induced gate noise with partial
26
correlation to the channel thermal noise, as shown in Fig. 1.11 (b). The source noise voltage is
given by
Svd;v?d = 4kT?2tnoiVdseffI
d
; (1.94)
and
?tnoi = RNOIB
"
1 +TNOIB ?Leff
? V
gteff
EsatLeff
?2#
; (1.95)
where RNOIB = 0:37 is model parameter. The drain noise current is given by
Sid;i?d = 4kT VdseffI
d
[Gds +?tnoi(Gm +Gmbs)]2; (1.96)
and
?tnoi = RNOIA
"
1 +TNOIA?Leff
? V
gteff
EsatLeff
?2#
; (1.97)
where RNOIB = 0:577 is model parameter.
However, BSIM4 noise model is not accurate. Fig. 1.12 shows comparison of Sid;i?d for the
data and BSIM holistic model for the gate length of 0.18 ?m device. Device width of 10 ?m,
and the number of fingers is 8. Data is obtained from Georgia Institute of Technology. Fig. 1.13
shows the noise parameters for the data and BSIM model. The results from BSIM model deviate
from the data. A more accurate noise modeling is needed.
27
0 50 100 150 200
0
0.5
1
1.5
2
x 10
?21
I
DS
    (mA/mm)
S
id,id
*
    (A
2
/Hz)
data
BSIM model
V
ds
 = 1 V 
0.18 mm device, W = 10 mm, Nf = 8 
Figure 1.12: Comparison of Sid;i?d for the data and BSIM holistic model for 0.18 ?m device. W
= 10 ?m, Nf = 8.
0 50 100 150 200
0
0.5
1
1.5
2
2.5
3
3.5
I
DS
    (mA/mm)
NF
min
    (dB)
data
BSIM model
f = 5 GHz 
0.18 mm device, W = 10 mm, Nf = 8.
V
DS
 = 1 V 
0 50 100 150 200
0
200
400
600
800
1000
I
DS
    (mA/mm)
R
n
    (
W
) 
data
BSIM model
f = 5 GHz 
V
DS
 = 1 V 
0.18 mm device, W = 10 mm, Nf = 8.
0 50 100 150 200
0
0.5
1
1.5
2
I
DS
    (mA/mm)
G
opt
    (mS)
data
BSIM model
f = 5 GHz 
V
DS
 = 1 V 
0.18 mm device, W = 10 mm, Nf = 8.
0 50 100 150 200
?2
?1.5
?1
?0.5
0
I
DS
    (mA/mm)
B
opt
    (mS)
data
BSIM model
f = 5 GHz 
V
DS
 = 1 V 
0.18 mm device, W = 10 mm, Nf = 8.
Figure 1.13: Comparison of noise parameters for the data and BSIM holistic model for 0.18 ?m
device. W = 10 ?m, Nf = 8.
28
1.4.2 Gate Noise Voltage and Drain Noise Current Modeling
v
h
i
h
Figure 1.14: MOSFET noise model: Pospieszalski model
Di erent from gate and drain noise current representation, another widely accepted noise
model in the GaAs community is the Pospieszalski model, which is based on the hybrid repre-
sentation, as shown in Fig. 1.14 [27]. While the gate current noise in the van der Ziel model is
frequency dependent and correlated to drain current noise, the Pospieszalski model uses an input
voltage noise source Svh;v?h , which is frequency independent. An output noise current ih is used
in Pospieszalski model. Svh;v?h is proportional to the non-quasi-statistic channel resistance Rgs.
Sih;i?h is proportional to the output conductance gds. Gate temperature Tg and drain temperature
Td are used in the model, function as coe cients as in van der Ziel model. Further, this model
29
assumes that two noise sources have negligible correlation
Svh;v?h = 4kTgRgs; (1.98)
Sih;i?h = 4kTdgds; (1.99)
Svh;i?h = 0: (1.100)
Further investigations showed that this assumption is well satisfied in GaAs devices. However,
no study has shown that it is valid for MOSFET devices. In this dissertation, Pospieszalski model
is successfully applied to MOSFET devices in chapter 6.
1.4.3 Role of Gate Resistance
The gate resistance Rg is associated with a thermal noise voltage of 4kTRg. This gate
thermal noise voltage is equivalent to an input noise current, an output noise current and a cor-
relation, as shown in Fig. 1.15,
Sig;i?g = 4kTRgjY11j2 = 4kTRg(!Cgs)2; (1.101)
Sid;i?d = 4kTRgjY21j2 = 4kTRgg2m; (1.102)
Sig;i?d = j4kTRg!gmCgs; (1.103)
c = Sig;i
?
dp
Sig;i?gSid;i?d = j1: (1.104)
(1.101) shows that the gate resistance leads to a gate noise current that proportional to f2, and
behaves like the induced gate noise. This gate resistance related gate noise current overwhelms
the induced gate noise for short channel devices. (1.102) shows that the gate resistance also leads
to a drain noise current. The gate resistance related gate and drain noise currents are correlated
30
as shown in (1.103). This indicates that reduction of the gate resistance Rg is really important
for obtain low noise in MOSFET.
Figure 1.15: Role of gate resistance noise to gate noise current, drain noise current, and their
correlation.
Although a metal silicide is added to the polysilicon gate to decrease its resistance, wide
devices with short channels might still show a significant gate resistance. The gate resistance
Rg consists of several parts: the resistance of the vias between metal1 and silicided polysilicon,
the e ective resistance of the silicide, and the contact resistance between silicide and polysilicon
[28]. For a single polysilicon gate finger connected with both sides [6],
Rg = 112RshWL + 12RshWextL + 12 RviaN
via
+ ?conWL; (1.105)
where Rsh is the silicide sheet resistance, Rvia is the resistance of the metal1-to-polysilicon via,
Nvia is the number of such vias, ?con is the silicide-to-polysilicon specific contact resistance. W ,
L, and Wext are depicted in Fig. 1.16. The factor 12 accounts for the distributed nature of the
gate resistance and the use of contacts on both sides of the gate.
31
Figure 1.16: Schematic layout of a single gate finger, showing the meaning of W , Wext, and L
in (1.105) [6].
Narrow fingers, double-sided contacting, guard ring and abundant contacting lead to reduc-
tion in Rg. Using multiple devices in parallel to obtain larger devices is also a way to reduce
Rg [4]. The width of finger, however, is optimized at 1 ?m for 90 nm technology node tran-
sistor [29]. Further reduction in the width of finger does not further reduce Rg. It is generally
accepted that the drain current noise and the gate resistance thermal noise are the dominant RF
noise sources of interest in scaled CMOS [30]. Since Rg is important especially for short channel
devices, accurate extraction of Rg plays a big role in compact noise modeling of modern CMOS,
which will be detailed addressed in chapter 7.
Fukui first proposed a set of empirical NFmin, Rn and Yopt equations for FETs based on
his observation of experimental data on MESFETs [31] [32] [33], which involve an empirical
Fukui?s noise figure coe cient Kf, and other ?constants,? and transistor gate resistance Rg and
transconductance gm. The noise figure coe cient has since been frequently used as a figure-of-
merit for comparing di erent technologies [34] [35] [36] [37] [38]. Recently, various equations
of NFmin, Rn and Yopt have been derived for CMOS with varying assumptions, by neglecting gate
resistance noise and/or induced gate noise [39] [40] [41], and by assuming a bias independent
ratio of ?gd0 to ?gm, which is problematic as detailed in chapter 8.
32
1.5 Dissertation Contributions
The following chapters provide detailed information about RF bipolar and CMOS noise in
terms of device physics. To achieve these goals, this dissertation tackles various areas including
microscopic noise simulation, Ge profile optimization in SiGe HBT device, noise characteriza-
tion, and compact noise modeling.
Chapter 1 gives an introduction of definitions and classifications of RF device noise and
noise parameters. Review of RF bipolar and CMOS noise models and the intrinsic noise sources
in RF bipolar and CMOS devices is also given in chapter 1.
Chapter 2 introduces di erent noise representations for a linear noisy two-port network.
The transformation matrices to other noise representations are given. Techniques of adding or
de-embedding a passive component to a linear two-port network are discussed. Noise sources
de-embedding for both MOSFET and SiGe HBT are given as examples which are repeatedly
used later in this dissertation.
Chapter 3 presents a new technique of simulating the spatial distribution of microscopic
noise contribution to the input noise current, voltage, and their correlation. The technique is first
demonstrated on a 50 GHz SiGe HBT. The spatial distributions by base majority holes, base
minority electrons, and emitter minority holes are analyzed, and compared to the compact noise
model. This technique is also applied to a 120 GHz MOSFET transistor. The spatial distribution
of drain noise current, gate noise current, and their correlation are analyzed.
Chapter 4 examines bipolar transistor noise modeling and noise physics using microscopic
noise simulation. Transistor terminal current and voltage noises resulting from velocity fluctu-
ations of electrons and holes in the base, emitter, collector, and substrate are simulated using
a new technique proposed in chapter 3, and compared with modeling results. Major physics
33
noise sources in bipolar transistor are qualitatively identified. The relevant importance as well
as model-simulation discrepancy is analyzed for each physical noise source.
Chapter 5 explores the RF noise physics and SiGe profile optimization for low noise using
microscopic noise simulation. A higher Ge gradient in a noise critical region near the EB junc-
tion reduces impedance field and hence minimum noise figure. A higher Ge gradient near the EB
junction, together with an unconventional Ge retrograding in the base to keep total Ge content
below stability, when optimized, can lead to significant noise improvement without sacrificing
peak fT and without any significant high injection fT rollo degradation.
In chapter 6, RF noise of 50 nm Le CMOS is simulated using hydrodynamic noise simula-
tion. Intrinsic noise sources for the Y- and H- noise representations are examined and models of
intrinsic noise sources are proposed. The relations between the Y- and H- noise representations
for MOSFETs are examined, and the importance of correlation for both representations is quan-
tified. The theoretical values of H- noise representation model parameters are derived for the first
time for long channel devices. The H- noise representation correlation is shown theoretically to
have a zero imaginary part. The H- noise representation has the inherent advantage of a more
negligible correlation, which makes circuit design and simulation easier. Chapter 6 also exper-
imentally extracts the H-representation noise sources using noise parameters measured on 0.25
?m RF CMOS devices. A simple yet e ective model is proposed to model the H-representation
noise sources as a function of bias. Excellent modeling results are achieved for all of the noise
parameters up to 26 GHz, at all biases.
The gate resistance is important as well as the drain noise current for noise modeling of
scaled MOSFET. Accurately extract the gate resistance becomes an important issue. Chapter 7
explains the frequency and bias dependence of the e ective gate resistance by considering the
34
e ect of gate-to-body capacitance, gate to source/drain overlap capacitances, fringing capac-
itances, and Non-Quasi-Static (NQS) e ect. A new method of separating the physical gate
resistance and the NQS channel resistance is proposed. Separating the gate-to-source parasitic
capacitances from the gate-to-source inversion capacitance is found to be necessary for accurate
modeling of all of the Y-parameters.
Chapter 8 examines the di erences between the gd0 and gm referenced drain current excess
noise factors in CMOS transistors as a function of channel length and bias. The technology
scaling are discussed for 0.25 ?m process, 0.18 ?m process and 0.12 ?m process. Using standard
linear noisy two-port theory, a simple derivation of noise parameters is presented. The results
are compared with the well known Fukui?s empirical FET noise equations. Experimental data
on a 0.18 ?m CMOS process are measured and used to evaluate the simple model equations.
New figures-of-merit for minimum noise figure is proposed. The amount of drain current noise
produced to achieve one GHz fT is shown to fundamentally determine the noise capability of
the intrinsic transistor.
Finally Chapter 9 concludes the work in this dissertation.
35
CHAPTER 2
NOISE NETWORK ANALYSIS AND DE-EMBEDDING
This chapter introduces di erent noise representations for a linear noisy two-port network.
The transformation matrices to other noise representations are given. Techniques of adding or
de-embedding passive components to a linear two-port network are discussed. For example,
the open-short de-embedding procedure is needed for measurement data to move the reference
plane to the device terminals. Noise sources de-embedding for both MOSFET and SiGe HBT
are given as examples which are repeatedly used later in this dissertation.
2.1 Noise Representations
A noisy two-port network can be described by a noiseless two-port network with input
noise voltages or currents, and output noise voltages or currents. In general, there are four
noise representations, including chain noise representation, Y- noise representation, Z- noise
representation, and H- noise representation.
2.1.1 Chain Noise Representation (ABCD- Noise Representation)
Chain noise representation, or ABCD- noise representation, describes the noise of a two-
port network with an input noise voltage va, an input noise current ia, and their correlation, as
shown in Fig. 2.1. The power spectral densities (PSD) of va, ia, and their correlation are Sva;v?a ,
36
Sia;i?a, and Sia;v?a , respectively. The chain noise matrix is defined as
CA =
2
64 Sva;v?a Sva;i?a
Sia;v?a Sia;i?a
3
75 (2.1)
v
a
i
a
Noiseless
Two-Port
 Y
V
1
I
1
V
2
I
2
+
_
+
_
Noisy
Two-Port
Figure 2.1: The chain noise representation of a linear noisy two-port network.
Chain noise representation is the most convenient because it is directly related to the noise
parameters NFmin, Rn and Yopt = Gopt+jBopt by [11]. The noise factor for a noisy linear two-port
as shown in Fig. 2.2 is [42] [43]
F = Si=NiS
o=No
;
= NoG
pNi
; (2.2)
= Ni +N
0
i
Ni ; (2.3)
= 1 + N
0
i
Ni; (2.4)
37
where Gp = So=Si is the power gain of the two-port, Ni is the input noise power delivered to the
noisy two-port due to source noise current is, and N0i is the noise power delivered to the noisy
two-port due to va and ia.
v
a
i
a
Noiseless
Two-Port
 Y
V
1
I
1
V
2
I
2
+
_
+
_
i
s
Z
s
Z
i
i
n
+i
n
'
Z
L
Figure 2.2: Noisy linear two-port network.
If Zi denotes the input admittance of the two-port shown in Fig. 2.2, the noise current
delivered by the source to the noise free two-port is
in =  is ZsZ
i +Zs
; (2.5)
and
Ni =< in;i?n > <(Zi); (2.6)
=< is;i?s >
??
?? Zs
Zi +Zs
??
??
2
<(Zi); (2.7)
= 4kTGs jZsj
2
jZi +Zsj2<(Zi) f; (2.8)
where Zs is the source impedance, and Ys = 1=Zs is the source admittance with a real part of
Gs and an imaginary part of Bs. The noise current delivered to the noise free two-port by the
38
correlated noise voltage and noise current of the noisy two-port is
i0n =  va 1Z
i +Zs
 ia ZsZ
i +Zs
; (2.9)
and
N0i =< i0n;i0?n > <(Zi); (2.10)
=
"
< va;v?a > 1jZ
i +Zsj2
+ < ia;i?a >
??
?? Zs
Zi +Zs
??
??
2
+ 2<
?
< ia;v?a > ZsjZ
i +Zsj2
?#
<(Zi);
(2.11)
=?Sva;v?a +Sia;i?ajZsj2 + 2< Sia;v?aZs?? 1jZ
i +Zsj2
<(Zi) f: (2.12)
Substituting (2.8) and (2.12) in (2.4),
F = 1 + Sva;v
?a +Sia;i?ajZsj2 + 2<
 S
ia;v?aZs
?
4kTGsjZsj2 ; (2.13)
= 1 + Sva;v
?ajYsj2 +Sia;i?a + 2<
 S
ia;v?aY?s
?
4kTGs ; (2.14)
Let Sia;v?a = Gu +jBu, we have
F = 1 + Sva;v
?ajGs +jBsj2 +Sia;i?a + 2<((Gu +jBu)(Gs  jBs))
4kTGs ; (2.15)
= 1 + Sva;v
?a (G2s +B2s) +Sia;i?a + 2(GuGs +BuBs)
4kTGs : (2.16)
39
To find out the optimum Bs to minimize noise factor F, fiFfiBs = 0,
2Sva;v?aBs + 2Bu
4kTGs = 0; (2.17)
hence the optimum source susceptance Bopt is
Bopt =  BuS
va;v?a
: (2.18)
To find out the optimum Gs to minimize noise factor F, fiFfiGs = 0,
 Sia;i?a +G2sSva;v?a  B2sSva;v?a  2BuBs = 0; (2.19)
Substituting Bs = Bopt in,
 Sia;i?a +G2sSva;v?a + B
2u
Sva;v?a = 0; (2.20)
hence the optimum source conductance Gopt is
Gopt =
vu
utSia;i?a
Sva;v?a  
B2u
S2v
a;v?a
: (2.21)
Substituting Gs and Bs using their optimum values Gopt and Bopt in (2.16), the minimum noise
factor Fmin is
Fmin = 1 +
q
Sva;v?aSia;i?a  B2u +Gu
2kT : (2.22)
40
Note that Gu = <(Sia;v?a ), and Bu = =(Sia;v?a ), the noise parameters NFmin, Rn, Gopt, and
Bopt finally are [43]
Fmin = 1 +
pS
va;v?aSia;i?a  [=(Sia;v?a )]2 +<(Sia;v?a )
2kT ; (2.23)
= 1 + 2Rn
?
Gopt + <(Sia;v
?a )
Sva;v?a
?
; (2.24)
NFmin = 10 log10(Fmin); (2.25)
Rn = Sva;v
?a
4kT ; (2.26)
Gopt =
s
Sia;i?a
Sva;v?a  
?=(S
ia;v?a )
Sva;v?a
 2
; (2.27)
Bopt =  =(Sia;v
?a )
Sva;v?a ; (2.28)
where < and = stand for the real and the imaginary parts of a factor, respectively.
Solved from (2.24), (8.17), 8.18, and (8.19), the chain noise representation parameters
Sva;v?a , Sia;i?a, and Sia;v?a , can be obtained using the noise parameters NFmin, Rn and Yopt by [11],
Sva;v?a = 4kTRn; (2.29)
Sia;i?a = 4kTRn??Yopt??2 ; (2.30)
Sia;v?a = 2kT (Fmin  1)  4kTRnYopt; (2.31)
or in the format of noise matrix,
CA = 4kT
2
64 Rn Fmin 12  RnY?opt
Fmin 1
2  RnYopt RnjYoptj
2
3
75: (2.32)
41
2.1.2 Y- Noise Representation
The Y- noise representation describes the noise of a two-port network with an input noise
current i1, an output noise current i2, and their correlation, as shown in Fig. 2.3. The PSD?s of i1,
i2, and their correlation are Si1;i?1 , Si2;i?2 , and Si2;i?1 , respectively. The Y- noise matrix is defined
as
CY =
2
64 Si1;i?1 Si1;i?2
Si2;i?1 Si2;i?2
3
75 (2.33)
The output of microscopic noise simulation tool TAURUS are Y- noise representation parameters
[44]. Y- noise representation is also commonly used in compact noise modeling of both RF
bipolar and MOSFET transistors, as detailed later in section 1.3.2 and 1.4.1.
I
2
Noiseless
Two-Port
Y
I
1
V
2
i
1
i
2
V
1
Noisy
Two-Port
Figure 2.3: The Y- noise representation of a linear noisy two-port network.
Conversions between the chain noise representation parameters and the Y- noise represen-
tation parameters can be derived as follows. We denote Y as total admittance matrix. The ac
42
I  V relations including noise for the representations shown in Fig. 2.1 and Fig. 2.3 are
0
B@ I1  ia
I2
1
CA=
2
64 Y11 Y12
Y21 Y22
3
75?
0
B@ V1  va
V2
1
CA; (2.34)
0
B@ I1  i1
I2  i2
1
CA=
2
64 Y11 Y12
Y21 Y22
3
75?
0
B@ V1
V2
1
CA: (2.35)
Equating the noise terms of the two representations for both I1 and I2, we find the relations
between (i1;i2) and (va;ia),
i1 = ia  Y11va; (2.36)
i2 =  Y21va; (2.37)
and
va =  1Y
21
i2; (2.38)
ia = i1  Y11Y
21
i2; (2.39)
where Y11 and Y21 are elements of Y matrix. Therefore, the Y- noise representation parameters
Si1;i?1 , Si2;i?2 , and Si2;i?1 , can be derived using the chain noise representation parameters Sva;v?a ,
43
Sia;i?a, and Sia;v?a as
Si1;i?1 = Sia;i?a +jY11j2Sva;v?a  2<(Y?11Sia;v?a ); (2.40)
Si2;i?2 = jY21j2Sva;v?a; (2.41)
Si2;i?1 = Y21Y?11Sva;v?a  Y21S?i
a;v?a
: (2.42)
Alternatively, the chain noise representation parameters Sva;v?a , Sia;i?a, and Sia;v?a , can be derived
using the Y- noise representation parameters Si1;i?1 , Si2;i?2 , and Si2;i?1 as
Sva;v?a = 1jY
21j2
Si2;i?2; (2.43)
Sia;i?a = Si1;i?1 +
??
??Y11
Y21
??
??
2
Si2;i?2  2<
?Y
11
Y21Si2;i?1
?
; (2.44)
Sia;v?a = Y11jY
21j2
Si2;i?2  1Y?
21
S?i
2;i?1
: (2.45)
2.1.3 Z- Noise Representation
The Z- noise representation describes the noise of a two-port network with an input noise
voltage v1, an output noise voltage v2, and their correlation, as shown in Fig. 2.4. The PSD?s
of v1, v2, and their correlation are Sv1;v?1 , Sv2;v?2 , and Sv1;v?2 , respectively. The Z- noise matrix is
defined as
CZ =
2
64 Sv1;v?1 Sv1;v?2
Sv2;v?1 Sv2;v?2
3
75 (2.46)
The output of microscopic noise simulation tool DESSIS are Z- noise representation parameters
[45]. The simulation results in this work are done using DESSIS.
44
I
2
Noiseless
Two-Port
Y
I
1
V
2
v
1
v
2
V
1
Noisy
Two-Port
Figure 2.4: The Z- noise representation of a linear noisy two-port network.
Conversions between the chain noise representation parameters and the Z- noise represen-
tation parameters can be derived as follows. The ac I  V relations including noise for the
representations shown in Fig. 2.1 and Fig. 2.4 are
0
B@ I1  ia
I2
1
CA=
2
64 Y11 Y12
Y21 Y22
3
75?
0
B@ V1  va
V2
1
CA; (2.47)
0
B@ I1
I2
1
CA=
2
64 Y11 Y12
Y21 Y22
3
75?
0
B@ V1  v1
V2  v2
1
CA: (2.48)
Equating the noise terms of the two representations for both I1 and I2, we find the relations
between (v1;v2) and (va;ia),
v1 = va  Y22Y
11Y22  Y12Y21
ia; (2.49)
v2 = Y21Y
11Y22  Y12Y21
ia; (2.50)
45
and
va = v1 + Y22Y
21
v2; (2.51)
ia = Y11Y22  Y12Y21Y
21
v2: (2.52)
Therefore, the Z- noise representation parameters Sv1;v?1 , Sv2;v?2 , and Sv1;v?2 , can be derived using
the chain noise representation parameters Sva;v?a , Sia;i?a, and Sia;v?a as
Sv1;v?1 = Sva;v?a +
??
?? Y22
Y11Y22  Y12Y21
??
??
2
Sia;i?a  2<
? Y
22
Y11Y22  Y12Y21Sia;v?a
?
; (2.53)
Sv2;v?2 =
??
?? Y21
Y11Y22  Y12Y21
??
??
2
Sia;i?a; (2.54)
Sv1;v?2 = Y
?
21
Y?11Y?22  Y?12Y?21S
?
ia;v?a  
Y22Y?21
jY11Y22  Y12Y21j2Sia;i
?a: (2.55)
Alternatively, the chain noise representation parameters Sva;v?a , Sia;i?a, and Sia;v?a , can be derived
using the Z- noise representation parameters Sv1;v?1 , Sv2;v?2 , and Sv1;v?2 as
Sva;v?a = Sv1;v?1 +
??
??Y22
Y21
??
??
2
Sv2;v?2 + 2<
?Y?
22
Y?21Sv1;v
?
2
?
; (2.56)
Sia;i?a =
??
??Y11Y22  Y12Y21
Y21
??
??
2
Sv2;v?2; (2.57)
Sia;v?a = Y
?
22(Y11Y22  Y12Y21)
jY21j2 Sv2;v
?
2 +
Y11Y22  Y12Y21
Y21 S
?
v1;v?2: (2.58)
2.1.4 H- Noise Representation
The H- noise representation describes a noisy two-port network with an input noise voltage
vh, an output noise current ih, and their correlation, as shown in Fig. 2.5. The PSD?s of vh, ih,
46
and their correlation are Svh;v?h , Sih;i?h, and Svh;i?h, respectively. The H- noise matrix is defined as
CH =
2
64 Svh;v?h Svh;i?h
Sih;v?h Sih;i?h
3
75 (2.59)
H- noise representation is popular for compact noise modeling of GaAs MESFETs and HEMTs.
As we will show in chapter 6, the H- noise representation is also advantageous for CMOS tran-
sistors. Therefore we are more concerned with the conversions between Y- noise representation
parameters and H- noise representation parameters.
v
h
Noiseless
Two-Port
 Y
V
1
I
1
V
2
I
2
+
_
+
_
Noisy
Two-Port
i
h
Figure 2.5: The H- noise representation of a linear noisy two-port network.
The I  V relations including noise in Fig. 2.3 and Fig. 2.5 are given by:
0
B@ I1  i1
I2  i2
1
CA=
2
64 Y11 Y12
Y21 Y22
3
75?
0
B@ V1
V2
1
CA; (2.60)
0
B@ I1
I2  ih
1
CA=
2
64 Y11 Y12
Y21 Y22
3
75?
0
B@ V1  vh
V2
1
CA: (2.61)
47
Solving (2.60) and (2.61), i1 and i2 are related to vh and ih as
i1 =  Y11vh (2.62)
i2 = ih  Y21vh; (2.63)
and
vh =  1Y
11
i1 (2.64)
ih = i2  Y21Y
11
i1: (2.65)
Therefore, the Y- noise representation parameters Si1;i?1 , Si2;i?2 , and Si1;i?2 , can be derived
using the H- noise representation parameters Svh;v?h , Sih;i?h, and Sih;v?h as
Si1;i?1 = jY11j2Svh;v?h; (2.66)
Si2;i?2 = Sih;i?h +jY21j2Svh;v?h  2<(Y21Svh;i?h); (2.67)
Si1;i?2 = Y11Y?21Svh;v?h  Y11Svh;i?h: (2.68)
Alternatively, the H- noise representation parameters Svh;v?h , Sih;i?h, and Sih;v?h , can be derived
using the Y- noise representation parameters Si1;i?1 , Si2;i?2 , and Si1;i?2 as
Svh;v?h = 1jY
11j2
Si1;i?1; (2.69)
Sih;i?h = Si2;i?2 +
??
??Y21
Y11
??
??
2
Si1;i?1  2<(Y21Y
11
Si1;i?2 ); (2.70)
Svh;i?h = Y
?
21
jY11j2  
1
Y11Si1;i?2: (2.71)
48
2.2 Transformation to Other Noise Representations
The ABCD-, Y-, Z-, and H- noise representations can be transformed to another by the
matrix operation:
C0 = T ?C ?Ty; (2.72)
where C and C0 are the original and resulting noise correlation matrices respectively, T is the
transformation matrix given in Table 2.1, and Ty is the transpose conjugate of T. The ABCD, Y,
Z and H two-port network parameters are used in Table 2.1. The conversion of ABCD, Y, Z and
H parameters are given in Table 2.2.
Original Representation
CY CZ CA CH
C0Y
? 1 0
0 1
 ? Y
11 Y12
Y21 Y22
 ?  Y
11 1
 Y21 0
 ?  Y
11 0
 Y21 1
 
C0Z
? Z
11 Z12
Z21 Z22
 ? 1 0
0 1
 ? 1  Z
11
0  Z21
 ? 1  Z
12
0  Z22
 
C0A
? 0 A
12
1 A22
 ? 1  A
11
0  A21
 ? 1 0
0 1
 ? 1 A
12
0 A22
 
C0H
?  h
11 0
 h21 1
 ? 1  h
12
0  h22
 ? 1  h
11
0  h21
 ? 1 0
0 1
 
Table 2.1: Transformation matrices to calculate other noise representations
49
Y Z A H S
Y
Y11
Y12
Y21
Y22
Z22
 Z Z
12
 Z Z
21
 ZZ
11
 Z
A22
A12
  A
A12
 1
A12
A11
A12
1
h11
 h12
h11
h21
h11
 H
h11
Y0 1 S11+S22  S1+S11+S22+ S
Y0  2S121+S11+S22+ S
Y0  2S211+S11+S22+ S
Y0 1+S11 S22  S1+S11+S22+ S
Z
Y22
 Y Y
12
 Y Y
21
 YY
11
 Y
Z11
Z12
Z21
Z22
A11
A21
 A
A21
1
A21
A22
A21
 H
h22
h12
h22
 h21
h22
1
h22
Z0 1+S11 S22  S1 S11 S22+ S
Z0 2S121 S11 S22+ S
Z0 2S211 S11 S22+ S
Z0 1 S11+S22  S1 S11 S22+ S
A
 Y22
Y21
 1
Y21
  Y
Y21
 Y11
Y21
Z11
Z21
 Z
Z21
1
Z21
Y22
Z21
A11
A12
A21
A22
  H
h21
 h11
h21
 h22
h21
 1
h21
1+S11 S22  S
2S21
Z0 1+S11+S22+ S2S21
Y0 1 S11 S22+ S2S21
1 S11+S22  S
2S21
H
1
Y11
 Y12
Y11
Y21
Y11
 Y
Y11
 Z
Z22
Z12
Z22
 Z21
Z22
1
Z22
A12
A22
 A
A22
 1
A22
A21
A22
h11
h12
h21
h22
Z0 1+S11+S22+ S1 S11+S22  S
2S12
1 S11+S22  S
2S21
1 S11+S22  S
Y0 1 S11 S22+ S1 S11+S22  S
S
Y0(Y0 Y11+Y22)  Y
Y0(Y11+Y22+Y0)+ Y
 2Y12Y0
Y0(Y11+Y22+Y0)+ Y
 2Y21Y0
Y0(Y11+Y22+Y0)+ Y
Y0(Y0+Y11 Y22)  Y
Y0(Y11+Y22+Y0)+ Y
Z0(Z11 Z22 Z0)+ Z
Z0(Z11+Z22+Z0)+ Z
2Z12Z0
Z0(Z11+Z22+Z0)+ Z
2Z21Z0
Z0(Z11+Z22+Z0)+ Z
Z0(Z22 Z11 Z0)+ Z
Z0(Z11+Z22+Z0)+ Z
A11+A12=Z0 A21=Z0 A22
A11+A12=Z0+A21=Z0+A22
2 A
A11+A12=Z0+A21=Z0+A22
2
A11+A12=Z0+A21=Z0+A22
 A11+A12=Z0 A21=Z0+A22
A11+A12=Z0+A21=Z0+A22
h11 h22 1+ H
h11+h22+1+ H
2h12
h11+h22+1+ H
 2h21
h11+h22+1+ H
h11 h22 1  H
h11+h22+1+ H
S11
S12
S21
S22
 Y = Y11Y22  Y12Y21,  Z = Z11Z22  Z12Z21,  H = h11h22  h12h21,  A = A11A22  A12A21.
Table 2.2: Conversions between two-port network parameters.
2.3 Adding Noisy Passive Components to a Noisy Two-Port Network
If the noise of the intrinsic two-port network is known, in order to calculate the noise of
a complex network, one needs to start from the noise of the intrinsic two-port network, then
procedurally add the noise of other noisy passive components to the intrinsic, which is called
the ?adding? procedure. Reversely speaking, if the noise of a complex network is known, one
needs to remove the noise of each noisy passive component to calculate the noise of the intrinsic
network, which is called the ?de-embedding? procedure. Both the two-port network parameters
and noise parameters are involved in either the adding procedure or the de-embedding procedure.
Here only the adding procedure is discussed. The de-embedding procedure is just a reverse
50
process. Basically, there are two kinds of cases to add noisy passive components to a noisy
two-port network.
In transistor noise modeling, the raw data measured includes pad and interconnect. One
common case is to add noisy passive components in parallel with a two-port network, as shown
in Fig. 2.6. The added noisy passive components are denoted as Y1, Y2, and Y3, with thermal
noise current of 4kT<(Y1), 4kT<(Y2), and 4kT<(Y3), respectively.
I
2
Noiseless
Two-Port
Y
I
1
V
2
i
1
i
2
V
1
Y
1
Y
3
Y
2
Y
total
4kTReY
1
4kTReY
3
4kTReY
2
Figure 2.6: Adding noisy passive components parallel to a linear noisy two-port network.
The Y-parameter matrix of the noisy two-port network is denoted as Y . The Y-parameter
matrix of after adding the passive components is
Ytotal = Y +
2
64 Y1 +Y2  Y2
 Y2 Y3 +Y2
3
75 (2.73)
51
Denote the input and output noise currents of the Y- noise representation after adding the passive
components as i01 and i02. The I  V relations including noise is Fig. 2.6 are given by:
0
B@ I1  i1  Y1V1  iY1  Y2(V2  V1)  iY2
I2  i2  Y3V2  iY3 +Y2(V2  V1) +iY2
1
CA=
2
64 Y11 Y12
Y21 Y22
3
75?
0
B@ V1
V2
1
CA; (2.74)
0
B@ I1  i01
I2  i02
1
CA=
2
64 Ytotal11 Ytotal12
Ytotal21 Ytotal22
3
75?
0
B@ V1
V2
1
CA; (2.75)
(2.76)
where
SiY1;i?Y
1
= 4kT<(Y1); (2.77)
SiY2;i?Y
2
= 4kT<(Y2); (2.78)
SiY3;i?Y
3
= 4kT<(Y3): (2.79)
Equating the noise terms for both I1 and I2, we find the relations between (i1;i2) and (i01;i02),
i01 = i1 +iY1 +iY2; (2.80)
i02 = i2 +iY3  iY2; (2.81)
52
and
Si01;i0?1 = Si1;i?1 + 4kT<(Y1) + 4kT<(Y2); (2.82)
Si02;i0?2 = Si2;i?2 + 4kT<(Y3) + 4kT<(Y2); (2.83)
Si01;i0?2 = Si1;i?2  4kT<(Y2); (2.84)
or in the format of noise matrix
CtotalY = CY + 4kT ?<
2
64 Y1 +Y2  Y2
 Y2 Y3 +Y2
3
75; (2.85)
where CY is the Y- noise matrix for the noisy two-port, and CtotalY is the Y- noise matrix after
adding the passive components to the noisy two-port.
The other common case is to add noisy passive components in series with the two-port
network terminals, as shown in Fig. 2.7. The added noisy passive components are denoted as Z1,
Z2, and Z3, with thermal noise voltage of 4kT<(Z1), 4kT<(Z2), and 4kT<(Z3), respectively.
The Z-parameter matrix of the noisy two-port network is denoted as Z. The Z-parameter
matrix of after adding the passive components is
Ztotal = Z +
2
64 Z1 +Z2 Z2
Z2 Z3 +Z2
3
75 (2.86)
53
I
2
Noiseless
Two-Port
Z
I
1
V
2
v
1
v
2
V
1
Z
2
Z
1
Z
3
Z
total
4kTReZ
1
4kTReZ
3
4kTReZ
2
Figure 2.7: Adding noisy passive components in series with a linear noisy two-port network.
Denote the input and output noise currents of the Z- noise representation after adding the passive
components as v01 and v02. The I  V relations including noise is Fig. 2.7 are given by:
0
B@ V1  v1  Z1I1  vZ1  Z2(I1 +I2)  vZ2
V2  v2  Z3I2  vZ3 +Z2(I1 +I2) +vZ2
1
CA=
2
64 Z11 Z12
Z21 Z22
3
75?
0
B@ I1
I2
1
CA; (2.87)
0
B@ V1  v01
V2  v02
1
CA=
2
64 Ztotal11 Ztotal12
Ztotal21 Ztotal22
3
75?
0
B@ I1
I2
1
CA; (2.88)
(2.89)
54
where
SvZ1;v?Z
1
= 4kT<(Z1); (2.90)
SvZ2;v?Z
2
= 4kT<(Z2); (2.91)
SvZ3;v?Z
3
= 4kT<(Z3): (2.92)
Equating the noise terms for both V1 and V2, we find the relations between (v1;v2) and (v01;v02),
v01 = v1 +vZ1 +vZ2; (2.93)
v02 = v2 +vZ3 +vZ2; (2.94)
and
Sv01;v0?1 = Sv1;v?1 + 4kT<(Z1) + 4kT<(Z2); (2.95)
Sv02;v0?2 = Sv2;v?2 + 4kT<(Z3) + 4kT<(Z2); (2.96)
Sv01;v0?2 = Sv1;v?2 + 4kT<(Z2); (2.97)
or in the format of noise matrix
CtotalZ = CZ + 4kT ?<
2
64 Z1 +Z2 Z2
Z2 Z3 +Z2
3
75; (2.98)
where CZ is the Z- noise matrix for the noisy two-port, and CtotalZ is the Z- noise matrix after
adding the passive components to the noisy two-port.
55
2.4 Open/Short De-embedding
The equivalent circuit diagram used for open-short de-embedding method is shown in
Fig. 2.8, including both the parallel parasitics Yp1, Yp2, Yp3, and the series parasitics ZL1, ZL2
and ZL3 surrounding the transistor [7]. Denote the S-parameters of the measurement as Smeas,
the S-parameters of the open de-embedding structure as Sopen, and the S-parameters of the short
de-embedding structure as Sshort. Using the relations between Y- and S- parameters in Table 2.2,
the Y-parameters of the measurement, the open and short de-embedding structure, Ymeas, Yopen
and Yshort are obtained. Open and short de-embedding are performed for both Y-parameters and
noise parameters to move the reference plane to the device terminals. The resulting Y-parameters
and noise parameters are for the transistor. The MATLAB programming for Y-parameters and
noise parameters open-short de-embedding is given in Appendix A.
Figure 2.8: Equivalent circuit diagram used for open-short de-embedding method, including
both the parallel parasitics Yp1, Yp2, Yp3, and the series parasitics ZL1, ZL2 and ZL3 surrounding
the transistor [7].
56
2.4.1 Open De-embedding of Y-parameters and Noise Parameters
The Y-parameter for the open de-embedded transistor Yod is [7]
Yod = Ymeas  Yopen: (2.99)
The short de-embedding structure also needs to be open de-embedded. The Y-parameter for the
open de-embedded short de-embedding structure Yos is [7]
Yos = Yshort  Yopen: (2.100)
Denote the noise parameters for measurement as NFmin, Rn and ?opt, where
?opt = Y0  YoptY
0  Yopt
: (2.101)
Yopt can be thus obtained by ?opt as
Yopt = Y0 1  ?opt1 +?
opt
: (2.102)
The chain noise representation matrix of the measurement CA;meas can be thus obtained using
(2.32). To perform open-short de-embedding, CA;meas needs to be transformed to the Y-noise
representation matrix CY;meas using (2.72),
CY;meas = TA Y ?CA;meas ?TyA Y; (2.103)
57
and TA Y is given by Table 2.1:
TA Y =
2
64  Ymeas11 1
 Ymeas21 0
3
75; (2.104)
where Ymeas11 and Ymeas21 are elements of Ymeas matrix. Therefore, the Y- noise representation
matrix for open de-embedded transistor CY;od is
CY;od = CY;meas  4kT<[Yod]: (2.105)
2.4.2 Short De-embedding of Y-parameters and Noise Parameters
The Z-parameter for the short de-embedded transistor Z is [7]
Z = Zod  Zos; (2.106)
where Zod and Zos are Z-parameter matrices of the open de-embedded transistor and the short
de-embedding structure, respectively. Zod and Zos are obtained from Yod and Yos using Table 2.2.
For short de-embedding of the noise parameters, we need to start with the Z-noise repre-
sentation matrix of the open de-embedded transistor CZ;od,
CZ;od = TY Z ?CY;od ?TyY Z; (2.107)
58
and TY Z is given by Table 2.1:
TY Z =
2
64 Zod11 Zod12
Zod21 Zod22
3
75; (2.108)
where Zod11, Zod12, Zod21, and Zod22 are elements of Zod matrix. The Z-noise representation matrix
of the open-short de-embedded transistor CZ is thus obtained,
CZ = CZ;od  4kT<[Zos]: (2.109)
Fig. 2.9 ? Fig. 2.16 show the bias and frequency dependence of the noise parameters NFmin,
Rn, and Yopt of raw measurement data, open de-embedding, and open-short de-embedding data.
The results show that the short de-embedding is important for noise parameters de-embedding,
and cannot be neglected.
2.4.3 Problems Encountered in MATLAB Programming for Open-Short De-embedding
The open-short de-embedding process is realized in MATLAB. The conversions of di er-
ent noise representations can be accomplished using MATLAB matrices operation. However,
unexpected imaginary part are obtained for some elements in the matrix which should be real
numbers theoretically. Here, measurement data of 0.12 ?m process measured in IBM is used as
an example. Vgs = 0.685 V, Vds = 1.5 V. At f = 28 GHz, CA for raw data is
CA =
2
64 0:39291636000000 0:01285241162005  0:02039784627097i
0:01285241162005 + 0:02039784627097i 0:00166866364322
3
75:
(2.110)
59
4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
Frequency    (GHz)
NF
min
    (dB)
raw data
open de?embeded
open?short de?embeded
L = 0.18 mm
W = 10 mm
N
f
 = 8  
I
DS
 = 148 mA/mm 
Figure 2.9: NFmin v.s. frequency. IDS = 148 ?A=?m. VDS = 1 V.
The very first step is to transform chain noise representation matrix CA to Y- noise representation
matrix CY using (2.103). The transform matrice T is
T =
2
64  0:02092695425297  0:06129572408422i 1
 0:06717448393087 + 0:14736069526087i 0
3
75: (2.111)
When realizing (2.103) in MATLAB, if the following code is used,
CA = [Sva, Siava?; Siava, Sia];
T = [-Y11, 1; -Y21, 0]; T_conjtrans = T?;
CY = T * CA * T_conjtrans;
Si1 = CY(1,1); Si2 = CY(2,2); Si1i2 = CY(1,2);
60
0 20 40 60 80 100 120 140
0
1
2
3
4
5
I
DS
    (mA/mm)
NF
min
    (dB)
raw data
open de?embeded
open?short de?embededf = 10 GHz 
L = 0.18 mm
W = 10 mm
N
f
 = 8  
Figure 2.10: NFmin v.s. IDS normalized by size of device. f = 10 GHz. VDS = 1 V.
the resulting Si1 is (0.00278463150577 - 0.00000000000000i), with neglegible imaginary part,
which is theoretically wrong. The origin of the problem lies in the complex number operation
in MATLAB. Let x be a complex number, and y be a real number. In MATLAB programming,
x*x?*y gives a real number. However, x*y*x? gives a complex number with an imaginary part.
Although the produced imaginary part is negligible for one step of calculation, the induced error
cannot be neglected after multiple steps of similar operations. For example, the resulting the
open-short de-embedded NFmin for the transistor using matrix operation is (1.45838190834363
+ 0.01511935061286i), which has considerable imaginary part. Therefore MATLAB matrix
operation cannot be directly used. Instead, detailed operations for each element of a matrix are
applied:
CA = [Sva, Siava?; Siava, Sia];
T = [-Y11, 1; -Y21, 0]; T_conjtrans = T?;
61
4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Frequency    (GHz)
R
n
/50
W
 
raw data
open de?embeded
open?short de?embeded
I
DS
 = 148 mA/mm 
L = 0.18 mm
W = 10 mm
N
f
 = 8  
Figure 2.11: Rn v.s. frequency. IDS = 148 ?A=?m. VDS = 1 V.
CY(1,1) = (abs(T(1,1)))^2*CA(1,1) + (abs(T(1,2)))^2*CA(2,2)...
+ 2*real(T_conjtrans(1,1)*T(1,2)*CA(2,1));
CY(1,2) = T(1,1)*T_conjtrans(1,2)*CA(1,1)+T(1,2)*T_conjtrans(1,2)*CA(2,1)...
+T(1,1)*T_conjtrans(2,2)*CA(1,2)+T(1,2)*T_conjtrans(2,2)*CA(2,2);
CY(2,1) = CY(1,2)?;
CY(2,2) = (abs(T(2,1)))^2*CA(1,1) + (abs(T(2,2)))^2*CA(2,2)...
+ 2*real(T_conjtrans(2,2)*T(2,1)*CA(1,2));
Si1 = CY(1,1); Si2 = CY(2,2); Si1i2 = CY(1,2);
The resulting Si1 is 0.00278463150577, which has no imaginary part. After multiple steps, the
open-short de-embedded NFmin for the transistor is 1.45574769257762, which is slightly lower
than the real part of the result using matrix operation.
62
0 20 40 60 80 100 120 140
0
0.5
1
1.5
2
2.5
3
3.5
4
I
DS
    (mA/mm)
R
n
/50
W
 
raw data
open de?embeded
open?short de?embeded
f = 10 GHz 
L = 0.18 mm
W = 10 mm
N
f
 = 8  
Figure 2.12: Rn v.s. IDS normalized by size of device. f = 10 GHz. VDS = 1 V.
2.5 Transistor Internal Noise De-embedding
MOSFET transistor ig and id noise de-embedding procedure and SiGe HBT transistor ib
and ic noise de-embedding procedure are discussed in this section. The techniques are repeatedly
used in later chapters of this dissertation.
2.5.1 MOSFET Transistor ig and id Noise De-embedding
The equivalent circuit of the transistor is shown in Fig. 2.17. Here Rg is the gate electrode
resistance, and Rs and Rd are the source and drain series resistances. Rg, Rs and Rd all have
the usual 4kTR thermal noise voltage. Rgs is the non-quasi-static (NQS) channel resistances.
gds is the output conductance. gm is transconductance. Cgs and Cgd are the gate to source and
gate to drain capacitances. Cdb is the drain to body junction capacitance, and Rdb is the body
63
4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
6
7
8
Frequency    (GHz)
G
opt
    (mS)
raw data
open de?embeded
open?short de?embeded
I
DS
 = 148 mA/mm 
L = 0.18 mm
W = 10 mm
N
f
 = 8  
Figure 2.13: Gopt v.s. frequency. IDS = 148 ?A=?m. VDS = 1 V.
resistance of the drain to body junction. Rdb has the usual 4kTR thermal noise. The equivalent
circuit parameters are extracted using the method described in [9]. Note that Rgs, and gds do not
have the usual 4kTR thermal noise. Instead, ig and id, the Y-noise representation parameters,
are used to describe all of the noise from the intrinsic transistor.
Here we choose to define ig and id as the Y-representation input and output noise current
for the level II block shown in Fig. 2.17. The level II block consists of Rg, Cgs, the gm controlled
source and gds, and is the core part for noise modeling. The level I block is defined as the
combination of the level II block with the branch of Cgd, and the branch of Cdb and Rdb. Next
we need to extract the power spectral densities (PSD) of ig, id, and their correlation, which we
64
0 20 40 60 80 100 120 140
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
I
DS
    (mA/mm)
G
opt
    (mS)
raw data
open de?embeded
open?short de?embeded
f = 10 GHz 
L = 0.18 mm
W = 10 mm
N
f
 = 8  
Figure 2.14: Gopt v.s. IDS normalized by size of device. f = 10 GHz. VDS = 1 V.
denote as SIIi
g;i?g
, SIIi
d;i?d
, and SIIi
g;i?d
. They can also be written using matrix notation as:
CYII 4=
2
64 SIIig;i?g SIIig;i?d
SIIi
d;i?g
SIIi
d;i?d
3
75; (2.112)
where CYII is also referred to as the Y-representation noise matrix for the level II block.
Firstly, the thermal resistances outside of the level I block, Rg, Rs and Rd, need to be
removed. Denote the Z-parameters of the level I block as ZI, which is related to Z as
ZI = Z  Z1; (2.113)
65
4 6 8 10 12 14 16 18 20
?18
?16
?14
?12
?10
?8
?6
?4
?2
0
Frequency    (GHz)
B
opt
    (mS)
raw data
open de?embeded
open?short de?embeded
I
DS
 = 148 mA/mm 
L = 0.18 mm
W = 10 mm
N
f
 = 8  
Figure 2.15: Bopt v.s. frequency. IDS = 148 ?A=?m. VDS = 1 V.
where
Z1 =
2
64 Rs +Rg Rs
Rs Rs +Rd
3
75: (2.114)
Using the open-short de-embedded transistor Z- noise representation matrix CZ, the Z- noise
representation matrix of the level I block CZI is
CZI = CZ  4kT<[Z1]: (2.115)
The next step is to remove the branch of Cgd, and the branch of Cdb and Rdb to obtain the
Y- noise representation matrix of the level II block CYII. Y-parameters matrix of the level I block
YI can be obtained from ZI using Table 2.2. Therefore Y-parameters matrix of the level II block
66
0 20 40 60 80 100 120 140
?10
?8
?6
?4
?2
0
I
DS
    (mA/mm)
B
opt
    (mS)
raw data
open de?embeded
open?short de?embeded
f = 10 GHz 
L = 0.18 mm
W = 10 mm
N
f
 = 8  
Figure 2.16: Bopt v.s. IDS normalized by size of device. f = 10 GHz. VDS = 1 V.
YII is,
YII = YI  Y1; (2.116)
Y1 =
2
64 j!Cgd  j!Cgd
 j!Cgd j!Cgd + j!Cdb1+j!CdbRdb
3
75: (2.117)
The Y-representation noise matrix for the level I block, CYI can be obtained from CZI as,
CYI = TZ Y ?CZI ?TZ Yy; (2.118)
67
g
R
gs
C
gs
R
 
ds
g
/S   B /S   B
G
D
 
 
gs
v
gd
C
d
R
+
-
s
R
db
C
4
d
kTR
4
g
kTR
4
s
kTR
level II
level I
,
+ -
db
R
+
- 4
db
kTR
+ -
m                gs
g  e      v
 ??   - j
*
II
g  g
i  i
S
*
II
d  d
i  i
S
,
Figure 2.17: The small signal equivalent circuit model used with Y-representation noise sources.
and TZ Y is given by Table 2.1:
TZ Y =
2
64 YI11 YI12
YI21 YI22
3
75; (2.119)
where YI11, YI12, YI21, and YI22 are elements of YI matrix. Therefore the Y- noise representation
matrix of the level II CYII is
CYII = CYI  4kT<[Y1]: (2.120)
68
Thus, the ig and id noise currents of MOSFET transistor are finally de-embedded from
measurement data,
SIIi
g;i?g
= CYII (1;1); (2.121)
SIIi
d;i?d
= CYII (2;2); (2.122)
SIIi
g;i?d
= CYII (1;2): (2.123)
Fig. 2.18 ? Fig. 2.20 shows the bias dependence of Y- noise current sources for the whole
transistor and the intrinsic transistor for 0.24 ?m gate length MOSFET transistor. W = 4 ?m,
number of finger Nf is 128. The gate resistance Rg is extracted using the advanced parameter
extraction method in chapter 7. Rg = 0.6  . Both the input and output Y- noise representa-
tion currents decreases after deembedding to the intrinsic device. The imaginary part of their
correlation is also less for the intrinsic device.
2.5.2 SiGe HBT Transistor ib and ic Noise De-embedding
The process of SiGe HBT transistor ib and ic noise de-embedding is similar to the pro-
cedures in section 2.5.1. The thermal noise of a SiGe HBT transistor is simulated using 2-D
DESSIS v9.0 simulation tool [45]. The output of DESSIS simulation tool is the Y- parame-
ter and the Z- noise representation parameters Sv1;v?1 , Sv2;v?2 and Sv2;v?1 (Sv1;v?2 for DESSIS v7.0).
Firstly we are interested in calculating the noise parameters NFmin, Rn and Yopt, which inevitably
involves the calculation of chain noise representation parameters Sva;v?a , Sia;i?a, and Sia;v?a from Z-
noise representation parameters.
69
0 100 200 300 400 500
0
0.2
0.4
0.6
0.8
1
x 10
?22
I
DS
    (mA/mm)
S
ig,ig*
    (A
2
/Hz)
whole transistor
intrinsic transistor
Data: L = 0.24 mm, W = 4 mm, Nf = 128 
V
ds
 = 1.2 V 10 GHz 
Figure 2.18: Y- noise representation input noise current for the whole and the intrinsic MOSFET
transistor.
Denote Y- parameters of the output of DESSIS simulation as Y , the Z- noise representation
matrix of the output of DESSIS simulation as CZ. The chain noise representation matrix CA is
CA = TZ A ?CZ ?TyZ A; (2.124)
and TZ A is given by Table 2.1:
TZ A =
2
64 1  A12
0  A21
3
75; (2.125)
70
0 100 200 300 400 500
0
0.2
0.4
0.6
0.8
1
x 10
?20
I
DS
    (mA/mm)
S
id,id*
    (A
2
/Hz)
whole transistor
intrinsic transistor
V
ds
 = 1.2 V 10 GHz 
Data: L = 0.24 mm, W = 4 mm, Nf = 128 
Figure 2.19: Y- noise representation output noise current for the whole and the intrinsic MOS-
FET transistor.
where A12 and A21 are elements of ABCD matrix A, which can be converted from Y using
Table 2.2. The noise parameters NFmin, Rn and Yopt can then be obtained using (2.24) ? (8.19)
directly.
Secondly, we are interested in ib and ic noise currents of SiGe HBT transistor. The equiv-
alent circuit for the simulated SiGe HBT transistor is the same as Fig. 1.2 in chapter 1, which
includes base resistance rb with usual 4kTR thermal noise voltage, and the intrinsic transistor
whose noise is described using Y- noise representation parameters Sib;i?b, Sic;i?c and Sic;i?b.
71
0 100 200 300 400 500
?1
?0.5
0
0.5
1
I
DS
    (mA/mm)
?
(C
ig,id*
)
whole transistor
intrinsic transistor
V
ds
 = 1.2 V 10 GHz 
Data: L = 0.24 mm, W = 4 mm, Nf = 128 
0 100 200 300 400 500
?1
?0.5
0
0.5
1
I
DS
    (mA/mm)
`
(C
ig,id*
)
whole transistor
intrinsic transistor
V
ds
 = 1.2 V 10 GHz 
Data: L = 0.24 mm, W = 4 mm, Nf = 128 
Figure 2.20: Y- noise representation correlation for the whole and the intrinsic MOSFET tran-
sistor.
Through circuit analysis, Fig. 1.2 and Fig. 2.1 show
0
B@ I1  ib
I2  ic
1
CA=
2
64 Yint11 Yint12
Yint21 Yint22
3
75?
0
B@ V1  vb  I1rb
V2
1
CA; (2.126)
0
B@ I1  ia
I2
1
CA=
2
64 Y11 Y12
Y21 Y22
3
75?
0
B@ V1  va
V2
1
CA: (2.127)
Y11, Y12, Y21 and Y22 are Y- parameters for the whole transistor Y that includes both rb and the
intrinsic transistor. Yint11 , Yint12 , Yint21 and Yint22 are elements of the intrinsic transistor Y- parameters
matrix Yint . The intrinsic Y- parameters Yint relates to whole Y- parameters Y as,
Yint11 = Y111  Y
11rb
; (2.128)
Yint12 = Y121  Y
11rb
; (2.129)
Yint21 = Y211  Y
11rb
; (2.130)
Yint22 = Y221  Y
11rb
 rb(Y11Y22  Y12Y21)1  Y
11rb
: (2.131)
72
From (2.126), we have,
I1 = ib +Yint11 (V1  vb)  I1rbYint11 +Yint12 V2; (2.132)
= ib1 +Yint
11 rb
+ Y
int
11
1 +Yint11 rb (V1  vb) +
Yint12
1 +Yint11 rbV2; (2.133)
= ib1 +Yint
11
+Y11(V1  vb) +Y12V2; (2.134)
and
I2 = ic +Yint21 (V1  vb  I1rb) +Yint22 V2: (2.135)
Substituting (2.134) in (2.135),
I2 = ic +Yint21 (V1  vb)  Y
int
21 ibrb
1 +Yint11 rb  Y
int
21 Y11(V1  vb)rb  Y
int
21 Y12V2rb +Y
int
22 V2; (2.136)
= ic +Yint21 (1  Y11rb)(V1  vb)  Y21ibrb +V2(Yint22  Y12Yint21 rb); (2.137)
= ic +Y21(V1  vb)  Y21ibrb +V2
? Y
22
1  Y11rb  
rb(Y11Y22  Y12Y21)
1  Y11rb  
Y12Y21rb
1  Y11rb
 
