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 modelsimulation 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 gatetobody capacitance, gate to source/drain overlap capacitances,
fringing capacitances, and NonQuasiStatic (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 twoport 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 figuresofmerit 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 ECS0119623 and ECS0112923,
the Semiconductor Research Corporation under SRC #2001NJ937, 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 stylefile 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 Timedelay and Phasedelay 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 DEEMBEDDING 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 TwoPort Network . . . . . . . 50
2.4 Open/Short Deembedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.4.1 Open Deembedding of Yparameters and Noise Parameters . . . . . . 57
2.4.2 Short Deembedding of Yparameters and Noise Parameters . . . . . . 58
2.4.3 Problems Encountered in MATLAB Programming for OpenShort De
embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5 Transistor Internal Noise Deembedding . . . . . . . . . . . . . . . . . . . . . 63
ix
2.5.1 MOSFET Transistor ig and id Noise Deembedding . . . . . . . . . . . 63
2.5.2 SiGe HBT Transistor ib and ic Noise Deembedding . . . . . . . . . . 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 Yrepresentation Noise Sources . . . . . . . . . . . . . . . . . . . . . 146
6.4.2 Hrepresentation 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 HRepresentation 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 FigureofMerit for NFmin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
8.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
9 CONCLUSIONS 237
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BIBLIOGRAPHY 242
APPENDICES 247
A MATLAB PROGRAMMING FOR OPENSHORT 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 twoport. . . . . . . . . . . 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 smallsignal equivalent circuit for intrinsic bipolar device. . . . . . . . . . 11
1.6 Timedelay noise model in [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.7 Phasedelay 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 twoport network. . . . . . . . 37
2.2 Noisy linear twoport network. . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 The Y noise representation of a linear noisy twoport network. . . . . . . . . 42
2.4 The Z noise representation of a linear noisy twoport network. . . . . . . . . 45
2.5 The H noise representation of a linear noisy twoport network. . . . . . . . . 47
2.6 Adding noisy passive components parallel to a linear noisy twoport network. . 51
2.7 Adding noisy passive components in series with a linear noisy twoport network. 54
2.8 Equivalent circuit diagram used for openshort deembedding 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 Yrepresentation 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 2D gate noise current concentration CSig;i?g at 5 GHz. Vds = 1 V. Vgs = 0.5 V. . 97
xv
3.12 2D gate noise current concentration CSig;i?g at 5 GHz. Vds = 1 V. Vgs = 1 V. . . 97
3.13 2D drain noise current concentration CSid;i?
d
at 5 GHz. Vds = 1 V. Vgs = 0.5 V. 98
3.14 2D drain noise current concentration CSid;i?
d
at 5 GHz. Vds = 1 V. Vgs = 1 V. . 98
3.15 2D real part of noise current correlation concentration <(CSig;i?
d
) at 5 GHz. Vds
= 1 V. Vgs = 0.5 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.16 2D real part of noise current correlation concentration <(CSig;i?
d
) at 5 GHz. Vds
= 1 V. Vgs = 1 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.17 2D imaginary part of noise current correlation concentration =(CSig;i?
d
) at 5
GHz. Vds = 1 V. Vgs = 0.5 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.18 2D 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) = 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
Signaltonoise ratio describes the ratio of useful signal power and the unwanted noise
power. When a combination of signal and noise go through a noisy twoport network, as shown
in Fig. 1.1, both the signal and unwanted noise will be amplified at the same factor. In addition,
the twoport network adds its own noise. Therefore, the signaltonoise ratio becomes smaller
after a noisy twoport network. Noise factor F is defined as the signaltonoise ratio at the input
divided by the signaltonoise 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 twoport network. [10]
The noise figure of a twoport 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 twoport.
1.2.1 Minimum Noise Figure NFmin
The minimum noise figure NFmin is a very important parameter for noise. As selfexplained
in its name, NFmin determines the minimum noise figure for a noisy twoport network. NF
reaches its minimum NFmin when Ys = Yopt. It indicates the attribute of the noisy twoport. 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 GummelPoon,
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 threedimensional 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 Yparameters, 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 secondorder 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 spacechargeregion (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 smallsignal 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 Timedelay and Phasedelay Model
Timedelay noise model is proposed by M. Rudolph in 1999 using commonemitter 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 timedelay 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 phasedelay noise model is proposed by G.F. Niu in 2001 using commonbase con
figuration [3]. The essence of the phasedelay noise model is shown in Fig. 1.7. The collector
current shows shot noise only because the electron current being injected into the collectorbase
13
Figure 1.6: Timedelay 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 collectorbase 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 emitterinjected 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: Phasedelay noise model in [3].