; (2.138)
= ic +Y21(V1  vb)  Y21ibrb +Y22V2: (2.139)
73
From (2.127), we have,
I1 = ia +Y11(V1  va) +Y12V2; (2.140)
I2 = Y21(V1  va) +Y22V2: (2.141)
(2.141)-(2.139), we have
va = vb  icY
21
+ibrb: (2.142)
(2.140)-(2.134), and using the result of (2.142), we have
ia = ib1 +Yint
11 rb
+Y11(va  vb); (2.143)
= ib1 +Yint
11 rb
 Y11Y
21
ic +Y11ibrb; (2.144)
= ib  Y11Y
21
ic: (2.145)
Finally, Fig. 1.2 can be transformed to the form of the chain noise representation Fig. 2.1,
va = vb +ibrb  1Y
21
ic; (2.146)
ia = ib  Y11Y
21
ic; (2.147)
= ib  ich
21
; (2.148)
74
where
h21 = Y21Y
11
= Y
int
21
Yint11 = h
int
21: (2.149)
The resulting Sva;v?a , Sia;i?a and Sia;v?a are
Sva;v?a = Svb;v?b + 1jY
21j2
Sic;i?c +Sib;i?br2b  2<( rbY
21
Sic;i?b ); (2.150)
Sia;i?a = Sib;i?b +
??
??Y11
Y21
??
??
2
Sic;i?c  2<
?Y
11
Y21Sic;i?b
?
; (2.151)
Sia;v?a = Y11jY
21j2
Sic;i?c +Sibrb  1Y?
21
S?i
c;i?b
 rbh
21
Sic;i?b: (2.152)
On the contrary, Fig. 1.2 can be transformed from the form of the chain noise representation
Fig. 2.1,
ic =  Yinternal21 (va  vb  iarb); (2.153)
=  Y211  Y
11rb
(va  vb  iarb); (2.154)
ib = ia  Yinternal11 (va  vb  iarb); (2.155)
= 11  Y
11rb
ia  Y111  Y
11rb
(va  vb): (2.156)
75
The resulting Sib;i?b, Sic;i?c and Sic;v?b are
Sib;i?b = 1j1  Y
11rbj2
(Sia;i?a +jY11j2(Sva;v?a  4kTrb)  2<(Y?11Sia;v?a )); (2.157)
Sic;i?c = j Y211  Y
11rb
j2(Sva;v?a  4kTrb +Sia;i?ar2b  2rb<Sia;v?a ); (2.158)
Sic;i?b = 1j1  Y
11rbj2
(Y21rbSia;i?a +Y21Y?11(Sva;v?a  4kTrb)  Y21S?i
a;v?a
 Y21Y?11rbSia;v?a ):
(2.159)
The base resistance rb for each bias is determined using semi-circle fitting method [46]. Plot
=(h11) versus <(h11), fit the data using a semi-circle, rb is determined using the high frequency
intercept with the <(h11) axis. Using (2.157) ? (2.159), the ib and ic noise currents of SiGe HBT
transistor are thus obtained.
Fig. 2.21 and Fig. 2.22 shows Y- noise current sources for the whole transistor and the
intrinsic transistor for DESSIS simulation results of 8HP 0:12 ? 1 ?m2 SiGe HBT transistor
at 40 GHz. Both the input noise current and output noise current become less for the intrinsic
device. The absolute value of Y- noise representation correlation decreases after de-embedding
to the intrinsic device.
2.6 Importance of Terminal Series Resistances to Noise parameters
The gate electrode resistance Rg for MOSFET transistors, or the base resistance rb for SiGe
HBT transistors is the input series resistance to the intrinsic device. The input series resistance
is the most important for noise parameters, since its thermal noise contribution is amplified by
the two-port network. On the contrary, the output series resistance is the least important for
76
1 10
10
?24
10
?23
J
C
    (mA/mm
2
)
S
ib,ib*
    (A
2
/Hz)
whole transistor
intrinsic transistor
1 10
10
?23
10
?22
10
?21
J
C
    (mA/mm
2
)
S
ic,ic*
    (A
2
/Hz)
whole transistor
intrinsic transistor
Figure 2.21: Y- noise representation input and output noise currents for the whole and the intrin-
sic SiGe HBT transistor.
0 5 10 15
?1.4
?1.2
?1
?0.8
?0.6
?0.4
?0.2
0
x 10
?23
J
C
    (mA/mm
2
)
?
S
ic,ib*
    (A
2
/Hz)
whole transistor
intrinsic transistor
1 10
?10
?22
?10
?23
?10
?24
J
C
    (mA/mm
2
)
`
S
ic,ib*
    (A
2
/Hz)
whole transistor
intrinsic transistor
Figure 2.22: Y- noise representation correlation for the whole and the intrinsic SiGe HBT tran-
sistor.
77
0 100 200 300 400 500 600 700
0
0.2
0.4
0.6
0.8
1
I
DS
    (mA/mm)
NF
min
    (dB)
f =5 GHz Simulation: intrinsic
Adding R
s
 and R
d
Adding R
s
, R
d
 and R
g
V
DS
=1 V 
R
s
=R
d
=45 W 
R
g
=120 W 
intrinsic 
adding R
s
 and R
d
 
adding R
s
, R
d
 and R
g
 
Simulation: L = 50 nm, W = 2 mm
Figure 2.23: NFmin vs IDS with and without Rg, Rs and Rd at 5 GHz.
0 100 200 300 400 500 600 700
0
1000
2000
3000
4000
5000
6000
I
DS
    (mA/mm)
R
n
    (
W
)
Simulation: intrinsic
Adding R
s
, R
d
 and R
g
f =5 GHz V
DS
=1 V 
intrinsic 
Simulation: L = 50 nm, W = 2 mm
adding R
s
, R
d
 and R
g
 
R
g
=120 W 
R
s
=R
d
=45 W 
Figure 2.24: Rn vs IDSwith and without Rg, Rs and Rd at 5 GHz.
78
0 100 200 300 400 500 600 700
0
1
2
3
4
x 10
?5
G
opt
    (S)
Simulation: intrinsic
Adding R
s
, R
d
 and R
g
f =5 GHz V
DS
=1 V 
intrinsic 
Simulation: L = 50 nm, W = 2 mm
adding R
s
, R
d
 and R
g
 
R
g
=120 W 
R
s
=R
d
=45 W 
I
DS
    (mA/mm) 
Figure 2.25: Gopt vs IDS with and without Rg, Rs and Rd at 5 GHz.
0 100 200 300 400 500 600 700
?7
?6
?5
?4
?3
?2
?1
0
x 10
?5
B
opt
    (S)
Simulation: intrinsic
Adding R
s
, R
d
 and R
g
f =5 GHz V
DS
=1 V 
intrinsic 
Simulation: L = 50 nm, W = 2 mm
adding R
s
, R
d
 and R
g
 
R
g
=120 W 
R
s
=R
d
=45 W 
I
DS
    (mA/mm) 
Figure 2.26: Bopt vs IDS with and without Rg, Rs and Rd at 5 GHz.
79
noise parameters. The drain electrode resistance Rd for MOSFET transistors and the collector
resistance rc for SiGe HBT transistors are the output series resistance to the intrinsic device.
The source electrode resistance Rs for MOSFET transistors, or the emitter resistance re for
SiGe HBT transistors, contributes thermal noise to the input terminal, the output terminal, and
their correlation. Therefore they are less important for noise parameters compared to the input
series resistance.
Fig. 2.23 shows NFmin simulated at 5 GHz versus IDS. We find that Rs is the major reason
for the increase of NFmin compared to the intrinsic NFmin, yet it is a function of the Rg and Rs
values. Fig. 2.24 shows Rn vs IDS, and Fig. 2.25 and Fig. 2.26 show Gopt and Bopt vs IDS at 5
GHz.
2.7 Summary
Di erent noise representations for a linear noisy two-port network are introduced. The
transformation matrices to other noise representations are given for ABCD-, Y-, Z-, and H-
noise representations. Techniques of adding or de-embedding a passive component to a linear
two-port network are discussed. Noise sources de-embedding for both MOSFET and SiGe HBT
are given for repeatedly use in later chapters.
80
CHAPTER 3
MICROSCOPIC NOISE CONTRIBUTIONS
This chapter presents a new technique of simulating the spatial distribution of microscopic
noise contribution to the input noise current, voltage, and their correlation. The technique is
demonstrated on a 50 GHz SiGe HBT. A strong ?noise crowding? e ect is observed in the
spatial distribution of noise concentrations due to base majority holes. The spatial distributions
by base majority holes, base minority electrons, and emitter minority holes are analyzed, and
compared to the compact noise model. This technique is also applied to a 120 GHz MOSFET
transistor. The spatial distribution of drain noise current, gate noise current, and their correlation
are analyzed.
3.1 Introduction
One of the key concerns in optimizing SiGe HBTs is to minimize noise, which requires
methods of simulating transistor noise parameters for a given device design. One method is
to simulate transistor s-parameters, extract parameters of an equivalent circuit, and then calcu-
late the noise parameters using a circuit-level transistor noise model [47]. The accuracy of this
method is limited by the accuracy of the transistor noise model used. The other method is micro-
scopic noise simulation. The terminal voltage noise is obtained by summing the responses of the
terminal voltage to carrier velocity fluctuations, and hence current density fluctuations at each
grid cell, which is the basic element for equation solutions, using Shockley?s impedance field
approach [48], which has recently become available in TCAD tools. The results of microscopic
noise simulation are typically given by the spatial distribution of either the open circuit noise
81
voltages or the short circuit noise currents. For comparison with measurements, however, the
input noise current ia and the input noise voltage va for a chain representation shown in Fig. 2.1
in chapter 2 are the most convenient [47] [10]. The spectral densities of ia, va and ia;v?a directly
relate to circuit-level noise parameters: minimum noise figure NFmin, noise resistance Rn, and
the optimal source admittance Yopt by (2.24) ? (8.19) in chapter 2.
This chapter presents a new technique of simulating the spatial distribution of microscopic
noise contributions to the input noise current, voltage and their correlation, and results obtained
on a 50 GHz SiGe HBT technology [49]. The technique facilities the identification of major
noise sources within the transistor physical structure, leading to device-level optimization, such
as doping profile, Ge profile, and/or device layout, with respect to the noise parameters.
3.2 Microscopic Noise Simulation
Shockley?s impedance field approach is illustrated in Fig. 3.1 [48]. Velocity fluctuation
(thermal agitation of carriers) causes current density fluctuation fiIn=p(r). Current density fluc-
tuations at each location propagate towards the contact through Zn=p(r;rContact). Noise voltage
fluctuation results at each contact with fiV (rContact) [45].
The local noise source CSi are proportional to carrier density and di usivity,
CnSi = 4qnDn; (3.1)
CpSi = 4qpDp; (3.2)
where superscripts n and p denote electron and hole respectively. CSi has a unit of A2/Hz/cm3.
n and p are electron and hole concentrations. Dn and Dp are electron and hole di usivity, re-
spectively. The impedance field eZ(r;rContact) from local noise source to terminal noise voltage
82
( ) 
Contactpn
rrZ ,
( ) rI
pn
? 
( ) 
Contact
rV?
Figure 3.1: Impedance field method
is,
eZn(r;rContact) = 1
q rZn(r;rContact); (3.3)
eZp(r;rContact) = 1
q rZp(r;rContact); (3.4)
where subscripts n and p denote electron and hole respectively. jeZn(r;rContact)j2 andjeZp(r;rContact)j2
have a unit of V2/A2.
83
The terminal noise voltage power spectral density is obtained by integrating the ?noise
concentration? over the device volume,
Sv =
Z
 
CSvd ; (3.5)
=
Z
 
CnSvd +
Z
 
CpSvd ; (3.6)
CnSv = eZn(r;rContact)CnSi eZ?n (r;rContact); (3.7)
CpSv = eZp(r;rContact)CpSi eZ?p (r;rContact); (3.8)
where CSv is the ?concentration,? or volume density of Sv, and has a unit of V2=Hz/cm3.
3.3 New Technique: Microscopic Noise Contribution of Chain Noise Representation Pa-
rameters
Consider the transistor as a noisy linear two port. The open circuit noise voltage parameters
are obtained by integrating the ?noise concentration? over the device volume
Sn =
Z
 
CSnd ; (3.9)
where n is v1;v?1, v2;v?2, or v1;v?2. For instance, CSv1;v?
1
is the ?concentration,? or volume den-
sity of Sv1;v?1 , and has a unit of V2=Hz=cm3. CSv1;v?
1
, CSv2;v?
2
and CSv1;v?
2
are solved in TCAD
tools including DESSIS [45] and TAURUS [44]. In principle, the boundary conditions can be
modified to directly solve for the ?concentration? of the chain representation noise parameters
Sva;v?a , Sia;i?a and Sia;v?a . This, however, has not been implemented in TCAD tools. We propose
here an alternative that uses postprocessing of CSv1;v?
1
, CSv2;v?
2
and CSv1;v?
2
, and requires no code
84
development by TCAD vendors. The impedance representation noise parameters Sv1;v?1 , Sv2;v?2
and Sv1;v?2 can be transformed to the chain representation noise parameters Sva;v?a , Sia;i?a and Sia;v?a
using transformation matrix in Table 2.1 in chapter 2 [43],
CA = TZ A ?CZ ?TyZ A; (3.10)
2
64 Sva;v?a Sva;i?a
Sia;v?a Sia;i?a
3
75= T
Z A ?
2
64 Sv1;v?1 Sv1;v?2
Sv2;v?1 Sv2;v?2
3
75?Ty
Z A; (3.11)
and
TZ A =
2
64 1  A11
0  A21
3
75; (3.12)
where A11 and A21 are elements of the ABCD parameter matrix A.
An inspection of (3.11) shows that the transform is linear. Substituting Sv1;v?1 , Sv2;v?2 and
Sv1;v?2 expressed in the integral form of (3.9) into (3.11), the concentration of the chain represen-
tation noise parameters, CSva;v?a , CSia;i?
a , and CSia;v?a are obtained as
2
64 CSva;v?a CSva;i?a
CSia;v?a CSia;i?
a
3
75= T
Z A ?
2
64 CSv1;v?1 CSv1;v?2
CSv2;v?
1
CSv2;v?
2
3
75?Ty
Z A: (3.13)
Integration of CSva;v?a , CSia;i?
a , and CSia;v?a over the whole device gives the transistor Sva;v
?a , Sia;i?a
and Sia;v?a , respectively.
85
3.4 Spatial Distribution of Microscopic Noise Contributions in RF SiGe HBT Transistor
The technique is applied to noise analysis of a 50 GHz SiGe HBT [49]. DESSIS from ISE
is used for noise simulation. The device structure is constructed based on device layout. The
doping and Ge profiles were determined using SIMS. A set of physical models suitable for HBT
simulation were selected, and the model coe cients were calibrated to reproduce the measured
dc I  V characteristics and high frequency s-parameters. The carrier noise temperature is
assumed to be the same as the lattice temperature. The DESSIS simulation input deck and
TECPLOT mcr file can be found in B.1 in Appendix B.
3.4.1 Input Noise Voltage Sva;v?a
Fig. 3.2 shows the spatial distribution of the input noise voltage concentration, CSva;v?a . The
transistor is biased at a relatively low JC of 0.1 mA=?m2, and the operating frequency is 2 GHz.
Observe that CSva;v?a is the highest in the SiGe base, indicating that transistor Sva;v?a mainly comes
from the SiGe base. This provides guidelines to development of better circuit-level transistor
noise model. That is, the noise sources originate from the EB junction.
To identify the individual contributions from electrons and holes, the spatial distributions of
the CSva;v?a due to electrons and holes are plotted in Figs. 3.3 and 3.4, respectively. It can be seen
that the electron contributions mainly come from the minority electrons in the base, and the hole
contributions mainly come from the majority holes in the base. The CSva;v?a due to electrons is
nearly uniform along the x-direction inside the neutral base. However, the CSva;v?a due to holes is
highly nonuniform along the x-direction, indicating a strong ?noise crowding? e ect. The CSva;v?a
due to holes decreases from the emitter periphery towards the emitter center. The overall CSva;v?a
86
X (?m)
Y
(
?
m
)
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
0
0.1
0.2
0.3
0.4
0.5
0.6
S
va
Conc. (V
2
/Hz/cm
3
)
2.3417E-03
1.8913E-03
1.4410E-03
9.9070E-04
5.4038E-04
9.0064E-05
total contribution 
(electron+hole)
E
B
C
JC=0.1 mA/?m
2
Figure 3.2: 2D distribution of the total noise concentration CSva;v?a at 2 GHz. JC=0.1 mA=?m2.
is relatively uniform along the emitter width direction, simply because the electron contribution
dominates (73%).
However, as JC increases to 0.5 mA=?m2, the hole contribution to CSva;v?a becomes more
dominant, and counts for 60% of the total Sva;v?a . This results in considerable crowding of the
total CSva , as shown in Fig. 3.5. Interestingly, the electron contribution to CSva;v?
a remains uniform
laterally inside the neutral base, despite the higher JC and hence more severe dc and ac current
crowding e ect. The crowding in the hole contribution becomes stronger as JC increases.
A logical and interesting question is how the microscopic noise simulation results compare
to circuit-level compact noise modeling results. We consider here the SPICE noise model used
in [47] and [10] as introduced in chapter 1. The base current shot noise 2qIB, the collector
current shot noise 2qIC, and the thermal noise 4kTrb are accounted for using the equivalent
circuit shown in Fig. 1.4. The compact noise model is lumped in nature, and does not take into
87
X (?m)
Y
(
?
m
)
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
0
0.1
0.2
0.3
0.4
0.5
0.6
S
va
Conc. (V
2
/Hz/cm
3
)
2.1545E-03
1.7401E-03
1.3258E-03
9.1150E-04
4.9718E-04
8.2864E-05
 
electron contribution
E
B
C
JC=0.1 mA/?m 
2 
Figure 3.3: 2D distribution of electron noise concentration CSva;v?a at 2 GHz. JC=0.1 mA=?m2.
account distributive e ects. Roughly speaking, the 2qIC collector current shot noise results from
minority electrons in the base, the 2qIB base current shot noise results from minority holes in
the emitters [50], and the 4kTrb base resistance thermal noise results from the majority holes in
the neutral base. Sva;v?a is then obtained by taking SPICE model equations (1.11) ? (1.13) into
(2.150) derived in chapter 2 [47] [12],
Sva;v?a = 2qICjY
21j2
+ 2qIBr2b + 4kTrb; (3.14)
Qualitatively, the compact model captures the two major contributors to Sva;v?a , base major-
ity holes, and base minority electrons. Furthermore, the simulated majority hole contribution is
equal to 4kTrb, provided that rb is extracted from h11 [46]. However, the electron contribution,
2qIC=jY21j2, di ers from microscopic results by 16% at JC=0.1 mA=?m2, and 7% at JC=0.5
mA=?m2. This is clearly caused by the lumped nature of the compact noise model, which cannot
88
X (?m)
Y
(
?
m
)
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
0
0.1
0.2
0.3
0.4
0.5
0.6
S
va
Conc. (V
2
/Hz/cm
3
)
3.9089E-04
3.1572E-04
2.4055E-04
1.6538E-04
9.0206E-05
1.5034E-05
E
B
C
hole contribution
 
JC=0.1 mA/?m
2
Figure 3.4: 2D distribution of hole noise concentration CSva;v?a at 2 GHz. JC=0.1 mA=?m2.
take into account 2D distributive e ect. The results suggest that 2D distributive e ect should be
modeled to obtain more accurate Sva;v?a .
3.4.2 Input Noise Current Sia;i?a
Fig. 3.6 ? Fig. 3.8 show the spatial distribution of CSia;i?a , as well as the individual contribu-
tions from electrons and holes, respectively. JC=0.1 mA=?m2. Most of the electron contribution
comes from the base minority electrons. While for the hole contribution, both the emitter mi-
nority holes and the base majority holes are important. The hole contribution to Sia;i?a increases
from 63% to 81% of the total Sia;i?a as JC increases from 0.1 to 0.5 mA=?m2. Sia;i?a is given by
taking SPICE model equations (1.11) ? (1.13) into (2.151) [47] [12],
Sia;i?a = 2qIB + 2qICjh
21j2
; (3.15)
89
X (?m)
Y
(
?
m
)
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
0
0.1
0.2
0.3
0.4
0.5
0.6
S
va
Conc. (V
2
/Hz/cm
3
)
6.9397E-04
5.6051E-04
4.2706E-04
2.9360E-04
1.6015E-04
2.6691E-05
E
B
C
total contribution
JC=0.5 mA/?m
2
Figure 3.5: 2D distribution of the total noise concentration CSva;v?a at 2 GHz. JC=0.5 mA=?m2.
Observe that the compact model does not have any contribution from the majority holes in the
base, which is significant according to noise simulation results. On the other hand, the simulated
electron contribution to Sia;i?a, is well predicted by 2qIC=jh21j2, within 5% accuracy. The hole
contribution to Sia;i?a, however, is overestimated by 2qIB by as high as 16% at JC=0.5 mA=?m2.
3.4.3 Input Noise Voltage and Current Correlation Sia;v?a
Fig. 3.9 shows the real part of CSia;v?a at JC = 0.1 mA=?m2. The simulation results show that
the electron contribution mainly comes from the base minority electrons, and the hole contribu-
tion mainly comes from the emitter minority holes. Fig. 3.10 shows the imaginary part of CSia;v?a .
Its electron contribution also mainly comes from the base minority electrons. The hole contribu-
tion, however, mainly comes from the base majority holes and shows a strong ?noise crowding?
90
X (?m)
Y
(
?
m
)
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
0
0.1
0.2
0.3
0.4
0.5
0.6
S
ia
Conc. (A
2
/Hz/cm
3
)
8.9130E-12
7.1989E-12
5.4849E-12
3.7709E-12
2.0568E-12
3.4281E-13
total contribution 
(electron+hole)
E
B
C
JC=0.1 mA/?m
2
Figure 3.6: 2D distribution of noise concentration CSia;i?a at 2 GHz. JC=0.1 mA=?m2.
e ect. However, the total CSia;v?a is dominated by the base electron contribution, which counts for
87% of the total <(Sia;v?a ) and 95% of the total =(Sia;v?a ). As JC increases to 0.5 mA=?m2, the
electron contribution for CSia;v?a becomes less dominant, and counts for 63% of the total <(Sia;v?a )
and 81% of the total =(Sia;v?a ).
The input noise voltage and current correlation Sia;v?a is predicted by taking SPICE model
equations (1.11) ? (1.13) into (2.152) [47] [12],
Sia;v?a = 2qIC Y11jY
21j2
+ 2qIBrb; (3.16)
Note that the compact model does not have any hole contribution at all, which can be important
according to noise simulation. For <(Sia;v?a ), the electron contribution predicted by (3.16) devi-
ates from noise simulation by 14% at JC=0.1 mA=?m2, and by 4% at JC = 0.5 mA=?m2. For
=(Siav?a ), the deviation is 9% and 69% at JC = 0.1 and 0.5 mA?m2, respectively. A significant
91
X (?m)
Y
(
?
m
)
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
0
0.1
0.2
0.3
0.4
0.5
0.6
S
ia
Conc. (A
2
/Hz/cm
3
)
7.6371E-12
6.1684E-12
4.6997E-12
3.2311E-12
1.7624E-12
2.9373E-13
electron contribution
E
B
C
JC=0.1 mA/?m
2
Figure 3.7: 2D distribution of electron contribution to noise concentration CSia;i?a at 2 GHz.
JC=0.1 mA=?m2.
source of deviation is due to the hole contribution, which does not exist in the compact model,
but can become important at higher JC.
92
X (?m)
Y
(
?
m
)
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
0
0.1
0.2
0.3
0.4
0.5
0.6
S
ia
Conc. (A
2
/Hz/cm
3
)
8.8933E-12
7.1830E-12
5.4728E-12
3.7625E-12
2.0523E-12
3.4205E-13
hole contribution
E
B
C
JC=0.1 mA/?m
2
Figure 3.8: 2D distribution of hole contribution to noise concentration CSia;i?a at 2 GHz. JC=0.1
mA=?m2.
X (?m)
Y
(
?
m
)
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
0
0.1
0.2
0.3
0.4
0.5
0.6
ReS
iava*
Conc. (VA/Hz/cm
3
)
3.3262E-08
2.6082E-08
1.8901E-08
1.1721E-08
4.5405E-09
-2.6400E-09
total contribution 
(electron+hole)
E
B
C
JC=0.1 mA/?m
2
Figure 3.9: 2D distribution of the total noise concentration <(CSia;v?a ) at 2 GHz. JC=0.1
mA=?m2.
93
X (?m)
Y
(
?
m
)
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
0
0.1
0.2
0.3
0.4
0.5
0.6
ImS
iava*
Conc. (VA/Hz/cm
3
)
1.2425E-07
9.9873E-08
7.5492E-08
5.1111E-08
2.6730E-08
2.3498E-09
total contribution 
(electron+hole)
E
B
C
JC=0.1 mA/?m
2
Figure 3.10: 2D distribution of the total noise concentration =(CSia;v?a ) at 2 GHz. JC=0.1
mA=?m2.
94
3.5 Spatial Distribution of Microscopic Noise Contributions in RF MOSFET Transistor
The technique is applied to noise analysis of a 50nm Le MOSFET transistor. The device
structure is constructed based on reported 90 nm CMOS literature and the ITRS roadmap. dc
I  V , Y-parameters, and noise parameters are simulated using hydrodynamic transport models.
The simulator used is DESSIS 9.0 from ISE [45]. The Lombardi surface mobility model and the
default carrier energy relaxation time is used. The simulated I  V and gm characteristics are
comparable to reported data on 90 nm CMOS devices with similar structures. The transistor has
a 70 nm poly gate length, a 46 nm metallurgical channel length, and an e ective oxide thickness
of 1.2 nm. The channel doping is retrograded from the surface toward the bulk, and halos are
used for suppressing short channel e ect.
Di erent from section 3.4, the new technique is applied to obtain the noise concentration of
Y-noise representation parameters including gate noise current Sig;i?g, drain noise current Sid;i?d,
and their correlation Sig;i?d. The impedance representation noise parameters Sv1;v?1 , Sv2;v?2 and
Sv1;v?2 can be transformed to the Y -noise representation parameters Sig;i?g, Sid;i?d and Sig;i?d using
transformation matrix in Table 2.1 in chapter 2 [43],
CY = TZ Y ?CZ ?TyZ Y; (3.17)
2
64 Sig;i?g Sig;i?d
Sid;i?g Sid;i?d
3
75= T
Z Y ?
2
64 Sv1;v?1 Sv1;v?2
Sv2;v?1 Sv2;v?2
3
75?Ty
Z Y; (3.18)
and
TZ Y =
2
64 Y11 Y12
Y21 Y22
3
75; (3.19)
95
where Y11, Y12, Y21 and Y22 are elements of the Y parameter matrix Y .
An inspection of (3.18) shows that the transform is linear. Substituting Sv1;v?1 , Sv2;v?2 and
Sv1;v?2 expressed in the integral form of (3.9) into (3.18), the concentration of the chain represen-
tation noise parameters, CSig;i?g , CSid;i?
d
, and CSig;v?
d
are obtained as
2
64 CSig;i?g CSig;i?d
CSid;i?g CSid;i?
d
3
75= T
Z Y ?
2
64 CSv1;v?1 CSv1;v?2
CSv2;v?
1
CSv2;v?
2
3
75?Ty
Z Y: (3.20)
Integration of CSig;i?g , CSid;i?
d
, and CSig;v?
d
over the whole device gives the transistor Sig;i?g, Sid;i?d
and Sig;i?d, respectively.
3.5.1 Gate Noise Current Sig;i?g
Fig. 3.11 and Fig. 3.12 show the spatial distribution of CSig;i?g at 5 GHz. Vds = 1 V and Vgs
= 0.5 V and 1 V. CSig;i?g is the highest near the source side under the gate, and increases with
increasing Vgs.
3.5.2 Drain Noise Current Sid;i?d
Fig. 3.13 and Fig. 3.14 show the spatial distribution of CSid;i?
d
at 5 GHz. Vds = 1 V and
Vgs = 0.5 V and 1 V. Similar to CSig;i?g , CSid;i?
d
is the highest near the source side under the gate.
Moreover, another peak value of CSid;i?
d
occurs near the interface of the bulk and the source.
CSid;i?
d
increases with increasing Vgs.
96
X 
Y 
-0 .0 2 -0 .0 1 0 0. 01 0. 02 
-0 .0 1 
-0 . 005 
0 
0. 005 
0. 01 
0. 015 
0. 02 
0. 025 
S 
ig 
C onc . (  A 
2 
/H z/ cm 
3 
) 
1. 2722E -1 0 
1. 1026E -1 0 
9. 3295E -1 1 
7. 6332E -1 1 
5. 9369E -1 1 
4. 2407E -1 1 
2. 5444E -1 1 
8. 4814E -1 2 
F r  a m  e0 0 1  
G
S
D
Figure 3.11: 2-D gate noise current concentration CSig;i?g at 5 GHz. Vds = 1 V. Vgs = 0.5 V.
X 
Y 
-0 .0 2 -0 .0 1 0 0. 01 0. 02 
-0 .0 1 
-0 . 005 
0 
0. 005 
0. 01 
0. 015 
0. 02 
0. 025 
S 
ig 
C onc . (  A 
2 
/H z/ cm 
3 
) 
1. 6679E -1 0 
1. 4455E -1 0 
1. 2232E -1 0 
1. 0008E -1 0 
7. 7837E -1 1 
5. 5598E -1 1 
3. 3359E -1 1 
1. 1120E -1 1 
F r  a m  e0 0 1  
G
S
D
Figure 3.12: 2-D gate noise current concentration CSig;i?g at 5 GHz. Vds = 1 V. Vgs = 1 V.
97
X 
Y 
-0 .0 3 -0 .0 2 -0 .0 1 0 0. 01 0. 02 0. 03 
-0 . 005 
0 
0. 005 
0. 01 
0. 015 
0. 02 
0. 025 
0. 03 
0. 035 
S 
id 
C onc . (  A 
2 
/H z/ cm 
3 
) 
2. 4772E -0 6 
1. 3427E -0 6 
7. 2773E -0 7 
3. 9443E -0 7 
2. 1379E -0 7 
1. 1587E -0 7 
6. 2804E -0 8 
3. 4040E -0 8 
1. 8450E -0 8 
1. 0000E -0 8 
Fr am e 0  01 
G
S
D
Figure 3.13: 2-D drain noise current concentration CSid;i?
d
at 5 GHz. Vds = 1 V. Vgs = 0.5 V.
X 
Y 
-0 .0 3 -0 .0 2 -0 .0 1 0 0. 01 0. 02 0. 03 
-0 . 005 
0 
0. 005 
0. 01 
0. 015 
0. 02 
0. 025 
0. 03 
0. 035 
S 
id 
C onc . (  A 
2 
/H z/ cm 
3 
) 
4. 3894E -0 6 
2. 2325E -0 6 
1. 1355E -0 6 
5. 7756E -0 7 
2. 9377E -0 7 
1. 4942E -0 7 
7. 5998E -0 8 
3. 8655E -0 8 
1. 9661E -0 8 
1. 0000E -0 8 
Fr am e 0  01 
G
S
D
Figure 3.14: 2-D drain noise current concentration CSid;i?
d
at 5 GHz. Vds = 1 V. Vgs = 1 V.
98
X 
Y 
-0 .0 2 -0 .0 1 0 0. 01 0. 02 
-0 .0 1 
-0 . 005 
0 
0. 005 
0. 01 
0. 015 
0. 02 
0. 025 
Re S 
ig id * 
C onc . (  A 
2 
/H z/ cm 
3 
) 
8. 8666E -1 2 
3. 3456E -1 2 
-2 . 1754E -1 2 
-7 . 6964E -1 2 
-1 . 3217E -1 1 
-1 . 8738E -1 1 
-2 . 4259E -1 1 
-2 . 9780E -1 1 
F r  a m  e0 0 1  
G
S
D
Figure 3.15: 2-D real part of noise current correlation concentration <(CSig;i?
d
) at 5 GHz. Vds =
1 V. Vgs = 0.5 V.
X 
Y 
-0 .0 2 -0 .0 1 0 0. 01 0. 02 
-0 .0 1 
-0 . 005 
0 
0. 005 
0. 01 
0. 015 
0. 02 
0. 025 
Re S 
ig id * 
C onc . (  A 
2 
/H z/ cm 
3 
) 
1. 9972E -1 1 
1. 2020E -1 1 
4. 0688E -1 2 
-3 . 8826E -1 2 
-1 . 1834E -1 1 
-1 . 9786E -1 1 
-2 . 7737E -1 1 
-3 . 5688E -1 1 
F r  a m  e0 0 1  
G
S
D
Figure 3.16: 2-D real part of noise current correlation concentration <(CSig;i?
d
) at 5 GHz. Vds =
1 V. Vgs = 1 V.
99
X 
Y 
-0 .0 2 -0 .0 1 0 0. 01 0. 02 
-0 .0 1 
-0 . 005 
0 
0. 005 
0. 01 
0. 015 
0. 02 
0. 025 
Im S 
ig id * 
Co nc . (  A 
2 
/H z/ cm 
3 
) 
1. 7413E -0 8 
1. 4862E -0 8 
1. 2311E -0 8 
9. 7593E -0 9 
7. 2081E -0 9 
4. 6569E -0 9 
2. 1057E -0 9 
-4 . 4548E -1 0 
F r  a m  e0 0 1  
G
S
D
Figure 3.17: 2-D imaginary part of noise current correlation concentration =(CSig;i?
d
) at 5 GHz.
Vds = 1 V. Vgs = 0.5 V.
X 
Y 
-0 .0 2 -0 .0 1 0 0. 01 0. 02 
-0 .0 1 
-0 . 005 
0 
0. 005 
0. 01 
0. 015 
0. 02 
0. 025 
Im S 
ig id * 
Co nc . (  A 
2 
/H z/ cm 
3 
) 
2. 9843E -0 8 
2. 5297E -0 8 
2. 0752E -0 8 
1. 6206E -0 8 
1. 1660E -0 8 
7. 1147E -0 9 
2. 5691E -0 9 
-1 . 9765E -0 9 
F r  a m  e0 0 1  
G
S
D
Figure 3.18: 2-D imaginary part of noise current correlation concentration =(CSig;i?
d
) at 5 GHz.
Vds = 1 V. Vgs = 1 V.
100
3.5.3 Drain and Gate Noise Current Correlation Sig;i?d
Fig. 3.15 ? Fig. 3.18 show the spatial distribution of <(CSig;i?
d
) and =(CSig;i?
d
) at 5 GHz. Vds
= 1 V and Vgs = 0.5 V and 1 V. <(CSig;i?
d
) is quite small compared to =(CSig;i?
d
). The overall
integration of <(CSig;i?
d
) is negative for both Vgs?. =(CSig;i?
d
) is the highest near the source side
under the gate. Another peak value of =(CSig;i?
d
) occurs near the interface of the bulk and the
source. =(CSid;i?
d
) increases with increases with increasing Vgs.
3.6 Summary
We have presented a new technique of simulating the spatial distribution of microscopic
noise contribution to the input noise current, voltage, as well as their cross-correlations. The
technique is first demonstrated on a 50 GHz SiGe HBT. The spatial contributions by base ma-
jority holes, base minority electrons, and emitter minority holes are analyzed, and compared to
results from a compact noise model. A strong crowding e ect is observed in the spatial distribu-
tion of noise concentrations due to base majority holes. The results suggest that 2D distributive
e ect needs to be taken into account in future compact noise model development.
The technique is also applied to a 46 nm Le MOSFET transistor. The spatial distribution
of the Y- noise representation parameters CSig;i?g , CSid;i?
d
, <(CSig;i?
d
) and =(CSig;i?
d
) are analyzed.
The region under the gate near the source side is the most important for all of the Y- noise
representation parameters.
101
CHAPTER 4
BIPOLAR NOISE MODELING
This chapter examines bipolar transistor noise modeling and noise physics using micro-
scopic noise simulation. Transistor terminal current and voltage noises resulting from velocity
fluctuations of electrons and holes in the base, emitter, collector, and substrate are simulated us-
ing a new technique, and compared with modeling results. Major physics noise sources in bipolar
transistor are qualitatively identified. The relevant importance as well as model-simulation dis-
crepancy is analyzed for each physical noise source. The results are then used to propose a new
noise model.
4.1 Introduction
Mixed-signal and RFIC design demands compact transistor models that can accurately
model not only the dc and ac parameters, but also transistor noise parameters, including mini-
mum noise figure NFmin, optimal source (noise matching) admittance Yopt, and noise resistance
Rn. NFmin, Yopt and Rn are fundamentally determined by the input noise voltage and current for
the chain representation of a noisy linear two-port, as shown in Fig. 2.1 in chapter 2. Fig. 1.4 in
chapter 1 shows the essence of SPICE noise modeling in major CAD tools. The noise physics ac-
counted for include: base resistance thermal noise, base current shot noise, and collector current
shot noise, all of which are essentially macroscopic approximations of the microscopic di usion
noise due to velocity fluctuations of electrons and holes. Fig. 4.1 shows the chain noise param-
eters comparison of measured data from IBM and compact noise model at JC = 0.01 mA=?m2.
The compact noise modeling is good for low current density. The accuracy of such compact
102
noise modeling, however, becomes worse at higher current densities required for high speed [3].
Fig. 4.2 shows the chain noise parameters comparison of measured data and compact noise
model at JC = 0.63 mA=?m2. The compact model deviates from the measured data, and the
di erence increases dramatically with increasing frequency. An improvement on the compact
noise model becomes necessary for high current density and high frequency.
5 10 15
10
?19
10
?18
Freq     (GHz)
S
va
    (V
2
/Hz)
measured     
compact model
5 10 15
10
?23
10
?22
Freq     (GHz)
S
ia
    (A
2
/Hz)
measured     
compact model
5 10 15
10
?24
10
?23
10
?22
10
?21
10
?20
Freq     (GHz)
?
(S
iava*
)    (VA/Hz)
measured     
compact model
5 10 15
10
?21
10
?20
Freq     (GHz)
`
(S
iava*
)    (VA/Hz)
measured     
compact model
Figure 4.1: Chain noise parameter: measured vs compact model. JC=0.01 mA=?m2.
However, measured data itself cannot give us an e cient way to improve the compact
noise model, in the reason that the measured data cannot give us detailed information about
di erent noise sources in the device. Microscopic noise simulation available in recent years
makes it possible to have a close look of device noise from the structure level. By comparing
the chain noise parameters at high JC of 0.65 mA=?m2, as shown in Fig. 4.3, we observed that
103
5 10 15
10
?19
10
?18
Freq     (GHz)
S
va
    (V
2
/Hz)
measured
compact model
5 10 15
10
?22
10
?21
Freq     (GHz)
S
ia
    (A
2
/Hz)
measured
compact model
5 10 15
10
?21
10
?20
Freq     (GHz)
?
(S
iava*
)    (VA/Hz)
measured
compact model
5 10 15
10
?21
10
?20
Freq     (GHz)
`
(S
iava*
)    (VA/Hz)
measured
compact model
J
C
 = 0.63 mA/mm
2
 
J
C
 = 0.63 mA/mm
2
 
J
C
 = 0.63 mA/mm
2
 
J
C
 = 0.63 mA/mm
2
 
Figure 4.2: Chain noise parameter: measured v.s. compact model, JC=0.63 mA=?m2.
the simulation result complies to the measured data with a much better trend, which makes it
feasible to examine the compact noise model with the microscopic noise simulation results.
By means of microscopic noise simulation and the technique in chapter 3, this chapter
examines the noise physics accounted for in the noise model. Regional contribution analysis are
performed to verify the origins of noise in the device and compared to compact noise model, and
resulted from an e ort to improve bipolar transistor noise modeling.
4.2 Technical Approach
4.2.1 Microscopic Input Noise Concentration
In microscopic noise simulation, the two-port open circuit noise voltage parameters Sv1;v?1 ,
Sv2;v?2 and Sv1;v?2 are obtained by integrating the ?noise concentration? CSv1;v?
1
, CSv2;v?
2
, and CSv1;v?
2
104
5 10 15 20
10
?19
10
?18
Freq    (GHz)
S
va
    (V
2
/Hz)
simulation
compact model
5 10 15 20
10
?22
10
?21
Freq    (GHz)
S
ia
    (A
2
/Hz)
simulation
compact model
5 10 15 20
10
?21
10
?20
Freq    (GHz)
?
(S
iava*
)    (VA/Hz)
simulation
compact model
5 10 15 20
10
?21
10
?20
Freq    (GHz)
`
(S
iava*
)    (VA/Hz)
simulation
compact model
J
C
 = 0.65 mA/mm
2
 
J
C
 = 0.65 mA/mm
2
 
J
C
 = 0.65 mA/mm
2
 J
C
 = 0.65 mA/mm
2
 
Figure 4.3: Chain noise parameter: simulation v.s. compact model, JC=0.65 mA=?m2..
over the device volume. CSv1;v?
1
, CSv2;v?
2
, and CSv1;v?
2
are solved in TCAD tools, including TAURUS
[44] and DESSIS [45]. Each noise concentration consists of an electron contribution and a hole
contribution, which account for electron and hole velocity fluctuations, respectively,
2
64 CSv1;v?1 CSv1;v?2
CSv2;v?
1
CSv2;v?
2
3
75=
2
64 C
e
Sv1;v?1 C
e
Sv1;v?2
CeS
v2;v?1
CeS
v2;v?2
3
75+
2
64 C
h
Sv1;v?1 C
h
Sv1;v?2
ChS
v2;v?1
ChS
v2;v?2
3
75; (4.1)
where superscripts e and h stand for electron and hole contributions, respectively. The ?noise
concentration? for the chain representation, CSia;i?a , CSva;v?
a , and CSia;v?a and their electron and
105
hold contributions can then be obtained using the technique proposed in chapter 3 [51].
2
64 CSva;v?a CSva;i?a
CSia;v?a CSia;i?
a
3
75=
2
64 CeSva;v?a CeSva;i?a
CeS
ia;v?a
CeS
ia;i?a
3
75+
2
64 ChSva;v?a ChSva;i?a
ChS
ia;v?a
ChS
ia;i?a
3
75; (4.2)
2
64 CeSva;v?a CeSva;i?a
CeS
ia;v?a
CeS
ia;i?a
3
75= T
Z A ?
2
64 C
e
Sv1;v?1 C
e
Sv1;v?2
CeS
v2;v?1
CeS
v2;v?2
3
75?Ty
Z A; (4.3)
2
64 ChSva;v?a ChSva;i?a
ChS
ia;v?a
ChS
ia;i?a
3
75= T
Z A ?
2
64 C
h
Sv1;v?1 C
h
Sv1;v?2
ChS
v2;v?1
ChS
v2;v?2
3
75?Ty
Z A; (4.4)
where TZ A is the transform matrix from Z- noise representation to chain noise representation as
in (3.12). Integration of CSia;i?a , CSva;v?
a , and CSia;v?a over the whole device gives transistor Sia;i
?a,
Sva;v?a and Sia;v?a . The electron and hole contributions of Sia;i?a, Sva;v?a and Sia;v?a are obtained
similarly.
4.2.2 Macroscopic Input Noise
Through noise circuit analysis, Fig. 1.4 can be transformed to the form of Fig. 2.1 by (3.14),
(3.15), and (3.16) derived in chapter 3 [12]. The resulting Sva;v?a , Sia;i?a and Sia;v?a are
Sva;v?a = 2qICjY
21j2
+ 2qIBr2b + 4kTrb; (4.5)
Sia;i?a = 2qIB + 2qICjh
21j2
; (4.6)
Sia;v?a = 2qIC Y11jY
21j2
+ 2qIBrb; (4.7)
where h21 = Y21=Y11. The Y parameters are for the whole transistor that includes both rb and
the intrinsic transistor.
106
4.2.3 Microscopic and Macroscopic Connections
Physically speaking, the 4kTrb terms in the model equations account for velocity fluctu-
ations of holes in the base. One can therefore compare the 4kTrb related terms in the model
equations with the integration of the hole contribution of the noise concentration in the base.
Similarly, the 2qIB terms account for emitter minority hole velocity fluctuation, and the 2qIC
terms account for base minority electron velocity fluctuation [50]. Thus, connections between
compact noise model and microscopic noise simulation can be established for Sva;v?a , Sia;i?a and
Sia;v?a , as shown in Table 4.1. Here the superscripts e and h stand for electron and hole contribu-
tions, respectively.
Model Simulation
Sev
a;v?a
2qIC=jY21j2 Rbase CeS
va;v?a
d 
Shv
a;v?a
2qIBr2b Remitter ChS
va;v?a
d 
4kTrb Rbase ChS
va;v?a
d 
Sei
a;i?a
2qIC=jh21j2 Rbase CeS
ia;i?a
d 
Shi
a;i?a
2qIB Remitter ChS
ia;i?a
d 
Sei
a;v?a
2qICY11=jY21j2 Rbase CeS
ia;v?a
d 
Shi
a;v?a
2qIBrb Remitter ChS
ia;v?a
d 
Table 4.1: Connections between noise modeling and simulation for Sva;v?a , Sia;i?a and Sia;v?a .
4.3 Chain Representation Parameters
Noise simulation is performed for a 50 GHz SiGe HBT from 1 to 20 GHz using DESSIS
[49]. The emitter area AE=0.5?1?m2. The doping and Ge profiles were determined using
SIMS. A set of physical models suitable for HBT simulation were selected, and the model co-
e cients were calibrated to reproduce measured dc I  V characteristics and high frequency
107
s-parameters. The carrier noise temperature is assumed to be the same as the lattice temperature.
The simulated CSv1;v?
1
, CSv2;v?
2
, and CSv1;v?
2
are converted to CSia;i?a , CSva;v?
a , and CSia;v?a using (3.13).
Their electron and hole contributions are converted using (4.3) and (4.4). We now examine the
modeling results using the simulation results as a reference. No attempt is made to ?tune? the
noise simulation to match measured noise data, which will require careful de-embedding of par-
asitics not included in the simulated structure. The simulated bias and frequency dependences,
however, still qualitatively match measured data, for all noise parameters.
4.3.1 Sva;v?a , Sia;i?a and Sia;v?a
Fig. 4.4 (a) compares the modeled and simulated Sva;v?a , Sev
a;v?a
and Shv
a;v?a
for JC=0.01
mA=?m2. The electron contribution Sev
a;v?a
dominates over the hole contribution Shv
a;v?a
. Note
that the model slightly underestimates Sev
a;v?a
, and significantly underestimates Shv
a;v?a
. The sim-
ulated Sev
a;v?a
and Shv
a;v?a
are both frequency dependent. Despite inaccurate modeling of Shv
a;v?a
,
the total Sva;v?a is well modeled, because of the dominance of Sev
a;v?a
. At a higher JC of 0.65
mA/?m2, however, the hole contribution dominates over the electron contribution, as shown in
Fig. 4.4 (b). An inspection of Figs. 4.4 (a) and (b) immediately shows that with increasing JC,
Sev
a;v?a
decreases, while Shv
a;v?a
stays about the same. The model underestimates Shv
a;v?a
, and over-
estimates Sev
a;v?a
. Observe that the simulated Shv
a;v?a
is frequency dependent, while the modeled
Shv
a;v?a
(4kTrb) is frequency independent.
Noise concentration contours at 2 GHz are shown for CeS
va;v?a
and ChS
va;v?a
in Figs. 4.5 and
4.6, respectively. JC=0.65 mA/?m2. Observe that both CeS
va;v?a
and ChS
va;v?a
are the highest in
the SiGe base, indicating that transistor Sva;v?a mainly comes from the SiGe base. This provides
guidelines to future noise model development, that is, the transistor noise mainly originates from
108
5 10 15 20
10
- 17
10
- 16
Freq    (GHz)
       (a)          
S
va
, 
S
vae
, and 
S
vah
   (V
2
/Hz)
      hole  
contribution
   electron 
contribution
total
Impedance Field Simulation
Compact Noise Model       
J
C
 = 0.01 mA/?m
2
 
5 10 15 20
10
- 18
10
- 17
Freq    (GHz)
        (b)          
S
va
, 
S
vae
, and 
S
vah
    (V
2
/Hz)
      hole  
contribution
   electron 
contribution
total
Impedance Field Simulation
Compact Noise Model       
J
C
 = 0.65 mA/?m
2
 
Figure 4.4: Sva;v?a , Sev
a;v?a
, and Shv
a;v?a
vs frequency at (a) JC=0.01 mA=?m2. (b) JC=0.65
mA/?m2.
the EB junction. This contradicts the conventional wisdom that the collector current shot noise
originates from passage of electrons through the reverse biased CB junction. In the intrinsic
base, and along the x-direction, CeS
va;v?a
is uniform, while ChS
va;v?a
is highly nonuniform, and shows
a strong ?base noise crowding? e ect.
Fig. 4.7 (a) shows the integrals of CeS
va;v?a
in the base, emitter, collector, and p-substrate,
together with the 2qIC related term in the model. JC=0.65 mA/?m2. Note that the model ac-
counts for only the base contribution, which is reasonable, since the simulated base electron
contribution overwhelmingly dominates over other electron contributions. The 2qIC descrip-
tion, however, overestimates Sev
a;v?a
, and thus a better description is required. Fig. 4.7 (b) shows
the integrals of ChS
va;v?a
in the base, emitter, collector, and p-substrate. Also shown are the 2qIB
109
X(?m)
Y(
?
m)
1.5 2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
S
va
Conc. (V
2
/Hz/cm
3
)
1.13714341E-04
9.18461982E-05
6.99780558E-05
4.81099134E-05
2.62417709E-05
4.37362849E-06
E
B
C
electron contribution
JC=0.65 mA/?m
2
Figure 4.5: 2D distribution of CeS
va;v?a
at 2 GHz, JC=0.65 mA/?m2.
(emitter holes) and 4kTrb (base holes) related terms accounted for in the model. The collector
and substrate hole noises are indeed negligible. The noise from the base majority holes domi-
nates over the noise from the emitter minority holes. The base majority hole noise contribution
is more than predicted by 4kTrb, and frequency dependent as well. The noise from the emitter
minority holes increases with frequency, and is underestimated by 2qIB related term.
Fig. 4.8 (a) compares modeled and simulated Sia;i?a, Sei
a;i?a
, and Shi
a;i?a
for JC=0.01 mA=?m2.
Sei
a;i?a
increases with frequency and is slightly underestimated by the model. Shi
a;i?a
increases dra-
matically with frequency, and is significantly underestimated. At a higher JC of 0.65 mA/?m2,
however, the Sei
a;i?a
discrepancy between model and simulation becomes much more pronounced,
110
X(?m)
Y(
?
m)
1.5 2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
S
va
Conc. (V
2
/Hz/cm
3
)
3.13664075E-04
2.53344060E-04
1.93024046E-04
1.32704032E-04
7.23840172E-05
1.20640029E-05
E
B
C
hole contribution
JC=0.65 mA/?m
2
Figure 4.6: 2D distribution of ChS
va;v?a
at 2 GHz, JC=0.65 mA/?m2.
as shown in Fig. 4.8 (b). Thus, for Sia;i?a, 2qIC is not a good description for base minority elec-
tron noise. Like for Shv
a;v?a
, the frequency dependence for Shi
a;i?a
is not accounted for in the model.
Shi
a;i?a
dominates at lower frequencies, while Sei
a;i?a
becomes dominant at higher frequencies.
Fig. 4.9 (a) shows the integrals of CeS
ia;i?a
in the base, emitter, collector, and p-substrate.
JC=0.65 mA/?m2. The model only accounts for the base electron contribution, a 2qIC=jh21j2
term. Like for other noise parameters, the base minority electron contribution for Sei
a;i?a
is poorly
modeled by the 2qIC related term. Fig. 4.9 (b) shows the regional contributions of Shi
a;i?a
. The
model accounts for only the emitter hole contribution through the 2qIB term. Even though the
collector and substrate hole contributions are indeed negligible, the base hole contribution is not
negligible at higher frequencies. This emitter contribution constitutes the main discrepancy for
the total Shi
a;i?a
between modeling and simulation, and shows frequency dependence.
111
0 5 10 15 20
10
- 26
10
 - 25
10
- 24
10
- 23
10
- 22
10
 - 21
10
- 20
10
- 19
10
- 18
10
- 17
base electrons     
emitter electrons  
collector electrons
substrate electrons
2qI
C
/|Y
21
|
2
   
0 5 10 15 20
10
- 24
10
- 23
10
- 22
10
- 21
10
- 20
10
- 19
10
 - 18
10
- 17
base holes     
emitter holes  
collector holes
substrate holes
4kTr
b
         
2qI
B
r
b
2
     
Regional contributions of 
S
vae
    (V
2
/Hz)
Regional contributions of 
S
vah
    (V
2
/Hz)
Freq    (GHz)         
 (a)          
Freq    (GHz)         
 (b)          
   base   
electrons 
2qI
C
/|Y
21
|
2
 
 emitter  
electrons 
substrate 
electrons 
collector 
electrons 
J
C
 = 0.65 mA/?m
2
 
J
C
 =  0.65 mA/?m
2
 
substrate 
   holes  
collector 
   holes  
2qI
B
r
b
2
 
emitter
 holes 
4kTr
b
 
base  
holes 
Figure 4.7: Regional contributions of Sev
a;v?a
(a) and Shv
a;v?a
(b) at JC=0.65 mA/?m2.
Similar analysis is performed for Sia;v?a . The results also show that the noise from the
base minority electrons is poorly described by the model. Similar problems exist with 4kTrb
description of the base hole noise, and 2qIC description of the base minority electron noise.
4.3.2 NFmin, Yopt and Rn
NFmin, Yopt and Rn are obtained from Sva;v?a , Sia;i?a and Sia;v?a by (2.24) ? (8.19) derived in
chapter 2 [11].
To compare the impact of electron and hole noise on circuit-level noise parameters, we
examine NFemin and NFhmin, defined as the NFmin that the transistor would have when only electron
velocity or only hole velocity fluctuates, respectively. NFemin is obtained by substituting Sev
a;v?a
,
Sei
a;i?a
and Sei
a;v?a
into (2.24). NFhmin is obtained similarly. Since NFmin is not a linear function
112
5 10 15 20
10
- 26
10
- 25
10
- 24
10
- 23
Freq    (GHz)
       (a)          
S
ia
, 
S
iae
, and 
S
iah
    (A
2
/Hz)
      hole  
contribution
   electron 
contribution
total
Impedance Field Simulation
Compact Noise Model       
J
C
 = 0.01 mA/?m
2
 
5 10 15 20
10
 - 24
10
- 23
Freq    (GHz)
        (b)          
S
ia
, 
S
iae
, and 
S
iah
    (A
2
/Hz)
      hole  
contribution
   electron 
contribution
total
Impedance Field Simulation
Compact Noise Model       
J
C
 = 0.65 mA/?m
2
 
Figure 4.8: Sia;i?a, Sei
a;i?a
, and Shi
a;i?a
vs frequency at (a) JC=0.01 mA=?m2. (b) JC=0.65 mA/?m2.
of Sva;v?a , Sia;i?a and Sia;v?a , NFmin 6= NFemin + NFhmin. Yeopt is similarly defined and obtained from
substituting Sev
a;v?a
, Sei
a;i?a
and Sei
a;v?a
into (8.18) and (8.19). Like NFmin, Yopt 6= Yeopt +Yhopt.
Since Rn = Sva;v?a=4kT, which is a linear function, Rn = Ren+Rhn. The problems with Sva;v?a
modeling directly translate into Rn inaccuracy as shown in Fig. 4.10 (a). At low JC, NFmin and
NFemin are well described by the model since Sva;v?a , Sia;i?a and Sia;v?a are well modeled at this
bias. At high JC, which is shown in Fig. 4.10 (b), however, they are both overestimated, and the
discrepancies increase dramatically with frequency. NFhmin is poorly modeled at high JC. Note
that the frequency dependence of NFhmin is not modeled.
Similarly, Yopt is well modeled at low bias. However, at JC=0.65 mA/?m2, neither Yopt
nor Yeopt or Yhopt is well modeled, as shown in Figs. 4.11 (a) and (b). The discrepancies increase
with frequency. Again, the frequency dependence of Yhopt is not accounted for by the model. The
113
0 5 10 15 20
10
- 32
10
 - 31
10
- 30
10
- 29
10
- 28
10
- 27
10
- 26
10
- 25
10
 - 24
10
- 23
10
- 22
base electrons     
emitter electrons  
collector electrons
substrate electrons
2qI
C
/|h
21
|
2
   
0 5 10 15 20
10
 - 31
10
- 30
10
- 29
10
 - 28
10
 - 27
10
- 26
10
- 25
10
- 24
10
- 23
base holes     
emitter holes  
collector holes
substrate holes
2qI
B
          
Regional contributions of 
S
iae
    (A
2
/Hz)
Regional contributions of 
S
iah
    (A
2
/Hz)
Freq    (GHz)         
(a)          
Freq    (GHz)         
 (b)          
J
C
 = 0.65 mA/?m
2
 
2qI
C
/|h
21
|
2
 
   base   
electrons 
collector 
electrons 
  emitter 
electrons 
substrate 
electrons 
emitter 
 holes  
base  
holes 
2qI
B
 
J
C
 = 0.65 mA/?m
2
 
substrate 
   holes  
collector 
   holes  
Figure 4.9: Regional contribution of Sei
a;i?a
(a) and Shi
a;i?a
(b) at JC=0.65 mA/?m2.
discrepancies of Rn, NFmin and Yopt are all fundamentally caused by the inaccurate modeling of
Sva;v?a , Sia;i?a and Sia;v?a . In particular, the description of base minority electron noise using 2qIC
is clearly responsible for the inaccuracy of the electron contributions, and the description of base
majority hole noise using 4kTrb is responsible for the inaccuracy of the hole contributions.
4.4 Intrinsic Base and Collector Noise
As we have discussed above, the main noise sources are from base electrons, base holes,
and emitter holes. The integrations of CeS
va;v?a
, CeS
ia;i?a
and CeS
ia;v?a
in base region are transformed to
intrinsic transistor SBEi
b;i?b
, SBEi
c;i?c
and SBEi
c;i?b
by (2.157), (2.158) and (2.159) derived in chapter 2. The
superscript BE represents the contribution from base electrons. Similarly, we obtain intrinsic
114
5 10 15 20 
100 
150 
200 
250 
300 
350 
400 
450 
500 
550 
Freq    (GHz) 
(a) 
R 
n 
,  
R 
n e 
, and  
R 
n h 
     
( 
? 
) 
R 
n 
h 
R 
n 
e 
R 
n 
Impedance Field Simulation 
Compact Noise Model         
5 10 15 20 
1 
2 
3 
4 
5 
6 
Freq    (GHz) 
(b) 
NF 
min 
, NF 
min e 
, and NF 
min h 
    (dB) 
 NF 
min 
h 
NF 
min 
e 
NF 
min 
Impedance Field Simulation 
Compact Noise Model         
J 
C 
 = 0.65 mA/ ? m 
2 
  
J 
C 
 = 0.65 mA/ ? m 
2 
  
Figure 4.10: (a) Rn, and (b) NFmin vs frequency. JC=0.65 mA/?m2.
transistor Sib;i?b, Sic;i?c and Sic;i?b from base holes with superscript BH and from emitter holes with
superscript EH. In the compact model, SEHi
b;i?b
is modeled as 2qIB, Sib;i?b from base region are
not counted for. SBEi
c;i?c
is modeled as 2qIC, Sic;i?c from base and emitter holes are not modeled.
Sib;i?b and Sic;i?c are not correlated to each other. Thus the connections between compact noise
model and microscopic noise simulation can be established for Sib;i?b, Sic;i?c and Sic;i?b as shown
in Table 4.2. Yint11 , Yint12 , Yint21 and Yint22 are elements of intrinsic transistor Y parameter matrix Yint.
Sib;i?b Sic;i?c Sic;i?b
SPICE model 2qIB 2qIC 0
van Vliet model 4kT<Yint11  2qIB 4kT<Yint22 + 2qIC 2kT(Yint21 +Yint?12 )  2qIC
Table 4.2: SPICE model and van Vliet model for intrinsic transistor Sib;i?b, Sic;i?c and Sic;i?b.
115
5 10 15 20
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10
- 3
Freq    (GHz)
(a)
G
opt
, 
G
opte
, and 
G
opth
    
(
?
-1
)   
 
Impedance Field Simulation
Compact Noise Model       
5 10 15 20
 16
 14
 12
 10
 8
 6
 4
 2
0
x 10
- 4
Freq    (GHz)
(b)
 
 
Impedance Field Simulation
Compact Noise Model       
G
opt
G
opt
h
   G
opt
e
J
C
 = 0.65 mA/?m
2
 
J
C
 = 0.65 mA/?m
2
 
B
opt
, 
B
opte
, and 
B
opth
(
?
-1
)   
B
opt
e
B
opt
h
B
opt
Figure 4.11: (a) Gopt, and (b) Bopt vs frequency. JC=0.65 mA/?m2.
Fig. 4.12 (a) shows Sib;i?b and its contributions SBEi
b;i?b
, SBHi
b;i?b
and SEHi
b;i?b
, respectively at a low
JC of 0.01 mA=?m2. Sib;i?b is dominated by SEHi
b;i?b
and well modeled by 2qIB at low frequency.
However, as frequency increases, SBEi
b;i?b
and SBHi
b;i?b
increases dramatically and become dominant.
Moreover, SEHi
b;i?b
increases with frequency and can not be well modeled by 2qIB at high fre-
quencies. Fig. 4.12 (b) shows Sib;i?b and its contributions at a higher bias of JC=0.65 mA=?m2.
Similarly, Sib;i?b is dominated by emitter holes at lower frequencies and by base electrons and
holes, which are not counted for in the compact noise model, at higher frequencies. This sug-
gests that the compact noise model for Sib;i?b should be improved by grasping the frequency
dependence at high frequency range for both high and low JC?s.
116
0 5 10 15 20
10
-28
10
- 27
10
- 26
10
 - 25
10
 - 24
freq    (Hz)
S
ib
    (A
2
/Hz)
base electron
base hole
emitter hole
2qI
B
base & emitter hole
total
4kT? Y
11
int
 - 2qI
B
J
C
 = 0.01 mA/?m
2
 
0 5 10 15 20
10
-25
10
 - 24
10
 - 23
10
 - 22
freq    (Hz)
S
ib
    (A
2
/Hz)
base electron
base hole
emitter hole
2qI
B
base & emitter hole
total
4kT? Y
11
int
 - 2qI
B
J
C
 = 0.65 mA/?m
2
 
(a) (b)
Figure 4.12: Regional contributions of internal input noise current Sib;i?b (a) JC=0.01 mA/?m2.
(b) JC=0.65 mA/?m2.
Besides the compact noise model, the simulated intrinsic transistor input and output noise
currents are also compared with van Vliet noise model as introduced in chapter 1 [13]. The van
Vliet model equations are given in (1.14), (1.15), and (1.16).
Fig. 4.12 shows that 4kT<(Yint11 )  2qIB grasp the frequency dependence at both low and
high JC?s. However, it is more close to the overall hole contribution SEHi
b;i?b
+ SEHi
b;i?b
than for the
total Sib;i?b. The base electron contribution SEHi
b;i?b
is only important at high JC, yet there has not
been a good model for it.
Fig. 4.13 (a) shows Sic;i?c and its contributions from base electrons, base holes and emitter
holes at JC=0.01 mA=?m2. At this bias, Sic;i?c is dominated by base electrons, which is slightly
117
underestimated by 2qIC. As JC increases to 0.65 mA=?m2 as shown in Fig. 4.13 (b), contri-
bution from base holes becomes comparable to that from base electrons, and both of them are
decreasing with frequency. Contribution from emitter holes is totally negligible at both biases.
Apparently, Sic;i?c which is modeled by base minority electrons complies with simulation results
well at low bias. However, at high bias, noise from majority carriers in the base plays an impor-
tant role and makes total Sic;i?c deviates from 2qIC. This deviation was also claimed at high bias
in [52]. However, [52] made the wrong comparison. It compared 2qIC with the output noise
current of the whole transistor, that can be expressed as,
Si2;i?2 = 2qIC + 4kTrbjY21j2 + 2qIBrbjY21j2; (4.8)
which has already included the hole contribution as shown in Fig. 4.14. Moreover, in low in-
jection the apparent deviation from the compact model for drift di usion noise in low bias as
claimed in [52] is not observed in our study.
Similar to Sib;i?b analysis, comparison of Sic;i?c and 4kT<Yint22 + 2qIC is also shown in
Fig. 4.13. the van Vliet model does not show any improvement to the frequency dependence
of SBEi
c;i?c
. Further, the base hole contribution SBHi
c;i?c
needs to be modeled at high JC. The emitter
hole contribution SEHi
c;i?c
is negligible at both biases.
Fig. 4.15 and Fig. 4.16 shows the correlation term Sic;i?b at low and high JC, respectively.
In the compact noise model Sic;i?c and Sib;i?b have no correlation. The simulation result, however,
Fig. 4.15 shows that Sic;i?b is negligible at low frequency but noneligible at high frequency at low
JC. <Sic;i?b is positive and slightly dominated by <SBHi
c;i?b
over <SBEi
c;i?b
, which has a negative sign.
=Sic;i?b and its contributions are all negative. =SBEi
c;i?b
slightly dominates over =SBHi
c;i?b
. SEHi
c;i?b
is
negligible. At high JC, as shown in Fig. 4.16, Sic;i?b can not be neglected for the whole frequency
118
0 5 10 15 20
10
 - 29
10
 - 28
10
 - 27
10
- 26
10
 - 25
10
 - 24
10
 - 23
freq    (Hz)
S
ic
    (A
/Hz)
base electron
base hole
emitter hole
2qI
C
base & emitter hole
total
4kT? Y
22
int
+2qI
C
J
C
 = 0.01 mA/?m
2
 
0 5 10 15 20
10
 - 23
10
 - 22
10
 - 21
freq    (Hz)
S
ic
    (A
2
/Hz)
base electron
base hole
emitter hole
2qI
C
base & emitter hole
total
4kT? Y
22
int
+2qI
C
J
C
 = 0.65 mA/?m
2
 
(a) (b)
Figure 4.13: Regional contributions of internal output noise current Sic;i?c (a) JC=0.01 mA/?m2.
(b) JC=0.65 mA/?m2.
span. Both <Sic;i?b and =Sic;i?b are dominated by their base hole contribution at low frequency.
SBHi
c;i?b
and SBEi
c;i?b
are comparable at high frequencies. SEHi
c;i?b
can still be neglected.
Comparison of Sic;i?b and 2kT(Yint21 +Yint?12 ) 2qIC is also shown in Fig. 4.15 and Fig. 4.16.
At low bias, the van Vliet model grasps the frequency dependence of Sic;i?b, yet slightly underes-
timates both the real and the imaginary part. Its imaginary part is more close to =SBEi
c;i?b
. At high
bias, however, the van Vliet model deviated from Sic;i?b a lot. Hence, compared to compact noise
model, [13] has its advantage of better frequency dependence description at low JC, where mi-
nority carrier noise dominates. However, as JC increases, where majority carrier noise becomes
comparable to minority carrier noise, [13] does not do a better job than the compact noise model.
119
10
?2
10
?1
10
?28
10
?27
10
?26
10
?25
10
?24
10
?23
10
?22
J
C
    (mA/mm
2
)
Output noise current S
i2
    (A
2
/Hz)
total S
i2
electron
hole
compact model
2qI
C
4kTr
b
|Y
21
|
2
2qI
B
r
b
2
|Y
21
|
2
f = 2 GHz 
Figure 4.14: Output noise current of whole transistor Si2;i?2 at 2 GHz.
4.5 Summary
We have examined bipolar transistor noise modeling for each physical noise source using
microscopic noise simulation. Regional analysis is performed for the chain representation noise
parameters. The base majority hole noise contribution is shown to be larger than modeled using
4kTrb and frequency dependent for all noise parameters. The 2qIB related terms underestimates
the emitter hole noise, especially for higher frequencies. The base minority electron contribution
is poorly modeled by the 2qIC related terms for all noise parameters, particularly for higher JC
required for high speed. Further, regional analysis for intrinsic transistor input and output noise
current is performed. The input noise current consists not only the emitter hole contribution
corresponding to 2qIB, but also the base electron and hole contribution which are frequency
120
0 5 10 15 20
10
-29
10
 - 28
10
 - 27
10
- 26
10
- 25
freq    (Hz)
? 
(S
icib*
)    (A
2
/Hz)
base electron(negative)
base hole
emitter hole
base & emitter hole
total
2kT[? (Y
21
int
+Y
12
int*
) - g
m0
]
0 5 10 15 20
 - 10
- 24
- 10
 - 25
 10
 - 26
 - 10
- 27
- 10
 - 28
 - 10
 - 29
freq    (Hz)
? 
(S
icib*
)    (A
2
/Hz)
base electron
base hole
emitter hole
base & emitter hole
total
2kT? (Y
21
int
+Y
12
int*
)
J
C
 = 0.01 mA/?m
2
 
J
C
 = 0.01 mA/?m
2
 
(a) (b)
-
Figure 4.15: Regional contributions of internal noise current correlation Sic;i?b. JC=0.01
mA/?m2. (a) <Sic;i?b. (b) =Sic;i?b.
dependent and should be counted for especially at high frequencies. At higher JC, the output
noise current consists not only the base electron contribution corresponding to 2qIC, but also the
base hole contribution that not counted for in the compact noise model. Moreover, the frequency
dependence of base electron contribution is not described. The correlation term which is not
modeled in the compact noise model should be considered for higher JC and higher frequency.
This chapter also compared the intrinsic transistor input and output noise current with a
noise model that derived from the transport theory of density fluctuations that applies to three
dimensional device. The comparison shows that this model has a better description of frequency
121
(a) (b)
0 5 10 15 20
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
x 10
 -23
freq    (Hz)
? 
(S
icib*
)    (A
2
/Hz)
base electron
base hole
emitter hole
base & emitter hole
total
2kT[? (Y
21
int
+Y
12
int*
) - g
m0
]
0 5 10 15 20
 - 10
- 22
 10
 - 23
 10
- 24
 - 10
- 25
freq    (Hz)
? 
(S
icib*
)    (A
2
/Hz)
base electron
base hole
emitter hole
base & emitter hole
total
2kT? (Y
21
int
+Y
12
int*
)
J
C
 = 0.65 mA/?m
2
 
J
C
 = 0.65 mA/?m
2
 
-
-
Figure 4.16: Regional contributions of internal noise current correlation Sic;i?b. JC=0.65
mA/?m2. (a) <Sic;i?b. (b) =Sic;i?b.
dependence than the compact noise model at low bias. However, as for higher JC, it has no
advantage over the compact noise model.
122
CHAPTER 5
SIGE PROFILE OPTIMIZATION FOR LOW NOISE
This chapter explores the RF noise physics and SiGe profile optimization for low noise
using microscopic noise simulation. A higher Ge gradient in a noise critical region near the
EB junction reduces impedance field and hence minimum noise figure. A higher Ge gradient
near the EB junction, together with an unconventional Ge retrograding in the base to keep total
Ge content below stability, when optimized, can lead to significant noise improvement without
sacrificing peak fT and without any significant high injection fT rollo degradation.
5.1 Introduction
RF noise is an important aspect of RF devices as it sets the sensitivity of a wireless re-
ceiver. At a given technology generation, the base resistance is primarily limited by the max-
imum amount of base dopants that can be kept in place after device fabrication, and hence
limited by thermal cycle. SiGe profile, however, can be optimized to reduce minimum noise
figure [53] [54]. In previous work, the profile optimization was made by simulating device y-
parameters, and then calculating the minimum noise figure NFmin using a set of approximate
noise modeling equations [54] [55]. Those equations rely on simplified equivalent circuit, and
simplified noise source description, which become less valid at higher RF frequencies [12],
particularly for scaled devices with higher speed. In some cases, unphysical noise results are
obtained, preventing a meaningful optimization.
The purpose of this work is to investigate SiGe HBT noise physics and related SiGe profile
optimization using a more physical approach ? microscopic noise simulation. Using techniques
123
described in chapter 3, we can calculate the transistor equivalent input noise current or voltage
as integration of their corresponding noise concentration, in the same way the total number of
electrons is calculated as integration of electron concentration. This enables us to examine how
SiGe profile a ects the input noise current or voltage, the noise concentration profile, the local
noise source, as well as the propagation of local noise source towards the input, which we address
below. The results are then used to optimize SiGe profile for low noise under constant SiGe film
stability constraint. We use here a hypothetical SiGe HBT structure similar to those 200 GHz
HBTs reported in the literature [56] [57].
5.2 SiGe Profile Impact
From the power spectral densities of the input noise current, voltage, and their correlation
as Sia;i?a, Sva;v?a and Sia;v?a , the minimum noise figure NFmin, the noise resistance Rn, and optimum
source admittance Yopt are given (2.24) ? (8.19) in chapter 2. We first examine how SiGe pro-
file a ects NFmin, Sva;v?a , Sia;i?a, <(Sia;v?a ) and =(Sia;v?a ) using two ?conventional? sample SiGe
profiles shown in Fig. 5.1 (a). Profile I has a constant Ge gradient in the base. Compared to
profile I, profile II has a higher Ge gradient near the EB junction, but a flat Ge fraction near the
CB junction to not create any Ge retrograding inside the base. Profile II has 33% more total Ge.
Noise simulations are then performed using DESSIS [45], from 1 to 60 GHz, across a wide bias
range. Energy balance equations are solved to account for non-equilibrium transport in these
scaled devices. The DESSIS simulation input deck and MATLAB programming are given in
B.2 and B.3 in Appendix B.
Fig. 5.1 (b) shows the Gummel curves of the two profiles. The base current density JB is
the same for both profiles, as expected, because of identical emitter structure. Profile II gives
124
higher collector current density JC, and hence higher ?. fT is also slightly higher for profile II,
as shown in Fig. 5.2. A peak fT over 200 GHz is reached at JC = 10 mA=?m2. Fig. 5.3 shows
NFmin versus JC at 40 GHz. A clear improvement of NFmin can be observed for profile II. Gopt
is less for profile II. Profile I and II have similar Rn and Bopt.
0.14 0.15 0.16 0.17 0.18 0.19
0
0.05
0.1
0.15
0.2
0.25
0.3
Depth    (?m)
Ge Profile
Profile I
Profile II
0.75 0.8 0.85 0.9
10
- 4
10
- 2
10
0
10
2
V
BE
    (V)
J
B
 and 
J
C
    (mA/
?
m
2
)
Profile I
Profile II
J
C
 
J
B
 
(a) 
(b) 
E
B
C
Figure 5.1: (a) Ge profile I and II. (b) Gummel curves for profile I and II.
5.2.1 Distributive Transit Time Analysis
Distributive transit time analysis as a function of JC are performed to find out the reason
of fT improvement. Details of distributive transit time analysis can be found in [10]. The
spatial distribution of the total transit time is simulated, in terms of the so called di erential
transit time ?di . In an ideal 1-D bipolar transistor, at any position x, ?di (x) ? x represents the
local contribution to the total transit time due to minority carrier charge storage from depth x to
(x +  x). ?di has a unit of ps/?m, and its integration from emitter to collector gives the total
transit time ?ec [10]. The cuto frequency fT is related to ?ec by fT = 1=2??ec.
125
0.1 1 10 50
0
50
100
150
200
250
J
C
    (mA/mm
2
)
f
T
    (GHz)
Profile I
Profile II
Figure 5.2: fT vs JC for Ge profile I and II.
Fig. 5.4 shows the simulated di erential transit time ?di profile for Ge profile I and II for
JC = 2 mA=?m2. The ?di profile improvement of Ge profile II over profile I mainly lies in
the emitter and the base. Since the emitter of profile I and II are the same, the improvement of
?di of profile II in the emitter is the result of improved dc current gain ?, which is induced by
the additional Ge in the base of profile II. The improvement of ?di of profile II in the base is
the result of the enhanced Ge gradient of profile II near the emitter-base junction. However, this
improvement is slightly alleviated by the additional ?di induced by the Ge grading transition of
profile II [58].
126
1 10
0
0.5
1
J
C
    (mA/mm
2
)
NF
min
    (dB)
Profile I
Profile II
1 10
0
5
10
J
C
    (mA/mm
2
)
R
n
/50
W
Profile I
Profile II
1 10
0
0.5
1
1.5
J
C
    (mA/mm
2
)
G
opt
    (mS)
Profile I
Profile II
1 10
?0.6
?0.4
?0.2
0
J
C
    (mA/mm
2
)
B
opt
    (mS)
Profile I
Profile II
40 GHz 
40 GHz 
40 GHz 
40 GHz 
Profile I 
Profile II 
Profile II 
Profile I 
Profile II 
Profile I 
Profile II 
Profile I 
Figure 5.3: Noise parameters vs JC for profile I and II at 40 GHz.
5.2.2 Input Noise Voltage and Current
As NFmin is determined by Sia;i?a, Sva;v?a , and the real and imaginary parts of Sia;v?a , as shown
in (2.24) in chapter 2, we compare Sia;i?a, Sva;v?a , and real and imaginary parts of Sia;v?a for the
two profiles in Fig. 5.5, as a function of JC, at 40 GHz. The comparisons are similar at other
frequencies. Note that the input noise voltage Sva;v?a and the imaginary part of the correlation
=(Sia;v?a ) are approximately the same for both profiles in the whole bias range. This explains
similar Rn and Bopt for profile I and II from (8.17) and (8.19) in chapter 2. The input noise
current Sia;i?a, and the real part of the correlation <(Sia;v?a ), however, are much lower for profile
II. It is not clear if the Sia;i?a reduction or the <(Sia;v?a ) reduction, or both, is responsible for the
NFmin reduction in profile II. To find this out, we plot the two terms of Fmin  1 in Fig. 5.6.
An inspection of (2.24) immediately shows that the first term is determined by Sia;i?a, Sva;v?a and
127
0.15 0.155 0.16 0.165 0.17 0.175 0.18 0.185 0.19
0
10
20
30
40
50
Depth    (mm)
t
diff
    (ps/
m
m)
Profile I
Profile II
J
C
 = 2 mA/mm
2
 
     EB       
Boundary      
      CB     
Boundary     
Ge grading  
transitions 
Figure 5.4: 1-D center cut of ?di for profile I and II. JC = 2 mA=?m2.
=(Sia;v?a ), while the second term is determined by <(Sia;v?a ). The first term clearly dominates
over the second term. Because Sva;v?a and =(Sia;v?a ) are the same for both profiles, the smaller
input noise current Sia;i?a is the primary reason for the NFmin reduction in profile II. Similarly, we
find Sia;i?a is the primary reason for reduction of Gopt in profile II from (8.18) in chapter 2. This
suggests that we can focus on Sia;i?a in understanding the impact of SiGe profile on NFmin.
5.3 New Approach: Regional Electron and Hole Contributions
Fig. 5.7 shows the regional contributions of Sva;v?a . ?base;n? and ?base;p? denote base
electron and hole contributions, respectively. ?emitter;n? is used to denote emitter electron
contribution. The collector contribution is negligible. Sva;v?a is dominated by base electron con-
tribution at low JC, and dominated by base hole contribution at JC higher than 2 mA=?m2. The
base hole contribution is pretty much determined by the base doping, and does not change much
128
1 10
0
2
4
6
8
x 10
?18
J
C
    (mA/mm
2
)
S
va,va*
    (V
2
/Hz)
Profile I
Profile II
1 10
0
2
4
x 10
?24
J
C
    (mA/mm
2
)
S
ia,ia*
    (A
2
/Hz)
Profile I
Profile II
1 10
0
2
4
x 10
?22
J
C
    (mA/mm
2
)
?
(S
ia,va*
)    (VA/Hz)
Profile I
Profile II
1 10
0
2
4
6
x 10
?21
J
C
    (mA/mm
2
)
`
(S
ia,va*
)    (VA/Hz)
Profile I
Profile II
Profile II 
Profile I 
40 GHz 40 GHz 
Profile I 
Profile II 
40 GHz 
Profile I 
Profile II 
40 GHz 
Profile I 
Profile II 
Figure 5.5: Sia;i?a, Sva;v?a and their correlation vs JC. f = 40 GHz.
with bias. The base electron contributions of Sva;v?a for profile I and II are approximately the
same, despite the Ge profile di erence. This is qualitatively consistent with first order noise
models [54].
Fig. 5.8 shows the regional contributions of Sia;i?a. At lower and moderate JC?s where
NFmin is low, Sia;i?a is dominated by the base electron contribution, which is responsible for the
reduction of Sia;i?a for profile II. At lower JC, the emitter hole contribution of Sia;i?a is negligible
because of the high ?, unlike in the 50 GHz HBTs discussed in [55]. Thus, the higher ? and
hence smaller 2qIB is not the reason for the reduced Sia;i?a in profile II at lower JC. At higher JC
near peak fT , however, the emitter hole contribution becomes comparable to the base electron
contribution. The base hole contribution of Sia;i?a also comes into the picture at high JC. The
base hole contributions of Sia;i?a are almost the same for the two profiles. The main reason for
129
1 10
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
J
C
    (mA/mm
2
)
(F
min
?1) Contributions
Profile I
Profile II
40 GHz 
2kT 
?(S
ia,va*
) 
2kT 
[S
ia,ia*
S
va,va*
?(`(S
ia,va*
)
2
]
1/2
 
Figure 5.6: (Fmin  1) contributions. f = 40 GHz.
the better noise performance of profile II at low JC of interest to low noise is thus the smaller
base electron contribution of Sia;i?a.
5.3.1 Noise Critical Region and Ge Profile Impact
We now analyze the spatial distribution of the noise concentration for Sia;i?a due to electrons
and holes. Using techniques in chapter 3, Sia;i?a is the volume integration of the input noise
current noise concentration CSia;i?a , which has electron contribution CnS
ia;i?a
and hole contribution
130
1 10
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x 10
?18
J
C
    (mA/mm
2
)
S
va,va*
 Contributions    (V
2
/Hz)
40 GHz 
base, p 
base, n 
emitter, n 
solid lines: Profile I 
dash lines: Profile II 
Figure 5.7: Comparison of Regional contributions of Sva;v?a . f=40 GHz.
ChS
ia;i?a
,
Sia;i?a =
Z
 
CSia;i?ad ; (5.1)
=
Z
 
CnS
ia;i?a
dV +
Z
V
CpS
ia;i?a
d ; (5.2)
CSia;i?a = CnS
ia;i?a
+CpS
ia;i?a
: (5.3)
The input noise current noise concentration has a unit of A2/Hz/cm3. Fig. 5.9 (a) shows the
1-D center cut of the input noise current noise concentrations due to electrons and holes, CnS
ia;i?a
and CpS
ia;i?a
. Integration of CnS
ia;i?a
over volume gives the total input noise current due to electrons.
Similarly, the integration of CpS
ia;i?a
over volume gives the total input noise current due to holes.
JC = 2 mA=?m2, frequency is 40 GHz. First, the electron contribution dominates over the hole
131
1 10
0
0.5
1
1.5
2
2.5
3
3.5
x 10
?24
J
C
    (mA/mm
2
)
S
ia,ia*
 Contributions    (A
2
/Hz)
40 GHz 
base, n 
emitter, p 
solid lines: Profile I 
dash lines: Profile II 
base, p 
Figure 5.8: Comparison of Regional contributions of Sia;i?a. f=40 GHz.
contribution. It is clear that most of the input noise current comes from near the EB junction,
where CnS
ia;i?a
is highest. The primary reason for the smaller Sia;i?a of profile II is its smaller CnS
ia;i?a
near the EB junction.
The noise concentration is given by the product of a local noise source which is proportional
to carrier density, and the impedance field, which describes noise propagation,
CnS
ia;i?a
= CnSijGn;iaj2; (5.4)
CpS
ia;i?a
= CpSijGp;iaj2; (5.5)
132
where jGn;iaj2 and jGp;iaj2 are the electron and hole impedance field from local noise source to
input noise current concentration, respectively, with a unit of A2/A2. Fig. 5.9 (b) shows 1-D cen-
ter cut of local electron noise current source CnSi, and the electron impedance field jGn;iaj2. The
CnS
ia;i?a
di erence between the two profiles is clearly dominated by the di erence in impedance
field, rather than the local noise source. The fundamental reason for the smaller Sia;i?a and NFmin
in profile II is thus the reduced base impedance field, which means less noise current produced
at the transistor input (base) for the same amount of local current density fluctuations. Observe
that the high impedance field occurs over 10 nm at the beginning of the neutral base, where Ge
ramps up for both profiles. The higher Ge gradient in profile II in this ?noise critical? region
clearly has led to Sia;i?a and NFmin reduction. To not have any retrograding of Ge in the base,
Ge fraction is kept constant after the Ge peak in profile II, leading to more total Ge, which is
undesired from a SiGe film stability standpoint. A logical question is if the benefit of reduced
impedance field over the 10 nm ?noise critical? region can be maintained if Ge is retrograded
after the peak to keep the total Ge content the same. This is indeed the case, and can be used for
SiGe profile optimization at constant stability.
5.4 Optimization Under Constant Stability
We now increase the Ge gradient in the noise critical region where the impedance field for
input noise current is high, while keeping the total Ge content the same to maintain SiGe film
stability. Inevitably, for su ciently high peak Ge, we are forced to have Ge retrograding in
the base, which is usually avoided in conventional SiGe profile design, as the retrograding can
introduce a retarding field. Inspection of simulation details, however, shows that a retrograding
of Ge can indeed be used in the later part of the base, near the CB junction, without degrading
133
0.15 0.155 0.16 0.165 0.17 0.175 0.18 0.185 0.19
0
0.2
0.4
0.6
0.8
1
1.2
x 10
?9
Depth    (mm)
C
S
ia,ia
*
n
 and C
S
ia,ia
*
p
   (A
2
/Hz/cm
3
)
Profile I
Profile II
J
C
=2 mA/mm
2
 
f = 40 GHz 
     EB  
Boundary 
        CB    
Boundary      
C
S
ia,ia*
n
 
        noise   
critical region 
C
S
ia,ia*
p
 
Figure 5.9: 1-D center cut of CnS
ia;i?a
and CpS
ia;i?a
for Ge profile I and II. JC = 2 mA=?m2. f = 40
GHz.
fT if designed properly. The fallo of p-type base doping near the CB junction helps in part as
it generates an accelerating field. In this case, profile III has a slightly higher peak Ge fraction.
An optimized profile example using Ge retrograding in the base is given in Fig. 5.11 (a) ? profile
III, together with the reference profile ? profile I.
Fig. 5.11 (b) compares the Gummel curves of profile I, II and III. JB is the same, and JC
is higher for the optimized profile III. Despite the Ge retrograding in the later part of the base,
profile III shows a higher peak fT than profile I, as shown in Fig. 5.12. The fT rollo , however,
occurs at a slightly lower JC. The rollo slope for profile III is similar to that for profile I,
because the smaller Ge retrograding gradient partially o sets the ?earlier? retrograding, an e ect
di erent from the SiGe profile design tradeo discussed in [55]. Fig. 5.13 shows 1-D center cut
of ?di for profile I, II and III at JC = 2 mA=?m2.
134
10
?20
10
?19
10
?18
C
S
i
n
    (A
2
/Hz/cm
3
)
Profile I
Profile II
0.15 0.155 0.16 0.165 0.17 0.175 0.18 0.185 0.19
0
2
4
6
8
10
12
14
16
18
x 10
9
Depth    (mm)
|G
n,ia
|
2
    (A
2
/A
2
)
J
C
=2 mA/mm
2
 
f = 40 GHz 
     EB  
Boundary 
        CB    
Boundary      
        noise   
critical region 
Figure 5.10: 1-D center cut of CnSi and jGn;iaj2 for profile I and II. JC = 2 mA=?m2. f = 40
GHz.
Fig. 5.14 shows 1-D center cut of electron and hole noise concentrations of Sia;i?a. Profile
III shows lower CnS
ia;i?a
than profile I in the ?noise critical? region, despite the Ge retrograding in
the base required for stability, due to reduced impedance field. In terms of reducing the input
noise current, Profile III is as e ective as profile II, which is over stability limit, and does not
have Ge retrograding in the base.
Fig. 5.15 shows NFmin?JC for profile I, II and III at 10 GHz and 60 GHz. Fig. 5.16 shows
NFmin versus frequency for profile I, II and III at JC = 2 and 10 mA=?m2. The overall noise
improvement is about the same as that achieved by profile II, which has 33 % more total Ge and
is over stability limit. With profile III, we have increased fT , ? and decreased NFmin without
sacrificing SiGe film stability. The high injection fT rollo degradation has been kept minimum
by minimizing the gradient of Ge retrograding.
135
0.14 0.15 0.16 0.17 0.18 0.19 
0 
0.05 
0.1 
0.15 
0.2 
0.25 
0.3 
Depth     ( ? m) 
Ge Profile 
Profile  I 
Profile II 
Profile III 
0.75 0.8  0.85 0.9  
10 
- 4 
10 
-  2 
10 
0 
10 
2 
V 
BE 
    (V) 
J 
B 
 and  
J 
C 
    (mA/ 
? 
m 
2 
) 
J 
C 
  
J 
B 
  
(a)  (b)  
Figure 5.11: (a) Constant stability Ge profiles: profile I, II and III. (b) Gummel curves for profile
I, II and III.
0.1 1 10 50
0
50
100
150
200
250
J
C
    (mA/mm
2
)
f
T
    (GHz)
Profile I
Profile II
Profile III
Figure 5.12: fT vs JC for profile I, II and III.
136
0.15 0.155 0.16 0.165 0.17 0.175 0.18 0.185 0.19
0
10
20
30
40
50
Depth    (mm)
t
diff
    (ps/
m
m)
Profile I
Profile II
Profile III
J
C
 = 2 mA/mm
2
 
     EB       
Boundary      
      CB     
Boundary     
Ge grading  
transitions 
Figure 5.13: 1-D center cut of ?di for profile I, II and III. JC = 2 mA=?m2.
0.15 0.155 0.16 0.165 0.17 0.175 0.18 0.185 0.19
0
0.2
0.4
0.6
0.8
1
1.2
x 10
?9
Depth    (mm)
C
S
ia,ia
*
n
 and C
S
ia,ia
*
p
   (A
2
/Hz/cm
3
)
Profile I
Profile II
Profile III
J
C
=2 mA/mm
2
 
f = 40 GHz 
     EB  
Boundary 
        CB    
Boundary      
C
S
ia,ia*
n
 
        noise   
critical region 
C
S
ia,ia*
p
 
Figure 5.14: 1-D center cut of CnS
ia;i?a
and CpS
ia;i?a
for profile I, II and III. f = 40 GHz. JC = 2
mA=?m2.
137
1 10
0  
0.2
0.4
0.6
0.8
1  
1.2
1.4
1.6
1.8
2  
J
C
    (mA/mm
2
)
NF
min
    (dB)
Profile I
Profile II
Profile III
10 GHz 
60 GHz 
Figure 5.15: NFmin vs JC at 10 and 60 GHz.
0 10 20 30 40 50 60
0
0.5
1
1.5
2
Frequency    (GHz)
NF
min
    (dB)
Profile I
Profile II
Profile III
J
C
=10 mA/mm
2
 