In commonbase 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)
Commonbase noise sources ic and ie can be easily converted to commonemitter 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 commonemitter 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 timedelay model and phasedelay model ultimately
give the same noise model expressions. At low frequency, the timedelay model and phasedelay
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 wellknown 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 gateto
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 KlaassenPrins 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 KlaassenPrins equation is extensively
used to calculate the noise for long channel MOSFETs [17] [18] [6] [19]. The quasistatic 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 quasiFermi 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 gatesource 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 KlaassenPrins 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 zerofield 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 KlaassenPrins 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 pincho 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 pincho 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
chargebased 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 shortchannel 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 shortchannel 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 nonquasistatistic 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 metal1topolysilicon via,
Nvia is the number of such vias, ?con is the silicidetopolysilicon 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, doublesided 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 figureof
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 twoport network.
The transformation matrices to other noise representations are given. Techniques of adding or
deembedding a passive component to a linear twoport network are discussed. Noise sources
deembedding 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 modelsimulation 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 Hrepresentation noise sources using noise parameters measured on 0.25
?m RF CMOS devices. A simple yet e ective model is proposed to model the Hrepresentation
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 gatetobody capacitance, gate to source/drain overlap capacitances, fringing capac
itances, and NonQuasiStatic (NQS) e ect. A new method of separating the physical gate
resistance and the NQS channel resistance is proposed. Separating the gatetosource parasitic
capacitances from the gatetosource inversion capacitance is found to be necessary for accurate
modeling of all of the Yparameters.
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 twoport 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 figuresofmerit 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 DEEMBEDDING
This chapter introduces di erent noise representations for a linear noisy twoport network.
The transformation matrices to other noise representations are given. Techniques of adding or
deembedding passive components to a linear twoport network are discussed. For example,
the openshort deembedding procedure is needed for measurement data to move the reference
plane to the device terminals. Noise sources deembedding for both MOSFET and SiGe HBT
are given as examples which are repeatedly used later in this dissertation.
2.1 Noise Representations
A noisy twoport network can be described by a noiseless twoport 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
TwoPort
Y
V
1
I
1
V
2
I
2
+
_
+
_
Noisy
TwoPort
Figure 2.1: The chain noise representation of a linear noisy twoport 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 twoport
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 twoport, Ni is the input noise power delivered to the
noisy twoport due to source noise current is, and N0i is the noise power delivered to the noisy
twoport due to va and ia.
v
a
i
a
Noiseless
TwoPort
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 twoport network.
If Zi denotes the input admittance of the twoport shown in Fig. 2.2, the noise current
delivered by the source to the noise free twoport 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 twoport by the
38
correlated noise voltage and noise current of the noisy twoport 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 twoport 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
TwoPort
Y
I
1
V
2
i
1
i
2
V
1
Noisy
TwoPort
Figure 2.3: The Y noise representation of a linear noisy twoport 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 twoport 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
TwoPort
Y
I
1
V
2
v
1
v
2
V
1
Noisy
TwoPort
Figure 2.4: The Z noise representation of a linear noisy twoport 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 twoport 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
TwoPort
Y
V
1
I
1
V
2
I
2
+
_
+
_
Noisy
TwoPort
i
h
Figure 2.5: The H noise representation of a linear noisy twoport 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 twoport 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 twoport network parameters.
2.3 Adding Noisy Passive Components to a Noisy TwoPort Network
If the noise of the intrinsic twoport network is known, in order to calculate the noise of
a complex network, one needs to start from the noise of the intrinsic twoport 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 ?deembedding? procedure. Both the twoport network parameters
and noise parameters are involved in either the adding procedure or the deembedding procedure.
Here only the adding procedure is discussed. The deembedding procedure is just a reverse
50
process. Basically, there are two kinds of cases to add noisy passive components to a noisy
twoport 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 twoport 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
TwoPort
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 twoport network.
The Yparameter matrix of the noisy twoport network is denoted as Y . The Yparameter
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 twoport, and CtotalY is the Y noise matrix after
adding the passive components to the noisy twoport.