J
C
=2 mA/mm
2
 
Figure 5.16: NFmin vs frequency at JC = 2 and 10 mA=?m2.
138
5.5 Summary
We have explored RF noise physics in advanced SiGe HBTs using microscopic noise simu-
lation. We have shown that SiGe profile primarily a ects the minimum noise figure through the
input noise current, and identified the small region near the EB junction as where most of the
input noise current originates. A higher Ge gradient in this region helps reducing the impedance
field for the input noise current. At constant SiGe film stability, increasing the Ge gradient in
the noise critical region ultimately necessitates retrograding of Ge inside the neutral base, and
the gradient of such Ge retrograding needs to be optimized within stability limit to minimize
high injection fT rollo degradation. An example of successful SiGe profile optimization using
unconventional Ge retrograding inside the base has been presented.
139
CHAPTER 6
MODELING OF INTRINSIC NOISE IN CMOS
In this chapter, RF noise of 50 nm Le CMOS is simulated using hydrodynamic noise sim-
ulation. Intrinsic noise sources for the Y- and H- noise representations are examined and models
of intrinsic noise sources are proposed. The relations between the Y- and H- noise representa-
tions for MOSFETs are examined, and the importance of correlation for both representations is
quantified. The theoretical values of H- noise representation model parameters are derived for
the first time for long channel devices. The H- noise representation correlation is shown theoret-
ically to have a zero imaginary part. The H- noise representation has the inherent advantage of a
more negligible correlation, which makes circuit design and simulation easier.
The H-representation noise sources are experimentally extracted using noise parameters
measured on 0.25 ?m RF CMOS devices. A simple yet e ective model is proposed to model the
H-representation noise sources as a function of bias. Excellent modeling results are achieved for
all of the noise parameters up to 26 GHz, at all biases.
6.1 Introduction
Recent CMOS scaling has led to significant RF performance improvement. One of the ma-
jor concerns is RF noise. A popular noise representation for MOSFET is the Y-representation,
which describes the short-circuit input and output noise currents, ig and id, as shown in Fig. 1.9
in section 1.4.1. For GaAs MESFETs and HEMTs, however, the H-representation is more pop-
ular, and is represented by the Pospieszalski model [27], as shown in Fig. 1.14 in section 1.4.2.
140
This chapter first investigates the RF noise performance of 50 nm Le CMOS transistors us-
ing hydrodynamic microscopic noise simulation [59]. The simulation results are then used to
analyze the intrinsic noise sources for the Y- noise representation and the H- noise representa-
tion. This chapter will show that the Y- representation noise sources can be modeled using the
Y-parameters, and the H-representation noise sources can be modeled using the H-parameters.
We will also show that the correlation between noise sources has negligible impact on transistor
noise parameters for both noise representations. This chapter further examines the relationships
between MOSFET Y- and H- noise representations, and derives a set of theoretical equations for
conversion between the two noise models described above. The theoretical values of H- repre-
sentation model parameters are derived for the first time for long channel devices. The H- noise
representation correlation is shown theoretically to have a zero imaginary part. We further show
that the H- noise representation has the inherent advantage of a more negligible correlation for
noise parameter modeling.
6.2 Technical Approach
The device structure is constructed based on reported 90 nm CMOS literature and the ITRS
roadmap. DC I  V , y-parameters, and noise parameters are simulated using hydrodynamic
transport models. The simulator used is DESSIS 9.0 from ISE [45]. The Lombardi surface
mobility model and the default carrier energy relaxation time is used. The simulated I V and gm
characteristics are comparable to reported data on 90 nm CMOS devices with similar structures.
The simulations are performed from 1 to 40 GHz. The transistor has a 70 nm poly gate length,
a 50 nm metallurgical channel length, and an e ective oxide thickness of 1.2 nm. The channel
doping is retrograded from the surface toward the bulk, and halos are used for suppressing short
141
channel e ect. Due to a limitation of the simulator in handling terminal resistances during noise
simulation, only the intrinsic device is simulated. The terminal resistances are then added to
the intrinsic device through standard linear noise circuit analysis. The error introduced is quite
small when compared to self-consistent mixed-mode device and circuit simulation results. The
DESSIS simulation input deck and MATLAB Programming are given in Appendix C.
6.3 Simulation Results
6.3.1 DC I  V Curves
Fig. 6.1 shows IDS and gm vs Vgs at Vds = 1 V. ID;sat = 1341 ?A/?m at Vgs = 1 V. Fig. 6.2
shows the output curves for Vgs = 0.1 ? 1 V, with step of 0.1 V.
6.3.2 Noise Parameters
Fig. 6.3 shows NFmin simulated at 5 GHz versus IDS. The simulation is performed for a 2
?m wide finger. The rg value is estimated assuming double side gate contact, and a gate sheet
resistance of 16  =?, which gives 40  lateral resistance. We assume an additional 80  for
other gate resistance components. rs=rd=45  for W = 2 ?m is from the ITRS roadmap for 90
nm CMOS (which have Le between 50 and 70 nm) [60]. Experimental measurement of deep
submicron CMOS noise has proven challenging, which is certainly the case for 90 nm CMOS
processes with below 1 dB NFmin. The measured NFmin for an experimental 90 nm CMOS [8]
is plotted for comparison. The simulated NFmin is still below measurement data. The electrical
channel length di erence between the simulated and experimental structures, models of mobility
and microscopic noise source density, uncertainty in terminal resistances and other parasitics in
the test structure could all contribute to the simulation-data discrepancy. We note
142
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10
?1
10
0
10
1
10
2
10
3
10
4
V
gs
    (V)
I
DS
    (
m
A/
m
m)
V
ds
 = 1 V  
Simulation: 50 nm L
eff
 device 
(a)
0 0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
x 10
?3
V
gs
    (V)
g
m
    (S)
Simulation: 50 nm L
eff
 device, W = 2 mm 
V
ds
 = 1 V  
(b)
Figure 6.1: (a) IDS, and (b) gm vs Vgs at Vds = 1 V.
143
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
200
400
600
800
1000
1200
1400
V
ds
    (V)
I
DS
    (
m
A/
m
m)
V
gs
 = 0.1 to 1 V, step: 0.1 V. 
Figure 6.2: Output Curve
0 100 200 300 400 500 600 700
0
0.2
0.4
0.6
0.8
1
I
DS
    (mA/mm)
NF
min
    (dB)
f =5 GHz
V
ds
=1 V 
r
s
=r
d
=45 W 
r
g
=120 W 
data 
Simulation 
Figure 6.3: Simulation vs data reported in [8]: NFmin at 5 GHz vs IDS.
144
that the measured NFmin is dependent on the number of gate fingers [8], which is not the case for
an ideal transistor. This may be another source of data-simulation discrepancy.
According to Fig. 6.3, the minimum of NFmin occurs when IDS is 100-200 ?A/?m, which
corresponds to moderate inversion operation, where the cuto frequency fT is rising rapidly and
near the peak (Fig. 6.4). The overall IDS dependence of NFmin is weak once moderate inversion
occurs. For low-noise amplifiers, the combination of a low NFmin and a high fT at low IDS is
desirable from a power consumption standpoint. Fig. 6.5 and Fig. 6.6 shows Gopt and Bopt at 5
GHz vs IDS and Fig. 6.7 shows Rn at 5 GHz vs IDS. Scaling enables high fT in the moderate
inversion region, and thus lower power CMOS LNA design.
0 100 200 300 400 500 600 700
0
50
100
150
I
DS
    (mA/mm)
f
T
    (GHz)
V
DS
 = 1 V
Figure 6.4: fT vs IDS.
145
0 100 200 300 400 500 600 700
0
1
2
3
4
x 10
?5
G
opt
    (S)
f =5 GHz V
ds
=1 V 
Simulation: L = 50 nm, W = 2 mm
R
g
=120 W 
R
s
=R
d
=45 W 
I
DS
    (mA/mm) 
Figure 6.5: Gopt vs IDS.
6.4 Intrinsic Noise Sources and Modeling
In general, two correlated noise sources are required to fully describe the noise behavior of
any linear noisy two port. For each linear two port parameter set (e.g. Y, Z, H, and ABCD),
there is a corresponding set of noise sources. The two noise sources (voltages, currents or a
combination) are in general frequency and bias dependent.
6.4.1 Y-representation Noise Sources
We first consider the noise sources for the Y-parameter set, the gate and drain current noises.
Fig. 6.8 shows Sid;i?d versus frequency. Sid;i?d is normalized by 4kTgm as opposed to 4kTgd0, as it
is more relevant for circuit design. Sid;i?d is nearly frequency independent for frequencies below
fT . At low IDS where LNAs are biased, Sid;i?d=4kTgm is not too much greater than its long
channel theoretical value (2=3).
146
0 100 200 300 400 500 600 700
?7
?6
?5
?4
?3
?2
?1
0
x 10
?5
B
opt
    (S)
f =5 GHz V
ds
=1 V 
Simulation: L = 50 nm, W = 2 mm
R
g
=120 W 
R
s
=R
d
=45 W 
I
DS
    (mA/mm) 
Figure 6.6: Bopt vs IDS.
0 100 200 300 400 500 600 700
0
1000
2000
3000
4000
5000
6000
I
DS
    (mA/mm)
R
n
    (
W
)
r
g
=120 W 
r
s
=r
d
=45 W 
f =5 GHz V
ds
=1 V 
Simulation: L = 50 nm, W = 2 mm
Figure 6.7: Rn vs IDS.
147
0 5 10 15 20
0
1
2
3
4
5
6
V
DS
 = 1 V
Freq    (GHz)
S
id,id*
/(4kTg
m,sat
)
I
DS
=41 mA/mm
I
DS
=134 mA/mm
I
DS
=275 mA/mm
I
DS
=1341 mA/mm
Figure 6.8: Sid;i?d=[4kTgm] vs frequency.
0 5 10 15 20
0
1
2
3
4
5
V
DS
 = 1 V
Freq    (GHz)
S
ig,ig*
/(4kT
?
Y
11
)
I
DS
=41 mA/mm
I
DS
=134 mA/mm
I
DS
=275 mA/mm
I
DS
=1341 mA/mm
Figure 6.9: Sig;i?g=[4kT<(Y11)] vs frequency.
148
Fig. 6.9 shows Sig;i?g normalized by 4kT<(Y11) versus frequency. Sig;i?g=[4kT<(Y11)] is
frequency independent, as found in long channel devices. The value of Sig;i?g=[4kT<(Y11)] is
close to 4 at lower IDS of interest to LNAs.
The normalized correlation c ? Sig;i?d=pSig;i?gSid;i?d shows a negligible real part, as found in
long channel devices (Fig. 6.10). The imaginary part of c is nearly frequency independent.
0 5 10 15 20
?0.2
0
0.2
0.4
0.6
0.8
1
V
DS
 = 1 V
Freq    (GHz)
?
C
S
ig,id*
 and 
`
C
S
ig,id*
I
DS
=41 mA/mm
I
DS
=134 mA/mm
I
DS
=275 mA/mm
I
DS
=1341 mA/mm
?C
S
ig,id*
 
`C
S
ig,id*
 
Figure 6.10: Normalized correlation c vs frequency.
A simplified yet accurate model for noise sources is desired for circuit design and circuit
simulation. We have shown that Sid;i?d and Sig;i?g can be readily modeled using gm and Y11 in these
50 nm Le devices, with simple coe cients that are constant for a given bias. The remaining
question is the correlation, which cannot be handled by certain simulators like SPICE. This
correlation can not be neglected for practical purposes in 50 nm Le CMOS. Figs. 6.11 and 6.12
show the NFmin, Rn and Yopt simulated with and without the correlation. From 10 ? 20 GHz, for
149
all biases of interest, the di erence between the noise parameters obtained with and without the
correlation is noticeable except for Rn.
0 10 20
0
0.2
0.4
0.6
0.8
V
DS
 = 1 V
Freq    (GHz)
NF
min
    (dB)
I
DS
=41 mA/mm
I
DS
=134 mA/mm
I
DS
=275 mA/mm
I
DS
=1341 mA/mm
0 10 20
0
500
1000
1500
Freq    (GHz)
R
n
    (
W
)
I
DS
=41 mA/mm
I
DS
=134 mA/mm
I
DS
=275 mA/mm
I
DS
=1341 mA/mm
symbols: with S
ig,id*
lines:                  
without S
ig,id*
      
V
DS
 = 1 V
symbols: with S
ig,id*
  
lines: without S
ig,id*
 
Figure 6.11: NFmin and Rn vs frequency, with and without correlation.
6.4.2 H-representation Noise Sources
A linear noisy two port can also be described using the H-parameter set, which involves an
input noise voltage and an output noise current, which are denoted as vh and ih. Note that ih is dif-
ferent from the id discussed above (the output noise current for Y-representation). Even though
the H-representation was not given in the original noise representation standards [43], it has been
successfully used for noise modeling of GaAs MESFETs and HEMTs. The Pospieszalski noise
model [27] falls into this category, which further assumed that vh and ih are uncorrelated.
Through circuit analysis, we convert the noise sources from Y- noise representation to H-
noise representation. Fig. 6.13 shows Svh;v?h versus frequency. In Pospieszalski?s model, it was
150
0 10 20
0
0.5
1
1.5
x 10
?4
Freq    (GHz)
?
Y
opt
    (
W
?1
)
I
DS
=41 mA/mm
I
DS
=134 mA/mm
I
DS
=275 mA/mm
I
DS
=1341 mA/mm
0 10 20
?5
?4
?3
?2
?1
0
x 10
?4
Freq    (GHz)
`
Y
opt
    (
W
?1
)
I
DS
=41 mA/mm
I
DS
=134 mA/mm
I
DS
=275 mA/mm
I
DS
=1341 mA/mm
symbols: with S
ig,id*
lines:                  
without S
ig,id*
      
V
DS
 = 1 V
symbols:           
with S
ig,id*
    
lines:             
without S
ig,id*
 
V
DS
 = 1 V
Figure 6.12: Yopt vs frequency, with and without correlation.
assumed that Svh;v?h is proportional to the NQS channel resistance rnqs, and therefore frequency
independent. Note that Svh;v?h is frequency independent as shown in Fig. 6.13, which complies
with the assumption in Pospieszalski?s model.
Svh;v?h normalized by 4kT<(h11) is shown in Fig. 6.14. This provides a natural way of
modeling Svh;v?h as long as the small signal model can correctly model h11. We only need to
model the Svh;v?h=[4kT<(h11)] ratio, which is a constant for a given bias. Svh;v?h=[4kT<(h11)] ?
5 at lower IDS, and decreases with increasing IDS as shown in Fig. 6.15.
Fig. 6.16 shows Sih;i?h=[4kT<(gm)] vs frequency, which is nearly frequency independent.
Fig. 6.17 shows the correlation between vh and ih. Compared to the Y-representation, the cor-
relation for the H-representation is much weaker for all of the biases and frequencies. In this
151
0 5 10 15 20
0
2
4
6
8
x 10
?19
Freq    (GHz)
S
vh,vh*
    (V
2
/Hz)
I
DS
=41 mA/mm
I
DS
=134 mA/mm
I
DS
=275 mA/mm
I
DS
=1341 mA/mm
V
DS
 = 1 V
Figure 6.13: Svh;v?h vs frequency.
sense, the H-representation is a better choice, comparing to the correlation for Y-representation
is large and non-negligible.
The NFmin, Rn and Yopt are calculated with and without the correlation for comparison. For
all practical purposes, the impact of the correlation on NFmin, Rn and =(Yopt) is negligible, as
shown in Figs. 6.18. The <(Yopt) shows a visible but small sensitivity to the correlation. This
suggests a new path to compact modeling of noise sources in 50 nm Le based on h11 and gm.
Only two parameters are needed for modeling noise at each bias, the ratio of Svh;v?h=[4kT<(h11)].
152
0 5 10 15 20
0
1
2
3
4
5
Freq    (GHz)
S
vh,vh*
/(4kT
?
h
11
)
I
DS
=41 mA/mm
I
DS
=134 mA/mm
I
DS
=275 mA/mm
I
DS
=1341 mA/mm  
V
DS
 = 1 V
Figure 6.14: Svh;v?h=[4kT<(h11)] vs frequency.
0 100 200 300 400 500 600 700
0
1
2
3
4
5
I
DS
   (mA/mm)
S
vh,vh*
/(4kT
?
h
11
)
V
ds
 = 1 V 
Figure 6.15: Svh;v?h=[4kT<(h11)] vs IDS.
153
0 5 10 15 20
0
0.5
1
1.5
2
2.5
Freq    (GHz)
S
ih,ih*
/(4kTg
m,sat
)
I
DS
=41 mA/mm
I
DS
=134 mA/mm
I
DS
=275 mA/mm
I
DS
=1341 mA/mm
V
DS
 = 1 V
Figure 6.16: Sih;i?h vs frequency.
0 5 10 15 20
?0.2
0
0.2
0.4
0.6
0.8
Freq    (GHz)
?
C
S
vh,ih*
 and 
`
C
S
vh,ih*
    
I
DS
=41 mA/mm
I
DS
=134 mA/mm
I
DS
=275 mA/mm
I
DS
=1341 mA/mm
`C
S
vh,ih*
 
?C
S
vh,ih*
 
V
DS
 = 1 V
Figure 6.17: Correlation of H-representation noise sources vs frequency.
154
0 10 20
0
0.2
0.4
0.6
0.8
Freq    (GHz)
NF
min
    (dB)
I
DS
=41 mA/mm
I
DS
=134 mA/mm
I
DS
=275 mA/mm
I
DS
=1341 mA/mm
0 10 20
0
500
1000
1500
Freq    (GHz)
R
n
    (
W
)
I
DS
=41 mA/mm
I
DS
=134 mA/mm
I
DS
=275 mA/mm
I
DS
=1341 mA/mm
symbols: with S
vh,ih*
lines:                  
without S
vh,ih*
      
V
DS
 = 1 V
symbols: with S
vh,ih*
  
lines: without S
vh,ih*
 
V
DS
 = 1 V
0 10 20
0
0.5
1
1.5
x 10
?4
Freq    (GHz)
?
Y
opt
    (
W
?1
)
I
DS
=41 mA/mm
I
DS
=134 mA/mm
I
DS
=275 mA/mm
I
DS
=1341 mA/mm
0 10 20
?5
?4
?3
?2
?1
0
x 10
?4
Freq    (GHz)
`
Y
opt
    (
W
?1
)
I
DS
=41 mA/mm
I
DS
=134 mA/mm
I
DS
=275 mA/mm
I
DS
=1341 mA/mm
                symbols:              
                with S
vh,ih*
       
lines: without S
vh,ih*
             
V
DS
 = 1 V
symbols:                
with S
vh,ih*
         
lines:                  
without S
vh,ih*
      
V
DS
 = 1 V
Figure 6.18: NFmin, Rn and Yopt with and without correlation between vh and ih.
155
6.5 Relations Between Y- and H- Noise Representations in MOSFETs
As discussed above, the widely used van der Ziel model [15] describes ig, id and their
correlation for a MOSFET operated in the saturation region as:
Sig;i?g = 4kT?ig<(Y11); (6.1)
Sid;i?d = 4kT?idgm; (6.2)
cY 4= Sig;i
?
dp
Sig;i?gSid;i?d = y +jx; (6.3)
where ?ig, ?id, and normalized correlation term cY are model parameters. Depending on pref-
erence, the zero Vds output conductance gd0 is sometimes used instead of gm in (6.2). For long
channel devices, ?ig = 4=3, ?id = 2=3, and cY = j0:4 (y = 0, x = 0:4). For short channel de-
vices, these parameters become bias dependent and deviate from their long channel values, and
there do not exist general expressions for these parameters. However, these parameters remain
frequency independent, and the correlation cY remains imaginary only, or y = 0, because of the
capacitive nature of the channel to gate coupling.
The H- noise representation describe the transistor noise with the An input noise voltage vh
and an output noise current ih. Note that ih is di erent from the id discussed above. Even though
it has been argued in [61] and [62] that the H- noise representation is inherently unsuitable for
MOSFETs, H- noise representation has been successfully applied the H-representation to noise
modeling in 50 nm MOSFETs using DESSIS simulation, as discussed in previous section in this
chapter [63]. The H- noise representation was further shown to have the advantage of having a
negligible correlation term, which is significant for circuit design [63]. A model for vh and ih
156
was also proposed [63]:
Svh;v?h = 4kT?vh<(h11) (6.4)
Sih;i?h = 4kT?ihgm (6.5)
cH 4= Svh;i
?
hp
Svh;v?hSih;i?h = a+jb; (6.6)
where ?vh, ?ih, and cH are bias dependent but frequency independent model parameters in gen-
eral. It was also observed that cH is largely real, and more negligible than cY for noise parame-
ters. The reasons for the observations, however, were not understood.
6.5.1 Relations Between Y- and H- Noise Representation Coe cients
Consider the simplified small signal equivalent circuit in Fig. 6.19 for a MOSFET operating
in saturation region. Rgs is the Non-Quasi-Static (NQS) channel resistance, which can be related
to gm through Rgs = 1=( gm).  = 5 for long channel. An inspection of Fig. 6.19 shows
<(h11) = Rgs, where < stands for taking the real part. The Y-parameter matrix is given by
Y =
2
64 j
!Cgs
A +
!2C2gsRgs
A 0
gm 0
3
75; (6.7)
where
A = 1 +!2C2gsR2gs = 1 +
? !
 !T
?2
; (6.8)
157
here !T = gm=Cgs. In practice, ! <<  !T , thus A ? 1 and
Y11 ? !2C2gsRgs +j!Cgs: (6.9)
gs
R
gs
C
m   gs
g  v
+ 
- 
gs
v
+ 
- 
+ 
-
1
V
2
V
1
I
2
I
Figure 6.19: Small signal equivalent circuit for intrinsic MOSFET.
Substituting (6.9) into (6.1),
Sig;i?g = 4kT?iggm
? !
!T
?2
; (6.10)
where ?ig = ?ig= . With Rgs = 1=( gm), (6.4) becomes:
Svh;v?h = 4kT?vh 1 g
m
= 4kT?vh 1g
m
; (6.11)
where ?vh = ?vh= .
Using two port noise circuit analysis, we have derived equations for conversions between
the two representations. Substituting (6.7), and (6.1) ? (6.3) into (2.69) ? (2.71) in section 2.1.4,
158
we have
Svh;v?h = 4kT? 1g
m
; (6.12)
Sih;i?h = 4kTgm
 
? +?id  2xp??id
?
; (6.13)
= 4kT?ihgm;
Svh;i?h = 4kT(?  xp??id) +jy ? 4kTp??id: (6.14)
The normalized H-representation correlation cH is defined by,
cH 4= Svh;i
?
hp
Svh;v?hSih;i?h (6.15)
= a+jb; (6.16)
and a and b are obtained by,
a = <(cH) =
p?  xp?
idq
? +?id  2xp??id
; (6.17)
b = =(cH) = y ?
p?
idq
? +?id  2xp??id
: (6.18)
Therefore given Y- noise representation model parameters ?ig, ?id and cY = y +jx, the H- noise
representation model parameters are obtained as:
?vh = ?ig; (6.19)
?ih = ?ig +?id  2x
q
?ig?id; (6.20)
159
a =
p?
ig  x
p?
idq
?ig +?id  2xp?ig?id
; (6.21)
b = y
p?
idq
?ig +?id  2xp?ig?id
: (6.22)
Since y, the real part of cY , is 0 physically due to capacitive gate to channel coupling, b = 0,
meaning that the corresponding H- noise representation correlation is purely real. As ?ig = ?vh,
we have ?ig = ?vh. From now on, we will use ? and ? for convenience.
For long channel device, ?vh = 4=3, ?ih = 0:6, which is less than ?id = 2=3, a = 0:2458,
which is smaller than x = 0:4, and b = 0.
Similarly, substituting (6.7), and (6.4) ? (6.6) into (2.66) ? (2.68) in section 2.1.4, we have,
Sig;i?g =
? !
!T
?2
? 4kT?gm (6.23)
Sid;i?d = 4kTgm
 
?ih +?  2ap?ih?
?
; (6.24)
= 4kT?idgm;
Sig;i?d = j !!
T
? 4kT?gm
" 
1  a
r?
ih
?
!
 jb
r?
ih
?
#
: (6.25)
The normalized correlation cY is obtained by,
cY 4= Sig;i
?
dp
Sig;i?gSid;i?d (6.26)
= y +jx; (6.27)
160
and x and y are obtained by
x = =(cY ) =
p?  ap?ih
q
?ih +?  2a
p
?ih?
; (6.28)
y = <(cY ) = b
p
?ihq
?ih +?  2a
p
?ih?
: (6.29)
Therefore, given H- noise representation parameters, Y- noise representation parameters can be
obtained by
?ig = ?vh; (6.30)
?id = ?ih +?  2ap?ih?; (6.31)
x =
p?  ap?
ihq
?ih +?  2ap?ih?
; (6.32)
y = b
p?
ihq
?ih +?  2ap?ih?
: (6.33)
Even if cH = 0 (a = 0, b = 0), meaning zero correlation for the H- noise representation, cY still
has an imaginary part according to (6.32) and (6.33)
cYjcH=0 = y +jx = j
s
?
?ih +?: (6.34)
6.5.2 Noise Parameters
We now examine the importance of correlation for noise parameter modeling, for Y- and
H- noise representations. The minimum noise figure NFmin, the noise resistance Rn, the real
and imaginary part of the optimal source admittance Gopt and Bopt, are directly determined by
161
the chain noise representation parameters Sva;v?a , Sia;i?a and Sia;v?a [11]. We now derive the chain
noise parameters Sva;v?a , Sia;i?a and Sia;v?a using Y- and H- noise representation model parameters.
Substituting (6.1) - (6.3) into (2.43), (2.44), and (2.45) in section 2.1.2, the chain noise
parameters Sva;v?a , Sia;i?a and Sia;v?a can be obtained using Y-representation model parameters by
Sva;v?a = 1jY
21j2
Sid;i?d; (6.35)
= 4kT 1g
m
?id; (6.36)
Sia;i?a = Sig;i?g +
??
??Y11
Y21
??
??
2
Sid;i?d  2<
?Y?
11
Y?21Sig;i
?
d
?
; (6.37)
= 4kTgm
? !
!T
?2 
?id +?  2xp?id?
?
; (6.38)
Sia;v?a = Y11jY
21j2
Sid;i?d  1Y?
21
Sig;i?d; (6.39)
= j4kT !!
T
 
?id  xp?id?
?
: (6.40)
Substituting (6.30) - (6.33) into (6.36), (6.38), and (6.40), the chain noise parameters Sva;v?a ,
Sia;i?a and Sia;v?a can be obtained using H-representation model parameters as well
Sva;v?a = 4kT 1g
m
 
?ih +?  2ap?ih?
?
; (6.41)
Sia;i?a = 4kTgm
? !
!T
?2
?ih; (6.42)
Sia;v?a = j4kT !!
T
 
?ih  ap?ih?
?
: (6.43)
Substituting (6.36), (6.38), and (6.40) in (2.24) ? (8.19) in section 2.1.1, the minimum
noise figure NFmin, the noise resistance Rn, the real and imaginary part of the optimal source
162
admittance Gopt and Bopt, are derived using Y- noise representation coe cients ?, ?id, and x,
NFmin = 10 log10
?
1 + 2 !!
T
q
(1  x2)?id?
?
; (6.44)
Rn = ?idg
m
; (6.45)
Gopt = gm !!
T
s
(1  x2) ??
id
; (6.46)
Bopt =  gm !!
T
?id  xp??id
?id : (6.47)
Substituting (6.41), (6.42), and (6.43) in (2.24) ? (8.19) in section 2.1.1, NFmin, Rn, Gopt
and Bopt are also derived using H- noise representation coe cients ?, ?ih, and a,
NFmin = 10 log10
?
1 + 2 !!
T
q
(1  a2)?ih?
?
: (6.48)
Rn = ?ih +?  2a
p?
ih?
gm ; (6.49)
Gopt = gm !!
T
p(1  a2)?
ih?
?ih +?  2ap?ih?
; (6.50)
Bopt =  gm !!
T
?ih  ap??ih
?ih +?  2ap?ih?
: (6.51)
6.6 Importance of Correlations
We now examine the importance of Y- and H- noise representation correlations to noise
parameters NFmin, Rn, Gopt and Bopt. By neglecting cY or cH, (6.44) and (6.48) reduce to
NFminjcY=0 = 10 log10
?
1 + 2 !!
T
p?
id?
?
; (6.52)
NFminjcH=0 = 10 log10
?
1 + 2 !!
T
p?
ih?
?
: (6.53)
163
Neglecting the correlation term will cause overestimation of NFmin for both Y- and H-representation.
Since
x2?id?  a2?ih? = (?id  ?ih)?; (6.54)
and ?id is normally greater than ?ih, cH is more negligible than cY for NFmin.
By neglecting cY or cH, (6.45) and (6.49) reduce to
RnjcY=0 = Rn; (6.55)
RnjcH=0 = ?ih +?g
m
: (6.56)
Rn does not change when neglecting cY , i.e., Rnjx=0 = Rn. An inspection of (6.31) shows that
?ih + ?  ?id = 2ap?ih?. Normally a > 0, hence RnjcH=0 > Rn, therefore neglecting cH will
overestimate Rn.
By neglecting cY or cH, (6.46) and (6.50) reduce to
Goptjx=0 = gm !!
T
s
?
?id ; (6.57)
Goptja=0 = gm !!
T
p?
ih?
?ih +?: (6.58)
Goptjx=0 overestimates Gopt. Since
Goptjx=0
Goptja=0 =
?ih +?q
?ih(?ih +?  2ap?ih?)
; (6.59)
164
and,
(?ih +?)2  [?ih(?ih +?  2ap?ih?)] = ?2 +??ih + 2a?ihp?ih?; (6.60)
> 0; (6.61)
we conclude that Goptjx=0 > Goptja=0. However, it is hard to determine the relationship between
Goptja=0 and Gopt theoretically.
By neglecting cY or cH, (6.47) and (6.51) reduce to
Boptjx=0 =  gm !!
T
; (6.62)
Boptja=0 =  gm !!
T
?ih
?ih +?: (6.63)
An inspection of (6.47), (6.62) shows that Boptjx=0 underestimates Bopt. Moreover, an inspection
of (6.62) and (6.63) shows that Boptjx=0 < Boptja=0. Comparing (6.51) and (6.63), we found that
Boptja=0  Bopt = a(?ih  ?)
p?
ih?
(?ih +?)(?ih +?  2ap?ih?)
: (6.64)
Therefore the di erence between Boptja=0 and Bopt determined by ?ih  ?.
It is su cient to simply define the induced errors by neglecting cY and cH for NFmin, Rn,
Gopt and Bopt to discuss the importance of Y- and H- noise representation correlations to the
noise parameters. Since NFmin is already in dB, we define the induced error of neglecting Y- and
H- noise presentation correlations cY and cH for NFmin as
 NFminjcY=H=0 = NFminjcY=H=0  NFmin: (6.65)
165
 NFmin is both frequency and bias dependent.  Rn, the induced error of neglecting cY and cH
for Rn, is defined as,
 RnjcY=H=0 = RnjcY=H=0  Rn: (6.66)
An inspection of (6.45), (6.49) and (6.66) shows that the error percentage term  Rn=Rn does
not depend on frequency. Similarly,  Gopt and  Bopt, the induced errors of neglecting cY and
cH for Gopt and Bopt, are defined as,
 GoptjcY=H=0 = GoptjcY=H=0  Gopt; (6.67)
 BoptjcY=H=0 = BoptjcY=H=0  Bopt: (6.68)
An inspection of (6.46) and (6.47) shows that the error percentage terms  Gopt=Gopt and  Bopt=Bopt
do not depend on frequency, although Gopt and Bopt are proportional to frequency.
We now consider the 50 nm Le NMOS intrinsic device used in this chapter [63]. The
small signal MOSFET model parameters are extracted, and used for deembedding to obtain
the noise parameters of the intrinsic MOSFET shown in Fig. 6.19. Model parameters for both
representations are extracted and used to verify the analytical conversion equations derived.
Fig. 6.20 quantifies the importance of cH and cY to NFminby plotting  NFmin vs frequency
for both representations. IDS = 41, 134, 275 and 1341 ?A/?m are used, which covers the whole
bias range of interest. With increasing IDS, ? decreases from 4.83 to 4.07,  decreases from 6.81
to 4.76, ? increases from 0.71 to 0.86, and ?id increases from 0.86 to 2.86. The correlation term
x ranges from 0.52 to 0.63. Accordingly, a decreases from 0.4 to 0.04, indicating cH becomes
166
more negligible as bias increases. ?ih increases from 0.70 to 2.08. For all biases and frequencies
of interest, neglecting cH results in little error in NFmin for all the biases and frequencies.
2 4 6 8 10 12 14 16 18 20
0
0.02
0.04
0.06
0.08
0.1
frequency    (GHz)
D
 NF
min
    (dB)
I
DS
=41 mA/mm, g
id
=0.86, a=4.83, y=6.81, q=0.71, c
Y
=j0.55
I
DS
=134 mA/mm, g
id
=1.2, a=4.23, y=5.70, q=0.74, c
Y
=j0.62
I
DS
=275 mA/mm, g
id
=1.5, a=4.06, y=5.13, q=0.79, c
Y
=j0.63
I
DS
=1341 mA/mm, g
id
=2.86, a=4.07, y=4.76, q=0.86, c
Y
=j0.52
c
Y
=0 
c
H
=0 
Figure 6.20: Importance of H- and Y- noise representation correlations:  NFmin vs frequency
for 50nm NMOS.
Since Rn does not change with frequency, Fig. 6.21 quantifies the importance of cH and cY
to Rn by plotting the error percentage term  Rn=Rn vs IDS for both representations at 5 GHz.
An inspection of (6.45) shows that neglecting cY has no change on Rn, i.e.,  RnjcY=0=Rn = 0,
as shown Fig. 6.21. Therefore, Y-noise representation is a better choice for Rn.  RnjcH=0=Rn >
1, indicates that neglecting cH overestimates Rn. Moreover  RnjcH=0=Rn decreases as bias
increases. It shows that cH is still negligible at higher biases for Rn.
Similarly, Fig. 6.22 and Fig. 6.23 quantify the importance of cH and cY to Gopt and Bopt by
plotting the error percentage terms  Gopt=Gopt and  Bopt=Bopt vs IDS for both representations.
167
0 200 400 600 800 1000 1200 1400
?10
0
10
20
30
40
50
60
I
DS
    (mA/mm)
D
 R
n
/R
n
    (%)
c
Y
 = 0 
5 GHz 
c
H
 = 0 
Figure 6.21: Importance of H- and Y- noise representation correlations:  Rn=Rn vs IDS for
50nm NMOS.
Frequency is 5 GHz. Fig. 6.22 shows that cY is not negligible for all biases for Gopt. cH be-
comes negligible as bias increases. At bias of interest IDS = 400 ?A/?m, the induced error for
neglecting cH is around 10%. Therefore, H- noise representation is still a good choice at higher
biases for Gopt. Fig. 6.23 shows that the induced error for neglecting cH is practically zero for
all biases. Therefore H- noise representation is a better choice for Bopt.
6.7 Extraction and Modeling of H-Representation RF Noise Sources in CMOS
It has been shown using microscopic noise simulation and simple equivalent circuit deriva-
tion that the H-representation provides certain advantages such as frequency independent noise
sources and negligible correlation [63], thus making easier noise analysis for circuit designers
and noise modeling for device modelers. In this section, we present experimental extraction and
168
0 200 400 600 800 1000 1200 1400
?30
?20
?10
0 
10
20
30
D
 G
opt
/G
opt
    (%)
I
DS
    (mA/mm)
c
Y
 = 0 
c
H
 = 0 
5 GHz 
Figure 6.22: Importance of H- and Y- noise representation correlations:  Gopt=Gopt vs IDS for
50nm NMOS.
modeling of the H-representation noise sources in a 0.25 ?m RF CMOS process. This section
will show that the extracted input noise voltage and output noise current can be successfully
modeled as simple functions of the channel resistance and transconductance respectively. The
parameters of these functions can be related to the biasing current and voltage in a straightfor-
ward manner. The new model yields excellent agreement with measured noise data, for all of
the noise parameters, including NFmin, Yopt, and Rn, from 2 ? 26 GHz, across a wide bias range.
6.7.1 Experimental Extraction
Noise parameters are measured on wafer from 2?26 GHz, using an ATN NP5 system. Open
and short de-embedding are performed for both Y-parameters and noise parameters to move
169
0 200 400 600 800 1000 1200 1400
?20
0
20
40
60
80
100
I
DS
    (mA/mm)
D
 B
opt
/B
opt
    (%)
5 GHz 
c
Y
 = 0 
c
H
 = 0 
Figure 6.23: Importance of H- and Y- noise representation correlations:  Bopt=Bopt vs IDS for
50nm NMOS.
the reference plane to the device terminals using techniques in section 2.4. The resulting Y-
parameters and noise parameters are for the transistor, the equivalent circuit of which is shown
in Fig. 6.24. The equivalent circuit parameters are extracted using the method described in [9].
Here we choose to define vh and ih as the H-representation input noise voltage and output noise
current for the level II block shown in Fig. 6.24. The level II block consists of Rgs, Cgs, the gm
controlled source and gds, and is the core part for noise modeling. The level I block is defined
as the combination of the level II block with Cgd, Rgd, Cdb and Rdb. Next we need to extract the
power spectral densities (PSD) of vh, ih, and their correlation, which we denote as SIIv
h;v?h
, SIIi
h;i?h
,
and SIIv
h;i?h
. They can also be written using matrix notation as:
CHII 4=
2
64 SIIvh;v?h SIIvh;i?h
SIIi
h;v?h
SIIi
h;i?h
3
75; (6.69)
170
g
R
gs
C
gs
R
j
m                gs
g  e      v
 ??   -
ds
g
/S   B /S   B
G
D
+ 
- 
gs
v
gd
C
d
R
X2B
X2D
s
R
db
C
4
d
kTR
4
g
kTR
4
s
kTR
level II
level I
*
II
,
h  h
v  v
S
*
II
,
h  h
i  i
S
X2B
gd
R
X2DX2B X2D
db
R
X2B
X2D 4
db
kTR
X2B X2D
Figure 6.24: The small signal equivalent circuit model used with H-representation noise sources.
where CHII is also referred to as the H-representation noise matrix for the level II block.
The Y- noise representation parameters matrix for block II, CYII, with elements SIIi
g;i?g
, SIIi
d;i?d
,
and SIIi
g;i?d
, are obtained using techniques in section 2.5.1. Next, we transform Y- noise represen-
tation matrix CYII to H- noise representation matrix CHII using transform matrix in Table 2.1:
CHII = TY H ?CYII ?TyY H (6.70)
TY H =
2
64  hII11 0
 hII21 1
3
75: (6.71)
6.7.2 Noise Source Modeling
The above extraction is applied to a 128 finger device from a 0.25 ?m RF CMOS process
measured in IBM. The designed length is 0.24 ?m. The device width is W = 4 ?m to minimize
gate resistance. Fig. 6.25 shows the measured and modeled Y-parameters versus frequency at
171
5 10 15 20 25
0
0.02
0.04
?
Y
11
    (S)
5 10 15 20 25
?5
0
5
x 10
?3
?
Y
12
    (S)
5 10 15 20 25
0
0.1
0.2
?
Y
21
    (S)
5 10 15 20 25
0
0.05
0.1
Frequency    (GHz)
?
Y
22
    (S)
5 10 15 20 25
0
0.1
0.2
`
Y
11
    (S)
5 10 15 20 25
?0.05
0
`
Y
12
    (S)
5 10 15 20 25
?0.1
?0.05
0
`
Y
21
    (S)
5 10 15 20 25
0
0.05
0.1
Frequency    (GHz)
`
Y
22
    (S)
L = 0.24 mm, W = 4 mm, Nf = 128 
symbols: data; lines: model. V
GS
 = 1.2 V, V
DS
 = 1.2 V. 
Figure 6.25: Data-model comparison of Y-parameter vs frequency at VGS = 1.2 V. VDS = 1.2 V.
VGS = 1:2 V and VDS = 1:2 V. All of the Y-parameters are well modeled. Fig. 6.26 shows the
Y-parameters at 10 GHz as a function of VGS. The biasing current dependence is well modeled
too.
Using the equivalent circuit parameters extracted, the Y-parameters and H-parameters for all
the blocks can be calculated using straightforward linear circuit analysis. The H-representation
noise matrix is then extracted using the procedures described in section 6.7.1. Fig. 6.27 shows
the extracted Svh;v?h and Sih;i?h as a function of frequency. VGS = 1.2 V, and VDS = 1.2 V. For
modeling purpose, we have normalized SIIv
h;v?h
by 4kTRgs, and normalized SIIi
h;i?h
by 4kTgm.
Observe that Svh;v?h and Sih;i?h are both frequency independent, which simplifies modeling. Thus,
for a given bias, we can define two coe cients ? and ? as follows:
172
0 0.5 1 1.5 2 2.5
2
4
6
x 10
?3
?
Y
11
    (S)
0 0.5 1 1.5 2 2.5
?1
0
1
x 10
?3
?
Y
12
    (S)
0 0.5 1 1.5 2 2.5
0
0.1
0.2
?
Y
21
    (S)
0 0.5 1 1.5 2 2.5
0
0.05
0.1
V
GS
    (V)
?
Y
22
    (S)
0 0.5 1 1.5 2 2.5
0.02
0.04
0.06
`
Y
11
    (S)
0 0.5 1 1.5 2 2.5
?0.03
?0.02
?0.01
`
Y
12
    (S)
0 0.5 1 1.5 2 2.5
?0.05
0
`
Y
21
    (S)
0 0.5 1 1.5 2 2.5
0.02
0.04
0.06
V
GS
    (V)
`
Y
22
    (S)
f = 10 GHz, V
DS
 = 1.2 V. symbols: data; lines: model. 
L = 0.24 mm, W = 4 mm, Nf = 128 
Figure 6.26: Data-model comparison of Y-parameter at f = 10 GHz. VDS = 1.2 V.
SIIv
h;v?h
4= 4kT?R
gs; (6.72)
SIIi
h;i?h
4= 4kT?
ihgm; (6.73)
where we express Svh;v?h using Rgs, and Sih;i?h using gm. The ? and ?ih coe cients can then be
extracted for each bias, and modeled as a function of bias, as detailed below.
Fig. 6.28 shows real and imaginary parts of the correlation. The normalized correlation co-
e cient is plotted. The normalized correlation coe cient is defined by CIIvh;ih? 4= SIIv
h;i?h
=
q
SIIv
h;v?h
SIIi
h;i?h
.
Overall, the correlation is small. We have compared the noise parameters calculated with and
173
5 10 15 20 25 
0 
0.5 
1 
1.5 
2 
Frequency    (GHz) 
S 
vh,vh* II 
/(4kTR 
gs 
) and  
S 
ih,ih* II 
/(4kTg 
m 
) 
S 
ih,ih* 
II 
  
4kTg 
m 
  
= 1.75  
4kTR 
gs 
  
S 
vh,vh* 
II 
  
= 0.52  
W = 4  ? m  L = 0.24  ? m  Nf = 128  
symbols: data; lines: model 
V 
DS 
 = 1.2 V,  V 
GS 
 = 1.2 V.  
Figure 6.27: SIIv
h;v?h
=(4kTRgs) and SIIi
h;i?h
=(4kTgm) (symbols) vs frequency. VGS = 1.2 V. VDS =
1:2 V.
without the correlation, and observed negligible di erence. This is consistent with previous mi-
croscopic noise simulation results [63]. We will thus neglect the correlation in the discussions
that follow.
Fig. 6.29 (a) shows the modeled and extracted NFmin and Rn at VGS = 1.2 V and VDS =1.2
V. Fig. 6.29 (b) shows the corresponding real and imaginary parts of Yopt. The correlation SIIv
h;i?h
is assumed to be zero in the modeling. Rn, NFmin, both real and imaginary parts of Yopt are well
fitted up to 26 GHz.
Fig. 6.30 shows extracted ? and ?ih as a function of VGS. For device modeling, we need
to model ? and ?ih as a function of bias. An inspection of experimental extraction data shows
that the bias dependence of ? and ?ih can be modeled through VGS and IDS using the following
174
5 10 15 20 25
?1
?0.5
0
0.5
1
Frequency    (GHz)
?
(C
vh,ih*II
) and 
`
(C
vh,ih*II
)
?(C
vh,ih*
II
)
`(C
vh,ih*
II
)
V
DS
 = 1.2 V, V
GS
 = 1.2 V. 
W = 4 mm 
L = 0.24 mm 
Nf = 128 
?(C
vh,ih*
II
) 
`(C
vh,ih*
II
) 
Figure 6.28: CIIv
h;i?h
vs frequency, VGS = 1.2 V. VDS = 1.2 V.
proposed equations:
? = ?0 +?1 ?IDS; (6.74)
and
?ih = ?ih;0 +?ih;1 ?VGS +?ih;2 ?VGS2; (6.75)
where ?0, ?1, ?ih;0, ?ih;1 and ?ih;2 are technology dependent parameters and can be easily deter-
mined once noise parameters are extracted. IDS has a unit of ?A/?m. From noise physics, we
expect these parameters to be independent of channel width, but dependent on channel length
and oxide thickness. For the device used at Vds = 1.2 V, ?0 = 0.4068, ?1 = 0.0011, ?ih;0 = 0.1774,
175
5 10 15 20 25
0
1
2
3
4
5
6
NF
min
    (dB)
0
0.05
0.1
0.15
0.2
0.25
0.3
R
n
/50
?
Frequency    (GHz)
symbols: data; lines: model 
V
DS
 = 1.2 V, V
GS
 = 1.2 V. 
Nf = 128 
L = 0.24 ?m 
W = 4 ?m 
5 10 15 20 25
0
20
40
60
G
opt
    (mS)
0
B
opt
    (mS)
- 150
- 100
 - 50
Frequency    (GHz)
L = 0.24 ?m 
W = 4 ?m 
Nf = 128 
symbols: data; lines: model 
V
DS
 = 1.2 V, V
GS
 = 1.2 V. 
 
(a)
(b)
NF 
min 
R 
n 
/50 ? 
B
opt
  G
opt
Figure 6.29: (a) NFmin and Rn vs frequency; (b) real and imaginary parts of Yopt vs frequency.
SIIv
h;i?h
= 0. VGS = 1.2 V. ? = 0.6, ?ih = 1.75. VDS = 1.2 V.
176
0.5 1 1.5 2 2.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
V
gs
    (V)
a
 = S
vh,vh
*
II
/(4kTr
gs
)
0
0.5
1
1.5
2
2.5
3
3.5
4
g
ih
 = S
ih,ih
*
II
/(4kTg
m
)
0.24 mm device, W = 4 mm, Nf = 128 
V
ds
 = 1.2 V 
symbols: data
lines: model
Figure 6.30: ? = SIIv
h;v?h
=(4kTRgs) and ?ih = SIIi
h;i?h
=(4kTgm) vs VGS. Symbols are extracted
values, lines are model results.
?ih;1 = 1.2974, and ?ih;2 = 0. The ? and ?ih calculated using (6.74) and (6.75) fit the extracted
data well, as shown in Fig. 6.30.
Note that ?, the SIIv
h;v?h
=(4kTRgs) ratio, is nearly flat at VGS slightly above Vth, then in-
creases with increasing VGS. However, the SIIv
h;v?h
=(4kTRgs) ratio is less than 1 for most biases.
This is di erent from noise simulation results using Shockley?s impedance field theory [63],
which show that ? is larger than 1. On the other hand, ?ih, the SIIi
h;i?h
=(4kTgm) ratio, increases
with increasing bias, which agrees with simulation [63].
Fig. 6.31 (a) shows the measured and modeled NFmin and Rn versus IDS at 10 GHz.
Fig. 6.31 (b) shows real and imaginary parts of Yopt versus IDS at 10 GHz. VDS = 1.2 V.
The correlation SIIv
h;i?h
is neglected. Excellent fitting is achieved for all of the noise parameters
across the whole biasing current range.
177
0 100 200 300 400 500
0
1
2
3
4
5
6
7
I
DS
    (mA/mm)
NF
min
    (dB)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
R
n
/50
W
0.24 mm device, W = 4 mm, Nf = 128 
V
ds
 = 1.2 V 
f = 10 GHz 
symbols: data; lines: model 
(a)
0 100 200 300 400 500
0
2
4
6
8
10
12
14
16
18
20
I
DS
    (mA/mm)
G
opt
    (mS)
?60
?50
?40
?30
?20
?10
0
B
opt
    (mS)
0.24 mm device, W = 4 mm, Nf = 128 
V
ds
 = 1.2 V 
f = 10 GHz 
symbols: data; lines: model 
(b)
Figure 6.31: (a) NFmin and Rn vs IDS; (b): real and imaginary parts of Yopt vs IDS. f = 10
GHz. VDS = 1.2 V.
178
0 20 40 60 80 100 120 140
0
2
4
6
8
10
I
DS
    (mA/mm)
NF
min
    (dB)
0
2
4
6
8
10
12
14
16
18
20
R
n
/50
W
0.24 mm device, W = 4 mm, Nf = 128 
V
ds
 = 0.2 V f = 10 GHz 
symbols: data 
lines: model 
(a)
0 20 40 60 80 100 120 140
0
5
10
15
20
I
DS
    (mA/mm)
G
opt
    (mS)
?60
?50
?40
?30
?20
?10
0
B
opt
    (mS)
0.24 mm device, W = 4 mm, Nf = 128 
V
ds
 = 0.2 V f = 10 GHz 
symbols: data 
lines: model 
(b)
Figure 6.32: (a) NFmin and Rn vs IDS; (b) real and imaginary parts of Yopt vs IDS. f = 10 GHz.
VDS = 0.2 V.
179
A logical question is if the ? and ?ih equations proposed apply to all of the VDS. In our
measurements, VGS is swept at VDS = 0.2 and 1.2 V. The resulting ?0, ?1, ?ih;0, ?ih;1 and ?ih;2
from the two di erent VDS are di erent for the devices used. It is possible that the ? and ?ih
equations proposed here may be less valid for another RF CMOS process, and new equations
will need to be developed using extracted data.
Fig. 6.32 (a) shows data-model comparison of NFmin and Rn vs IDS at 10 GHz for a VDS =
0:2 V. Fig. 6.32 (b) shows real and imaginary parts of Yopt vs IDS. The ?0 = 0:4068, ?1 = 0,
?ih;0 = 8:0535, ?ih;1 =  19:5941 and ?ih;2 = 13:9161 are extracted from Vds = 0.2 V. ?ih;0 and
?ih;2 increases with decreasing Vds, while ?ih;1 decreases with decreasing Vds. The model fits the
data very well without introducing additional equations.
6.8 Summary
We have presented microscopic RF noise simulation results on 50 nm Le CMOS devices,
and examined the compact modeling of intrinsic noise sources for both the Y-representation and
the H-representation. The correlation is shown to be smaller for the H-representation than for
the Y-representation. For practical biasing currents and frequencies, the correlation is negligible
for H-representation. Models for the noise sources are suggested.
Furthermore, we have examined the relations between the Y- and H-noise representations
for MOSFETs, and quantified the importance of correlation for both representations. The theo-
retical values of ?vh, ?ih and cH are derived for the first time for long channel devices, ?vh = 4=3,
?ih = 0:6, a = 0:2458, and b = 0. cH is shown theoretically to have a zero imaginary part.
We further show that Y-representation is a better choice for Rn, and the H-representation has the
180
inherent advantage of a more negligible correlation for NFmin, Gopt, and Bopt. Overall, the im-
portance of correlation is much more negligible for H-representation than for Y-representation.
This makes circuit design and simulation easier.
We have presented experimental extraction and modeling of H-representation noise sources
in a 0.25 ?m RF CMOS process. Excellent agreement is achieved between modeled and mea-
sured noise data, including all noise parameters, for Vds = 0.2 and 1.2 V, from 2 to 26 GHz. The
results suggest a new path to RF CMOS noise modeling.
181
CHAPTER 7
EFFECTIVE GATE RESISTANCE MODELING
Since Rg is important especially for short channel devices, accurate extraction of Rg plays
a big role in compact noise modeling of modern CMOS. This chapter explains the frequency
and bias dependence of the e ective gate resistance (real part of h11) by considering the e ect of
gate-to-body capacitance, gate to source/drain overlap capacitances, fringing capacitances, and
Non-Quasi-Static (NQS) e ect. A new method of separating the physical gate resistance and the
NQS channel resistance is proposed. Separating the gate-to-source parasitic capacitances from
the gate-to-source inversion capacitance is found to be necessary for accurate modeling of all of
the Y-parameters.
7.1 Introduction
Accurate extraction of e ective gate resistance Rg;e is important for RF CMOS modeling,
particularly in noise modeling [64] [65] [66]. The e ective gate resistance Rg;e often refers to
the sum of the gate electrode resistance Rg and the Non-Quasi-Static (NQS) channel resistance
Rnqs, as shown in the small signal equivalent circuit in Fig. 7.1. Rg does not depend on bias or
frequency, while Rnqs depends on bias [67].
Using the equivalent circuit in Fig. 7.1 , Rg;e = Rg +Rnqs is often extracted from the real
part of h11 (= 1=Y11) [64], which we denote as <(h11). Here < stands for the real part. The
source and drain series resistances Rs and Rd can be de-embedded using values determined from
dc I-V data. The extracted Rg;e should be independent of frequency, and decrease with increas-
ing Vgs. However, as we show below, measured <(h11) can be strongly frequency dependent,
182
gs
C
m   gs
g  v
O
r
S/B S/B
G
D
+ 
- 
gs
v
gd
C
g,eff
R
Figure 7.1: MOSFET small signal equivalent circuit model.
and does not decrease with Vgs. This was also observed in [68], where <(h11) of experimental
data is strongly frequency dependent from 1 to 4 GHz, particularly at Vgs slightly above thresh-
old voltage, where low-noise amplifiers are biased. Interestingly, the frequency dependence of
<(h11) is much weaker at both Vgs values well below Vth and Vgs values well above Vth. Further-
more, <(h11) is lowest at Vgs values well below Vth and well above Vth, but highest at moderate
Vgs values. These abnormal bias and frequency dependences of <(h11) cannot be explained by
the simple small signal equivalent circuit model in Fig. 7.1.
Fig. 7.2 shows the measured frequency dependence of <(h11) for a 0.18?m single-ended
gate contact CMOS device. Standard open/short de-embedding are performed on the S-parameters
measured using an HP8510C vector network analyzer from 2-20 GHz for a wide bias range. The
standard open/short de-embedding is a su cient de-embedding method for a frequency range
of 2-20 GHz [69]. The channel width W is 10 ?m. The number of fingers Nf is 8. Fig. 7.3
shows the bias dependence of <(h11). <(h11) increases with IDS at lower biases, but decreases
with IDS at higher biases. Moreover, the frequency dependence of <(h11) is the strongest at the
bias corresponding to the <(h11) peaks in Fig. 7.2 . This abnormal bias frequency dependence
of <(h11) has also been observed for devices with Nf = 16 and 32. However, only the device
183
with Nf = 8 is shown in this chapter as an example. The physical Rg extracted decreases with
increasing Nf , as expected.
4 6 8 10 12 14 16 18 20
0
20
40
60
80
100
Frequency    (GHz)
?
(h
11
)    (
W
)
V
gs
 = 0 V
V
gs
 = 0.2 V
V
gs
 = 0.3 V
V
gs
 = 0.4 V
V
gs
 = 0.5 V
V
gs
 = 0.6 V
V
gs
 = 0.7 V
V
gs
 = 0.8 V
V
gs
 = 0.9 V
V
gs
 = 1.0 V
Increasing I
DS
 
W = 10 mm
L = 0.18 mm
Nf = 8 
V
DS
 = 1 V 
Figure 7.2: <(h11) vs frequency for 0.18 ?m CMOS device, W = 10?m, Nf = 8. Vds=1 V.
Using the small signal model described in Fig. 7.1, we cannot obtain decent data-model
fitting, since the real part of h11 is independent of frequency. One possible way of producing
a frequency dependent <(h11) is to separate Rg and Rnqs using the small signal equivalent cir-
cuit model in [9], which is shown in Fig. 7.4. However, the data-model comparison using the
extraction method in [9], as shown in Fig. 7.5, shows that this model cannot yield a good fit of
the data either. The main di culty is that Cgd is the primary reason for the frequency depen-
dence of <(h11), while the value of Cgd is determined mainly by Y12, where Y12 is an element of
Y-parameter matrix for the whole device.
This chapter explains the above anomalous frequency and bias dependence of <(h11) in
saturation region where Vds > Vd;sat by including gate-to-body capacitance Cgb, the gate to
184
0 50 100 150 200
0
10
20
30
40
50
60
70
80
I
DS
    (mA/mm)
?
(h
11
)    (
W
)
3 GHz
5 GHz
10 GHz
15 GHz
20 GHz
increasing frequency 
V
DS
 = 1 V 
W = 10 mm  
L = 0.18 mm
Nf = 8      
Figure 7.3: <(h11) vs IDS for 0.18 ?m CMOS device, W = 10?m, Nf = 8. Vds=1 V.
source/drain overlap capacitance Cov;s and Cov;d, and the gate to source/drain fringing capaci-
tance Cfs and Cfd according to the equivalent circuit shown in Fig. 7.6. Note that Rnqs is part of
the intrinsic transistor, and Rnqs can also be used to model gate induced noise [63]. From a noise
standpoint, Rg has the noise power spectral density of 4kTR, while the noise associated with
Rnqs is described by the induced gate noise current. The bulk resistance component in series
with Cgb becomes important only when Cgb well dominates over other parasitic capacitances,
which is not the case from our extraction. Furthermore, this substrate resistance component is
fairly independent of gate biases, and thus cannot explain the observed behavior. Based on these
considerations, we will neglect the Rsub component in series with Cgb, and will only consider the
substrate resistance component in series with the drain-substrate junction. This method of de-
scribing gate resistance is similar to but di erent from the gate resistance option 3 in BSIM4 [5].
The key di erence is that the gate to body capacitance is placed directly between the G and B, as
185
g
R
gs
C
nqs
R
m   gs
g  v
O
r
S/B
S/B
G
D
+ 
- 
gs
v
gd
C
nqd
R
ds
C
Figure 7.4: CMOS small signal model in [9].
opposed to between G? and B. The gate-to-body capacitance charging occurs through movement
of majority carriers in the bulk, and thus does not experience the non-quasi-static delay due to
inversion charge formation in the channel. Another di erence is that the controling voltage of
the transconductance is the total voltage across the Rnqs and Cgs, and the transconductance term
is gm=(1 + j!?), which accounts for output NQS and charge partition e ects [18]. Cdb is the
drain-to-body junction capacitance , and Csub is the substrate capacitance.
7.2 h11 model
Fig. 7.7 shows the equivalent circuit for the h11 derivation, which is obtained by short-
ing the output of the circuit in Fig. 7.6. Rnqd is negligible for the device used. Rnqs, which
is used to describe the NQS e ect in the channel, decreases with increasing Vgs. Cgs is the
inversion charge capacitance that increases with Vgs normally, and slightly decreases with Vgs
due to the polysilicon-gate depletion e ect [70] [71]. Cp is the combination of the source side
peripheral capacitance Cperi;s and the drain side peripheral capacitance Cperi;d. Cperi;s includes
186
20 40 60 80 100 120 140 160 180 200
0
20
40
60
80
100
120
140
160
180
I
DS
    (mA/mm)
?
(h
11
)     (
W
)
3 GHz
5 GHz
10 GHz
15 GHz
20 GHz
symbols: data
solid lines: model in Fig. 1
V
ds
 = 1 V 
W = 10 mm  
L = 0.18 mm
Nf = 8       
Figure 7.5: Data-model comparison of <(h11) vs IDS for 0.18?m CMOS device, W = 10?m,
Nf = 8, using the small signal model in Fig. 7.4. Vds=1 V.
gate-to-body capacitance Cgb, gate-to-source overlap capacitance Cov;s, and gate-to-source fring-
ing capacitance Cfs. Cperi;d includes gate-to-drain overlap capacitance Cov;d, and gate-to-drain
fringing capacitance Cfd,
Cp = Cperi;s +Cperi;d; (7.1)
Cperi;s = Cgb +Cov;s + Cfs; (7.2)
Cperi;d = Cov;d + Cfd: (7.3)
The gate-to-drain capacitance Cgd is negligible in the saturation region. The source/drain series
resistances Rs and Rd can be extracted from dc I-V data, and de-embedded. Rs and Rd are
negligible for the devices used.
187
g
R
gs
C
nqs
R
1
m   gs
g  v
j??+
O
r
S/B S/B
G
D
 +
 -
gs
v
gd
C
 
nqd
R
  
 db
C
sub
R
'G
 sub
Cgb         ov,s        fs
C   +C   +C
ov,d         fd
C    +C
Figure 7.6: A more complete MOSFET small signal model.
An inspection of Fig. 7.7 gives the intrinsic h11 as
hintr11 = Rnqs + 1j!C
gs
; (7.4)
the real part of which is simply a frequency independent Rnqs, at least to first order, which
decreases with increasing Vgs.
g
R
gs
C
p
C
nqs
R
intr
11
h
11
h
Figure 7.7: h11 derivation illustration.
188
h11 is given by
h11 = Rg + 1j!C
p + 1Rnqs+ 1
j!Cgs
: (7.5)
The real and imaginary parts of h11 are
<(h11) = Rg + Rnqs 
1 + CpCgs
?2
+ (!CpRnqs)2
; (7.6)
=(h11) =  
 
1 + CpCgs
?
+!2CgsCpR2nqs
 
1 + CpCgs
?2
+!2C2pR2nqs
? 1!C
gs
: (7.7)
For convenience, we define a threshold frequency !1 as
!21 = 110R2
nqs(Cp==Cgs)2
; (7.8)
and another threshold frequency !2 as
!2 = 10!1: (7.9)
If ! < !1, or !2R2nqsC2p << (1 + CpCgs )2, (7.6) reduces to
<(h11) = Rg + Rnqs 
1 + CpCgs
?2; (7.10)
189
where (<(h11)  Rg) is independent of frequency. Here we denote the (<(h11)  Rg) value at
zero frequency as R1,
R1 = (<(h11)  Rg)j!=0 = Rnqs 
1 + CpCgs
?2 (7.11)
If !2R2nqsC2p >> (1 + CpCgs )2, or ! > !2, (7.6) and (7.7) reduce to
<(h11) = Rg + 1!2C2
pRnqs
; (7.12)
where (<(h11)  Rg) is proportional to 1=!2.
Since Rg is independent of frequency and bias, the frequency dependence of<(h11) directly
comes from the term (<(h11)  Rg). However, the frequency dependence of <(h11) depends
not only on the frequency dependence of (<(h11)  Rg), but also on the relative importance of
(<(h11)  Rg) compared to Rg. If Rg is much greater than the change of (<(h11)  Rg) over
the used frequency range, a relatively constant <(h11) can still be obtained.
The frequency dependence of (<(h11)  Rg) is illustrated in Fig. 7.8 and Fig. 7.9 in loga-
rithm and linear scales for both x and y axes, respectively. If the working frequency range lies
below !1, (<(h11)  Rg) is nearly a constant equal to R1 according to (7.10), and independent
of frequency. If the working frequency range lies between !1 and !2, the frequency dependence
of (<(h11)  Rg) is the most obvious on a linear scale, decreasing from 0.9R1 at !1 to 0.1R1 at
!2. If the working frequency range lies above !2, (<(h11) Rg) becomes inversely proportional
to !2, and decreases rapidly from 0.1R1 at !2 towards zero. When the working frequency range
is fixed, the decrease of !1 to !01 will result in more frequencies lying between !01 and !02, as
190
shown in Fig. 7.10. At the same time, (7.11) can be rewritten in terms of !1 as,
R1 = 110!2
1
?C2pRnqs: (7.13)
Compared to C2pRnqs, 110!2
1
is the dominant term for R1. Hence, R1 can be considered inversely
proportional to the threshold frequency !21. Therefore, as !1 decreases to !01, R1 increases as
shown in Fig. 7.10. As a result, in the working frequency range, (<(h11)  Rg) becomes more
frequency dependent, and vice versa.
0.1 1 10 100
10
?1
10
0
10
1
10
2
?
(h
11
)?R
g
    (
W
)
Frequency    (GHz)
w
1
2
= 
1 
10(C
p
//C
gs
)
2
R
nqs
2
 
w
2
2
= 
10 
(C
p
//C
gs
)
2
R
nqs
2
 
w
2
C
p
2
R
nqs
 
1 
R
1
 =  
R
nqs
 
(1+C
p
/C
gs
)
2
 
Figure 7.8: Frequency dependence of (<(h11)  Rg) in logarithm scale.
If Rnqs(Cp==Cgs) increases with increasing Vgs, !1 will decrease with increasing Vgs. As
a result, (<(h11)  Rg) becomes more frequency dependent with increasing Vgs. On the other
hand, if Rnqs(Cp==Cgs) decreases with increasing Vgs, !1 will increase with increasing Vgs. As
191
0 10 20 30 40 50 60 70 80
0
10
20
30
40
50
?
(h
11
)?R
g
    (
W
)
Frequency    (GHz)
w
1
2
= 
1 
10(C
p
//C
gs
)
2
R
nqs
2
 
w
2
2
= 
10 
(C
p
//C
gs
)
2
R
nqs
2
 
(1+C
p
/C
gs
)
2
 
R
nqs
 
w
2
C
p
2
R
nqs
 
1 
R
1
 =  
0.9R
1
0.1R
1
 
Figure 7.9: Frequency dependence of (<(h11)  Rg) in linear scale.
a result, (<(h11)  Rg) becomes less frequency dependent with increasing Vgs. Next, we extract
equivalent parameters, and use the extraction results to understand the observed <(h11) behavior.
7.3 Parameter Extraction
We extract Rg, Cp, Rnqs and Cgs through the following steps.
1. Determine an initial guess of Rg using semi-circle fitting.
Plot =(h11) versus <(h11), fit the data using a semi-circle, the high frequency intercept
with the <(h11) axis is used as an initial guess of Rg. This is the same as the extraction of
base resistance in bipolar devices [46].
2. Determine initial guesses of Cp, Cgs and Rnqs as follows.
192
0.1 1 10 100
0
10
20
30
40
50
?
(h
11
)?R
g
    (
W
)
Frequency    (GHz)
w
1
w
2
working frequency
         range   
w
2
?
w
1
?
R
1
? 
R
1
 
Figure 7.10: Influence of !1 on the frequency dependence of (<(h11)  Rg).
From (7.6), we have,
1
<(h11)  Rg = p2 +!
2 ?p1; (7.14)
p1 = C2pRnqs; (7.15)
p2 =
 
1 + CpCgs
?2
Rnqs : (7.16)
Moreover, from (7.6) and (7.7), we have,
 !=(h11)<(h11)  R
g
= q2 +!2 ?q1; (7.17)
q1 = CpRnqs; (7.18)
q2 =
1 + CpCgs
CgsRnqs: (7.19)
193
p1 and p2 can be extracted using 1<(h11) Rg vs !2 plot, and q1 and q2 can be extracted using
 !=(h11)<(h11) Rg vs !2 plot at each bias, as shown in Fig. 7.11.
0 5 10 15
x 10
21
0
0.05
0.1
w
2
    (1/s
2
)
1/(
?
(h
11
)?R
g
)    (S)
data
fitting
0 5 10 15
x 10
21
0
5
10
15
x 10
11
w
2
    (1/s
2
)
?
w
`
(h
11
)/[
?
(h
11
)?R
g
]    (1/s)
data
fitting
p
2
 
Slope: p
1
 
q
2
 
Slope: q
1
 
V
gs
 = 0.5 V.
V
ds
 = 1 V.
L = 0.18 mm
W = 10 mm
Nf = 8 
Figure 7.11: Extraction of p1, p2, q1 and q2 at Vgs = 0.5 V for 0.18 ?m device, W=10 ?m,
Nf = 8.
From (7.15), (7.16), (7.18), and (7.19), we can solve for Cp, Rnqs and Cgs as,
Cp = p1q
1
; (7.20)
Rnqs = q
2
1
p1; (7.21)
Cgs = 1 +
p1 + 4q
1q2
2q21q2 ?p1: (7.22)
These are our initial guesses of Cp, Rnqs and Cgs.
3. The Cp, Rnqs and Cgs values are refined by fitting <(h11) and =(h11) versus frequency for
each bias. Here the least mean square error method is used for numerical optimization.
194
4. (Cov;d + Cfd) is estimated from the intrinsic Y12, Yintr12 , by
Cov;d + Cfd =  =(Y
intr
12 )
! : (7.23)
(Cgb +Cov;s + Cfs) is then determined using (7.1) as
(Cgb +Cov;s + Cfs) = Cp  (Cov;d + Cfd): (7.24)
Fig. 7.12 shows the extracted capacitances for the same device used in Fig. 7.2 including
Cp, Cgs, Cperi;s, and Cperi;d versus Vgs. The gate electrode resistance Rg is 25  . For the Nf =
16 and 32 devices, Rg = 13 and 7  . Fig. 7.12 also shows the extracted Rnqs versus Vgs. Cgs
increases with increasing Vgs at first, then decreases with increasing Vgs after 0.8 V due to the
polysilicon-gate depletion e ect [70] [71]. Cp increases with increasing Vgs. Cperi;d is almost
independent of bias, while Cperi;s increases with increasing bias. Rnqs decreases with increasing
Vgs, as expected. Assuming the drain and source-side overlap and fringing capacitances are
approximately symmetric, Cgb can be roughly estimated by (Cperi;s  Cperi;d). Cgb is much
smaller than Cperi;d at lower Vgs, increases with Vgs, and saturates at high Vgs, as expected.
Fig. 7.13 shows !1, !2 and R1 vs Vgs calculated using (7.8), (7.9) and (7.11). For most
biases, the measured frequency range of 2-20 GHz lies between !1 and !2. As Vgs increases,
!1 begins to decrease first, at the same time, R1 begins to increase, for reasons detailed in
Section 7.2, indicating that (<(h11)  Rg) becomes more frequency dependent. !1 reaches
the lowest point at Vgs = 0.6 V, corresponding to the most frequency dependent <(h11) curve
in Fig. 7.2. After that, !1 begins to increase while R1 begins to decrease as Vgs increases.
Correspondingly, (<(h11)  Rg) becomes less frequency dependent again at higher biases.
195
0    0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 
0 
20 
40 
60 
80 
100 
V 
gs 
    (V) 
Capacitances    (fF) 
C 
p 
  
C 
gs 
  
W = 10  ? m  
L = 0.18  ? m  
Nf = 8  
V 
ds 
=1 V  
C 
peri,d 
=(C 
ov,d 
+C 
fd 
)  
C 
peri,s 
=(C 
gb 
+C 
ov,s 
+C 
fs 
)  
(C 
peri,s 
- C 
peri,d 
) 
0 
2000 
4000 
6000 
8000 
10000 
12000 
14000 
R 
nqs 
     
( 
? 
) 
R 
nqs 
  
Figure 7.12: Extracted capacitances Cp, Cperi;s, Cperi;d, and Cgs, and extracted Rnqs vs Vgs for
0.18 ?m device, W=10 ?m, Nf = 8.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 
1 
10 
100 
V 
gs 
    (V) 
Frequency    (GHz) 
? 
2 
  
? 
1 
  
2-20 GHz    
W = 10  ? m  
L = 0.18  ? m  
Nf = 8  
V 
ds 
 = 1 V  
0 
10 
20 
30 
40 
50 
R 
1 
     
( 
? 
) 
R 
1 
  
Figure 7.13: !1, !2 and R1 vs Vgs for 0.18 ?m device. W = 10 ?m, Nf = 8.
196
7.4 Results and Discussion
Fig. 7.14 compares the modeled and measured <(h11) at several biases as a function of
frequency. The model captures the frequency dependence of the measured <(h11) quite well. At
lower Vgs = 0.4 V, Rnqs = 1178  , the inversion capacitance Cgs = 7.4 fF is much smaller than
Cp = 73 fF, i.e. Cgs << Cp. The threshold frequency !1 = 6.4 GHz, !2 = 64 GHz and R1 =
10  . Hence, for a frequency range of 2 GHz to 20 GHz, most of the frequencies lie between
!1 and !2, but close to !1. Accordingly, (<(h11)  Rg) decreases from 9  at 4 GHz to 5  at
20 GHz, as shown in Fig. 7.15. Compared to Rg = 25  , the 4  decrease of (<(h11)  Rg) is
negligible. <(h11) shows only a slight decrease with increasing frequency as can be seen from
Fig. 7.14.
4 6 8 10 12 14 16 18 20
0
10
20
30
40
50
60
70
Frequency    (GHz)
?
(h
11
)    (
W
)
V
gs
=0 V  
V
gs
=0.4 V
V
gs
=0.6 V
V
gs
=0.8 V
V
gs
=0.9 V  
increasing I
DS
 
symbol: data 
solidline: model V
gs
 ? 0.6 V 
dashline: model V
gs
 > 0.6 V V
ds
=1 V 
W = 10 mm  
L = 0.18 mm
Nf = 8       
R
g
 
Figure 7.14: <(h11) vs frequency. Symbols are measurement data. Lines are modeling results.
197
4 6 8 10 12 14 16 18 20
0
5
10
15
20
25
30
35
40
Frequency    (GHz)
Modeled (
?
(h
11
)?R
g
)    (
W
)
W = 10 mm 
L = 0.18 mm 
Nf = 8 
w
2
 = 28 GHz 
w
1
 = 2.8 GHz 
R
g
 = 25 W 
V
gs
 = 0.6 V 
V
ds
 = 1 V 
V
gs
 = 0.4 V 
w
1
 = 6.4 GHz 
w
2
 = 64 GHz 
V
gs
 = 0.9 V 
w
1
 = 5.5 GHz 
w
2
 = 55 GHz 
Figure 7.15: Modeled (<(h11)  Rg) vs frequency for 0.18 ?m device. W = 10 ?m, Nf = 8.
Vgs = 0.4, 0.6, and 0.9 V.
At medium Vgs = 0.6 V, Rnqs = 1116  , Cp = 83 fF, and Cgs = 20 fF, Cgs is comparable
to Cp. Compared to Vgs = 0.4 V, !1 decreases to 2.8 GHz, !2 decreases to 28 GHz, while R1
increases to 41  . Hence, for a frequency range of 2 GHz to 20 GHz, most of the frequencies lie
between !1 and !2, and the frequency dependence is the most obvious. (<(h11) Rg) decreases
from 40  at 4 GHz, to 7  at 20 GHz, as shown in Fig. 7.15. As Rg = 25  , the overall <(h11)
shows an obvious decrease from 65  at 4 GHz to 32  at 20 GHz as can be seen from Fig. 7.14.
At a higher Vgs of 0.9 V, Rnqs = 703  , Cp = 94 fF, and Cgs = 15 fF, Cgs is comparable
to Cp. Compared to Vgs = 0.6 V, !1 increases to 5.5 GHz, !2 increases to 55 GHz, and R1
decreases to 13.5  . Hence, most of the frequencies (2-20 GHz) lie close to !1. (<(h11)  Rg)
becomes less frequency dependent, and decreases from 13  at 4 GHz to 6  at 20 GHz, as
198
shown in Fig. 7.15. As Rg = 25  , <(h11) slightly decreases from 38  at 4 GHz to 31  at 20
GHz as can be seen from Fig. 7.14.
Fig. 7.16 compares the modeled and measured <(h11) at several frequencies as a function
of IDS. The model captures the bias dependence of the measured <(h11) quite well. At 3 GHz,
which is close to the !1 for most biases, (7.10) holds. At lower Vgs, where Cp >> Cgs, (7.10)
reduces to
<(h11) = Rg + RnqsC
2gs
C2p : (7.25)
The bias dependence of <(h11) is complicated and not necessarily monotonic, because Rnqs,
Cgs and Cp are all functions of Vgs. Rnqs decreases with increasing Vgs as shown in Fig. 7.12.
Cgs increases with increasing Vgs at lower biases, does not change much with Vgs at medium
biases, and slightly decreases with increasing Vgs at higher biases. Cp slightly increases with
increasing Vgs. From (7.25), we observe that both the bias dependence of Rnqs and the bias
dependence of the Cgs=Cp ratio contribute to the bias dependence of <(h11). Fig. 7.17 shows
the bias dependence of the Cgs=Cp ratio for the device used. Cgs=Cp ratio increases with bias at
low Vgs, since the increase of Cgs is faster than the increase of Cp. At medium Vgs, the Cgs=Cp
ratio changes slightly, since the increases of Cgs and Cp are about the same. At high Vgs, while
Cgs decreases slightly and Cp increases slightly, the Cgs=Cp ratio decreases with increasing bias.
At lower Vgs, if Rnqs is the dominant changing parameter, <(h11) will decrease as Vgs
increases. If the Cgs=Cp ratio is the dominant changing parameter, <(h11) will increase with
Vgs. At medium Vgs, e.g. 0.6 to 0.8 V, where the Cgs=Cp ratio does not change much, and Rnqs
decreases with Vgs, <(h11) begins to decrease slightly with Vgs. At higher Vgs, e.g. 0.9 V, (7.25)
199
20 40 60 80 100 120 140 160 180 200
0
10
20
30
40
50
60
70
80
I
DS
    (mA/mm)
?
(h
11
)    (
W
)
3 GHz
5 GHz
10 GHz
15 GHz
20GHz
symbols: data 
solid lines: model 
V
ds
 = 1 V 
increasing frequency 
W = 10 mm  
L = 0.18 mm
Nf = 8       
R
g
 = 25 W 
Figure 7.16: <(h11) vs IDS. Symbols are measurement data. Lines are modeling results.
holds. Rnqs as well as the Cgs=Cp ratio decreases as Vgs increases. Therefore <(h11) is expected
to decrease as Vgs increases at higher Vgs.
Fig. 7.18 and Fig. 7.19 shows the data-model comparison for the Y-parameters at 3 GHz, 5
GHz, 10 GHz, 15 GHz and 20 GHz. Rg and (Cgb+Cov;s+Cfs) are de-embedded to obtain the Y-
parameters of the intrinsic circuit. The parameters of the intrinsic circuit are then extracted using
the method described in [9], with modifications to account for the di erences in the transcon-
ductance term. The Y-parameters fit quite well using the proposed method over all biases and at
all frequencies. This suggests that it is necessary to separately consider the (Cgb + Cov;s + Cfs)
and the inversion capacitance Cgs in order to accurately model all of the Y-parameters over all
biases.
200
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 
0 
0.05 
0.1 
0.15 
0.2 
0.25 
V 
gs 
    (V) 
C 
gs 
/C 
p 
W = 10  ? m  
L = 0.18  ? m  
Nf = 8  
V 
ds 
 = 1 V  
C 
gs 
/C 
p 
  
0 
2000 
4000 
6000 
8000 
10000 
12000 
14000 
R 
nqs 
     
( 
? 
) 
R 
nqs 
  
Figure 7.17: Cgs=Cp ratio vs Vgs for 0.18 ?m device. W = 10 ?m, Nf = 8.
7.5 Length and Width E ects
The anomalous frequency and bias dependence of <(h11) also exist in devices with di er-
ent channel length, as shown in Fig. 7.20. Theoretically, as channel length L decreases, Rnqs
decreases, Cgs decreases, the sum of peripheral capacitances Cp does not change with L, result-
ing in a decrease of R1 and an increase of !1, and hence less frequency dependence in the Rnqs
related term of <(h11). On the other hand, Rg has one component that increases with decreas-
ing L, and another component that decreases with decreasing L. Therefore, the corresponding
change in Rg with decreasing L depends on which component of Rg dominates. For the devices
shown in Fig. 7.20, Rg is slightly lower for the device with larger L. Therefore, <(h11) is less
frequency dependent for the device with smaller L in Fig. 7.20.
As device width W decreases, theoretically Rnqs increases, however Cgs and Cp decrease,
leading to an increase in R1 but no change in !1, and hence more frequency dependence in
201
0 0.5 1
0
2
4
?
(Y
11
)    (mS)
3 GHz
5 GHz
10 GHz
15 GHz
20 GHz
0 0.5 1
0
5
10
15
`
(Y
11
)    (mS)
0 0.5 1
?1.5
?1
?0.5
0
V
gs
    (V)
?
(Y
12
)    (mS)
0 0.5 1
?4
?2
0
V
gs
    (V)
`
(Y
12
)    (mS)
symbols: data
lines: model 
V
ds
 = 1 V 
Figure 7.18: The real and imaginary parts of Y11 and Y12 vs Vgs for 0.18?m CMOS device. W
= 10 ?m, Nf = 8. Symbols are measurement data. Lines are modeling results.
the Rnqs related term of <(h11). On the other hand, with decreasing W , one component of
Rg decreases, while another component of Rg increases. Therefore, the net change in Rg with
decreasing W depends on which component of Rg dominates. If the Rg increase is less than the
increase of R1 with decreasing W , or if Rg decreases with decreasing W , a stronger frequency
dependence in <(h11) can be expected.
As the number of fingers Nf decreases, both Rg and Rnqs increase, and both Cgs and Cp
decrease, leading to an increase in R1 but no change in !1, and hence a stronger frequency
dependence in the Rnqs related term of <(h11). However, theoretically Rg and R1 increase by
the same percentage with decreasing Nf , resulting in no change in the frequency dependence of
<(h11).
202
0 0.5 1
0
20
40
?
(Y
21
)    (mS)
3 GHz
5 GHz
10 GHz
15 GHz
20 GHz
0 0.5 1
?20
?10
0
`
(Y
21
)    (mS)
0 0.5 1
0
5
10
V
gs
    (V)
?
(Y
22
)    (mS)
0 0.5 1
0
5
10
V
gs
    (V)
`
(Y
22
)    (mS)
symbols: data
lines: model 
V
ds
 = 1 V 
Figure 7.19: The real and imaginary parts of Y21 and Y22 vs Vgs for 0.18?m CMOS device. W
= 10 ?m, Nf = 8. Symbols are measurement data. Lines are modeling results.
0 0.2 0.4 0.6 0.8 1 1.2
0
100
200
300
400
500
V
gs
    (V)
?
(h
11
)    (
W
)
increasing 
frequency  
W = 10 mm 
Nf = 1      
Solid lines: L = 0.5 mm
dash lines: L = 1 mm 
Figure 7.20: <(h11) vs Vgs for 0.5 ?m and 1 ?m CMOS device. W = 10 ?m, Nf = 1.
203
7.6 Summary
An anomalous frequency dependence and bias dependence of <(h11) is observed. <(h11)
decreases with frequency, and increases with Vgs at low biases. We have shown that both the
frequency dependence and bias dependence can be understood by considering the gate-to-body
capacitance and the parasitic gate-to-source capacitances as capacitances in parallel with the
series combination of the NQS resistance and inversion capacitance Cgs. A new parameter ex-
traction method is developed to separate the physical gate resistance and the NQS channel resis-
tance. The modeling results show excellent agreement with data, and suggest the importance of
modeling NQS e ect for RF CMOS even at frequencies well below fT of the technology. The
proposed model parameter extraction method can be used to facilitate MOSFET noise modeling
and more accurate Y-parameter modeling over a wide bias range.
204
CHAPTER 8
EXCESS NOISE FACTORS AND NOISE PARAMETER EQUATIONS FOR RF CMOS
This chapter examines the di erences between the gd0 and gm referenced drain current ex-
cess noise factors in CMOS transistors as a function of channel length and bias. The technology
scaling are discussed for 0.25 ?m process measured in IBM, 0.18 ?m process measured in Geor-
gia Institute of Technology and 0.12 ?m process measured in IBM. Using standard linear noisy
two-port theory, a simple derivation of noise parameters is presented. The results are compared
with the well known Fukui?s empirical FET noise equations. Experimental data on a 0.18 ?m
CMOS process are measured and used to evaluate the simple model equations. New figures-
of-merit for minimum noise figure is proposed. The amount of drain current noise produced to
achieve one GHz fT is shown to fundamentally determine the noise capability of the intrinsic
transistor.
8.1 Introduction
CMOS has recently become a technology for implementing lost cost RF system due to
its economy of scale and ability to integrate analog, digital and RF functions. For analog and
RF circuits, a deeper understanding of the drain current thermal noise at both the device and
circuit level is required. A primary figure-of-merit used is the so-called drain noise excess noise
factor, defined as Sid;i?d=4kTgd0, with gd0 being the output conductance at Vds = 0 V, and Sid;i?d
being the power spectral density (PSD) of drain current noise. As gd0 is used as a reference,
we will refer to this as the gd0 referenced excess noise factor, and denote it as ?gd0. For circuit
designers, however, the transconductance at the operating bias, gm, is a better reference for
205
defining excess noise factor, and we will refer to this as the gm referenced excess noise factor,
?gm = Sid;i?d=4kTgm. Here we examine the relationship between ?gd0 and ?gm using experimental
data, particular its bias and channel length dependence.
Ultimately, from a circuit perspective, we need to establish exactly how circuit level noise
parameters relate to device level parameters, including the minimum noise figure NFmin, the
noise resistance Rn, and the noise matching source admittance Yopt. Fukui?s equations have
been widely used in interpretation, understanding and modeling of noise properties of field-
e ect transistors (FETs), first in GaAs FETs and more recently in RF CMOS [34] [35] [36]
[37] [38]. Based on observation of experimental noise parameter data obtained on MESFETs
[31] [32] [33], Fukui first proposed a set of empirical equations for NFmin, Rn, and Zopt. These
equations involve an empirical Fukui?s noise figure coe cient Kf, and other ?constants.? Kf
has since been frequently used as a figure-of-merit for comparing the intrinsic noise performance
of di erent technologies [34] [36]. Recently, various equations of NFmin, Rn and Yopt have been
derived based on linear two-port theories and small signal equivalent circuits [40]. In this chapter,
the noise parameter equations from small signal equivalent circuit derivation are compared with
empirical Fukui?s equations to better understand the physical meanings of the various constants.
Noise measurements are then made on a 0.18?m CMOS process for model evaluation. The
results show that there does not exist a bias or channel length independent Fukui?s noise figure
coe cient for CMOS. The results are then used to develop new figures-of-merit for NFmin.
Experimental data are used to demonstrate the new NFmin figures-of-merit.
206
8.2 Excess Noise Factors
The PSD of drain current noise id can be expressed using either ?gd0 or ?gm
Sid;i?d =
?i
d;i?d
?
 f = 4kT?gmgm = 4kT?gd0gd0: (8.1)
The two excess noise factors are related by
?gm = ?gd0 gd0g
m
: (8.2)
In device modeling, ?gd0 is often preferred because it is less bias dependent [72]. Another perhaps
more important reason is that an analytical expression of ?gd0 is straightforward to derive using a
drift-di usion based noise source model, as was done in [15]. Given the weak bias dependence
of ?gd0, the bias dependence of ?gm should primarily come from the ratio of gd0=gm.
Fig. 8.1 shows the measured gd0=gm ratio versus Vgs for di erent channel length from a
0.13 ?m process. Similar results are obtained on 0.18 ?m process. Vds is chosen at 1.5 V to bias
the device in saturation. Observe in Fig. 8.1 that for long channel devices, gd0 = gm in strong
inversion (high Vgs), ?gd0 = ?gm, and di erentiating ?gd0 or ?gm does not make a di erence.
For short channel lengths of interest, however, the gd0=gm ratio increases linearly with Vgs.
If we assume a bias independent ?gd0, which remains to be verified, we should expect a strong
increase of ?gm with Vgs. Optimal biasing and sizing for low-noise amplifier optimization under
the assumption of a bias independent ?gm [40] is thus problematic.
With decreasing channel length, velocity saturation makes gm increasingly smaller than its
?long channel? behavior value, while gd0 does not su er from velocity saturation and remains
close to its long channel behavior, because Vds = 0 V. The gd0=gm ratio thus increases with
207
0 0.2 0.4 0.6 0.8 1 1.2
0
0.5
1
1.5
2
2.5
3
g
d0
/g
m
V
gs
    (V)
L = 0.12 mm
L = 0.18 mm
L = 0.24 mm
L = 0.56 mm
L = 1 mm
L = 2 mm
W = 1 mm, Nf = 32. 
decreasing L 
Figure 8.1: Measured ratio of gd0=gm vs Vgs for di erent channel lengths from a 0.13 ?m CMOS
process. Vds = 1.5 V.
decreasing channel length. A calculation of gd0=gm using the BSIM3v3 model equation with
and without velocity saturation confirms the above intuitive explanation.
Fig. 8.2 shows ?gd0 and ?gm extracted from noise parameter measurements for a 0.18 ?m
process. S-parameters and noise parameters were measured using a ATN NP-5B system on wafer
from 2 to 20 GHz, using open short de-embedding. Vds = 1 V. Gate resistance was extracted
from s-parameters, and further de-embedded for calculation of Sid;i?d. gm is extracted from y-
parameters (converted from s-parameters), and verified to be consistent with that obtained from
derivatives of Ids  Vgs. gd0 is extracted from Ids  Vds data, with a small Vds step of 0:05 V.
Devices with 8, 16 and 32 fingers were measured, and the resulting Sid;i?d is proportional to the
number of fingers. Note that ?gd0 decreases slightly with increasing bias, while ?gm increases
with increasing bias at a larger slope.
208
50 100 150 200
0
0.5
1
1.5
2
2.5
3
I
DS
    (mA/mm)
g
gm
 and 
g
gd0
L = 0.18 mm 
g
gm
g
gd0
g
gm
 
g
gd0
 
W = 10 mm,     N
f
 = 8 
V
ds
 = 1 V 
Figure 8.2: Measured ?gd0 and ?gm for a 0.18 ?m CMOS process. Vds = 1 V.
0 0.5 1 1.5 2 2.5
10
?2
10
?1
10
0
10
1
10
2
10
3
V
gs
    (V)
I
DS
    (
m
A/
m
m)
L = 0.24 mm, W = 4 mm, Nf = 128
L = 0.18 mm, W = 10 mm, Nf = 8
L = 0.12 mm, W = 5 mm, Nf = 30
Figure 8.3: IDS vs Vgs in saturation region for gate length of 0.24 ?m, 0.18 ?m, and 0.12 ?m
devices.
209
8.3 Technology Discussion of Excess Noise Factor
Fig. 8.3 shows IDS vs Vgs in saturation region for gate length of 0.24 ?m, 0.18 ?m, and
0.12 ?m devices. IDS increases with scaling. Fig. 8.4 show cuto frequency fT vs IDS and Vgs,
respectively. fT increases with decreasing gate length.
0 100 200 300 400 500 600 700
0
10
20
30
40
50
60
70
80
90
I
DS
    (mA/mm)
f
T
    (GHz)
L = 0.24 mm, W = 4 mm, Nf = 128
L = 0.18 mm, W = 10 mm, Nf = 8
L = 0.12 mm, W = 5 mm, Nf = 30
(a)
0 0.5 1 1.5
0
20
40
60
80
100
120
V
gs
    (V)
f
T
    (GHz)
L = 0.24 mm, W = 4 mm, Nf = 128
L = 0.18 mm, W = 10 mm, Nf = 8
L = 0.12 mm, W = 5 mm, Nf = 30
(b)
Figure 8.4: fT vs (a) IDS, and (b) Vgs for gate length of 0.24 ?m, 0.18 ?m, and 0.12 ?m devices.
Fig. 8.5 (a) shows Sid;i?d normalized by (W ?Nf) vs IDS and Sid;i?d vs Vgs for gate length
of 0.24 ?m, 0.18 ?m, and 0.12 ?m devices, respectively. Sid;i?d of 0.12 ?m gate length device is
210
the highest. The normalized Sid;i?d increases with scaling. gm normalized by (W ?Nf) vs IDS is
shown in Fig. 8.5 (b) for gate length of 0.24 ?m, 0.18 ?m, and 0.12 ?m devices. The normalized
gm increases with scaling.
0 50 100 150 200 250 300 350 400
0
1
2
3
4
5
6
x 10
?22
I
DS
    (mA/mm)
(S
id,id*
/(W
 
 Nf))
1/2
?
 10
?10
L = 0.24 mm, W = 4 mm, Nf = 128
L = 0.18 mm, W = 10 mm, Nf = 8
L = 0.12 mm, W = 5 mm, Nf = 30
0.25 mm process
0.12 mm process 
0.18 mm process
(a)
0 50 100 150 200 250 300 350 400
0
2
4
6
8
x 10
?4
I
DS
    (mA/mm)
g
m
/(W 
 
 Nf)    (S/
m
m)
0.12 mm process 
0.18 mm process
0.25 mm process 
(b)
Figure 8.5: (a) Sid;i?d, and (b) gm normalized by (W ? Nf) vs IDS for gate length of 0.24 ?m,
0.18 ?m, and 0.12 ?m devices.
211
Fig. 8.6 shows ?gm and ?gd0 vs IDS for gate length of 0.24 ?m, 0.18 ?m, and 0.12 ?m
devices. ?gm and ?gd0 do not necessarily increase or decrease with scaling, although normalized
Sid;i?d and gm increase with scaling as shown in Fig. 8.5.
0 100 200 300 400 500 600
0
1
2
3
4
I
DS
    (mA/mm)
g
g
m
 and 
g
g
d0
L = 0.24 mm, W = 4 mm, Nf = 128, V
ds
 = 1.2 V
L = 0.18 mm, W = 10 mm, Nf = 8 , V
ds
 = 1 V
L = 0.12 mm, W = 5 mm, Nf = 30, V
ds
 = 1 V
g
g
m
 
g
g
d0
 
Figure 8.6: ?gm and ?gd0 vs IDS for gate length of 0.24 ?m, 0.18 ?m, and 0.12 ?m devices.
8.4 Vds Dependence of Excess Noise Factor
Due to the limitation of measurement data, only gate length of 0.12 ?m device and 0.24 ?m
device are discussed here.
8.4.1 0.24 ?m device, W = 4 ?m, Nf = 128.
Fig. 8.7 (a) shows ?id and ?ih vs IDS and Vgs at Vds = 0.2 V and 1.2 V for 0.24 ?m device.
W = 4 ?m, Nf = 128. ?id is similar to but higher than ?ih for all biases. As Vds increases, both
?id and ?ih decrease. In section 6.7.1, modeling of ?ih is discussed for Vds = 0.2 and 1.2 V. ?id can
212
be similarly modeled.
?id = ?id;0 +?id;1 ?Vgs +?id;2 ?Vgs2; (8.3)
?id;0 = 8:1166, ?id;1 =  19:5604 and ?id;2 = 13:7592 for Vds = 0.2 V, and ?id;0 = 0:0810,
?id;1 = 1:4981 and ?id;2 = 0 for Vds = 1.2 V. Similar to analysis for ?ih, ?id;0 and ?id;2 increases
with decreasing Vds, while ?id;1 decreases with decreasing Vds.
0 50 100 150 200
0
2
4
6
8
10
I
DS
    (mA/mm)
g
id
 = S
id,id
*
/(4kTg
m
) and 
g
ih
 = S
ih,ih
*
/(4kTg
m
)
V
ds
 = 0.2 V
V
ds
 = 1.2 V
0.24 mm device
g
id
g
ih
 
W = 4 mm, Nf = 128  
(a)
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
0
1
2
3
4
5
6
7
8
9
V
gs
    (V)
g
id
 = S
id,id
*
/(4kTg
m
) and 
g
ih
 = S
ih,ih
*
/(4kTg
m
)
V
ds
 = 0.2 V
V
ds
 = 1.2 V
0.24 mm device, W = 4 mm, Nf = 128 
g
id
g
ih
 