The other common case is to add noisy passive components in series with the twoport
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 Zparameter matrix of the noisy twoport network is denoted as Z. The Zparameter
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
TwoPort
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 twoport 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 twoport, and CtotalZ is the Z noise matrix after
adding the passive components to the noisy twoport.
55
2.4 Open/Short Deembedding
The equivalent circuit diagram used for openshort deembedding 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 Sparameters of the measurement as Smeas,
the Sparameters of the open deembedding structure as Sopen, and the Sparameters of the short
deembedding structure as Sshort. Using the relations between Y and S parameters in Table 2.2,
the Yparameters of the measurement, the open and short deembedding structure, Ymeas, Yopen
and Yshort are obtained. Open and short deembedding are performed for both Yparameters and
noise parameters to move the reference plane to the device terminals. The resulting Yparameters
and noise parameters are for the transistor. The MATLAB programming for Yparameters and
noise parameters openshort deembedding is given in Appendix A.
Figure 2.8: Equivalent circuit diagram used for openshort deembedding 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 Deembedding of Yparameters and Noise Parameters
The Yparameter for the open deembedded transistor Yod is [7]
Yod = Ymeas Yopen: (2.99)
The short deembedding structure also needs to be open deembedded. The Yparameter for the
open deembedded short deembedding 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 openshort deembedding, CA;meas needs to be transformed to the Ynoise
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 deembedded transistor CY;od is
CY;od = CY;meas 4kT<[Yod]: (2.105)
2.4.2 Short Deembedding of Yparameters and Noise Parameters
The Zparameter for the short deembedded transistor Z is [7]
Z = Zod Zos; (2.106)
where Zod and Zos are Zparameter matrices of the open deembedded transistor and the short
deembedding structure, respectively. Zod and Zos are obtained from Yod and Yos using Table 2.2.
For short deembedding of the noise parameters, we need to start with the Znoise repre
sentation matrix of the open deembedded 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 Znoise representation matrix
of the openshort deembedded 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 deembedding, and openshort deembedding data.
The results show that the short deembedding is important for noise parameters deembedding,
and cannot be neglected.
2.4.3 Problems Encountered in MATLAB Programming for OpenShort Deembedding
The openshort deembedding 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
openshort deembedded 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
openshort deembedded 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 Deembedding
MOSFET transistor ig and id noise deembedding procedure and SiGe HBT transistor ib
and ic noise deembedding 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 Deembedding
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 nonquasistatic (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 Ynoise representation parameters,
are used to describe all of the noise from the intrinsic transistor.
Here we choose to define ig and id as the Yrepresentation 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 Yrepresentation 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 Zparameters 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 openshort deembedded 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. Yparameters matrix of the level I block
YI can be obtained from ZI using Table 2.2. Therefore Yparameters 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 Yrepresentation 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 Yrepresentation 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 deembedded 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 Deembedding
The process of SiGe HBT transistor ib and ic noise deembedding is similar to the pro
cedures in section 2.5.1. The thermal noise of a SiGe HBT transistor is simulated using 2D
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 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, Ynoise 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 HRepresentation RF Noise Sources in CMOS
It has been shown using microscopic noise simulation and simple equivalent circuit deriva
tion that the Hrepresentation 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 Hrepresentation 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 deembedding are performed for both Yparameters 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 Hrepresentation 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 Hrepresentation noise sources.
where CHII is also referred to as the Hrepresentation 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 Yparameters 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: Datamodel comparison of Yparameter vs frequency at VGS = 1.2 V. VDS = 1.2 V.
VGS = 1:2 V and VDS = 1:2 V. All of the Yparameters are well modeled. Fig. 6.26 shows the
Yparameters at 10 GHz as a function of VGS. The biasing current dependence is well modeled
too.
Using the equivalent circuit parameters extracted, the Yparameters and Hparameters for all
the blocks can be calculated using straightforward linear circuit analysis. The Hrepresentation
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: Datamodel comparison of Yparameter 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 datamodel 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 Yrepresentation and
the Hrepresentation. The correlation is shown to be smaller for the Hrepresentation than for
the Yrepresentation. For practical biasing currents and frequencies, the correlation is negligible
for Hrepresentation. Models for the noise sources are suggested.
Furthermore, we have examined the relations between the Y and Hnoise 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 Yrepresentation is a better choice for Rn, and the Hrepresentation 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 Hrepresentation than for Yrepresentation.