(b)
Figure 8.7: ?id and ?ih (a) vs IDS, and (b) vs Vgs at Vds = 0.2 V and 1.2 V for 0.24 ?m device.
W = 4 ?m, Nf = 128.
213
8.4.2 0.12 ?m Device, W = 5 ?m, Nf = 30.
Fig. 8.8 shows IDS vs Vgs at Vds = 1 V and 1.5 V for 0.12 ?m device. W = 5 ?m, Nf =
30. IDS slightly increases with increasing Vds. Fig. 8.9 shows IDS vs Vds at Vgs = 0.7, 1.0 and
1.5 V. IDS does not increase much with increasing VDS for Vgs = 0.7 and 1.0 V. For Vgs = 1.5
V, IDS increases at lower VDS, then saturates at higher VDS.
0.4 0.6 0.8 1 1.2 1.4 1.6
10
0
10
1
10
2
10
3
V
gs
    (V)
I
DS
    (
m
A/
m
m)
V
ds
 = 1.0 V
V
ds
 = 1.5 V
IBM 8rf: L = 0.12 mm, W = 5 mm, Nf = 30 
Figure 8.8: IDS vs Vgs at Vds = 1 V and 1.5 V for 0.12 ?m device. W = 5 ?m, Nf = 30.
0.5 1 1.5
0
100
200
300
400
500
600
700
V
ds
    (V)
I
DS
    (
m
A/
m
m)
Data: L = 0.12 mm, W = 5 mm, Nf = 30 
V
gs
 = 1.5 V 
V
gs
 = 1.0 V 
V
gs
 = 0.7 V 
Figure 8.9: IDS vs Vds at Vgs = 0.7, 1.0 and 1.5 V for gate length of 0.12 ?m device.
214
Fig. 8.10 shows the Sid;i?d, ?gm and ?gd0 vs Vgs at Vds = 1 V and 1.5 V. Fig. 8.11 shows the
Sid;i?d, ?gm and ?gd0 vs IDS at Vds = 1 V and 1.5 V.
0.4 0.6 0.8 1 1.2 1.4 1.6
0
1
2
3
4
5
6
7
x 10
?21
V
gs
    (V)
S
id,id
*
    (A
2
/Hz)
V
ds
 = 1.0 V
V
ds
 = 1.5 V
IBM 8rf: L = 0.12 mm, W = 5 mm, Nf = 30 
(a)
0.4 0.6 0.8 1 1.2 1.4 1.6
0
0.5
1
1.5
2
2.5
3
3.5
4
V
gs
    (V)
g
g
m
 and 
g
g
d0
V
ds
 = 1.0 V
V
ds
 = 1.5 V
IBM 8rf: L = 0.12 mm, W = 5 mm, Nf = 30 
(b)
Figure 8.10: (a) Sid;i?d, and (b)?gm and ?gd0 vs Vgs at Vds = 1 V and 1.5 V for 0.12 ?m device. W
= 5 ?m, Nf = 30.
Fig. 8.12 (a) shows Sid;i?d vs Vds at Vgs = 0.7, 1.0 and 1.5 V. Sid;i?d is almost flat over Vds at
Vgs = 0.7 V. For Vgs = 1.0 and 1.5 V, however, Sid;i?d increases with increasing Vds. Higher the
Vgs, higher the slope of Sid;i?d ? Vds curve. Fig. 8.12 (b) shows ?gd0 vs Vds at Vgs = 0.7, 1.0 and
1.5 V for gate length of 0.12 ?m device. ?gd0 slightly increases with increasing Vds, and is the
215
10
1
10
2
10
3
10
?21
I
DS
    (mA/mm)
S
i
d
,i
d*
    (A
2
/Hz)
V
ds
 = 1.0 V
V
ds
 = 1.5 V
IBM 8rf: L = 0.12 mm, W = 5 mm, Nf = 30 
(a)
0 100 200 300 400 500 600 700
0
0.5
1
1.5
2
2.5
3
3.5
4
I
DS
    (mA/mm)
g
g
m
 and 
g
g
d0
V
ds
 = 1.0 V
V
ds
 = 1.5 V
IBM 8rf: L = 0.12 mm, W = 5 mm, Nf = 30 
(b)
Figure 8.11: (a) Sid;i?d, and (b)?gm and ?gd0 vs IDS at Vds = 1 V and 1.5 V for 0.12 ?m device. W
= 5 ?m, Nf = 30.
lowest for Vgs = 1 V. Fig. 8.12 (c) shows ?gm vs Vds at Vgs = 0.7, 1.0 and 1.5 V for gate length
of 0.12 ?m device. ?gm is almost flat over Vds at Vgs = 0.7 and 1.0 V. For Vgs = 1.5 V, however,
?gm decreases in the linear region, then becomes flat in the saturation region.
216
0.5 1 1.5
0
1
2
3
4
5
x 10
?21
V
ds
    (V)
S
id,id
*
    (A
2
/Hz)
V
gs
 = 0.7 V 
V
gs
 = 1.0 V 
V
gs
 = 1.5 V 
Data: L = 0.12 mm, W = 5 mm, Nf = 30 
(a)
0.5 1 1.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
V
ds
    (V)
g
g
d0
V
gs
 = 0.7 VV
gs
 = 1.0 V
V
gs
 = 1.5 V
Data: L = 0.12 mm, W = 5 mm, Nf = 30
(b)
0.5 1 1.5
1
2
3
4
5
6
V
ds
    (V)
g
g
m
V
gs
 = 0.7 V 
V
gs
 = 1.0 V 
V
gs
 = 1.5 V 
Data: L = 0.12 mm, W = 5 mm, Nf = 30 
(c)
Figure 8.12: (a) Sid;i?d, (b) ?gd0, and (c) ?gm vs Vds at Vgs = 0.7, 1.0 and 1.5 V for gate length of
0.12 ?m device.
217
8.4.3 Simulation Results on 50 nm Le CMOS
In order to further investigate Vds dependence of Sid;i?d, ?gd0, and ?gm, 50 nm Le gate length
CMOS simulation results in chapter 6 are used. Fig. 8.13 shows Sid;i?d, ?gd0, and ?gm vs Vgs at Vds
= 0.1 V to 1.0 V with step of 0.1 V. Sid;i?d and ?gd0 increases with increasing Vds. ?gm decreases
with increasing Vds.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10
?26
10
?25
10
?24
10
?23
10
?22
10
?21
V
gs
    (V)
S
id,id
*
    (A
2
/Hz)
V
ds
 = 0.1 to 1 V
step = 0.1 V       
Simulation results: L = 50 nm 
(a)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
1
2
3
4
5
6
7
V
gs
    (V)
g
g
d0
V
ds
 = 0.1 to 1 V
step = 0.1 V       
Simulation results: L = 50 nm 
(b)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
5
10
15
V
gs
    (V)
g
g
m
V
ds
 = 0.1 to 1 V
step = 0.1 V       
Simulation results: L = 50 nm 
(c)
Figure 8.13: (a) Sid;i?d, (b) ?gd0, and (c) ?gm vs Vgs at Vds = 0.1 V to 1.0 V with step of 0.1 V for
50 nm Le CMOS simulation.
218
Fig. 8.14 shows Sid;i?d, ?gd0, and ?gm vs IDS at Vds = 0.1 V to 1.0 V with step of 0.1 V. Sid;i?d
vs IDS almost does not change for Vds above 0.3 V.
10
?2
10
0
10
2
10
4
10
?26
10
?25
10
?24
10
?23
10
?22
10
?21
I
ds
    (mA/mm)
S
id,id
*
    (A
2
/Hz)
V
ds
 = 0.1 to 1 V
step = 0.1 V       
Simulation results: L = 50 nm 
(a)
0 200 400 600 800 1000 1200 1400
0
1
2
3
4
5
6
7
I
ds
    (mA/mm)
g
g
d0
V
ds
 = 0.1 to 1 V
step = 0.1 V       
Simulation results: L = 50 nm 
(b)
0 200 400 600 800 1000 1200 1400
0
5
10
15
I
ds
    (mA/mm)
g
g
m
V
ds
 = 0.1 to 1 V
step = 0.1 V       
Simulation results: L = 50 nm 
(c)
Figure 8.14: (a) Sid;i?d, (b) ?gd0, and (c) ?gm vs IDS at Vds = 0.1 V to 1.0 V with step of 0.1 V for
50 nm Le CMOS simulation.
219
Fig. 8.15 shows simulation-model comparison of ?gm vs IDS and vs Vgs at Vds = 0.1 V to
1.0 V with step of 0.1 V. ?gm is modeled using (8.3). Excellent simulation-model agreement are
obtained. The Vds dependence of the model parameters ?id;0, ?id;1 and ?id;2 are shown in Fig. 8.16.
?id;0 and ?id;2 decreases as Vds increases. ?id;1 increases as Vds increases. The simulation results
complies with the measurement data analysis for the above 0.24 ?m device.
0 200 400 600 800 1000 1200 1400
0
5
10
15
I
ds
    (mA/mm)
g
g
m
V
ds
 = 0.1 to 1 V
step = 0.1 V       
Simulation results: L = 50 nm 
symbols: simulation
lines: model       
(a)
0 0.2 0.4 0.6 0.8 1
0
5
10
15
V
gs
    (V)
g
g
m
V
ds
 = 0.1 to 1 V
step = 0.1 V       
Simulation results: L = 50 nm 
symbols: simulation
lines: model       
(b)
Figure 8.15: ?gm (a) vs IDS, and (b) vs Vgs at Vds = 0.1 V to 1.0 V with step of 0.1 V for 50 nm
Le CMOS simulation.
220
?id;2, ?id;1, and ?id;0 can be further modeled as function of Vds.
?id;2 = 10[ 0:4475(log10 Vds)4 1:7225(log10 Vds)3 2:5302(log10 Vds)2 2:8588(log10 Vds) 0:3274]; (8.4)
?id;1 = 10[ 0:1552(log10 Vds)4 0:3067(log10 Vds)3 0:2801(log10 Vds)2 0:0093(log10 Vds)+1:5063]  30; (8.5)
?id;0 = 10[ 0:0698(log10 Vds)2 0:6374(log10 Vds) 0:5782]: (8.6)
The calulations using model equations (8.4) ? (8.6) are compared to model parameters ?id;2, ?id;1,
and ?id;0 in Fig. 8.16. Excellent agreement has been achieved. Therefore, ?gm at certain Vds and
Vgs can be modeled using 14 constant coe cients in (8.4) ? (8.6), together with (8.3).
0 0.2 0.4 0.6 0.8 1
?20
0
20
40
60
80
V
ds
    (V)
g
id,2
, 
g
id,1
 and 
g
id,0
 
g
id,2
g
id,1
g
id,0
Simulation results: L = 50 nm 
symbols: g
id,2
,  g
id,1
, and g
id,0
,
lines:       model 
Figure 8.16: ?id;0, ?id;1 and ?id;2 vs Vds for 50 nm Le CMOS simulation.
221
8.5 Noise Parameter Equations
Fig. 8.17 shows a simplified MOSFET equivalent circuit including gate resistance noise and
drain current noise. The Y matrix of the intrinsic device is denoted by Yintr. We first consider
only the intrinsic MOSFET without Rg, and consider only the drain current noise id.
V
1
I
2
I
1
V
2
i
d
- +
R
g
4kTR
g
Y
+ 
- 
C
gs
g
m
v
gs
C
gd
v
gs
Y
intr
Figure 8.17: MOSFET equivalent circuit with drain current noise and gate resistance noise.
We first convert id into va and ia, input voltage and current,
va =  idYintr
21
; (8.7)
ia =  idhintr
21
: (8.8)
222
For the dashed box in Fig. 8.17,
Yintr21 ? gm; (8.9)
Yintr11 = j!Ci; (8.10)
hintr21 = Y
intr
21
Yintr11 =
gm
j!Ci =
1
j
fT
f ; (8.11)
where Ci = Cgs +Cgd, and fT is cuto frequency. The PSDs of va, ia and their correlation are
then obtained as
Sva;v?a = < va;v
?a >
 f =
Sid;i?d
jYintr21 j2 ?
Sid;i?d
g2m ; (8.12)
Sia;i?a = < ia;i
?a >
 f =
Sid;i?d
jhintr21 j2 ?
? f
fT
?2
Sid;i?d; (8.13)
Sia;v?a = < ia;v
?a >
 f =
Sid;i?d
jYintr21 j2Y
intr
11 ? j
f
fT ?
Sid;i?d
gm : (8.14)
Now we add the gate resistance as shown in Fig. 8.17. The primary e ect is an increase in
Sva;v?a ,
Sva;v?a ? Sid;i
?
d
jYintr21 j2 + 4kTRg ?
Sid;i?d
g2m + 4kTRg: (8.15)
223
Sva;v?a , Sia;i?a, and Sia;v?a can then be used to calculate NFmin, Rn and Yopt using standard
equations in [11] as
NFmin = 10 log10
0
@1 + ff
T
s
Sid;i?d
kT Rg
1
A; (8.16)
Rn = ?gmg
m
+Rg = Sid;i
?
d=4kT
g2m +Rg; (8.17)
Gopt = gm ff
T
?
p?
gmgmRg
?gm +gmRg; (8.18)
Bopt =  gm ff
T
? ?gm?
gm +gmRg
: (8.19)
Zopt = Ropt +jXopt is also calculated from 1=Yopt as
Ropt = fTf
s
Rg
?gmgm =
gm
2?fCi
4kT
Sid;i?d ?
pR
g; (8.20)
Xopt = fTf ? 1g
m
= 1=2?fC
i
; (8.21)
Note that ?gm appears directly in the Rn, Ropt, and Xopt expressions. We can also write the NFmin
expression (8.16) by replacing Sid;i?d with 4kT?gmgm,
NFmin = 10 log10
?
1 + 2p?gm ff
T
pg
mRg
?
: (8.22)
224
8.6 Comparison with Fukui?s Equations
Based on experimental data in GaAs MESFETs, Fukui proposed the following empirical
equations [31] [32]
NFmin = 10 log10
?
1 +Kf ff
T
pg
mRg
?
; (8.23)
Rn = K2g2
m
; (8.24)
Ropt = K3
? 1
4gm +Rg
?
; (8.25)
Xopt = K4fC
gs
; (8.26)
where Kf, K2, K3 and K4 were proposed to be bias independent and channel length independent
[31]. Rn was later modified in [33] as
Rn = K
n
2
gm ; (8.27)
where Kn2 = 0:8.
An inspection of (8.23) and (8.22) immediately shows:
Kf = 2p?gm; (8.28)
which gives a meaning to Fukui?s noise figure coe cient. For long channel device operating in
saturation region (strong inversion), ?gm = ?gd0 = 2=3 [15], and Kf = 1:633. This is close to the
empirical Kf = 2 in [31] and [32], which was also proposed to be channel length independent
at the minimum NFmin bias point [31]. This is not the case for short channel CMOS, in which
225
Kf = 2p?gm becomes strongly bias dependent, as shown in Fig. 8.2. The bias dependence of
?gm is primarily due to the strong bias dependence of gd0=gm in short channel devices, as was
shown in Fig. 8.1. This indicates that there does not exist a bias independent or channel length
independent universal Fukui?s noise figure coe cient for RF CMOS. We therefore cannot use
(8.23) for low-noise optimization, as was done in [31] and [40].
Comparing (8.24), (8.27) and (8.17),
K2 = Sid;i
?
d
4kT = ?gmgm; (8.29)
Kn2 = Sid;i
?
d
4kTgm = ?gm (8.30)
for gm related terms. The Rg term was not included in Fukui?s Rn equation because of the low
Rg due to the use of metal gate in MESFETs, but is important for CMOS. Clearly neither Sid;i?d
nor ?gm is a constant. Instead, both Sid;i?d and ?gm should be bias and channel length dependent.
A comparison of (8.25) and (8.20) shows that the inverse frequency dependence is not
considered in Fukui?s Ropt equation. A comparison of (8.26) and (8.21) shows
K4 = 1=2?; (8.31)
which is indeed a constant. Note that Cgd was not included in (8.26). Table I summarizes the
?physical meanings? of K1 ? K4.
8.7 Model Validation
For validation, we compare measured and simulated noise parameters. Here we use the 8
finger device as an example. S-parameters and noise parameters are measured from 2 to 20 GHz.
226
Table 8.1: Comparison of Fukui empirical constants with our derivation.
Empirical equations [31] [32] Our derivation
NFmin Kf = 2 2p?gm
Rn K2 Sid;i?d4kT = ?gmgm
Kn2 = 0:8 [33] ?gm
(Rg not included) (Rg included)
Ropt K3 4g2m
pR
g
2?fCi?gm(1+4gmRg)
(f independent) (f dependent)
Zopt K4 1=2?
(Cgd not included) (Cgd included)
Vds is fixed at 1.5V, and Vgs is swept. Rg, gm, and fT are extracted from y-parameters. The Rs
and Rd extracted from dc measurements are negligibly small. Sid;i?d is extracted from measured
NFmin, Rn, and Yopt through standard noise de-embedding [43] [11].
For each parameter, comparisons are shown in Fig. 8.18 as a function of frequency at a fixed
Vgs of 0.7 V, and then in Fig. 8.19 as a function of bias at a fixed frequency of 5 GHz. Good
model-data correlation is achieved for both bias and frequency dependence of NFmin. A fairly
good correlation between model and data is observed for both bias and frequency dependence
of Rn. Rn is flat over frequency. With increasing Vgs, Rn decreases first and then stays nearly
constant, as expected from (8.17). Fairly good model-data correlation is observed for both bias
and frequency dependence of Gopt. Gopt is positive and linearly increases with frequency, as
expected from (8.18). Gopt is only weakly dependent on Vgs after gm and fT reach their peaks.
A larger discrepancy is observed at higher frequencies, which is related to the use of a simplified
equivalent circuit model. For frequencies below 5 GHz, the intended RF design frequencies for
a 0.18 ?m process, the model still works reasonably well over all biases.
227
0 5 10 15 20
0
1
2
3
4
NF
min
    (dB)
Frequency    (GHz)
0 5 10 15 20
0
1
2
3
4
R
n
/50
W
Frequency    (GHz)
Experimental Data
Model
0 5 10 15 20
0
1
2
3
4
5
6
Frequency    (GHz)
G
opt
    (mS)
0 5 10 15 20
?10
?8
?6
?4
?2
0
Frequency    (GHz)
B
opt
    (mS)
L = 0.18 mm 
W = 10 mm   
Nf = 8        
V
gs
 = 0.7 V
Figure 8.18: Model-data comparison of noise parameters vs frequency. Vgs = 0.7 V, Vds = 1 V.
8.8 Figure-of-Merit for NFmin
An inspection of (8.16) shows that it is the absolute value of the drain current noise Sid;i?d
that fundamentally determines NFmin. The Fukui?s noise figure coe cient, the Kf factor, which
is historically used as a figure-of-merit for comparing the noise figure capability of di erent
technologies, is less applicable to CMOS, as it is strongly bias dependent through ?gm.
Similarly, the ?gm excess noise factor cannot be used as a figure-of-merit for measuring the
minimum noise figure capability of a technology, even though it appears in (8.22). The product
of ?gm and gm simply leads us back to Sid;i?d. One can also decompose Sid;i?d into the product of
?gd0 and gd0, however, it is the Sid;i?d value that matters.
228
0.4 0.6 0.8 1
0
1
2
3
NF
min
    (dB)
V
gs
    (V)
0.4 0.6 0.8 1
0 
2 
4 
6 
8 
R
n
/50
W
V
gs
    (V)
Experimental Data
Model
0.4 0.6 0.8 1
0
1
2
3
V
gs
    (V)
G
opt
    (mS)
0.4 0.6 0.8 1
?6
?5
?4
?3
V
gs
    (V)
B
opt
    (mS)
L = 0.18 mm 
W = 10 mm   
Nf = 8        
f = 5 GHz
Figure 8.19: Model-data comparison of noise parameters vs Vgs, f = 5 GHz, Vds = 1 V.
To propose a figure-of-merit for measuring the intrinsic transistor low noise capability, we
rewrite (8.16) as
Fmin  1 = f ?KNF ?
rW
total ?Rg
kT ; (8.32)
where KNF is the proposed new figure-of-merit for NFmin
KNF =
q
Sid;i?d=Wtotal
fT ; (8.33)
and Wtotal is the total device width, Wtotal = W ? Nf. The normalization to Wtotal is made to
make KNF device width independent. The pWtotalRg term can be minimized through layout
229
techniques, while the KNF factor represents the noise capability of the intrinsic device, and
essentially represents the amount of noise current generated in order to achieve one GHz fT .
Fig. 8.20 show
q
Sid;i?d=(W ? Nf), fT , and the KNF factor vs log scale IDS and linear
scale IDS respectively for the 0.18 ?m process. Similarly, Fig. 8.21 show
q
Sid;i?d=(W ? Nf),
fT , and the KNF factor vs IDS for the 0.25 ?m process, the 0.18 ?m process, and the 0.12
?m process. Di erent normalizations are used to plot all quantities on the same scale. The
same noise measurements were made on the 0.25 ?m process and 0.12 ?m process, from which
Sid;i?d was extracted. Observe that with increasing IDS, both fT and Sid;i?d increase, as expected.
The KNF factor, which is a direct indicator of NFmin, decreases rapidly first as the device turns
on, reaches a minimum at a moderate IDS when Vgs is slightly above threshold voltage. This
corresponds to the bias for minimum NFmin, at which the lowest amount of noise is generated
for one GHz fT , or the same amount of fT is achieved with the least amount of noise.
With technology scaling, both Sid;i?d and fT increase as shown in Fig. 8.22 (a) and (b).
Only when the fT increase dominates over the Sid;i?d increase, NFmin improves (decreases) with
scaling. This di ers from the conventional wisdom that a higher fT in scaled device directly
leads to improved NFmin, a result from Fukui?s empirical NFmin equation. Fig. 8.22 (c) compares
the KNF factor of the 0.25 ?m process, 0.18 ?m process and 0.12 ?m process. Indeed, the KNF
factor, which directly determines intrinsic device NFmin, decreases (improves) with technology
scaling from 0.25 ?m, 0.18 ?m to 0.12 ?m, because the fT increase with scaling dominates the
drain current noise increase with scaling. The KNF factor does not include the Rg ?Wtotal e ect
by design to measure only intrinsic device noise figure. The Rg ?Wtotal term in (8.32), however,
can increase with scaling in a silicided poly gate process, which may ultimately limit overall
device NFmin, as detailed below. In order to compare technologies with di erent gate material
230
or devices with di erent layout, we define another noise figure-of-merit to include the e ect of
Rg ?Wtotal,
KNF;Rg = 1f
T
s
Sid;i?d
kT Rg = KNF
rR
gWtotal
kT ; (8.34)
and
Fmin = 1 +f ?KNF;Rg: (8.35)
Fig. 8.23 compares the KNF;Rg of three devices, one from the 0.18 ?m process with W = 10
?m, Nf = 8, and the other two from the 0.25 ?m process with W = 4 ?m, Nf = 128, and
the 0.12 ?m process with W = 5 ?m, Nf = 30. Note that the gate finger width is much larger
for the 0.18 ?m device. Rg ? Wtotal is 2000  ?m for the 0.18 ?m device, 307.2  ?m for the
0.25 ?m device, and 780  ?m for the 0.12 ?m device. Even though KNF , a measure of the
intrinsic device noise, is smaller in the 0.18 ?m device, KNF;Rg and hence NFmin are higher in
the 0.18 ?m device, because of the much smaller Rg ?Wtotal. The combination of a smaller gate
length L and a larger gate finger width W results in the higher Rg ?Wtotal in the 0.18 ?m device,
despite reduced gate sheet resistance (10.8  =2 for 0.18?m processes, 13.8  =2 for 0.25 ?m
processes, and 11.2  =2 for 0.12 ?m processes). A smaller finger gate width, e.g. 2 ?m, should
be used to decrease KNF;Rg and hence NFmin of the 0.18 ?m device.
231
10
?1
10
0
10
1
10
2
10
?23
10
?22
10
?21
I
DS
    (mA/mm)
(S
id,id*
/(W
 
 Nf))
1/2
?
 10
?10
, f
T
?
 10
?23
, and K
NF
L = 0.18 mm, W = 10 mm, Nf = 8 
(S
id,id*
/(W  Nf))
1/2
? 10
?10
 
f
T
? 10
?23
 
K
NF
 
(a)
0 20 40 60 80 100 120 140 160 180 200
10
?23
10
?22
10
?21
I
DS
    (mA/mm)
(S
id,id*
/(W
 
 Nf))
1/2
?
 10
?10
, f
T
?
 10
?23
, and K
NF
L = 0.18 mm, W = 10 mm, Nf = 8 
(S
id,id*
/(W  Nf))
1/2
? 10
?10
 
f
T
? 10
?23
 
K
NF
 
(b)
Figure 8.20:
q
Sid;i?d=(W ? Nf), fT , and KNF vs (a) log scale IDS, and (b) linear scale IDS for
the 0.18 ?m process, with Sid;i?d in unit of A2/Hz, W in unit of ?m, fT in unit of GHz, and KNF
in unit of A=
p
?mHz3.
232
0 100 200 300 400 500
0
1
2
3
4
x 10
?22
I
DS
    (mA/mm)
(S
id,id*
/(W
 
 Nf))
1/2
?
 10
?10
, f
T
?
 10
?23
, and K
NF
(S
id,id*
/(W  Nf))
1/2
? 10
?10
 
K
NF
 
f
T
? 10
?23
 
L = 0.24 mm, W = 4 mm, Nf = 128 
(a)
0 20 40 60 80 100 120 140 160 180 200
10
?23
10
?22
10
?21
I
DS
    (mA/mm)
(S
id,id*
/(W
 
 Nf))
1/2
?
 10
?10
, f
T
?
 10
?23
, and K
NF
L = 0.18 mm, W = 10 mm, Nf = 8 
(S
id,id*
/(W  Nf))
1/2
? 10
?10
 
f
T
? 10
?23
 
K
NF
 
(b)
0 100 200 300 400 500 600 700
0
1
2
3
4
5
6
7
8
x 10
?22
I
DS
    (mA/mm)(S
id,id*
/(W
 
 Nf))
1/2
?
 10
?10
, f
T
?
 10
?23
, and K
NF
(S
id,id*
/(W  Nf))
1/2
? 10
?10
 
K
NF
 
f
T
? 10
?23
 
L = 0.12 mm, W = 5 mm, Nf = 30 
(c)
Figure 8.21:
q
Sid;i?d=(W ? Nf), fT , and KNF vs IDS for (a) the 0.25 ?m process, (b) the 0.18
?m process, and (c) the 0.12 ?m process, with Sid;i?d in unit of A2/Hz, W in unit of ?m, fT in
unit of GHz, and KNF in unit of A=
p
?mHz3.
233
0 50 100 150 200 250 300 350 400
0
1
2
3
4
5
6
x 10
?22
I
DS
    (mA/mm)
(S
id,id*
/(W
 
 Nf))
1/2
?
 10
?10
L = 0.24 mm, W = 4 mm, Nf = 128
L = 0.18 mm, W = 10 mm, Nf = 8
L = 0.12 mm, W = 5 mm, Nf = 30
0.25 mm process
0.12 mm process 
0.18 mm process
(a)
0 50 100 150 200 250 300 350 400
0
10
20
30
40
50
60
70
80
90
I
DS
    (mA/mm)
f
T
    (GHz)
L = 0.24 mm, W = 4 mm, Nf = 128
L = 0.18 mm, W = 10 mm, Nf = 8
L = 0.12 mm, W = 5 mm, Nf = 30
(b)
0 50 100 150 200 250 300 350 400
0
0.5
1
1.5
2
2.5
3
x 10
?22
I
DS
    (mA/mm)
K
NF
    (A/(
m
m)
1/2
/Hz
3/2
)
L = 0.24 mm, W = 4 mm, Nf = 128
L = 0.18 mm, W = 10 mm, Nf = 8
L = 0.12 mm, W = 5 mm, Nf = 30
0.25 mm process 
0.18 mm process 
0.12 mm process 
(c)
Figure 8.22: (a)
q
Sid;i?d=(W ? Nf), (b) fT , and (c) KNF vs IDS comparison between a 0.25 ?m
process, a 0.18 ?m process, and a 0.12 ?m process.
234
0 50 100 150 200 250 300 350 400
0
0.02
0.04
0.06
0.08
0.1
I
DS
     (mA/mm)
K
NF,R
g
    (1/GHz)
L = 0.24 mm, W = 4 mm, Nf = 128
L = 0.18 mm, W = 10 mm, Nf = 8
L = 0.12 mm, W = 5 mm, Nf = 30
0.18 mm device, W = 10 mm 
0.24 mm device, W = 4 mm 
0.12 mm device, W = 5 mm 
Figure 8.23: KNF;Rg vs IDS comparison between a 0.18 ?m process device with W = 10 ?m, a
0.25 ?m process device with W = 4 ?m, and a 0.12 ?m process device with W = 5 ?m.
8.9 Summary
The di erence between gd0 and gm referenced excess noise factors in CMOS transistors is
examined. The technology scaling are discussed for 0.25 ?m process, 0.18 ?m process and 0.12
?m process. A simple set of analytical equations for NFmin, Rn and Yopt (or Zopt) is derived.
The equations are compared with Fukui?s empirical noise equations to identify the physical
meanings of various Fukui ?constants,? and validated using experimental data. The results show
that there does not exist a bias independent or channel length independent Fukui?s coe cient for
the well known NFmin equation. Instead, the amount of drain current noise produced to achieve
one GHz fT fundamentally determines the NFmin of the intrinsic device, and can be used as
a figure-of-merit to better measure the intrinsic noise figure capability of a technology. With
235
technology scaling from 0.25 ?m to 0.18 ?m, both fT and drain current noise increase. The fT
increase, however, dominates over the drain current noise increase, thus improving the minimum
noise figure of the intrinsic device. Another figure-of-merit is proposed to include the e ect of
gate resistance which facilitates layout optimization for low noise and evaluation of the relevant
importance of gate resistance noise with respect to drain current noise in determining NFmin.
236
CHAPTER 9
CONCLUSIONS
In this dissertation, detailed information about RF bipolar and CMOS noise in terms of
device physics were provided. To achieve these goals, this dissertation has tackled various areas
including microscopic noise simulation, Ge profile optimization in SiGe HBT device, noise
characterization, and compact noise modeling.
Chapter 1 gave an introduction of definitions and classifications of RF device noise and
noise parameters. Review of RF bipolar and CMOS noise models and the intrinsic noise sources
in RF bipolar and CMOS devices was also given in chapter 1. Di erent noise representations
for a linear noisy two-port network were introduced in chapter 2. The transformation matri-
ces to other noise representations were given for ABCD-, Y-, Z-, and H- noise representations.
Techniques of adding or de-embedding a passive component to a linear two-port network were
discussed. Noise sources de-embedding for both MOSFET and SiGe HBT were given for re-
peatedly use in later chapters.
In chapter 3, a new technique of simulating the spatial distribution of microscopic noise
contribution to the input noise current, voltage, as well as their cross-correlations were presented.
The technique was first demonstrated on a 50 GHz SiGe HBT. The spatial contributions by base
majority holes, base minority electrons, and emitter minority holes were analyzed, and compared
to results from a compact noise model. A strong crowding e ect was observed in the spatial
distribution of noise concentrations due to base majority holes. The results suggest that 2D
distributive e ect needs to be taken into account in future compact noise model development.
The technique was also applied to a 46 nm Le MOSFET transistor. The spatial distribution of
237
the Y- noise representation parameters CSig;i?g , CSid;i?
d
, <(CSig;i?
d
) and =(CSig;i?
d
) were analyzed.
The region under the gate near the source side is the most important for all of the Y- noise
representation parameters.
Bipolar transistor noise modeling for each physical noise source using microscopic noise
simulation were examined in chapter 4. Regional analysis was performed for the chain repre-
sentation noise parameters. The base majority hole noise contribution was shown to be larger
than modeled using 4kTrb and frequency dependent for all noise parameters. The 2qIB related
terms underestimates the emitter hole noise, especially for higher frequencies. The base minor-
ity electron contribution is poorly modeled by the 2qIC related terms for all noise parameters,
particularly for higher JC required for high speed. Further, regional analysis for intrinsic transis-
tor input and output noise current was performed. The input noise current consists not only the
emitter hole contribution corresponding to 2qIB, but also the base electron and hole contribution
which are frequency dependent and should be counted for especially at high frequencies. At
higher JC, the output noise current consists not only the base electron contribution correspond-
ing to 2qIC, but also the base hole contribution that not counted for in the compact noise model.
Moreover, the frequency dependence of base electron contribution is not described. The corre-
lation term which is not modeled in the compact noise model should be considered for higher
JC and higher frequency. Chapter 4 also compared the intrinsic transistor input and output noise
current with a noise model that derived from the transport theory of density fluctuations that
applied to three dimensional device. The comparison showed that this model has a better de-
scription of frequency dependence than the compact noise model at low bias. However, as for
higher JC, it has no advantage over the compact noise model.
238
RF noise physics in advanced SiGe HBTs using microscopic noise simulation was explored
in chapter 5. SiGe profile primarily a ects the minimum noise figure through the input noise
current, and identified the small region near the EB junction as where most of the input noise
current originates. A higher Ge gradient in this region helps reducing the impedance field for
the input noise current. At constant SiGe film stability, increasing the Ge gradient in the noise
critical region ultimately necessitates retrograding of Ge inside the neutral base, and the gradient
of such Ge retrograding needs to be optimized within stability limit to minimize high injection
fT rollo degradation. An example of successful SiGe profile optimization using unconventional
Ge retrograding inside the base was presented.
In chapter 6, microscopic RF noise simulation results on 50 nm Le CMOS devices were
presented, and the compact modeling of intrinsic noise sources for both the Y-representation
and the H-representation were examined. The correlation was shown to be smaller for the H-
representation than for the Y-representation. For practical biasing currents and frequencies, the
correlation is negligible for H-representation. Models for the noise sources were suggested. Fur-
thermore, the relations between the Y- and H-noise representations for MOSFETs were exam-
ined , and the importance of correlation for both representations were quantified. The theoretical
values of ?vh, ?ih and cH were derived for the first time for long channel devices, ?vh = 4=3,
?ih = 0:6, a = 0:2458, and b = 0. cH is shown theoretically to have a zero imaginary part. It was
further shown that Y-representation is a better choice for Rn, and the H-representation has the
inherent advantage of a more negligible correlation for NFmin, Gopt, and Bopt. Overall, the im-
portance of correlation is much more negligible for H-representation than for Y-representation.
This makes circuit design and simulation easier. Chapter 6 also presented experimental extrac-
tion and modeling of H-representation noise sources in a 0.25 ?m RF CMOS process. Excellent
239
agreement was achieved between modeled and measured noise data, including all noise param-
eters, for the whole bias range, from 2 to 26 GHz. The results suggest a new path to RF CMOS
noise modeling.
An anomalous frequency dependence and bias dependence of <(h11) was observed in chap-
ter 7. <(h11) decreases with frequency, and increases with Vgs at low biases. It was shown that
both the frequency dependence and bias dependence can be understood by considering the gate-
to-body capacitance and the parasitic gate-to-source capacitances as capacitances in parallel with
the series combination of the NQS resistance and inversion capacitance Cgs. A new parameter
extraction method was developed to separate the physical gate resistance and the NQS channel
resistance. The modeling results showed excellent agreement with data, and suggest the impor-
tance of modeling NQS e ect for RF CMOS even at frequencies well below fT of the technol-
ogy. The proposed model parameter extraction method can be used to facilitate MOSFET noise
modeling and more accurate Y-parameter modeling over a wide bias range.
The di erence between gd0 and gm referenced excess noise factors in CMOS transistors
was examined in chapter 8. The technology scaling were discussed for 0.25 ?m process, 0.18
?m process and 0.12 ?m process. A simple set of analytical equations for NFmin, Rn and Yopt
(or Zopt) was derived. The equations were compared with Fukui?s empirical noise equations to
identify the physical meanings of various Fukui ?constants,? and validated using experimental
data. The results showed that there does not exist a bias independent or channel length indepen-
dent Fukui?s coe cient for the well known NFmin equation. Instead, the amount of drain current
noise produced to achieve one GHz fT fundamentally determines the NFmin of the intrinsic de-
vice, and can be used as a figure-of-merit to better measure the intrinsic noise figure capability
of a technology. With technology scaling from 0.25 ?m to 0.18 ?m, both fT and drain current
240
noise increase. The fT increase, however, dominates over the drain current noise increase, thus
improving the minimum noise figure of the intrinsic device. Another figure-of-merit is proposed
to include the e ect of gate resistance which facilitates layout optimization for low noise and
evaluation of the relevant importance of gate resistance noise with respect to drain current noise
in determining NFmin.
241
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247
APPENDICES
248
APPENDIX A
MATLAB PROGRAMMING FOR OPEN-SHORT DEEMBEDDING IN CHAPTER 2
correction = 1; % correction = 0: matrix operation; 1: correction
k=1.38e-23;
To=290;
T=295;
dut = load(?DUT_Vgp685_Vd1p5_Fswp_DELSP?);
noise = load(?DUT_Vgp685_Vd1p5_Fswp_DELNP?);
open = load(?DUT_OPEN_840step2_SP.s2p?);
short = load(?DUT_SHORT_840step2_SP.s2p?);
for i=1:17
% S-parameters of the device:
fre(i)=dut(i,1);
mag=dut(i,2);
deg=dut(i,3)/180*pi;
s(1,1)=mag*(cos(deg)+j*sin(deg));
mag=dut(i,4);
deg=dut(i,5)/180*pi;
s(2,1)=mag*(cos(deg)+j*sin(deg));
mag=dut(i,6);
deg=dut(i,7)/180*pi;
s(1,2)=mag*(cos(deg)+j*sin(deg));
mag=dut(i,8);
deg=dut(i,9)/180*pi;
s(2,2)=mag*(cos(deg)+j*sin(deg));
% convert s-parameter to Y parameters
temp=50*((1+s(1,1))*(1+s(2,2))-s(1,2)*s(2,1));
y(1,1)=((1-s(1,1))*(1+s(2,2))+s(1,2)*s(2,1))/temp;
y(1,2)=-2*s(1,2)/temp;
y(2,1)=-2*s(2,1)/temp;
y(2,2)=((1+s(1,1))*(1-s(2,2))+s(1,2)*s(2,1))/temp;
% S-parameters of the open:
mag=open(i,2);
deg=open(i,3)/180*pi;
s(1,1)=mag*(cos(deg)+j*sin(deg));
mag=open(i,4);
deg=open(i,5)/180*pi;
s(2,1)=mag*(cos(deg)+j*sin(deg));
249
mag=open(i,6);
deg=open(i,7)/180*pi;
s(1,2)=mag*(cos(deg)+j*sin(deg));
mag=open(i,8);
deg=open(i,9)/180*pi;
s(2,2)=mag*(cos(deg)+j*sin(deg));
% convert s-parameter to Y parameters
temp=50*((1+s(1,1))*(1+s(2,2))-s(1,2)*s(2,1));
y_open(1,1)=((1-s(1,1))*(1+s(2,2))+s(1,2)*s(2,1))/temp;
y_open(1,2)=-2*s(1,2)/temp;
y_open(2,1)=-2*s(2,1)/temp;
y_open(2,2)=((1+s(1,1))*(1-s(2,2))+s(1,2)*s(2,1))/temp;
% S-parameters of the Short:
mag=short(i,2);
deg=short(i,3)/180*pi;
s(1,1)=mag*(cos(deg)+j*sin(deg));
mag=short(i,4);
deg=short(i,5)/180*pi;
s(2,1)=mag*(cos(deg)+j*sin(deg));
mag=short(i,6);
deg=short(i,7)/180*pi;
s(1,2)=mag*(cos(deg)+j*sin(deg));
mag=short(i,8);
deg=short(i,9)/180*pi;
s(2,2)=mag*(cos(deg)+j*sin(deg));
% convert s-parameter to Y parameters
temp=50*((1+s(1,1))*(1+s(2,2))-s(1,2)*s(2,1));
y_short(1,1)=((1-s(1,1))*(1+s(2,2))+s(1,2)*s(2,1))/temp;
y_short(1,2)=-2*s(1,2)/temp;
y_short(2,1)=-2*s(2,1)/temp;
y_short(2,2)=((1+s(1,1))*(1-s(2,2))+s(1,2)*s(2,1))/temp;
% 2. read in noise parameters of DUT
NFmin=noise(i,2);
NFmin_old(i)=NFmin;
NFmin=10^(NFmin/10.);
Rn=noise(i,5)*50;
Rn_old(i)=Rn;
mag=noise(i,3);
deg=noise(i,4)/180*pi;
Gama_opt=mag*(cos(deg)+j*sin(deg));
% convert the Gama_opt to Y_opt
Zopt=50*(1+Gama_opt)/(1.-Gama_opt);
250
Yopt=1./Zopt;
re_Yopt_old(i)=real(Yopt);
im_Yopt_old(i)=imag(Yopt);
% 3. Caluculate correlation matrix
Ca_dut(1,1)=Rn;
Ca_dut(1,2)=(NFmin-1)/2-Rn*conj(Yopt);
Ca_dut(2,1)=(NFmin-1)/2-Rn*Yopt;
Ca_dut(2,2)=Rn*abs(Yopt)*abs(Yopt);
Ca_dut=Ca_dut*2*k*To;
% 4. convert the Ca matrix into its Cy correlation matrix
T_dut=[-y(1,1) , 1; -y(2,1), 0];
%----------------------------------------------------------------------------
switch correction
case 0
Cy_dut=T_dut*Ca_dut*(T_dut?);
case 1
% Yan?s correction
T_dut_conj_trans = T_dut?;
Cy_dut(1,1) = (abs(T_dut(1,1)))^2*Ca_dut(1,1) ...
+ (abs(T_dut(1,2)))^2*Ca_dut(2,2)...
+ 2*real(T_dut_conj_trans(1,1)*T_dut(1,2)*Ca_dut(2,1));
Cy_dut(1,2) = T_dut(1,1)*T_dut_conj_trans(1,2)*Ca_dut(1,1)...
+T_dut(1,2)*T_dut_conj_trans(1,2)*Ca_dut(2,1)...
+T_dut(1,1)*T_dut_conj_trans(2,2)*Ca_dut(1,2)...
+T_dut(1,2)*T_dut_conj_trans(2,2)*Ca_dut(2,2);
Cy_dut(2,1) = Cy_dut(1,2)?;
Cy_dut(2,2) = (abs(T_dut(2,1)))^2*Ca_dut(1,1) ...
+ (abs(T_dut(2,2)))^2*Ca_dut(2,2)...
+ 2*real(T_dut_conj_trans(2,2)*T_dut(2,1)*Ca_dut(1,2));
end
%----------------------------------------------------------------------------
% 5. calculate the correlation matrix [Cy_open] of the open dummy structure
Cy_open=2*k*T*real(y_open);
% 6. subtract parallel parasitics from the Y_dut and Y_short
yi_dut=y-y_open;
yi_short=y_short-y_open;
% 7. deembed Cy_DUT from the parallel parasitic
Cyi_dut=Cy_dut-Cy_open;
% 8. convert the yi_dut to Zi_dut and Yi_short to Zi_short
temp=yi_dut(1,1)*yi_dut(2,2)-yi_dut(1,2)*yi_dut(2,1);
251
Zi_dut=[yi_dut(2,2), -yi_dut(1,2); -yi_dut(2,1), yi_dut(1,1)];
Zi_dut=Zi_dut/temp;
temp=yi_short(1,1)*yi_short(2,2)-yi_short(1,2)*yi_short(2,1);
Zi_short=[yi_short(2,2),-yi_short(1,2);-yi_short(2,1),yi_short(1,1)];
Zi_short=Zi_short/temp;
% 9. convert the Cyi_dut into Czi_dut
%----------------------------------------------------------------------------
switch correction
case 0
Czi_dut=Zi_dut*Cyi_dut*(Zi_dut?);
case 1
%Yan?s correction
Zi_dut_conj_trans = Zi_dut?;
Czi_dut(1,1) = (abs(Zi_dut(1,1)))^2*Cyi_dut(1,1) ...
+ (abs(Zi_dut(1,2)))^2*Cyi_dut(2,2)...
+2*real(Zi_dut_conj_trans(1,1)*Zi_dut(1,2)*Cyi_dut(2,1));
Czi_dut(1,2) = Zi_dut(1,1)*Zi_dut_conj_trans(1,2)*Cyi_dut(1,1)...
+Zi_dut(1,2)*Zi_dut_conj_trans(1,2)*Cyi_dut(2,1)...
+Zi_dut(1,1)*Zi_dut_conj_trans(2,2)*Cyi_dut(1,2)...
+Zi_dut(1,2)*Zi_dut_conj_trans(2,2)*Cyi_dut(2,2);
Czi_dut(2,1) = Czi_dut(1,2)?;
Czi_dut(2,2) = (abs(Zi_dut(2,1)))^2*Cyi_dut(1,1) ...
+ (abs(Zi_dut(2,2)))^2*Cyi_dut(2,2)...
+ 2*real(Zi_dut_conj_trans(2,2)*Zi_dut(2,1)*Cyi_dut(1,2));
end
%----------------------------------------------------------------------------
%10. calculate correlation matrix Czi_short after
% deembedding parallel parasitic
Czi_short=2*k*T*real(Zi_short);
%11. subtract series parasitics from Zi_dut to get
% Z parameter of the intrinsic transistor
Ztran=Zi_dut-Zi_short;
%12. De-embed Czi_dut from series parasitics to get
% the correlation matrix Cz of the intrinsic transistor
Cz=Czi_dut-Czi_short;
%13. convert the Ztran to its chain matrix Atrans
Atran=[Ztran(1,1), Ztran(1,1)*Ztran(2,2)-Ztran(1,2)*Ztran(2,1);
1, Ztran(2,2)];
Atran=Atran/Ztran(2,1);
252
%14. Transform Cz to Ca
Ta=[1, -Atran(1,1); 0, -Atran(2,1)];
%----------------------------------------------------------------------------
switch correction
case 0
Ca=Ta*Cz*(Ta?);
case 1
% Yan?s correction
Ta_conj_trans = Ta?;
Ca(1,1) = (abs(Ta(1,1)))^2*Cz(1,1) + (abs(Ta(1,2)))^2*Cz(2,2)...
+ 2*real(Ta_conj_trans(1,1)*Ta(1,2)*Cz(2,1));
Ca(1,2) = Ta(1,1)*Ta_conj_trans(1,2)*Cz(1,1)...
+Ta(1,2)*Ta_conj_trans(1,2)*Cz(2,1)...
+Ta(1,1)*Ta_conj_trans(2,2)*Cz(1,2)...
+Ta(1,2)*Ta_conj_trans(2,2)*Cz(2,2);
Ca(2,1) = Ca(1,2)?;
Ca(2,2) = (abs(Ta(2,1)))^2*Cz(1,1) + (abs(Ta(2,2)))^2*Cz(2,2)...
+ 2*real(Ta_conj_trans(2,2)*Ta(2,1)*Cz(1,2));
end
%----------------------------------------------------------------------------
%15. calculate the open-short deembedded NFmin, Yopt and Rn
temp=sqrt((Ca(1,1)*Ca(2,2)-(imag(Ca(1,2))^2)));
NFmin_new(i)=log10(1+1/k/T*((real(Ca(1,2)))+temp))*10;
im_NFmin_new(i)=imag(NFmin_new(i));
Yopt_new=(temp+j*imag(Ca(1,2)))/Ca(1,1);
re_Yopt_new(i)=real(Yopt_new);
im_Yopt_new(i)=imag(Yopt_new);
Zopt_new=1./Yopt_new;
mag_Gama_new(i)=abs(-(50-Zopt_new)/(50+Zopt_new));
ang_Gama_new(i)=angle(-(50-Zopt_new)/(50+Zopt_new))/pi*180;
Rn_new(i)=real(Ca(1,1)/2/k/T);
%----------------------------------------------------------------------------
%16. Yan: Calculate Sig, Sid, and correlation
temp=Ztran(1,1)*Ztran(2,2)-Ztran(1,2)*Ztran(2,1);
Ytran=[Ztran(2,2),-Ztran(1,2);-Ztran(2,1),Ztran(1,1)];
Ytran=Ytran/temp;
Ytran_conj_trans = Ytran?;
Cy(1,1) = (abs(Ytran(1,1)))^2*Cz(1,1) ...
+ (abs(Ytran(1,2)))^2*Cz(2,2)...
+2*real(Ytran_conj_trans(1,1)*Ytran(1,2)*Cz(2,1));
Cy(1,2) = Ytran(1,1)*Ytran_conj_trans(1,2)*Cz(1,1)...
+Ytran(1,2)*Ytran_conj_trans(1,2)*Cz(2,1)...
253
+Ytran(1,1)*Ytran_conj_trans(2,2)*Cz(1,2)...
+Ytran(1,2)*Ytran_conj_trans(2,2)*Cz(2,2);
Cy(2,1) = Cy(1,2)?;
Cy(2,2) = (abs(Ytran(2,1)))^2*Cz(1,1) ...
+ (abs(Ytran(2,2)))^2*Cz(2,2)...
+ 2*real(Ytran_conj_trans(2,2)*Ytran(2,1)*Cz(1,2));
Sig(i) = 2*Cy(1,1);
Sid(i) = 2*Cy(2,2);
Sigid(i) = 2*Cy(1,2);
Cigid(i) = Sigid(i)./sqrt(Sig(i).*Sid(i));
%----------------------------------------------------------------------------
%17. Yan: Calculate Svh, Sih, and correlation
Svh(i) = Sig(i)./(abs(Ytran(1,1))).^2;
Sih(i)= Sid(i) + Sig(i).*(abs(Ytran(2,1)./Ytran(1,1))).^2-...
2.*real(Ytran(2,1)./Ytran(1,1).*Sigid(i));
Svhih(i) = conj(Ytran(2,1))./(abs(Ytran(1,1))).^2.*Sig(i) -...
Sigid(i)./Ytran(1,1);
Cvhih(i) = Svhih(i)./sqrt(Svh(i).*Sih(i));
end
254
APPENDIX B
DESSIS INPUT DECK AND MATLAB PROGRAMMING FOR SIGE HBT NOISE
SIMULATION
B.1 5HP SiGe HBT
B.1.1 Mesh files
BND file
Oxide "DT" {rectangle[(2.3, 0.648) (2.8, 4.598)]}
Oxide "STI" {rectangle[(2.2, 0.248) (2.8, 0.648)]}
Oxide "STI2" {rectangle[(0.5, 0.248) (1.2, 0.648)]}
PolySi "PolySi" {polygon[(1.25, 0) (1.25, 0.068) (1.45, 0.068) (1.45, 0.148)
(1.95, 0.148) (1.95, 0.068) (2.15, 0.068) (2.15, 0)]}
Oxide "spacer1" {rectangle[(1.25, 0.068) (1.45,0.148)]}
Oxide "spacer2" {rectangle[(1.95, 0.068) (2.15, 0.148)]}
Silicon "Silicon1" {polygon[(0, 0.248) (0, 4.598) (2.3, 4.598) (2.3, 0.648)
(2.2, 0.648) (2.2, 0.248) (2.6, 0.248) (2.6, 0.24)
(0.8, 0.24) (0.8, 0.248) (1.2, 0.248) (1.2, 0.648)
(0.5, 0.648) (0.5, 0.248)]}
Silicon "Silicon2" {rectangle[(0.8, 0.148) (2.6, 0.1646)]}
SiliconGermanium "SiGe" {rectangle[(0.8, 0.1646) (2.6, 0.24)]}
Contact "Collector" {line[(0, 0.248) (0.47, 0.248)]}
Contact "Base1" {line[(0.8, 0.148) (1.2, 0.148)]}
Contact "Base2" {line[(2.2, 0.148) (2.6, 0.148)]}
Contact "Emitter" {line[(1.45, 0) (1.95, 0)]}
Contact "Psubstrate" {line[(0, 4.598) (2.3, 4.598)]}
CMD file
Title "BJT"
Definitions {
# Refinement regions
Refinement "all region"
{
MaxElementSize = (0.2 0.5)
MinElementSize = (0.05 0.05)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
}
Refinement "sige"
255
{
MaxElementSize = (0.05 0.005)
MinElementSize = (0.02 0.002)
RefineFunction = MaxTransDiff(Variable="xMoleFraction" Value=0.01)
}
Refinement "substrate region1"
{
MaxElementSize = (0.15 0.15)
MinElementSize = (0.08 0.08)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
}
Refinement "substrate region2"
{
MaxElementSize = (0.08 0.1)
MinElementSize = (0.03 0.005)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
}
Refinement "substrate region3"
{
MaxElementSize = (0.1 0.05)
MinElementSize = (0.05 0.005)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
}
Refinement "Oxide_shallow"
{
MaxElementSize = (0.05 0.05)
MinElementSize = (0.02 0.02)
}
Refinement "Oxide_DT"
{
MaxElementSize = (0.1 0.1)
MinElementSize = (0.05 0.05)
}
Refinement "Oxide_spacer"
{
MaxElementSize = (0.04 0.04)
MinElementSize = (0.02 0.01)
}
Refinement "Emitter"
{
MaxElementSize = (0.05 0.02)
MinElementSize = (0.01 0.005)
256
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
}
Refinement "eb_junction"
{
MaxElementSize = (0.05 0.02)
MinElementSize = (0.025 0.002)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
}
Refinement "cb_junctionup"
{
MaxElementSize = (0.05 0.05)
MinElementSize = (0.01 0.01)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
}
# Profiles
Constant "psubstrate"
{
Species = "BoronActiveConcentration"
Value = 2e+15
}
Constant "n_epi"
{
Species = "PhosphorusActiveConcentration"
Value = 5e+16
}
AnalyticalProfile "emitter"
{
Function = subMesh1D(datafile = "as.dat"
, Scale = 1,
Range = line[(0 0), (0.598 0)]
)
LateralFunction = Erf(Factor = 0)
}
AnalyticalProfile "collector"
{
Function = subMesh1D(datafile = "phos.xy"
, Scale = 1,
Range = line[(0 0), (0.598 7.6364e+17)]
)
LateralFunction = Erf(Factor = 0)
}
257
AnalyticalProfile "n_buried layer"
{
Function = subMesh1D(datafile = "bu_asyan.xy"
, Scale = 1, Range = line[(4.446e-1 1.5363805e+16), (2.720176 1.5363805e+16)]
)
LateralFunction = Erf(Factor=0)
}
AnalyticalProfile "intrinsic base"
{
Function = subMesh1D(datafile = "sims.dat"
, Scale = 1,
Range = line[(0 0), (0.598 0)]
)
LateralFunction = Erf(Factor = 0)
}
Constant "cc"
{ Species = "ArsenicActiveConcentration"
Value = 1e+20
}
Constant "base"
{
Species = "BoronActiveConcentration"
Value = 1e+16
}
Constant "extrinsic base"
{
Species = "BoronActiveConcentration"
Value = 1.5e+19
}
AnalyticalProfile "xMoleBase"
{
Function = subMesh1D(datafile = "xMol10.xy"
, Scale = 1,
Range = line[(0.1646 0), (0.2774 0)]
)
LateralFunction = Erf(Factor = 0)
}
}
Placements {
# Refinement regions
Refinement "all region"
258
{
Reference = "all region"
RefineWindow = rectangle [(0 0), (2.6 4.598)]
}
Refinement "substrate region1"
{
Reference = "substrate region1"
RefineWindow = rectangle [(0 0.248), (2.3 4.598)]
}
Refinement "emitter region"
{
Reference = "Emitter"
RefineWindow = rectangle [(1.25 0) (2.15 0.08)]
}
Refinement "sige region"
{
Reference = "sige"
RefineWindow = rectangle [(0.8 0.1646)(2.6 0.2774)]
}
Refinement "eb_junction"
{
Reference = "eb_junction"
RefineWindow = rectangle [(1.2 0.06), (2.6 0.16)]
}
Refinement "cb_junctionup"
{
Reference = "cb_junctionup"
RefineWindow = rectangle [(0 0.14), (2.35 1)]
}
Refinement "substrate region2"
{
Reference = "substrate region2"
RefineWindow = rectangle [(0 0.9), (2.6 1.2)]
}
Refinement "ST1"
{
Reference = "Oxide_shallow"
RefineWindow = rectangle [(0.5 0.24) (1.2 0.69)]
}
Refinement "ST2"
{
Reference = "Oxide_shallow"
259
RefineWindow = rectangle [(2.25 0.24) (2.6 0.69)]
}
Refinement "DT"
{
Reference = "Oxide_DT"
RefineWindow = rectangle[(2.3 0.248) (2.6 4.598)]
}
Refinement "spacer1"
{
Reference = "Oxide_spacer"
RefineWindow = rectangle [(1.25 0.068) (1.45 0.148)]
}
Refinement "spacer2"
{
Reference = "Oxide_spacer"
RefineWindow = rectangle [(1.95 0.068) (2.15 0.148)]
}
Refinement "substrate region3"
{
Reference = "substrate region3"
RefineWindow = rectangle [(0 2.5), (2.6 2.65)]
}
Refinement "patch"
{
Reference = "sige"
RefineWindow = rectangle [(0 0.248)(0.8 0.2774)]
}
# Profiles
Constant "psubstrate instance"
{
Reference = "psubstrate"
EvaluateWindow
{
Element = rectangle [(0 2.58), (2.3 4.598)]
DecayLength = 0
}
}
AnalyticalProfile "intrinsic base instance"
{
Reference = "intrinsic base"
260
ReferenceElement
{
Element = line [(0.8 0), (2.6 0)]
}
EvaluateWindow
{
Element = rectangle[(0.8 0), (2.6 0.598)]
}
}
Constant "collectorwhole instance"
{
Reference = "n_epi"
EvaluateWindow
{
Element = rectangle [(0 0), (2.8 2.598)]
DecayLength = 0
}
}
AnalyticalProfile "emitter instance"
{
Reference = "emitter"
ReferenceElement
{
Element = line [(1.25 0), (2.15 0)]
}
EvaluateWindow
{
Element = polygon[(1.25 0) (1.25 0.068) (1.45 0.068)
(1.45 0.598) (1.95 0.598) (1.95 0.068)
(2.15 0.068) (2.15 0)]
}
}
AnalyticalProfile "n_buried layer instance"
{
Reference = "n_buried layer"
ReferenceElement
{
Element = line[(0.5 0.4446) (2.3 0.4446)]
}
EvaluateWindow
{
Element = rectangle [(0.5 0.4446)(2.3 2.720176)]
261
DecayLength = 0
}
}
AnalyticalProfile "collector instance"
{
Reference = "collector"
ReferenceElement
{
Element = line [(1.35 0), (2.05 0)]
}
EvaluateWindow
{
Element = rectangle[(1.35 0)(2.05 0.598)]
}
}
Constant "extrinsic base left instance"
{
Reference = "extrinsic base"
EvaluateWindow
{
Element = rectangle [(0.8 0.148), (1.35 0.258)]
DecayLength = 0.010
}
}
Constant "extrinsic base right instance"
{
Reference = "extrinsic base"
EvaluateWindow
{
Element = rectangle [(2.05 0.148), (2.6 0.258)]
DecayLength = 0.010
}
}
Constant "Collector contact instance"
{
Reference = "cc"
EvaluateWindow
{
Element=rectangle[(0 0.248)(0.5 2.598)]
}
}
AnalyticalProfile "xMolBase instance"
262
{
Reference = "xMoleBase"
ReferenceElement
{
Element = line[(0.8 0.1646) (2.6 0.1646)]
Direction = positive
}
EvaluateWindow
{
Element = polygon[(0.8 0.1646) (0.8 0.248) (1.2 0.248)
(1.2 0.598) (2.2 0.598) (2.2 0.248)
(2.6 0.248) (2.6 0.1646)]
}
}
}
B.1.2 Noise Simulation CMD file
Device BJT {
Electrode {
{ Name="Emitter" Voltage=0 }
{ Name="Base1" Voltage=0 }
{ Name = "Base2" Voltage = 0}
{ Name="Collector" Voltage=0 }
{ Name = "Psubstrate" Voltage = 0}
}
File {
Grid = "msh10_msh.grd"
Doping = "msh10_msh.dat"
Current = "ac10ddall_des.plt"
Plot = "ac10ddall_des.dat"
}
Physics{
Areafactor= 1
EffectiveIntrinsicDensity(BandgapNarrowing( Slotboom) )
Mobility(
PhuMob
Highfieldsaturation
)
263
Fermi
Noise ( DiffusionNoise )
}
Physics (material = "Silicon") {
Recombination(
SRH( DopingDependence )
Auger
)
}
Physics (material = "PolySi") {
Recombination(
SRH( DopingDependence )
Auger
)
}
}
*----------------------------------------------------------------------*
*--End of Device{}
*----------------------------------------------------------------------*
Plot {
eDensity hDensity
TotalCurrent/Vector eCurrent/Vector hCurrent/Vector
ElectricField Potential SpaceCharge
Doping DonorConcentration AcceptorConcentration
SRH Auger
eQuasiFermi hQuasiFermi
eEparal hEparal
eMobility hMobility
eVelocity hVelocity
xMoleFraction
BandGap BandGapNarrowing
Affinity
ConductionBand ValenceBand
264
}
#NoisePlot {
# AllLNS AllLNVSD AllLNVXVSD GreenFunctions
#}
Math {
Extrapolate
NotDamped=200
Iterations=20
NewDiscretization
Derivatives
RelerrControl
Digits=6
}
File {
Output = "ac10ddall"
ACExtract="ac10ddall"
}
System {
BJT bjt (Base1=1 Base2 = 1 Collector=2 Emitter=0 Psubstrate=0)
Vsource_pset vb (1 0){ dc = 0 }
Vsource_pset vc (2 0){ dc = 0 }
}
Solve {
Coupled{Poisson Electron Hole }
Quasistationary (
InitialStep=0.1 Increment=1.4
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.75}
Goal {Parameter=vb.dc Voltage=0.75}
){
Coupled{Poisson Electron Hole }
}
save(fileprefix = "17510dd")
newcurrent = "ac10ddbias"
load(fileprefix = "17510dd")
Quasistationary (
265
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.76}
Goal {Parameter=vb.dc Voltage=0.76}
){
Coupled{Poisson Electron Hole }
}
save(fileprefix = "17610dd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.77}
Goal {Parameter=vb.dc Voltage=0.77}
){
Coupled{Poisson Electron Hole }
}
save(fileprefix = "17710dd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.78}
Goal {Parameter=vb.dc Voltage=0.78}
){
Coupled{Poisson Electron Hole }
}
save(fileprefix = "17810dd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.79}
Goal {Parameter=vb.dc Voltage=0.79}
){
Coupled{Poisson Electron Hole }
}
save(fileprefix = "17910dd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.80}
Goal {Parameter=vb.dc Voltage=0.80}
){
Coupled{Poisson Electron Hole }
266
}
save(fileprefix = "18010dd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.81}
Goal {Parameter=vb.dc Voltage=0.81}
){
Coupled{Poisson Electron Hole }
}
save(fileprefix = "18110dd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.82}
Goal {Parameter=vb.dc Voltage=0.82}
){
Coupled{Poisson Electron Hole }
}
save(fileprefix = "18210dd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.83}
Goal {Parameter=vb.dc Voltage=0.83}
){
Coupled{Poisson Electron Hole }
}
save(fileprefix = "18310dd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.84}
Goal {Parameter=vb.dc Voltage=0.84}
){
Coupled{Poisson Electron Hole }
}
save(fileprefix = "18410dd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.85}
267
Goal {Parameter=vb.dc Voltage=0.85}
){
Coupled{Poisson Electron Hole }
}
save(fileprefix = "18510dd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.86}
Goal {Parameter=vb.dc Voltage=0.86}
){
Coupled{Poisson Electron Hole }
}
save(fileprefix = "18610dd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.87}
Goal {Parameter=vb.dc Voltage=0.87}
){
Coupled{Poisson Electron Hole }
}
save(fileprefix = "18710dd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.88}
Goal {Parameter=vb.dc Voltage=0.88}
){
Coupled{Poisson Electron Hole }
}
save(fileprefix = "18810dd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.89}
Goal {Parameter=vb.dc Voltage=0.89}
){
Coupled{Poisson Electron Hole }
}
save(fileprefix = "18910dd")
Quasistationary (
268
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.90}
Goal {Parameter=vb.dc Voltage=0.90}
){
Coupled{Poisson Electron Hole }
}
save(fileprefix = "19010dd")
newcurrent = "ac10ddall"
load(fileprefix = "17510dd")
ACCoupled (
StartFrequency = 1e9 EndFrequency = 20e9
NumberofPoints =20 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "ac10ddall"
NoiseExtraction = "ac10ddall"
NoisePlot = "ac10ddall"
)
{Poisson Electron Hole }
load(fileprefix = "17610dd")
ACCoupled (
StartFrequency = 1e9 EndFrequency = 20e9
NumberofPoints = 20 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "ac10ddall"
NoiseExtraction = "ac10ddall"
NoisePlot = "ac10ddall"
)
{Poisson Electron Hole }
load(fileprefix = "17710dd")
ACCoupled (
StartFrequency = 1e9 EndFrequency = 20e9
NumberofPoints = 20 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "ac10ddall"
NoiseExtraction = "ac10ddall"
NoisePlot = "ac10ddall"
)
269
{Poisson Electron Hole }
load(fileprefix = "17810dd")
ACCoupled (
StartFrequency = 1e9 EndFrequency = 20e9
NumberofPoints = 20 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "ac10ddall"
NoiseExtraction = "ac10ddall"
NoisePlot = "ac10ddall"
)
{Poisson Electron Hole }
load(fileprefix = "17910dd")
ACCoupled (
StartFrequency = 1e9 EndFrequency = 20e9
NumberofPoints = 20 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "ac10ddall"
NoiseExtraction = "ac10ddall"
NoisePlot = "ac10ddall"
)
{Poisson Electron Hole }
load(fileprefix = "18010dd")
ACCoupled (
StartFrequency = 1e9 EndFrequency = 20e9
NumberofPoints = 20 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "ac10ddall"
NoiseExtraction = "ac10ddall"
NoisePlot = "ac10ddall"
)
{Poisson Electron Hole }
load(fileprefix = "18110dd")
ACCoupled (
StartFrequency = 1e9 EndFrequency = 20e9
NumberofPoints = 20 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "ac10ddall"
NoiseExtraction = "ac10ddall"
270
NoisePlot = "ac10ddall"
)
{Poisson Electron Hole }
load(fileprefix = "18210dd")
ACCoupled (
StartFrequency = 1e9 EndFrequency = 20e9
NumberofPoints = 20 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "ac10ddall"
NoiseExtraction = "ac10ddall"
NoisePlot = "ac10ddall"
)
{Poisson Electron Hole }
load(fileprefix = "18310dd")
ACCoupled (
StartFrequency = 1e9 EndFrequency = 20e9
NumberofPoints = 20 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "ac10ddall"
NoiseExtraction = "ac10ddall"
NoisePlot = "ac10ddall"
)
{Poisson Electron Hole }
load(fileprefix = "18410dd")
ACCoupled (
StartFrequency = 1e9 EndFrequency = 20e9
NumberofPoints = 20 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "ac10ddall"
NoiseExtraction = "ac10ddall"
NoisePlot = "ac10ddall"
)
{Poisson Electron Hole }
load(fileprefix = "18510dd")
ACCoupled (
StartFrequency = 1e9 EndFrequency = 20e9
NumberofPoints = 20 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
271
ACExtraction = "ac10ddall"
NoiseExtraction = "ac10ddall"
NoisePlot = "ac10ddall"
)
{Poisson Electron Hole }
load(fileprefix = "18610dd")
ACCoupled (
StartFrequency = 1e9 EndFrequency = 20e9
NumberofPoints = 20 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "ac10ddall"
NoiseExtraction = "ac10ddall"
NoisePlot = "ac10ddall"
)
{Poisson Electron Hole }
load(fileprefix = "18710dd")
ACCoupled (
StartFrequency = 1e9 EndFrequency = 20e9
NumberofPoints = 20 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "ac10ddall"
NoiseExtraction = "ac10ddall"
NoisePlot = "ac10ddall"
)
{Poisson Electron Hole }
load(fileprefix = "18810dd")
ACCoupled (
StartFrequency = 1e9 EndFrequency = 20e9
NumberofPoints = 20 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "ac10ddall"
NoiseExtraction = "ac10ddall"
NoisePlot = "ac10ddall"
)
{Poisson Electron Hole }
load(fileprefix = "18910dd")
ACCoupled (
StartFrequency = 1e9 EndFrequency = 20e9
NumberofPoints = 20 linear
272
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "ac10ddall"
NoiseExtraction = "ac10ddall"
NoisePlot = "ac10ddall"
)
{Poisson Electron Hole }
load(fileprefix = "19010dd")
ACCoupled (
StartFrequency = 1e9 EndFrequency = 20e9
NumberofPoints = 20 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "ac10ddall"
NoiseExtraction = "ac10ddall"
NoisePlot = "ac10ddall"
)
{Poisson Electron Hole }
}
B.1.3 Tecplot MCR file
Y parameters in this MCR file should be changed according to each bias and frequency.
#!MC 800
$!VarSet |MFBD| = ?/home/tcad2/cuiyan1/cuiyan/Dessis/5hpdop/pmisimu?
$!VarSet |MFBD1| = ?/home/tcad2/cuiyan1/cuiyan/Dessis/5hpdop/mesh?
$!Varset |fsel| = ?10dd10g?
$!Varset |f| = ?jcp25?
#p = total(t), ee, hh
$!Varset |p| = ?t?
$!Varset |num| = ?00?
#other noise model calculation
$!Varset |freq| = 2.00000000000000E+09
$!Varset |omega| =(2*PI*|freq|)
$!Varset |ReY11| = 5.30793553232469E-05
$!Varset |ImY11| = (|omega|*2.40555589212422E-14 )
$!Varset |ReY12| = -1.14676737756815E-06
$!Varset |ImY12| = (-2.13716228513946E-15*|omega|)
$!Varset |ReY21| = 4.08080901277576E-03
273
$!Varset |ImY21| = (-3.39993575257787E-14*|omega|)
$!Varset |ReY22| = 4.04802562859283E-06
$!Varset |ImY22| = ( 3.90143182411877E-15*|omega|)
#create 1.plt for Sv1
$!Newlayout
$!READDATASET ?"-ise:lay" "-ise:lc" "|MFBD1|/msh10_msh.grd"
"|MFBD|/ac|fsel||f|_bjt_1_00|num|_acgf_des.dat.gz"?
DATASETREADER = ?DF-ISE Loader?
$!ALTERDATA
EQUATION = ?{tLNVSD} = {LNVSD}?
$!ALTERDATA
EQUATION = ?{Sv1} = {|p|LNVSD}?
$!WRITEDATASET "|MFBD|/1.dat"
INCLUDEGEOM = NO
INCLUDECUSTOMLABELS = NO
VARPOSITIONLIST = [1-2,29]
BINARY = No
USEPOINTFORMAT = Yes
PRECISION = 9
#create 2.plt for Sv2
$!Newlayout
$!READDATASET ?"-ise:lay" "-ise:lc" "|MFBD1|/msh10_msh.grd"
"|MFBD|/ac|fsel||f|_bjt_2_00|num|_acgf_des.dat.gz"?
DATASETREADER = ?DF-ISE Loader?
$!ALTERDATA
EQUATION = ?{tLNVSD} = {LNVSD}?
$!ALTERDATA
EQUATION = ?{Sv2} = {|p|LNVSD}?
$!WRITEDATASET "|MFBD|/2.dat"
INCLUDEGEOM = NO
INCLUDECUSTOMLABELS = NO
VARPOSITIONLIST = [1-2,29]
BINARY = No
USEPOINTFORMAT = Yes
PRECISION = 9
#create 1_2.plt for ReSv12 and ImSv12
$!Newlayout
274
$!READDATASET ?"-ise:lay" "-ise:lc" "|MFBD1|/msh10_msh.grd"
"|MFBD|/ac|fsel||f|_bjt_1_2_00|num|_acgf_des.dat.gz"?
DATASETREADER = ?DF-ISE Loader?
$!ALTERDATA
EQUATION = ?{RetLNVXVSD} = {ReLNVXVSD}?
$!ALTERDATA
EQUATION = ?{ImtLNVXVSD} = {ImLNVXVSD}?
$!ALTERDATA
EQUATION = ?{ReSv12} = {Re|p|LNVXVSD}?
$!ALTERDATA
EQUATION = ?{ImSv12} = -{Im|p|LNVXVSD}?
$!WRITEDATASET "|MFBD|/1_2.dat"
INCLUDEGEOM = NO
INCLUDECUSTOMLABELS = NO
VARPOSITIONLIST = [1-2,15-16]
BINARY = No
USEPOINTFORMAT = Yes
PRECISION = 9
#combine Sv1, Sv2, Sv1v2 together
$!NEWLAYOUT
$!READDATASET ?"|MFBD|/1.dat" "|MFBD|/2.dat" "|MFBD|/1_2.dat" ?
READDATAOPTION = NEW
RESETSTYLE = YES
INCLUDEGEOM = NO
INCLUDECUSTOMLABELS = NO
VARLOADMODE = BYNAME
INITIALFRAMEMODE = TWOD
VARNAMELIST = ?"X" "Y" "Sv1" "Sv2" "ReSv12" "ImSv12"?
$!Varset |Dataset| = |Numzones|
$!Varset |Dataset| /= 3
$!alterdata equation = "{h2} = 0"
$!alterdata equation = "{rh12} = 0"
$!alterdata equation = "{ih12} = 0"
$!Loop |dataset|
$!Varset |Source1| = |Loop|
$!Varset |Source1| += |dataset|
$!Varset |Source2| = |Source1|
275
$!Varset |Source2| += |dataset|
$!Alterdata [|Loop|] equation = "{h2} = v4[|Source1|]"
$!Alterdata [|Loop|] equation = "{rh12} = v5[|Source2|]"
$!Alterdata [|Loop|] equation = "{ih12} = v6[|Source2|]"
$!endloop
$!varset |Deletezone| = |Dataset|
$!Varset |deletezone| += 1
$!Deletezones [|Deletezone| - |numzones|]
$!alterdata equation = "{Sv2} = {h2}"
$!Alterdata equation = "{ReSv12} = {rh12}"
$!Alterdata equation = "{ImSv12} = {ih12}"
$!WRITEDATASET "|MFBD|/all.dat"
INCLUDEGEOM = NO
INCLUDECUSTOMLABELS = NO
VARPOSITIONLIST = [1-6]
BINARY = no
USEPOINTFORMAT = yes
PRECISION = 9
$!NEWLAYOUT
$!READDATASET ?"|MFBD|/all.dat" ?
READDATAOPTION = NEW
RESETSTYLE = YES
INCLUDEGEOM = NO
INCLUDECUSTOMLABELS = NO
VARLOADMODE = BYNAME
VARNAMELIST = ?"X" "Y" "Sv1" "Sv2" "ReSv12" "ImSv12"?
$!Varset |abs2Y21| = (|ReY21|*|ReY21| + |ImY21|*|ImY21|)
$!Varset |abs2Y22| = (|ReY22|*|ReY22| + |ImY22|*|ImY22|)
$!Varset |Redelta0| = (|ReY11|*|ReY22|-|ImY11|*|ImY22|-|ReY12|*|ReY21|+|ImY12|*|ImY21|)
$!Varset |Imdelta0| = (|ReY11|*|ImY22|+|ReY22|*|ImY11|-|ReY12|*|ImY21|-|ReY21|*|ImY12|)
$!Varset |abs2delta0| = (|Redelta0|*|Redelta0| + |Imdelta0|*|Imdelta0|)
$!Varset |x| = (|Redelta0|*|ReY21|+|Imdelta0|*|ImY21|)
$!Varset |Redelta1| = (|x|/|abs2Y21|)
$!Varset |x| = (|Imdelta0|*|ReY21|-|Redelta0|*|ImY21|)
$!Varset |Imdelta1| = (|x|/|abs2Y21|)
$!Varset |abs2delta1| = (|Redelta1|*|Redelta1|+|Imdelta1|*|Imdelta1|)
276
$!Varset |x| = (|ReY22|*|ReY21|+|ImY22|*|ImY21|)
$!Varset |Redelta2| = (|x|/|abs2Y21|)
$!Varset |x| = (|ImY22|*|ReY21|-|ReY22|*|ImY21|)
$!Varset |Imdelta2| = (|x|/|abs2Y21|)
$!Varset |abs2delta2| = (|Redelta2|*|Redelta2|+|Imdelta2|*|Imdelta2|)
$!Varset |x| = (|Redelta0|*|ReY22|+|Imdelta0|*|ImY22|)
$!Varset |Redelta3| = (|x|/|abs2Y21|)
$!Varset |x| = (|Imdelta0|*|ReY22|-|Redelta0|*|ImY22|)
$!Varset |Imdelta3| = (|x|/|abs2Y21|)
#Sva, Sia
$!alterdata
equation = "{Sva} = {Sv1}+|abs2delta2|*{Sv2}+2*({ReSv12}*|Redelta2|+{ImSv12}*|Imdelta2|)"
$!alterdata
equation = "{Sia} = {Sv2}*|abs2delta1|"
$!alterdata
equation = "{ReSiava} = |Redelta1|*{ReSv12}+|Imdelta1|*{ImSv12}+|Redelta3|*{Sv2}"
$!alterdata
equation = "{ImSiava} = |Imdelta1|*{ReSv12}-|Redelta1|*{ImSv12}+|Imdelta3|*{Sv2}"
#Sin1, Sin2
$!Varset |abs2Y11| = (|ReY11|*|ReY11| + |ImY11|*|ImY11|)
$!Varset |abs2Y12| = (|ReY12|*|ReY12| + |ImY12|*|ImY12|)
$!Varset |Rex| = (|ReY11|*|ReY12|+|ImY11|*|ImY12|)
$!Varset |Imx| = (|ImY11|*|ReY12|-|ReY11|*|ImY12|)
$!Varset |Rey| = (|ReY21|*|ReY22|+|ImY21|*|ImY22|)
$!Varset |Imy| = (|ImY21|*|ReY22|-|ReY21|*|ImY22|)
$!Varset |Rez| = (|ReY21|*|ReY11|+|ImY21|*|ImY11|)
$!Varset |Imz| = (|ImY21|*|ReY11|-|ReY21|*|ImY11|)
$!Varset |Rew| = (|ReY22|*|ReY12|+|ImY22|*|ImY12|)
$!Varset |Imw| = (|ImY22|*|ReY12|-|ReY22|*|ImY12|)
$!Varset |Reu| = (|ReY22|*|ReY11|+|ImY22|*|ImY11|)
$!Varset |Imu| = (|ImY22|*|ReY11|-|ReY22|*|ImY11|)
$!Varset |Rev| = (|ReY21|*|ReY12|+|ImY21|*|ImY12|)
$!Varset |Imv| = (|ImY21|*|ReY12|-|ReY21|*|ImY12|)
277
$!alterdata
equation = "{Sin1} = |abs2Y11|*{Sv1}+|abs2Y12|*{Sv2} + 2*(|Rex|*{ReSv12}-|Imx|*{ImSv12})"
$!alterdata
equation = "{Sin2} = |abs2Y21|*{Sv1} + |abs2Y22|*{Sv2} + 2*(|Rey|*{ReSv12}-|Imy|*{ImSv12})"
$!alterdata
equation = "{ReSi2i1} = |Rez|*{Sv1} + |Rew|*{Sv2} + |Reu|*{ReSv12}+|Imu|*{ImSv12} + |Rev|*{ReSv12}-|Imv|*{ImSv12}"
$!alterdata
equation = "{ImSi2i1} = |Imz|*{Sv1} + |Imw|*{Sv2} + |Imu|*{ReSv12} -|Reu|*{ImSv12}+|Imv|*{ReSv12} + |Rev|*{ImSv12}"
$!WRITEDATASET "|MFBD|/final|fsel||f||p||num|.dat"
INCLUDEGEOM = NO
INCLUDECUSTOMLABELS = NO
VARPOSITIONLIST = [1-14]
BINARY = no
USEPOINTFORMAT = yes
PRECISION = 9
$!FIELDLAYERS SHOWMESH = NO
$!Fieldlayers showcontour = Yes
$!TWODAXIS YDETAIL{ISREVERSED = YES}
$!GLOBALCONTOUR LEGEND{SHOW = YES}
$!FIELD [1-18] CONTOUR{CONTOURTYPE = FLOOD}
$!ADDONCOMMAND
ADDONID = ?ISE TCAD ADD-on?
COMMAND = ?ORTHOSLICE X 1.75 Frame 001?
$!WRITEDATASET "|MFBD|/1dcut|fsel||f||p||num|.dat"
INCLUDEGEOM = NO
INCLUDECUSTOMLABELS = NO
BINARY = no
USEPOINTFORMAT = yes
PRECISION = 9
$!RemoveVar |MFBD|
278
B.2 8HP SiGe HBT
B.2.1 Mesh files
BND file
#8hp 2D structure
Oxide "DT1" {polygon[(2.05, 0.19) (2.05, 0.53) (2.17, 0.53)
(2.17, 4.30) (2.39, 4.30) (2.39, 0.19)]}
Oxide "DT2" {polygon[(-2.05, 0.19) (-2.05, 0.53) (-2.17, 0.53)
(-2.17, 4.30) (-2.39, 4.30) (-2.39, 0.19)]}
Oxide "STI1" {rectangle[(0.35, 0.19) (1.35, 0.53)]}
Oxide "STI2"{rectangle[(-0.35, 0.19) (-1.35, 0.53)]}
Oxide "spacer1" {polygon[(0.06, 0.15) (0.06, 0) (0.36, 0) (0.36, 0.05)
(0.12, 0.05) (0.12, 0.15)]}
Oxide "spacer2" {polygon[(-0.06, 0.15) (-0.06, 0) (-0.36, 0) (-0.36, 0.05)
(-0.12, 0.05) (-0.12, 0.15)]}
PolySi "PolySi" {rectangle[(-0.06, 0.15) (0.06, 0.04)]
}
PolySi "basesi1" {rectangle[(0.12, 0.15) (1.1, 0.05)]}
PolySi "basesi2" {rectangle[(-0.12, 0.15) (-1.1, 0.05)]}
Silicon "Silicon1" {polygon[(0.35, 0.19) (0.35, 0.53) (1.35, 0.53)
(1.35, 0.19) (2.05, 0.19) (2.05, 0.53)
(2.17, 0.53) (2.17, 4.30)
(-2.17, 4.30) (-2.17, 0.53)
(-2.05, 0.53) (-2.05, 0.19) (-1.35, 0.19)
(-1.35, 0.53) (-0.35, 0.53) (-0.35, 0.19)]
}
SiliconGermanium "SiGe" {rectangle[
(1.1, 0.15) (-1.1 0.19) ]}
Contact "Collector1" {line[(1.35, 0.19) (2.05, 0.19)]}
Contact "Collector2" {line[(-1.35, 0.19) (-2.05, 0.19)]}
Contact "Base1" {line[(0.36, 0.05) (1.1, 0.05)]}
Contact "Base2" {line[(-0.36, 0.05) (-1.1, 0.05)]}
Contact "Emitter" {line[(-0.06, 0.04) (0.06, 0.04)]}
Contact "Psubstrate" {line[(2.39, 4.3) (-2.39, 4.3)]}
CMD file
Title "BJT"
Definitions {
279
# Refinement regions
Refinement "all region"
{
MaxElementSize = (0.4 0.25)
MinElementSize = (0.2 0.05)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
}
Refinement "ccontact"
{
MaxElementSize = (0.15 0.1)
MinElementSize = (0.15 0.05)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
}
Refinement "cb1"
{
MaxElementSize = (0.05 0.02)
MinElementSize = (0.025 0.005)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
}
Refinement "sige"
{
MaxElementSize = (0.004 0.002)
MinElementSize = (0.002 0.001)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
}
Refinement "sige2"
{
MaxElementSize = (0.004 0.004)
MinElementSize = (0.002 0.002)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
}
Refinement "sige3"
{
MaxElementSize = (0.008 0.008)
MinElementSize = (0.004 0.004)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
}
Refinement "sige4"
{
MaxElementSize = (0.016 0.016)
MinElementSize = (0.008 0.008)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
280
}
Refinement "sige5"
{
MaxElementSize = (0.032 0.032)
MinElementSize = (0.016 0.016)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
}
Refinement "sige6"
{
MaxElementSize = (0.064 0.064)
MinElementSize = (0.032 0.032)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
}
Refinement "substrate region1"
{
MaxElementSize = (0.3 0.3)
MinElementSize = (0.15 0.15)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
}
Refinement "substrate region2"
{
MaxElementSize = (0.15 0.1)
MinElementSize = (0.075 0.04)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=1)
}
Refinement "Oxide_shallow"
{
MaxElementSize = (0.1 0.1)
MinElementSize = (0.05 0.01)
}
Refinement "Oxide_DT"
{
MaxElementSize = (0.2 0.2)
MinElementSize = (0.025 0.01)
}
Refinement "Oxide_spacer"
{
MaxElementSize = (0.015 0.01)
MinElementSize = (0.005 0.01)
}
Refinement "Emitter0"
{
281
MaxElementSize = (0.01 0.02)
MinElementSize = (0.002 0.01)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=0.1)
}
Refinement "Emitter1"
{
MaxElementSize = (0.02 0.02)
MinElementSize = (0.01 0.01)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=0.1)
}
Refinement "Emitter2"
{
MaxElementSize = (0.02 0.02)
MinElementSize = (0.02 0.01)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=0.1)
}
Refinement "Emitter3"
{
MaxElementSize = (0.08 0.04)
MinElementSize = (0.04 0.01)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=0.1)
}
# Profiles
Constant "psubstrate"
{
Species = "BoronActiveConcentration"
Value = 1e+15
}
Constant "n_epi"
{
Species = "PhosphorusActiveConcentration"
Value = 1e+16
}
AnalyticalProfile "collector"
{
Function = subMesh1D(datafile = "phos.dat"
, Scale = 1,
Range = line[(0 2.7940971e+15), (3.0 1.1610737e-195)]