This makes circuit design and simulation easier.
We have presented experimental extraction and modeling of Hrepresentation 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
gatetobody capacitance, gate to source/drain overlap capacitances, fringing capacitances, and
NonQuasiStatic (NQS) e ect. A new method of separating the physical gate resistance and the
NQS channel resistance is proposed. Separating the gatetosource parasitic capacitances from
the gatetosource inversion capacitance is found to be necessary for accurate modeling of all of
the Yparameters.
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 NonQuasiStatic (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 deembedded using values determined from
dc IV 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 lownoise 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 singleended
gate contact CMOS device. Standard open/short deembedding are performed on the Sparameters
measured using an HP8510C vector network analyzer from 220 GHz for a wide bias range. The
standard open/short deembedding is a su cient deembedding method for a frequency range
of 220 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 datamodel
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 datamodel 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
Yparameter 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 gatetobody 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 drainsubstrate 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 gatetobody capacitance charging occurs through movement
of majority carriers in the bulk, and thus does not experience the nonquasistatic 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
draintobody 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 polysilicongate 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: Datamodel 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.
gatetobody capacitance Cgb, gatetosource overlap capacitance Cov;s, and gatetosource fring
ing capacitance Cfs. Cperi;d includes gatetodrain overlap capacitance Cov;d, and gatetodrain
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 gatetodrain capacitance Cgd is negligible in the saturation region. The source/drain series
resistances Rs and Rd can be extracted from dc IV data, and deembedded. 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 semicircle fitting.
Plot =(h11) versus <(h11), fit the data using a semicircle, 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
polysilicongate 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 sourceside 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 220 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
220 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 (220 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 datamodel comparison for the Yparameters at 3 GHz, 5
GHz, 10 GHz, 15 GHz and 20 GHz. Rg and (Cgb+Cov;s+Cfs) are deembedded 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 Yparameters 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 Yparameters 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 gatetobody
capacitance and the parasitic gatetosource 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 Yparameter 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
twoport 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
ofmerit 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 figureofmerit used is the socalled 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 figureofmerit 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 twoport 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 figuresofmerit for NFmin.
Experimental data are used to demonstrate the new NFmin figuresofmerit.
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
driftdi 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 lownoise 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. Sparameters and noise parameters were measured using a ATN NP5B system on wafer
from 2 to 20 GHz, using open short deembedding. Vds = 1 V. Gate resistance was extracted
from sparameters, and further deembedded for calculation of Sid;i?d. gm is extracted from y
parameters (converted from sparameters), 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 simulationmodel 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 simulationmodel 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 lownoise 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. Sparameters 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 yparameters. 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 deembedding [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
modeldata 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 modeldata 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: Modeldata comparison of noise parameters vs frequency. Vgs = 0.7 V, Vds = 1 V.
8.8 FigureofMerit 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 figureofmerit 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 figureofmerit 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: Modeldata comparison of noise parameters vs Vgs, f = 5 GHz, Vds = 1 V.
To propose a figureofmerit 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 figureofmerit 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 figureofmerit 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 figureofmerit 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 figureofmerit 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 twoport 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 deembedding a passive component to a linear twoport network were
discussed. Noise sources deembedding 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 crosscorrelations 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 Yrepresentation
and the Hrepresentation were examined. The correlation was shown to be smaller for the H
representation than for the Yrepresentation. For practical biasing currents and frequencies, the
correlation is negligible for Hrepresentation. Models for the noise sources were suggested. Fur
thermore, the relations between the Y and Hnoise 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 Yrepresentation is a better choice for Rn, and the Hrepresentation 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 Hrepresentation than for Yrepresentation.
This makes circuit design and simulation easier. Chapter 6 also presented experimental extrac
tion and modeling of Hrepresentation 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
tobody capacitance and the parasitic gatetosource 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 Yparameter 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 figureofmerit 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 figureofmerit 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
BIBLIOGRAPHY
[1] J.C.J. Paasschens, ?Compact modeling of the noise of a bipolar transistor under DC and
AC current crowding conditions,? IEEE Transactions on Electron Devices, vol. 51, no. 9,
pp. 1483?1495, Sept. 2004.
[2] M. Rudolph, R. Doerner, L. Klapproth, and P. Heymann, ?An HBT noise model valid up
to transit frequency,? IEEE Electron Device Letters, vol. 20, no. 1, pp. 24?26, Jan. 1999.