)
LateralFunction = Erf(Factor = 0)
}
282
AnalyticalProfile "n_buri"
{
Function = subMesh1D(datafile = "asBuri.dat"
, Scale = 1,
Range = line[(0 2.2166889e+8), (3.0 2.4945547e+1)]
)
LateralFunction = Erf(Factor = 0)
}
AnalyticalProfile "emitter"
{
Function = subMesh1D(datafile = "as.dat"
, Scale = 1,
Range = line[(0 1e+21), (3.0 0)]
)
LateralFunction = Erf(Factor = 0)
}
AnalyticalProfile "intrinsic base"
{
Function = subMesh1D(datafile = "boron.dat"
, Scale = 1,
Range = line[(0 9.3631897e-224), (3.0 0)]
)
LateralFunction = Erf(Factor = 0)
}
Constant "cc"
{ Species = "ArsenicActiveConcentration"
Value = 1e21
}
Constant "extrinsic base"
{
Species = "BoronActiveConcentration"
Value = 5e20
}
AnalyticalProfile "xMoleBase"
{
Function = subMesh1D(datafile = "xmolg01.xy"
, Scale = 1,
Range = line[(0 0), (1.18 0)]
)
LateralFunction = Erf(Factor = 0)
}
}
283
Placements {
# Refinement regions
Refinement "all region"
{
Reference = "all region"
RefineWindow = rectangle [(-2.39 0), (2.39 4.30)]
}
Refinement "substrate region1"
{
Reference = "substrate region1"
RefineWindow = rectangle [(-2.17 2.30), (2.17 4.30)]
}
Refinement "base region1"
{
Reference = "cb1"
RefineWindow = polygon [(0.12 0.15), (0.12 0.05), (1.1 0.05),
(1.1 0.19), (-1.1 0.19), (-1.1 0.05),
(-0.12 0.05), (-0.12 0.15)]
}
Refinement "emitter region up"
{
Reference = "Emitter2"
RefineWindow = rectangle [(-0.06 0.04), (0.06 0.12)]
}
Refinement "emitter region middle"
{
Reference = "Emitter1"
RefineWindow = rectangle [(-0.06 0.12), (0.06 0.14)]
}
Refinement "emitter region down"
{
Reference = "Emitter0"
RefineWindow = rectangle [(-0.06 0.15), (0.06 0.14)]
}
Refinement "ccontact1"
{
Reference = "ccontact"
RefineWindow = rectangle [(-2.05 0.30)(-1.35 0.53)]
}
Refinement "ccontact2"
{
Reference = "ccontact"
284
RefineWindow = rectangle [(2.05 0.30)(1.35 0.53)]
}
Refinement "sige region6"
{
Reference = "sige6"
RefineWindow = rectangle [(0.5 0.5)(-0.5 0.6)]
}
Refinement "sige region5"
{
Reference = "sige5"
RefineWindow = rectangle [(0.4 0.15)(-0.4 0.50)]
}
Refinement "sige region4"
{
Reference = "sige4"
RefineWindow = rectangle [(0.36 0.15)(-0.36 0.4)]
}
Refinement "sige region3"
{
Reference = "sige3"
RefineWindow = rectangle [(0.14 0.15)(-0.14 0.3)]
}
Refinement "sige region2"
{
Reference = "sige2"
RefineWindow = rectangle [(0.12 0.15)(-0.12 0.21)]
}
Refinement "sige region"
{
Reference = "sige"
RefineWindow = rectangle [(0.08 0.15)(-0.08 0.19)]
}
Refinement "substrate region2"
{
Reference = "substrate region2"
RefineWindow = rectangle [(2.17 2.3), (-2.17 2.7)]
}
Refinement "spacer1"
{
Reference = "Oxide_spacer"
RefineWindow = polygon [(0.36, 0) (0.06, 0) (0.06 0.15) (0.12 0.15)
285
(0.12 0.05) (0.36 0)]
}
Refinement "spacer2"
{
Reference = "Oxide_spacer"
RefineWindow = polygon [(-0.36, 0) (-0.06, 0) (-0.06 0.15) (-0.12 0.15)
(-0.12 0.05) (-0.36 0)]
}
Refinement "ST1"
{
Reference = "Oxide_shallow"
RefineWindow = rectangle [(0.35, 0.19) (1.35, 0.53)]
}
Refinement "ST2"
{
Reference = "Oxide_shallow"
RefineWindow = rectangle [(-0.35, 0.19) (-1.35, 0.53)]
}
Refinement "DT1"
{
Reference = "Oxide_DT"
RefineWindow = polygon[(2.05, 0.19) (2.05, 0.53) (2.17, 0.53)
(2.17, 4.30) (2.39, 4.30) (2.39, 0.19)]
}
Refinement "DT2"
{
Reference = "Oxide_DT"
RefineWindow = polygon[(-2.05, 0.19) (-2.05, 0.53) (-2.17, 0.53)
(-2.17, 4.30) (-2.39, 4.30) (-2.39, 0.19)]
}
# Profiles
Constant "psubstrate instance"
{
Reference = "psubstrate"
EvaluateWindow
{
Element = rectangle [(2.17 2.30), (-2.17 4.30)]
DecayLength = 0
}
}
Constant "n_epi instance"
286
{
Reference = "n_epi"
EvaluateWindow
{
Element = polygon[(0.35, 0.19) (0.35, 0.53) (1.35, 0.53)
(1.35, 0.19) (2.05, 0.19) (2.05, 0.53)
(2.17, 0.53) (2.17, 2.30)
(-2.17, 2.30) (-2.17, 0.53)
(-2.05, 0.53) (-2.05, 0.19) (-1.35, 0.19)
(-1.35, 0.53) (-0.35, 0.53) (-0.35, 0.19)]
DecayLength = 0
}
}
AnalyticalProfile "collector instance"
{
Reference = "collector"
ReferenceElement
{
Element = line [(-0.12 0.04), (0.12 0.04)]
}
EvaluateWindow
{
Element = rectangle[(-0.12 0.04)(0.12 2.30)]
}
}
AnalyticalProfile "emitter instance"
{
Reference = "emitter"
ReferenceElement
{
Element = line [(-0.06 0.04), (0.06 0.04)]
}
EvaluateWindow
{
Element = rectangle[(-0.06, 0.53) (0.06, 0.04)]
DecayLength = 0
}
}
AnalyticalProfile "intrinsic base instance"
{
Reference = "intrinsic base"
ReferenceElement
287
{
Element = line [(-1.1 0.04), (1.1 0.04)]
}
EvaluateWindow
{
Element = rectangle[(-1.1 0.04), (1.1 0.53)]
}
}
Constant "extrinsic base left instance"
{
Reference = "extrinsic base"
EvaluateWindow
{
Element = rectangle [(-0.12 0.05), (-1.1 0.15)]
DecayLength = 0.005
}
}
Constant "extrinsic base right instance"
{
Reference = "extrinsic base"
EvaluateWindow
{
Element = rectangle [(0.12 0.05), (1.1 0.15)]
DecayLength = 0.005
}
}
AnalyticalProfile "n_buried layer instance"
{
Reference = "n_buri"
ReferenceElement
{
Element = line [(-2.17 0.04), (2.17 0.04)]
}
EvaluateWindow
{
Element = polygon[(0.35, 0.04) (0.35, 0.53) (1.35, 0.53)
(1.35, 0.19) (2.05, 0.19) (2.05, 0.53)
(2.17, 0.53) (2.17, 3)
(-2.17, 3) (-2.17, 0.53)
(-2.05, 0.53) (-2.05, 0.19) (-1.35, 0.19)
(-1.35, 0.53) (-0.35, 0.53) (-0.35, 0.04)]
DecayLength = 0
288
}
}
Constant "Collector contact instance left"
{
Reference = "cc"
EvaluateWindow
{
Element=rectangle[(-1.35 0.19)(-2.17 1)]
}
}
Constant "Collector contact instance right"
{
Reference = "cc"
EvaluateWindow
{
Element=rectangle[(1.35 0.19)(2.17 1)]
}
}
AnalyticalProfile "xMolBase instance"
{
Reference = "xMoleBase"
ReferenceElement
{
Element = line[(-1.1 0.04) (1.1 0.04)]
Direction = positive
}
EvaluateWindow
{
Element = rectangle[(-1.1 0.04) (1.1 0.19)]
}
}
}
B.2.2 Noise Simulation CMD file
Device BJT {
Electrode {
{ Name="Emitter" Voltage=0 }
{ Name="Base1" Voltage=0 }
{ Name = "Base2" Voltage = 0}
{ Name="Collector1" Voltage=0 }
289
{ Name="Collector2" Voltage=0 }
{ Name = "Psubstrate" Voltage = 0}
}
File {
Grid = "msh_msh.grd"
Doping = "msh_msh.dat"
Current = "achdet40g_des.plt"
Plot = "achdet40g_des.dat"
}
Physics{
Areafactor= 1
EffectiveIntrinsicDensity(BandgapNarrowing( Slotboom) )
Mobility(
PhuMob
Highfieldsaturation(CarrierTempDrive)
)
Fermi
Hydrodynamic(eTemp)
Noise ( DiffusionNoise(eTemperature) )
}
Physics (material = "Silicon") {
Recombination(
SRH( DopingDependence )
Auger
)
}
Physics (material = "PolySi") {
Recombination(
SRH( DopingDependence )
Auger
)
}
}
*----------------------------------------------------------------------*
*--End of Device{}
*----------------------------------------------------------------------*
290
Plot {
eDensity hDensity
TotalCurrent/Vector eCurrent/Vector hCurrent/Vector
ElectricField Potential SpaceCharge
Doping DonorConcentration AcceptorConcentration
SRH Auger
eQuasiFermi hQuasiFermi
eEparal hEparal
eMobility hMobility
eVelocity hVelocity
xMoleFraction
BandGap BandGapNarrowing
Affinity
ConductionBand ValenceBand
}
#NoisePlot {
# AllLNS AllLNVSD AllLNVXVSD GreenFunctions
#}
Math {
Extrapolate
NotDamped=200
Iterations=20
NewDiscretization
Derivatives
RelerrControl
Digits=6
}
File {
Output = "achdet40g"
ACExtract="achdet40g"
}
System {
291
BJT bjt (Base1=1 Base2 = 1 Collector1=2 Collector2=2 Emitter=0 Psubstrate=0)
Vsource_pset vb (1 0){ dc = 0 }
Vsource_pset vc (2 0){ dc = 0 }
}
Solve {
Coupled (Iterations=50) {Poisson }
Coupled { Poisson Electron Hole }
Coupled { Poisson Electron Hole ElectronTemperature}
Quasistationary (
InitialStep=0.025 Increment= 1.4
MinStep=1e-3 MaxStep=0.1
Goal {Parameter=vc.dc Voltage=1.75}
Goal {Parameter=vb.dc Voltage=0.75}
){
Coupled {Poisson Electron Hole ElectronTemperature}
}
save(fileprefix = "175hd")
newcurrent = "achdetbias"
load(fileprefix = "175hd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.76}
Goal {Parameter=vb.dc Voltage=0.76}
){
Coupled{Poisson Electron Hole ElectronTemperature}
}
save(fileprefix = "176hd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.77}
Goal {Parameter=vb.dc Voltage=0.77}
){
Coupled{Poisson Electron Hole ElectronTemperature}
}
save(fileprefix = "177hd")
Quasistationary (
InitialStep=1 Increment=1
292
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.78}
Goal {Parameter=vb.dc Voltage=0.78}
){
Coupled{Poisson Electron Hole ElectronTemperature}
}
save(fileprefix = "178hd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.79}
Goal {Parameter=vb.dc Voltage=0.79}
){
Coupled{Poisson Electron Hole ElectronTemperature}
}
save(fileprefix = "179hd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.80}
Goal {Parameter=vb.dc Voltage=0.80}
){
Coupled{Poisson Electron Hole ElectronTemperature}
}
save(fileprefix = "180hd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.81}
Goal {Parameter=vb.dc Voltage=0.81}
){
Coupled{Poisson Electron Hole ElectronTemperature}
}
save(fileprefix = "181hd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.82}
Goal {Parameter=vb.dc Voltage=0.82}
){
Coupled{Poisson Electron Hole ElectronTemperature}
}
293
save(fileprefix = "182hd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.83}
Goal {Parameter=vb.dc Voltage=0.83}
){
Coupled{Poisson Electron Hole ElectronTemperature}
}
save(fileprefix = "183hd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.84}
Goal {Parameter=vb.dc Voltage=0.84}
){
Coupled{Poisson Electron Hole ElectronTemperature}
}
save(fileprefix = "184hd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.85}
Goal {Parameter=vb.dc Voltage=0.85}
){
Coupled{Poisson Electron Hole ElectronTemperature}
}
save(fileprefix = "185hd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.86}
Goal {Parameter=vb.dc Voltage=0.86}
){
Coupled{Poisson Electron Hole ElectronTemperature}
}
save(fileprefix = "186hd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.87}
Goal {Parameter=vb.dc Voltage=0.87}
294
){
Coupled{Poisson Electron Hole ElectronTemperature}
}
save(fileprefix = "187hd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.88}
Goal {Parameter=vb.dc Voltage=0.88}
){
Coupled{Poisson Electron Hole ElectronTemperature}
}
save(fileprefix = "188hd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.89}
Goal {Parameter=vb.dc Voltage=0.89}
){
Coupled{Poisson Electron Hole ElectronTemperature}
}
save(fileprefix = "189hd")
Quasistationary (
InitialStep=1 Increment=1
MinStep=1e-3 MaxStep=1
Goal {Parameter=vc.dc Voltage=1.90}
Goal {Parameter=vb.dc Voltage=0.90}
){
Coupled{Poisson Electron Hole ElectronTemperature}
}
save(fileprefix = "190hd")
newcurrent = "achdet40g"
load(fileprefix = "175hd")
ACCoupled (
StartFrequency = 40e9 EndFrequency = 40e9
NumberofPoints = 1 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "achdet40g"
NoiseExtraction = "achdet40g"
NoisePlot = "achdet40g175"
295
)
{Poisson Electron Hole ElectronTemperature}
load(fileprefix = "176hd")
ACCoupled (
StartFrequency = 40e9 EndFrequency = 40e9
NumberofPoints = 1 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "achdet40g"
NoiseExtraction = "achdet40g"
NoisePlot = "achdet40g176"
)
{Poisson Electron Hole ElectronTemperature}
load(fileprefix = "177hd")
ACCoupled (
StartFrequency = 40e9 EndFrequency = 40e9
NumberofPoints = 1 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "achdet40g"
NoiseExtraction = "achdet40g"
NoisePlot = "achdet40g177"
)
{Poisson Electron Hole ElectronTemperature}
load(fileprefix = "178hd")
ACCoupled (
StartFrequency = 40e9 EndFrequency = 40e9
NumberofPoints = 1 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "achdet40g"
NoiseExtraction = "achdet40g"
NoisePlot = "achdet40g178"
)
{Poisson Electron Hole ElectronTemperature}
load(fileprefix = "179hd")
ACCoupled (
StartFrequency = 40e9 EndFrequency = 40e9
NumberofPoints = 1 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "achdet40g"
296
NoiseExtraction = "achdet40g"
NoisePlot = "achdet40g179"
)
{Poisson Electron Hole ElectronTemperature}
load(fileprefix = "180hd")
ACCoupled (
StartFrequency = 40e9 EndFrequency = 40e9
NumberofPoints = 1 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "achdet40g"
NoiseExtraction = "achdet40g"
NoisePlot = "achdet40g180"
)
{Poisson Electron Hole ElectronTemperature}
load(fileprefix = "181hd")
ACCoupled (
StartFrequency = 40e9 EndFrequency = 40e9
NumberofPoints = 1 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "achdet40g"
NoiseExtraction = "achdet40g"
NoisePlot = "achdet40g181"
)
{Poisson Electron Hole ElectronTemperature}
load(fileprefix = "182hd")
ACCoupled (
StartFrequency = 40e9 EndFrequency = 40e9
NumberofPoints = 1 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "achdet40g"
NoiseExtraction = "achdet40g"
NoisePlot = "achdet40g182"
)
{Poisson Electron Hole ElectronTemperature}
load(fileprefix = "183hd")
ACCoupled (
StartFrequency = 40e9 EndFrequency = 40e9
NumberofPoints = 1 linear
Node(1 2) Exclude(vb vc)
297
ObservationNode(1 2)
ACExtraction = "achdet40g"
NoiseExtraction = "achdet40g"
NoisePlot = "achdet40g183"
)
{Poisson Electron Hole ElectronTemperature}
load(fileprefix = "184hd")
ACCoupled (
StartFrequency = 40e9 EndFrequency = 40e9
NumberofPoints = 1 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "achdet40g"
NoiseExtraction = "achdet40g"
NoisePlot = "achdet40g184"
)
{Poisson Electron Hole ElectronTemperature}
load(fileprefix = "185hd")
ACCoupled (
StartFrequency = 40e9 EndFrequency = 40e9
NumberofPoints = 1 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "achdet40g"
NoiseExtraction = "achdet40g"
NoisePlot = "achdet40g185"
)
{Poisson Electron Hole ElectronTemperature}
load(fileprefix = "186hd")
ACCoupled (
StartFrequency = 40e9 EndFrequency = 40e9
NumberofPoints = 1 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "achdet40g"
NoiseExtraction = "achdet40g"
NoisePlot = "achdet40g186"
)
{Poisson Electron Hole ElectronTemperature}
load(fileprefix = "187hd")
ACCoupled (
StartFrequency = 40e9 EndFrequency = 40e9
298
NumberofPoints = 1 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "achdet40g"
NoiseExtraction = "achdet40g"
NoisePlot = "achdet40g187"
)
{Poisson Electron Hole ElectronTemperature}
load(fileprefix = "188hd")
ACCoupled (
StartFrequency = 40e9 EndFrequency = 40e9
NumberofPoints = 1 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "achdet40g"
NoiseExtraction = "achdet40g"
NoisePlot = "achdet40g188"
)
{Poisson Electron Hole ElectronTemperature}
load(fileprefix = "189hd")
ACCoupled (
StartFrequency = 40e9 EndFrequency = 40e9
NumberofPoints = 1 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "achdet40g"
NoiseExtraction = "achdet40g"
NoisePlot = "achdet40g189"
)
{Poisson Electron Hole ElectronTemperature}
load(fileprefix = "190hd")
ACCoupled (
StartFrequency = 40e9 EndFrequency = 40e9
NumberofPoints = 1 linear
Node(1 2) Exclude(vb vc)
ObservationNode(1 2)
ACExtraction = "achdet40g"
NoiseExtraction = "achdet40g"
NoisePlot = "achdet40g190"
)
{Poisson Electron Hole ElectronTemperature}
}
299
B.3 MATLAB Programming for Simulation Results
This is MATLAB Programming for 8HP DESSIS simulation results. The MATLAB pro-
gramming is similar for 5HP SiGe HBT DESSIS simulation results.
B.3.1 Main file
close all; clear all; clc;
q = 1.6e-19;
kt = 0.0259*q;
datapath = ?D:\Yan\research\8hp\noisedata?;
cd(datapath);
filename = {?hdetall?,?hd2etall?,?g05hdetall?};
legname = {?design I?, ?design II?, ?design III?};
x1 = 20;
fileNumber=length(filename);
datasel =1; %1: bias dependence, 2: frequency dependence
for filsel = [1:3],
load(filename{filsel});
rbrange = (num_of_freq-5):num_of_freq;
Jc = Ic./0.12.*1e3; Jb = Ib./0.12.*1e3;
nx = x1;
for n = nx;
switch datasel
case 1 %bias dependence
sv12x = conj(sv12); sv12eex = conj(sv12ee); sv12hhx = conj(sv12hh);
SV = [sv1(:,n) sv12x(:,n) conj(sv12x(:,n)) sv2(:,n)];
SVee = [sv1ee(:,n) sv12eex(:,n) conj(sv12eex(:,n)) sv2ee(:,n)];
SVhh = [sv1hh(:,n) sv12hhx(:,n) conj(sv12hhx(:,n)) sv2hh(:,n)];
Y = [Y11(:,n) Y12(:,n) Y21(:,n) Y22(:,n)]; Z = z_from_Y(Y);
for x = 1: num_of_bias,
Y11f = Y11(x,:); Y12f = Y12(x,:);Y21f = Y21(x,:);Y22f = Y22(x,:);
h11f = 1./Y11f;
Yf = [conj(Y11f?) conj(Y12f?) conj(Y21f?) conj(Y22f?)];
Zf = Z_from_Y(Yf);
Z11f = Zf(:,1); Z12f = Zf(:,2); Z21f = Zf(:,3); Z22f = Zf(:,4);
rbh(x) = rb_from_h11(h11f(rbrange));
rb(x) = rbh(x); re(x) = 0; rc(x) = 0;
end
numend = num_of_bias;
case 2 %frequency dependence
300
sv12x = conj(sv12); sv12eex = conj(sv12ee); sv12hhx = conj(sv12hh);
SV = [conj(sv1(n,:)?) conj(sv12x(n,:)?) sv12x(n,:)? conj(sv2(n,:)?)];
SVee = [conj(sv1ee(n,:)?) conj(sv12eex(n,:)?) ...
sv12eex(n,:)? conj(sv2ee(n,:)?)];
SVhh = [conj(sv1hh(n,:)?) conj(sv12hhx(n,:)?) ...
sv12hhx(n,:)? conj(sv2hh(n,:)?)];
Y = [conj(Y11(n,:)?) conj(Y12(n,:)?) conj(Y21(n,:)?) conj(Y22(n,:)?)];
Z = z_from_Y(Y);
h11 = 1./Y(:,1); rbh(n) = rb_from_h11(h11(rbrange));
Z11f = Z(:,1); Z12f = Z(:,2); Z21f = Z(:,3); Z22f = Z(:,4);
rb(n) = rbh(n); re(n) = 0; rc(n) = 0;
numend = num_of_freq;
end
%----------------------------------------------------------------------------
for x = 1:numend,
y = Y(x,:); z = Z(x,:); a = a_from_y(y);
cz = 0.5.*SV(x,:); ca = c_from_z_to_a(cz, a); cy = c_from_a_to_y(ca, y);
nf = nf_from_ca(ca, 50);
svb(x) = 2*cz(1); svc(x) = 2*cz(4);
svbvcr(x) = 2*real(cz(2)); svbvci(x) = 2*imag(cz(2));
cvbvcr(x) = svbvcr(x)/sqrt(svc(x)*svb(x));
cvbvci(x) = svbvci(x)/sqrt(svc(x)*svb(x));
sva(x) = 2*ca(1); sia(x) = 2*ca(4);
siavar(x) = 2*real(ca(3)); siavai(x) = 2*imag(ca(3));
ciavar(x) = siavar(x)/sqrt(sia(x)*sva(x));
ciavai(x) = siavai(x)/sqrt(sia(x)*sva(x));
sib(x) = 2*cy(1); sic(x) = 2*cy(4);
sicibr(x) = 2*real(cy(3)); sicibi(x) = 2*imag(cy(3));
cicibr(x) = sicibr(x)/sqrt(sib(x)*sic(x));
cicibi(x) = sicibi(x)/sqrt(sib(x)*sic(x));
nfmin(x) = nf(1); rn(x) = nf(2); Yopt(x) = nf(3);
czee = 0.5.*SVee(x,:); caee = c_from_z_to_a(czee, a);
cyee = c_from_a_to_y(caee, y); nfee = nf_from_ca(caee, 50);
svbee(x) = 2*czee(1); svcee(x) = 2*czee(4);
svbvcree(x) = 2*real(czee(2)); svbvciee(x) = 2*imag(czee(2));
cvbvcree(x) = svbvcree(x)/sqrt(svcee(x)*svbee(x));
cvbvciee(x) = svbvciee(x)/sqrt(svcee(x)*svbee(x));
svaee(x) = 2*caee(1); siaee(x) = 2*caee(4);
siavaree(x) = 2*real(caee(3)); siavaiee(x) = 2*imag(caee(3));
ciavaree(x) = siavaree(x)/sqrt(siaee(x)*svaee(x));
ciavaiee(x) = siavaiee(x)/sqrt(siaee(x)*svaee(x));
301
sibee(x) = 2*cyee(1); sicee(x) = 2*cyee(4);
sicibree(x) = 2*real(cyee(3)); sicibiee(x) = 2*imag(cyee(3));
cicibree(x) = sicibree(x)/sqrt(sibee(x)*sicee(x));
cicibiee(x) = sicibiee(x)/sqrt(sibee(x)*sicee(x));
nfminee(x) = nfee(1); rnee(x) = nfee(2); Yoptee(x) = nfee(3);
czhh = 0.5.*SVhh(x,:); cahh = c_from_z_to_a(czhh, a);
cyhh = c_from_a_to_y(cahh, y); nfhh = nf_from_ca(cahh, 50);
svbhh(x) = 2*czhh(1); svchh(x) = 2*czhh(4);
svbvcrhh(x) = 2*real(czhh(2)); svbvcihh(x) = 2*imag(czhh(2));
cvbvcrhh(x) = svbvcrhh(x)/sqrt(svchh(x)*svbhh(x));
cvbvcihh(x) = svbvcihh(x)/sqrt(svchh(x)*svbhh(x));
svahh(x) = 2*cahh(1); siahh(x) = 2*cahh(4);
siavarhh(x) = 2*real(cahh(3)); siavaihh(x) = 2*imag(cahh(3));
ciavarhh(x) = siavarhh(x)/sqrt(siahh(x)*svahh(x));
ciavaihh(x) = siavaihh(x)/sqrt(siahh(x)*svahh(x));
sibhh(x) = 2*cyhh(1); sichh(x) = 2*cyhh(4);
sicibrhh(x) = 2*real(cyhh(3)); sicibihh(x) = 2*imag(cyhh(3));
cicibrhh(x) = sicibrhh(x)/sqrt(sibhh(x)*sichh(x));
cicibihh(x) = sicibihh(x)/sqrt(sibhh(x)*sichh(x));
nfminhh(x) = nfhh(1); rnhh(x) = nfhh(2); Yopthh(x) = nfhh(3);
end
for x = 1:numend,
y = Y(x,:); z = Z(x,:);, a = a_from_y(y);
switch datasel
case 1
rbx = rb(x); Ibx = Ib(x); Icx = Ic(x);
rex = re(x); rcx = rc(x);
case 2
rbx = rb(n); Ibx = Ib(n); Icx = Ic(n);
rex = re(n); rcx = rc(n);
end
zb = [rbx+rex rex rex rex+rcx]; czb = 2*kt.*zb;
zi = z-zb; yi = y_from_z(zi); ai = a_from_y(yi);
if x ==1, yix = yi(3); end
cz = 0.5.*SV(x,:); ca = c_from_z_to_a(cz, a);
cy = c_from_a_to_y(ca, y);
czi = cz - czb; cai = c_from_z_to_a(czi, ai);
cyi = c_from_a_to_y(cai, yi);
sibi(x) = 2*cyi(1); sici(x) = 2*cyi(4);
sicibri(x) = 2*real(cyi(3)); sicibii(x) = 2*imag(cyi(3));
cicibri(x) = sicibri(x)./sqrt(sibi(x).*sici(x));
302
cicibii(x) = sicibii(x)./sqrt(sibi(x).*sici(x));
czhh = 0.5.*SVhh(x,:); cahh = c_from_z_to_a(czhh, a);
cyhh = c_from_a_to_y(cahh, y);
czihh = czhh - czb; caihh = c_from_z_to_a(czihh, ai);
cyihh = c_from_a_to_y(caihh, yi);
sibihh(x) = 2*cyihh(1); sicihh(x) = 2*cyihh(4);
sicibrihh(x) = 2*real(cyihh(3)); sicibiihh(x) = 2*imag(cyihh(3));
cicibrihh(x) = sicibrihh(x)./sqrt(sibihh(x).*sicihh(x));
cicibiihh(x) = sicibiihh(x)./sqrt(sibihh(x).*sicihh(x));
czee = 0.5.*SVee(x,:); caee = c_from_z_to_a(czee, a);
cyee = c_from_a_to_y(caee, y);
cziee = czee; caiee = c_from_z_to_a(cziee, ai);
cyiee = c_from_a_to_y(caiee, yi);
sibiee(x) = 2*cyiee(1); siciee(x) = 2*cyiee(4);
sicibriee(x) = 2*real(cyiee(3)); sicibiiee(x) = 2*imag(cyiee(3));
cicibriee(x) = sicibriee(x)./sqrt(sibiee(x).*siciee(x));
cicibiiee(x) = sicibiiee(x)./sqrt(sibiee(x).*siciee(x));
sibs(x) = 2*q*Ibx; sics(x) = 2*q*Icx; sicibrs(x) = 0; sicibis(x) = 0;
cysi = 0.5*[sibs(x), sicibrs(x) - j*sicibis(x), ...
sicibrs(x) + j*sicibis(x), sics(x)];
casi = c_from_y_to_a(cysi, ai);
czsi = c_from_a_to_z(casi, zi); czs = czsi + czb;
cas = c_from_z_to_a(czs, a); nfs = nf_from_ca(cas, 50);
svas(x) = 2*cas(1); sias(x) = 2*cas(4);
siavars(x) = real(2*cas(3)); siavais(x) = imag(2*cas(3));
nfmins(x) = nfs(1); rns(x) = nfs(2); Yopts(x) = nfs(3);
sibv(x) = 4*kt*real(yi(1)) - 2*q*Ibx;
sicv(x) = 4*kt*real(yi(4)) + 2*q*Icx;
sicibrv(x) = 2*kt*real(yi(3)+y(2)?-yix);
sicibiv(x) = 2*kt*imag(yi(3)+y(2)?);
cyvi = 0.5*[sibv(x), sicibrv(x) - j*sicibiv(x), ...
sicibrv(x) + j*sicibiv(x), sicv(x)];
cavi = c_from_y_to_a(cyvi, ai); czvi = c_from_a_to_z(cavi, zi);
czv = czvi + czb;
cav = c_from_z_to_a(czv, a); nfv = nf_from_ca(cav, 50);
svav(x) = 2*cav(1); siav(x) = 2*cav(4);
siavarv(x) = real(2*cav(3)); siavaiv(x) = imag(2*cav(3));
nfminv(x) = nfv(1); rnv(x) = nfv(2); Yoptv(x) = nfv(3);
303
end
end
end
B.3.2 Z_from_Y.m
function Z = Z_from_Y(Y)
%Z = Z_from_Y(Y)
z0 = 50;
Y11 = Y(:,1);
Y12 = Y(:,2);
Y21 = Y(:,3);
Y22 = Y(:,4);
Y_delta = Y11.*Y22 - Y12.*Y21;
Z11 = Y22./Y_delta;
Z12 = -Y12./Y_delta;
Z21 = -Y21./Y_delta;
Z22 = Y11./Y_delta;
Z = [Z11, Z12, Z21, Z22];
B.3.3 rb_from_h11.m
function rb=rb_from_h11(h11)
%rb=rb_from_h11(h11)
rb=circle(h11);
B.3.4 circle.m
function rb=circle(h11)
%rb=circle(h11)
ydata=imag(h11); ydata=ydata(:);
xdata=real(h11); xdata=xdata(:);
[ymin, y_ind]=min(ydata);
nsize=size(ydata);
para0=[xdata(y_ind), abs(ymin)];
newPara=fminsearch(?myCostFunc?, para0,[],[xdata ydata])
rb=newPara(1)-newPara(2);
B.3.5 myCostFunc.m
function cost=myCostFunc(para, data)
%para(1) is x0, para(2) is r
cost=sum((sqrt(data(:,2).^2+(data(:,1)-para(1)).^2)-para(2)).^2);
304
B.3.6 c_from_z_to_a.m
function C_A = C_from_Z_to_A(C_Z, A)
%C_A = C_from_Z_to_A(C_Z, A)
k = size(A, 1);
for i = 1:k;
CZ = [C_Z(i,1), C_Z(i,2); C_Z(i,3), C_Z(i,4)];
A_temp = [A(i,1), A(i,2); A(i,3), A(i,4)];
Trans = [1, -A_temp(1,1); 0, -A_temp(2,1)];
Trans_conj_trans = [Trans(1,1)?, Trans(2,1)?; Trans(1,2)?, Trans(2,2)?];
CA = Trans*CZ*Trans_conj_trans;
C_A(i,1) = (abs(Trans(1,1)))^2*CZ(1,1) + (abs(Trans(1,2)))^2*CZ(2,2)...
+ 2*real(Trans_conj_trans(1,1)*Trans(1,2)*CZ(2,1));
C_A(i,2) = Trans(1,1)*Trans_conj_trans(1,2)*CZ(1,1)...
+Trans(1,2)*Trans_conj_trans(1,2)*CZ(2,1)...
+Trans(1,1)*Trans_conj_trans(2,2)*CZ(1,2)...
+Trans(1,2)*Trans_conj_trans(2,2)*CZ(2,2);
C_A(i,3) = C_A(i,2)?;
C_A(i,4) = (abs(Trans(2,1)))^2*CZ(1,1) + (abs(Trans(2,2)))^2*CZ(2,2)...
+ 2*real(Trans_conj_trans(2,2)*Trans(2,1)*CZ(1,2));
end
B.3.7 c_from_a_to_y.m
function C_Y = C_from_A_to_Y(C_A, Y)
%C_Y = C_from_A_to_Y(C_A, Y)
k = size(Y, 1);
for i = 1:k;
CA = [C_A(i,1), C_A(i,2); C_A(i,3), C_A(i,4)];
Y_temp = [Y(i,1), Y(i,2); Y(i,3), Y(i,4)];
Trans = [-Y_temp(1,1),1; -Y_temp(2,1),0];
Trans_conj_trans = [Trans(1,1)?, Trans(2,1)?; Trans(1,2)?, Trans(2,2)?];
CY = Trans*CA*Trans_conj_trans;
C_Y(i,1) = (abs(Trans(1,1)))^2*CA(1,1) + (abs(Trans(1,2)))^2*CA(2,2)...
+ 2*real(Trans_conj_trans(1,1)*Trans(1,2)*CA(2,1));
C_Y(i,2) = Trans(1,1)*Trans_conj_trans(1,2)*CA(1,1)...
+Trans(1,2)*Trans_conj_trans(1,2)*CA(2,1)...
+Trans(1,1)*Trans_conj_trans(2,2)*CA(1,2)...
+Trans(1,2)*Trans_conj_trans(2,2)*CA(2,2);
C_Y(i,3) = C_Y(i,2)?;
C_Y(i,4) = (abs(Trans(2,1)))^2*CA(1,1) + (abs(Trans(2,2)))^2*CA(2,2)...
+ 2*real(Trans_conj_trans(2,2)*Trans(2,1)*CA(1,2));
305
end
B.3.8 nf_from_ca.m
%function nf = nf_from_ca(ca,Z0);
function nf = nf_from_ca(ca,Z0);
k=1.38066e-023;
T=300;
kt = k*T;
sia = 2*ca(:,4);
siava = 2*ca(:,3);
sva = 2*ca(:,1);
gva1 = 4*kt/sva;
rn1 = 1/gva1/Z0;
gia1 = sia/(4*kt);
yc1 = siava/sva;
gc1 = real(yc1);
bc1 = imag(yc1);
gso1 = sqrt(gva1*gia1-bc1^2);
bso1 = -bc1;
yopt1 = gso1+j*bso1;
gammaopt1 = (1-yopt1*Z0)/(1+yopt1*50);
fmin1 = 1+2*(gso1+gc1)/gva1;
nfmin1 = 10*log10(fmin1);
nf = [nfmin1 rn1 yopt1];
B.3.9 y_from_z.m
function Y = Y_from_Z(Z)
%Y = Y_from_Z(Z)
Z11 = Z(:,1);
Z12 = Z(:,2);
Z21 = Z(:,3);
Z22 = Z(:,4);
delta = Z11.*Z22 - Z12.*Z21;
Y11 = Z22./delta;
Y12 = -Z12./delta;
Y21 = -Z21./delta;
Y22 = Z11./delta;
Y = [Y11, Y12, Y21, Y22];
306
B.3.10 a_from_y.m
function A = A_from_Y(Y);
%from Y parameter to ABCD = [A B C D], A = A_from_Y(Y)
z0 = 50;
Y11 = Y(:,1);
Y12 = Y(:,2);
Y21 = Y(:,3);
Y22 = Y(:,4);
Y_delta = Y11.*Y22 - Y12.*Y21;
A11 = -Y22./Y21;
A12 = -1./Y21;
A21 = -Y_delta./Y21;
A22 = -Y11./Y21;
A = [A11, A12, A21, A22];
307
APPENDIX C
DESSIS INPUT DECK AND MATLAB PROGRAMMING FOR 50 NM Le MOSFET NOISE
SIMULATION
C.1 Mesh files
C.1.1 BND file
Oxide "leftox" {rectangle[(-0.081,-0.15 ) (-0.025, 0)]}
PolySi "gatepoly" {rectangle[(-0.025, -0.001) (0.025, -0.15)]}
Oxide "rightox" {rectangle[(0.025, -0.15) (0.081, 0)]}
Oxide "gateox" {rectangle[(-0.025, 0) (0.025, -0.001)]}
Silicon "chanelsi" {rectangle[(-0.525, 0) (0.525, 1)]}
Contact "drain" {line[(0.081, 0) (0.525, 0)]}
Contact "gate" {line[(-0.022, -0.15) (0.022, -0.15)]}
Contact "source" {line[(-0.525, 0) (-0.081, 0)]}
Contact "bulk" {line[(-0.525, 1) (0.525, 1)]}
C.1.2 CMD file
Title "nmos"
Definitions {
Refinement "all region"
{
MaxElementSize = (0.05 0.1)
MinElementSize = (0.0025 0.01)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=0.1)
}
Refinement "oxide"
{
MaxElementSize = (0.04 0.04)
MinElementSize = (0.005 0.01)
}
Refinement "source"
{
MaxElementSize = (0.05 0.005)
MinElementSize = (0.025 0.0025)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=0.01)
}
Refinement "source1"
{
308
MaxElementSize = (0.1 0.01)
MinElementSize = (0.05 0.005)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=0.1)
}
Refinement "gate"
{
MaxElementSize = (0.04 0.04)
MinElementSize = (0.005 0.01)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=0.1)
}
Refinement "gateoxide"
{
MaxElementSize = (0.001 0.00025)
MinElementSize = (0.001 0.00025)
}
Refinement "drain"
{
MaxElementSize = (0.0025 0.005)
MinElementSize = (0.001 0.001)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=0.001)
}
Refinement "refine"
{
MaxElementSize = (0.001 0.001)
MinElementSize = (0.0005 0.0005)
RefineFunction = MaxTransDiff(Variable="DopingConcentration" Value=0.001)
}
Refinement "interface"
{
MaxElementSize = (0.01 0.005)
MinElementSize = (0.0025 0.005)
}
# Profiles
Constant "bulkboron"
{
Species = "BoronActiveConcentration"
Value =1e+15
}
Constant "bulkarsen"
{
309
Species = "ArsenicActiveConcentration"
Value =1e+5
}
Constant "npoly"
{
Species = "ArsenicActiveConcentration"
Value =1e+21
}
Constant "channeln"
{
Species = "ArsenicActiveConcentration"
Value =1e+12
}
Constant "channelp"
{
Species = "BoronActiveConcentration"
Value =1e17
}
Constant "channelp2"
{
Species = "BoronActiveConcentration"
Value =1e+18
}
AnalyticalProfile "bulkn"
{
Species = "ArsenicActiveConcentration"
Function = gauss(peakpos=0, PeakVal =1.5e21,
ValatDepth = 1e20,
depth = 0.015
)
lateralfunction = gauss(standarddeviation = 0.002)
}
AnalyticalProfile "bulkn1"
{
Species = "ArsenicActiveConcentration"
Function = gauss(peakpos=0, PeakVal =1.5e21,
ValatDepth = 1e20,
depth = 0.043
)
lateralfunction = gauss(standarddeviation = 0.0095) #0.00613758)
}
AnalyticalProfile "bulkpg"
310
{
Species = "BoronActiveConcentration"
Function = gauss(peakpos=0, PeakVal =1e19,
ValatDepth = 1e18,
depth = 0.015
)
lateralfunction = gauss(standarddeviation = 0.002)
}
AnalyticalProfile "bulkpg1"
{
Species = "BoronActiveConcentration"
Function = gauss(peakpos=0, PeakVal =1e19,
ValatDepth = 1e18,
depth = 0.043
)
lateralfunction = gauss(standarddeviation = 0.0095) #0.00613758)
}
AnalyticalProfile "bulkp"
{
Species = "BoronActiveConcentration"
Function = gauss(peakpos=0, PeakVal=6.5e+18,
ValatDepth = 3e18
depth = 0.005700
)
lateralfunction = gauss(standarddeviation = 0.006)
}
AnalyticalProfile "bulkp2"
{
Species = "BoronActiveConcentration"
Function = gauss(peakpos=0, PeakVal =1e18,
ValatDepth = 1e15,
depth = 0.4
)
lateralfunction = gauss(standarddeviation = 0.002)
}
AnalyticalProfile "bulkp1"
{
Species = "BoronActiveConcentration"
Function = gauss(peakpos=0, PeakVal=1.35e18,
ValatDepth = 1.2e18
depth = 0.030
)
311
lateralfunction = gauss(standarddeviation = 0.06)
}
}
Placements {
# Refinement regions
Refinement "all region"
{
Reference = "all region"
RefineWindow = rectangle [(-0.525 -0.15), (0.525 1)]
}
Refinement "leftoxide"
{
Reference = "oxide"
RefineWindow = rectangle [(-0.081 0), (-0.025 -0.15)]
}
Refinement "rightoxide"
{
Reference = "oxide"
RefineWindow = rectangle [(0.025 0), (0.081 -0.15)]
}
Refinement "source1 instant"
{
Reference = "source1"
RefineWindow = rectangle [(-0.525 0), (0.525 0.045)]
}
Refinement "source instant"
{
Reference = "source"
RefineWindow = rectangle [(-0.102 0), (0.102 0.045)]
}
Refinement "gate"
{
Reference = "gate"
RefineWindow = rectangle [(-0.025 -0.15) (0.025 -0.001)]
}
Refinement "gaterefine"
{
Reference = "gateoxide"
RefineWindow = rectangle [(-0.025 0) (0.025 -0.001)]
}
312
Refinement "undergate"
{
Reference = "drain"
RefineWindow = rectangle [(-0.052 0) (0.052 0.045)]
}
Refinement "interface1"
{
Reference = "interface"
RefineWindow = rectangle [(-0.525 0.044), (0.525 0.047)]
}
Refinement "interface2"
{
Reference = "interface"
RefineWindow = rectangle [(-0.081 0.001), (0.081 -0.002)]
}
Refinement "underrefine"
{
Reference = "refine"
RefineWindow = rectangle [(-0.025 0) (0.025 0.025)]
}
# Profiles
Constant "bulkarsen instance"
{
Reference = "bulkarsen"
EvaluateWindow
{
Element = rectangle [(-0.525 0), (0.525 1)]
DecayLength = 0
}
}
Constant "bulkboron instance"
{
Reference = "bulkboron"
EvaluateWindow
{
Element = rectangle [(-0.525 0), (0.525 1)]
DecayLength = 0
}
}
Constant "channelboron instance"
{
Reference = "channelp"
313
EvaluateWindow
{
Element = rectangle [(-0.525 0.045), (0.525 0.05)]
# direction = positive
DecayLength = 0.20
}
}
Constant "npoly instance"
{
Reference = "npoly"
EvaluateWindow
{
Element = rectangle [(-0.025 -0.15), (0.025 -0.001)]
DecayLength = 0
}
}
AnalyticalProfile "sourcen"
{
Reference = "bulkn"
ReferenceElement
{
Element = line[(-0.525 0) (-0.025 0)]
Direction =positive
}
EvaluateWindow
{
Element = rectangle[(-0.525 0) (0 0.045)
]
}
}
AnalyticalProfile "drain"
{
Reference = "bulkn"
ReferenceElement
{
Element = line[(0.025 0) (0.525 0)]
Direction =positive
}
EvaluateWindow
{
Element = rectangle[(0 0) (0.525 0.045)
]
314
}
}
AnalyticalProfile "sourcep"
{
Reference = "bulkpg"
ReferenceElement
{
Element = line[(-0.525 0) (-0.025 0)]
Direction =positive
}
EvaluateWindow
{
Element = rectangle[(-0.525 0) (0 0.045)
]
}
}
AnalyticalProfile "drainp"
{
Reference = "bulkpg"
ReferenceElement
{
Element = line[(0.025 0) (0.525 0)]
Direction =positive
}
EvaluateWindow
{
Element = rectangle[(0 0) (0.525 0.045)
]
}
}
AnalyticalProfile "sourcenl"
{
Reference = "bulkn1"
ReferenceElement
{
Element = line[(-0.525 0) (-0.081 0)]
Direction =positive
}
EvaluateWindow
{
Element = rectangle[(-0.525 0)
(-0.025 0.045) ]
315
}
}
AnalyticalProfile "drainl"
{
Reference = "bulkn1"
ReferenceElement
{
Element = line[(0.081 0) (0.525 0)]
Direction =positive
}
EvaluateWindow
{
Element = rectangle[(0.025 0) (0.525 0.045)
]
}
}
AnalyticalProfile "sourcepl"
{
Reference = "bulkpg1"
ReferenceElement
{
Element = line[(-0.525 0) (-0.081 0)]
Direction =positive
}
EvaluateWindow
{
Element = rectangle[(-0.525 0)
(-0.025 0.045) ]
}
}
AnalyticalProfile "draipl"
{
Reference = "bulkpg1"
ReferenceElement
{
Element = line[(0.081 0) (0.525 0)]
Direction =positive
}
EvaluateWindow
{
Element = rectangle[(0.025 0) (0.525 0.045)
]
316
}
}
AnalyticalProfile "undergateboron1"
{
Reference = "bulkp"
ReferenceElement
{
Element = line[(-0.008 0) (-0.006 0)]
Direction =positive
}
EvaluateWindow
{
Element = rectangle[(-0.525 0)(0.525 1)
]
}
}
AnalyticalProfile "undergateboron2"
{
Reference = "bulkp"
ReferenceElement
{
Element = line[(0.006 0) (0.008 0)]
Direction =positive
}
EvaluateWindow
{
Element = rectangle[(-0.525 0)(0.525 1)
]
}
}
AnalyticalProfile "undergateboron1_1"
{
Reference = "bulkp1"
ReferenceElement
{
Element = line[(-0.025 0.045) (-0.024 0.045)]
}
EvaluateWindow
{
Element = rectangle[(-0.525 0)(0.525 1)
]
}
317
}
AnalyticalProfile "undergateboron2_1"
{
Reference = "bulkp1"
ReferenceElement
{
Element = line[(0.024 0.045) (0.025 0.045)]
}
EvaluateWindow
{
Element = rectangle[(-0.525 0)(0.525 1)
]
}
}
}
C.2 Noise Simulation CMD file
Device nmos {
Electrode {
{ Name="drain" Voltage=0 }
{ Name="source" Voltage=0 }
{ Name = "gate" Voltage = 0 }
{ Name="bulk" Voltage=0 }
}
File {
Grid = "msh_msh.grd"
Doping = "msh_msh.dat"
Current = "noiseqmhdetvdswp_des.plt"
Plot = "noiseqmhdetvdswp_des.dat"
}
Physics{
Areafactor= 1
EffectiveIntrinsicDensity( Slotboom )
Hydrodynamic(eTemp)
Mobility(
dopingdependence(Masetti)
enormal(Lombardi)
318
Highfieldsaturation(CarrierTempDrive)
)
eQCvanDort
Fermi
Noise ( DiffusionNoise ( eTemperature ))
}
}
*----------------------------------------------------------------------*
*--End of Device{}
*----------------------------------------------------------------------*
Plot {
eDensity hDensity
TotalCurrent/Vector eCurrent/Vector hCurrent/Vector
ElectricField Potential SpaceCharge
Doping DonorConcentration AcceptorConcentration
SRH Auger
eQuasiFermi hQuasiFermi
eEparal hEparal
eMobility hMobility
eVelocity hVelocity
xMoleFraction
BandGap BandGapNarrowing
Affinity
ConductionBand ValenceBand
}
Math {
Extrapolate
NotDamped=200
Iterations=20
NewDiscretization
Derivatives
RelerrControl
Digits=6
319
}
File {
Output = "noiseqmhdetvdswp"
ACExtract="noiseqmhdetvdswp"
}
System {
nmos NMOS (drain=2 gate=1 source=0 bulk=0)
Vsource_pset vg (1 0){ dc = 0 }
Vsource_pset vd (2 0){ dc = 0 }
}
Solve {
Coupled (Iterations=50) {poisson}
Coupled { poisson Electron }
Coupled {poisson Electron ElectronTemperature}
Quasistationary (
initialstep = 0.2 MinStep=1e-1 MaxStep=1
Goal {Parameter=vd.dc Voltage=0}
Goal {Parameter=vg.dc Voltage=0.1}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg0.1acqmhd")
load(fileprefix = "vd0vg0.1acqmhd")
Quasistationary (
initialstep = 0.5 MinStep=1e-1 MaxStep=1
Goal {Parameter=vg.dc Voltage=0.2}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg0.2acqmhd")
Quasistationary (
initialstep = 0.5 MinStep=1e-1 MaxStep=1
Goal {Parameter=vg.dc Voltage=0.3}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg0.3acqmhd")
Quasistationary (
320
initialstep = 0.5 MinStep=1e-1 MaxStep=1
Goal {Parameter=vg.dc Voltage=0.4}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg0.4acqmhd")
Quasistationary (
initialstep = 0.5 MinStep=1e-1 MaxStep=1
Goal {Parameter=vg.dc Voltage=0.5}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg0.5acqmhd")
Quasistationary (
initialstep = 0.5 MinStep=1e-1 MaxStep=1
Goal {Parameter=vg.dc Voltage=0.6}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg0.6acqmhd")
Quasistationary (
initialstep = 0.5 MinStep=1e-1 MaxStep=1
Goal {Parameter=vg.dc Voltage=0.7}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg0.7acqmhd")
Quasistationary (
initialstep = 0.5 MinStep=1e-1 MaxStep=1
Goal {Parameter=vg.dc Voltage=0.8}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg0.8acqmhd")
Quasistationary (
initialstep = 0.5 MinStep=1e-1 MaxStep=1
Goal {Parameter=vg.dc Voltage=0.9}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg0.9acqmhd")
Quasistationary (
321
initialstep = 0.5 MinStep=1e-1 MaxStep=1
Goal {Parameter=vg.dc Voltage=1}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg1acqmhd")
newcurrent = "vd0vg0p1acqmhd_"
Coupled {poisson Electron ElectronTemperature}
load(fileprefix = "vd0vg0.1acqmhd")
Quasistationary (
initialstep = 0.05 Increment = 1 MinStep=1e-2 MaxStep=0.1
Goal {Parameter=vd.dc Voltage=1}
){
ACCoupled (
StartFrequency = 1e9 EndFrequency =4e10
NumberOfPoints = 10 linear
Node(1 2) Exclude(vd vg)
ObservationNode(1 2)
ACExtraction = "acqmhdetvdswp"
NoiseExtraction = "acqmhdetvdswp"
NoisePlot = "acqmhdetvdswp"
) {Poisson Electron ElectronTemperature}
}
newcurrent = "vd0vg0p2acqmhd_"
load(fileprefix = "vd0vg0.2acqmhd")
Quasistationary (
initialstep = 0.05 Increment = 1 MinStep=1e-2 MaxStep=0.1
Goal {Parameter=vd.dc Voltage=1}
){
ACCoupled (
StartFrequency = 1e9 EndFrequency =4e10
NumberOfPoints = 10 linear
Node(1 2) Exclude(vd vg)
ObservationNode(1 2)
ACExtraction = "acqmhdetvdswp"
NoiseExtraction = "acqmhdetvdswp"
NoisePlot = "acqmhdetvdswp"
) {Poisson Electron ElectronTemperature}
}
newcurrent = "vd0vg0p3acqmhd_"
load(fileprefix = "vd0vg0.3acqmhd")
322
Quasistationary (
initialstep = 0.05 Increment = 1 MinStep=1e-2 MaxStep=0.1
Goal {Parameter=vd.dc Voltage=1}
){
ACCoupled (
StartFrequency = 1e9 EndFrequency =4e10
NumberOfPoints = 10 linear
Node(1 2) Exclude(vd vg)
ObservationNode(1 2)
ACExtraction = "acqmhdetvdswp"
NoiseExtraction = "acqmhdetvdswp"
NoisePlot = "acqmhdetvdswp"
) {Poisson Electron ElectronTemperature}
}
newcurrent = "vd0vg0p4acqmhd_"
load(fileprefix = "vd0vg0.4acqmhd")
Quasistationary (
initialstep = 0.05 Increment = 1 MinStep=1e-2 MaxStep=0.1
Goal {Parameter=vd.dc Voltage=1}
){
ACCoupled (
StartFrequency = 1e9 EndFrequency =4e10
NumberOfPoints = 10 linear
Node(1 2) Exclude(vd vg)
ObservationNode(1 2)
ACExtraction = "acqmhdetvdswp"
NoiseExtraction = "acqmhdetvdswp"
NoisePlot = "acqmhdetvdswp"
) {Poisson Electron ElectronTemperature}
}
newcurrent = "vd0vg0p5acqmhd_"
load(fileprefix = "vd0vg0.5acqmhd")
Quasistationary (
initialstep = 0.05 Increment = 1 MinStep=1e-2 MaxStep=0.1
Goal {Parameter=vd.dc Voltage=1}
){
ACCoupled (
StartFrequency = 1e9 EndFrequency =4e10
NumberOfPoints = 10 linear
Node(1 2) Exclude(vd vg)
ObservationNode(1 2)
ACExtraction = "acqmhdetvdswp"
323
NoiseExtraction = "acqmhdetvdswp"
NoisePlot = "acqmhdetvdswp"
) {Poisson Electron ElectronTemperature}
}
newcurrent = "vd0vg0p6acqmhd_"
load(fileprefix = "vd0vg0.6acqmhd")
Quasistationary (
initialstep = 0.05 Increment = 1 MinStep=1e-2 MaxStep=0.1
Goal {Parameter=vd.dc Voltage=1}
){
ACCoupled (
StartFrequency = 1e9 EndFrequency =4e10
NumberOfPoints = 10 linear
Node(1 2) Exclude(vd vg)
ObservationNode(1 2)
ACExtraction = "acqmhdetvdswp"
NoiseExtraction = "acqmhdetvdswp"
NoisePlot = "acqmhdetvdswp"
) {Poisson Electron ElectronTemperature}
}
newcurrent = "vd0vg0p7acqmhd_"
load(fileprefix = "vd0vg0.7acqmhd")
Quasistationary (
initialstep = 0.05 Increment = 1 MinStep=1e-2 MaxStep=0.1
Goal {Parameter=vd.dc Voltage=1}
){
ACCoupled (
StartFrequency = 1e9 EndFrequency =4e10
NumberOfPoints = 10 linear
Node(1 2) Exclude(vd vg)
ObservationNode(1 2)
ACExtraction = "acqmhdetvdswp"
NoiseExtraction = "acqmhdetvdswp"
NoisePlot = "acqmhdetvdswp"
) {Poisson Electron ElectronTemperature}
}
newcurrent = "vd0vg0p8acqmhd_"
load(fileprefix = "vd0vg0.8acqmhd")
Quasistationary (
initialstep = 0.05 Increment = 1 MinStep=1e-2 MaxStep=0.1
Goal {Parameter=vd.dc Voltage=1}
){
324
ACCoupled (
StartFrequency = 1e9 EndFrequency =4e10
NumberOfPoints = 10 linear
Node(1 2) Exclude(vd vg)
ObservationNode(1 2)
ACExtraction = "acqmhdetvdswp"
NoiseExtraction = "acqmhdetvdswp"
NoisePlot = "acqmhdetvdswp"
) {Poisson Electron ElectronTemperature}
}
newcurrent = "vd0vg0p9acqmhd_"
load(fileprefix = "vd0vg0.9acqmhd")
Quasistationary (
initialstep = 0.05 Increment = 1 MinStep=1e-2 MaxStep=0.1
Goal {Parameter=vd.dc Voltage=1}
){
ACCoupled (
StartFrequency = 1e9 EndFrequency =4e10
NumberOfPoints = 10 linear
Node(1 2) Exclude(vd vg)
ObservationNode(1 2)
ACExtraction = "acqmhdetvdswp"
NoiseExtraction = "acqmhdetvdswp"
NoisePlot = "acqmhdetvdswp"
) {Poisson Electron ElectronTemperature}
}
newcurrent = "vd0vg1acqmhd_"
load(fileprefix = "vd0vg1acqmhd")
Quasistationary (
initialstep = 0.05 Increment = 1 MinStep=1e-2 MaxStep=0.1
Goal {Parameter=vd.dc Voltage=1}
){
ACCoupled (
StartFrequency = 1e9 EndFrequency =4e10
NumberOfPoints = 10 linear
Node(1 2) Exclude(vd vg)
ObservationNode(1 2)
ACExtraction = "acqmhdetvdswp"
NoiseExtraction = "acqmhdetvdswp"
NoisePlot = "acqmhdetvdswp"
) {Poisson Electron ElectronTemperature}
}
325
}
C.3 MATLAB Programming for Simulation Results
C.3.1 Main file
close all; clear all; clc;
q = 1.6e-19;
kt = 0.0259*q;
datapath = ?D:\Yan\research\nmos\50nm\vdswpdata?;
cd(datapath);
filename = {?vdswpvg0p1?, ?vdswpvg0p2?, ?vdswpvg0p3?,...
?vdswpvg0p4?, ?vdswpvg0p5?, ?vdswpvg0p6?,...
?vdswpvg0p7?, ?vdswpvg0p8?, ?vdswpvg0p9?, ?vdswpvg1?};
x1 = 1;
Vdtmp = [0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225...
0.25 0.275 0.3 0.325 0.35 0.375 0.4 0.425 0.45...
0.475 0.5 0.525 0.55 0.575 0.6 0.625 0.65 0.675...
0.7 0.725 0.75 0.775 0.8 0.825 0.85 0.875 0.9...
0.925 0.95 0.975 1.0];
for vdsel = [1:length(Vdtmp)],
Vdx = Vdtmp(vdsel);
fileNumber=length(filename);
datasel = 1; %1: bias dependence, 2: frequency dependence
for filsel = [1:10],
load(filename{filsel});
Jd = Id./Area.*1e6; Jg = Ig./Area.*1e6;
nx = x1;
for n = [nx]; %frequency or bias point selection.
switch datasel
case 1 %bias dependence
sv12x = conj(sv12); sv12eex = conj(sv12ee); sv12hhx = conj(sv12hh);
SV = [sv1(:,n) sv12x(:,n) conj(sv12x(:,n)) sv2(:,n)];
SVee = [sv1ee(:,n) sv12eex(:,n) conj(sv12eex(:,n)) sv2ee(:,n)];
SVhh = [sv1hh(:,n) sv12hhx(:,n) conj(sv12hhx(:,n)) sv2hh(:,n)];
Y = [Y11(:,n) Y12(:,n) Y21(:,n) Y22(:,n)]; Z = z_from_Y(Y);
numend = num_of_bias;
case 2 %frequency dependence
sv12x = conj(sv12); sv12eex = conj(sv12ee); sv12hhx = conj(sv12hh);
SV = [conj(sv1(n,:)?) conj(sv12x(n,:)?) sv12x(n,:)? conj(sv2(n,:)?)];
SVee = [conj(sv1ee(n,:)?) conj(sv12eex(n,:)?) sv12eex(n,:)? conj(sv2ee(n,:)?)];
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SVhh = [conj(sv1hh(n,:)?) conj(sv12hhx(n,:)?) sv12hhx(n,:)? conj(sv2hh(n,:)?)];
Y = [conj(Y11(n,:)?) conj(Y12(n,:)?) conj(Y21(n,:)?) conj(Y22(n,:)?)];
Z = z_from_Y(Y); S = s_from_y(Y);
numend = num_of_freq;
Igx = Ig(x1);Idx = Id(x1);
clear Ig; clear Id;
Ig = Igx; Id = Idx;
end
for x = 1:numend,
y = Y(x,:); z = Z(x,:); a = a_from_y(y);
cz = 0.5.*SV(x,:); ca = c_from_z_to_a(cz, a);
cy = c_from_a_to_y(ca, y);
nf = nf_from_ca(ca, 50);
ch = c_from_y_to_h(cy, y);
svb(x) = 2*cz(1); svc(x) = 2*cz(4);
svbvcr(x) = 2*real(cz(2)); svbvci(x) = 2*imag(cz(2));
cvbvcr(x) = svbvcr(x)/sqrt(svc(x)*svb(x));
cvbvci(x) = svbvci(x)/sqrt(svc(x)*svb(x));
sva(x) = 2*ca(1); sia(x) = 2*ca(4);
siavar(x) = 2*real(ca(3)); siavai(x) = 2*imag(ca(3));
ciavar(x) = siavar(x)/sqrt(sia(x)*sva(x));
ciavai(x) = siavai(x)/sqrt(sia(x)*sva(x));
sib(x) = 2*cy(1); sic(x) = 2*cy(4);
sicibr(x) = 2*real(cy(3)); sicibi(x) = 2*imag(cy(3));
cicibr(x) = sicibr(x)/sqrt(sib(x)*sic(x));
cicibi(x) = sicibi(x)/sqrt(sib(x)*sic(x));
svh(x) = 2*ch(1); sih(x) = 2*ch(4);
svhihr(x) = 2*real(ch(2)); svhihi(x) = 2*imag(ch(2));
cvhihr(x) = svhihr(x)/sqrt(svh(x)*sih(x));
cvhihi(x) = svhihi(x)/sqrt(svh(x)*sih(x));
nfmin(x) = nf(1); rn(x) = nf(2); Yopt(x) = nf(3);
czee = 0.5.*SVee(x,:); caee = c_from_z_to_a(czee, a);
cyee = c_from_a_to_y(caee, y); nfee = nf_from_ca(caee, 50);
svbee(x) = 2*czee(1); svcee(x) = 2*czee(4);
svbvcree(x) = 2*real(czee(2)); svbvciee(x) = 2*imag(czee(2));
cvbvcree(x) = svbvcree(x)/sqrt(svcee(x)*svbee(x));
cvbvciee(x) = svbvciee(x)/sqrt(svcee(x)*svbee(x));
svaee(x) = 2*caee(1); siaee(x) = 2*caee(4);
siavaree(x) = 2*real(caee(3)); siavaiee(x) = 2*imag(caee(3));
ciavaree(x) = siavaree(x)/sqrt(siaee(x)*svaee(x));
327
ciavaiee(x) = siavaiee(x)/sqrt(siaee(x)*svaee(x));
sibee(x) = 2*cyee(1); sicee(x) = 2*cyee(4);
sicibree(x) = 2*real(cyee(3)); sicibiee(x) = 2*imag(cyee(3));
cicibree(x) = sicibree(x)/sqrt(sibee(x)*sicee(x));
cicibiee(x) = sicibiee(x)/sqrt(sibee(x)*sicee(x));
nfminee(x) = nfee(1); rnee(x) = nfee(2); Yoptee(x) = nfee(3);
czhh = 0.5.*SVhh(x,:); cahh = c_from_z_to_a(czhh, a);
cyhh = c_from_a_to_y(cahh, y); nfhh = nf_from_ca(cahh, 50);
svbhh(x) = 2*czhh(1); svchh(x) = 2*czhh(4);
svbvcrhh(x) = 2*real(czhh(2)); svbvcihh(x) = 2*imag(czhh(2));
cvbvcrhh(x) = svbvcrhh(x)/sqrt(svchh(x)*svbhh(x));
cvbvcihh(x) = svbvcihh(x)/sqrt(svchh(x)*svbhh(x));
svahh(x) = 2*cahh(1); siahh(x) = 2*cahh(4);
siavarhh(x) = 2*real(cahh(3)); siavaihh(x) = 2*imag(cahh(3));
ciavarhh(x) = siavarhh(x)/sqrt(siahh(x)*svahh(x));
ciavaihh(x) = siavaihh(x)/sqrt(siahh(x)*svahh(x));
sibhh(x) = 2*cyhh(1); sichh(x) = 2*cyhh(4);
sicibrhh(x) = 2*real(cyhh(3)); sicibihh(x) = 2*imag(cyhh(3));
cicibrhh(x) = sicibrhh(x)/sqrt(sibhh(x)*sichh(x));
cicibihh(x) = sicibihh(x)/sqrt(sibhh(x)*sichh(x));
nfminhh(x) = nfhh(1); rnhh(x) = nfhh(2); Yopthh(x) = nfhh(3);
end
end
end
C.3.2 c_from_y_to_h.m
function x = c_from_y_to_h(cy, Y);
%function x = c_from_y_to_h(cy, Y);
Y11 = Y(1); Y21 = Y(3);
sin1 = cy(1); sin2 = cy(4); sin1in2 = cy(2); sin2in1 = cy(3);
sv = sin1./(abs(Y11)).^2;
si= sin2 + sin1.*(abs(Y21./Y11)).^2-...
2.*real(Y21./Y11.*sin1in2);
svi = conj(Y21)./(abs(Y11)).^2.*sin1 -...
sin1in2./Y11;
x = [sv svi conj(svi) si];
328