[3] G. F. Niu, J. D. Cressler, W.E. Ansley, C. Webster, and D. Harame, ?A unified approach
to RF and microwave noise parameter modeling in bipolar transistors,? IEEE Transactions
on Electron Devices, vol. 48, no. 11, pp. 2568?2574, Nov. 2001.
[4] A.J. Scholten, L.F. Tiemeijer, R.J. Havens, R. de Kort, R. van Langevelde, and D.B.M.
Klaassen, ?RF noise modeling & characterization,? 2006 IEEE MTTS International Mi
crowave Symposium, June 2006.
[5] Xuemei Xi, Mohan Dunga, Jin He, Weidong Liu, Kanyu M. Cao, Xiaodong Jin, Je J.
Ou, Mansun Chan, Ali M. Niknejad, and Chenming Hu, BSIM4 manual, BSIM Research
Group at UC Berkeley, Mar. 2004.
[6] A. Scholten, L.F. Tiemeijer, R. van Langevelde, R.J. Havens, A.T.A. Zegersvan Duijn
hoven, and V.C. Venezia, ?Noise modeling for RF CMOS circuit simulation,? IEEE Trans
actions on Electron Devices, vol. 50, no. 3, pp. 618?632, Mar. 2003.
[7] M.C.A.M. Koolen, J.A.M. Geelen, and M.P.J.G. Versleijen, ?An improved deembedding
technique for onwafer highfrequency characterization,? Proceeding of IEEE Bipolar Cir
cuit and Technology, pp. 188?191, Sept. 1991.
[8] D. Becher, G. Banerjee, R. Basco, C. Hung, K. Kuhn, and WeiKai Shih, ?Noise Perfor
mance of 90 nm CMOS Technology,? IEEE MTTS International Microwave Symposium
Digest, pp. 17?20, June 2004.
[9] Andreas Pascht, Markus Gr?oing, Dirk Wiegner, and Manfred Berroth, ?Smallsignal and
temperature noise model for MOSFETs,? IEEE Transactions on Microwave Theory and
Techniques, vol. 50, no. 8, pp. 1927?1934, Aug. 2002.
[10] J.D. Cressler and G.F. Niu, SiliconGermanium heterojunction bipolar transistors, Artech
House, 2003.
[11] H.A. Haus, W.R. Atkinson, W.H. Fonger, W.W. Mcleod, G.M. Branch, W.A. Harris, E.K.
Stodola, W.B. Davenport Jr., S.W. Harrison, and T.E. Talpey, ?Representation of noise in
linear twoports,? Proceeding of IRE, vol. 48, pp. 69?74, 1960.
242
[12] Yan Cui, Guofu Niu, and D.L. Harame, ?An examination of bipolar transistor noise mod
eling and noise physics using microscopic noise simulation,? Proceeding of IEEE Bipo
lar/BiCMOS Circuit and Technology, pp. 225?228, Sept. 2003.
[13] K.M. Van Vliet, ?General Transistor Theory of Noise in PN JunctionLike Devices?I.
ThreeDimensional Green?s Function Formulation,? SolidState Electronics, vol. 15, no.
10, pp. 1033?1053, Oct. 1972.
[14] A. van der Ziel, ?Thermal noise in field e ect transistors,? Proc. IRE, vol. 50, pp. 1808?
1812, Aug. 1962.
[15] A. van der Ziel, Noise in SolidState Devices and Circuits, John Wiley & Sons, 1986.
[16] F.M. Klaassen and J. Prins, ?Thermal noise of MOS transistors,? Philips Res. Rep., vol.
22, pp. 505?514, 1967.
[17] A. van der Ziel, Noise Sources, Characterization, Measurement, Englewood Cli s, NJ:
PrenticeHall, 1970.
[18] Yannis Tsividis, Operation and Modeling of The CMOS Transistor, McGrawHill, 2
edition, 1999.
[19] C.H. Chen and M.J. Deen, ?High frequency noise of MOSFETs i modeling,? Solid State
Electron, vol. 42, pp. 2069?2081, 1998.
[20] J.C.J. Paasschens, A.J. Scholten, and R. van Langevelde, ?Generalizations of the Klaassen
Prins equation for calculating the noise of semiconductor devices,? IEEE Transactions on
Electron Devices, vol. 52, pp. 2463?2472, Nov. 2005.
[21] F. Bonani and G. Ghione, Noise in Semiconductor Devices, Berlin, Germany: Springer
Verlag, 2001.
[22] K.M. Van Vliet, A. Friedmann, R.J.J. Zijlstra, A. Gisolf, and A. van der Ziel, ?Noise in
single injection diodes. I. A survey of methods,? Journal of Applied Physics, vol. 46, pp.
1804?1813, 1975.
[23] K.M. Van Vliet, A. Friedmann, R.J.J. Zijlstra, A. Gisolf, and A. van der Ziel, ?Noise in
single injection diodes. II. Applications,? Journal of Applied Physics, vol. 46, pp. 1814?
1823, 1975.
[24] D.K. Ferry, J.R. Barker, and C. Jacoboni, Physics of Nonlinear Transport in Semiconduc
tors, New York: Plenum, 1980.
[25] J.P. Nougier, ?Fluctuations and noise of hot carriers in semiconductor materials and de
vices,? IEEE Transactions on Electron Devices, vol. 41, no. 12, pp. 2035?2049, Dec. 1994.
243
[26] C. H. Chen and M. J. Deen, ?Channel noise modeling of deepsubmicron MOSFETs,?
IEEE Transactions on Electron Devices, vol. 49, no. 8, pp. 1484?1487, Aug. 2002.
[27] M. W. Pospieszalski, ?Modeling of noise parameters of MESFET?s and MODFET?s and
their frequency and temperature dependence,? IEEE Transactions on Microwave Theory
and Techniques, vol. 37, no. 9, pp. 508?509, Sept. 1989.
[28] A. Litwin, ?Overlooked interfacial silicidepolysilicon gate resistance in MOS transistors,?
IEEE Transactions on Electron Devices, vol. 48, pp. 2179?2181, 2001.
[29] Takaaki Tatsumi, ?Geometry Optimization of Sub100nm Node RF CMOS Utilizing Three
Dimensional TCAD Simulation,? IEEE European SolidState Device Research Confer
ence, pp. 319?322, Sept. 2006.
[30] A. Scholten, L.F. Tiemeijer, R. van Langevelde, R.J. Havens, A.T.A. Zegersvan Duijn
hoven, and V.C. Venezia, ?Noise modeling for RF CMOS circuit simulation,? IEEE Trans
actions on Electron Devices, vol. 50, no. 3, pp. 618?632, Mar. 2003.
[31] Hatsuaki Fukui, ?Optimal noise figure of microwave GaAs MESFET?s,? IEEE Transac
tions on Electron Devices, vol. 26, no. 7, pp. 1032?1037, July 1979.
[32] Hatsuaki Fukui, ?Design of microwave GaAs MESFET?s for broadband lownoise am
plifiers,? IEEE Transactions on Microwave Theory and Techniques, vol. 27, no. 7, pp.
643?650, July 1979.
[33] Hatsuaki Fukui, ?Addendum to ?design of microwave GaAs MESFET?s for broadband
lownoise amplifiers?,? IEEE Transactions on Microwave Theory and Techniques, vol. 29,
no. 10, pp. 1119, Oct. 1981.
[34] M.C. King, M.T. Yang, C.W. Kuo, Yun Chang, and A. Chin, ?RF noise scaling trend
of MOSFETs from 0.5 ?m to 0.13 ?m technology nodes,? IEEE MTTS International
Microwave Symposium Digest, vol. 1, pp. 9?12, June 2004.
[35] S. Tehrani, V. Nair, C. E. Weitzel, and G. Tam, ?The e ects of parasitic capacitance on
the noise figure of MESFETs,? IEEE Transactions on Electron Devices, vol. 35, no. 5, pp.
703?706, May 1988.
[36] C. L. Lau, M. Feng, T. R. Lepkowski, G. W. Wang, Y. Chang, and C. Ito, ?Halfmicrometer
gatelength ionimplanted GaAs MESFET with 0.8dB noise figure at 16 GHz,? IEEE
Electron Device Letters, vol. 10, no. 9, pp. 409?411, Sept. 1989.
[37] Lawrence E. Larson, ?Silicon technology tradeo s for radiofrequency/mixedsignal
?systemsonachip?,? IEEE Transactions on Electron Devices, vol. 50, no. 3, pp. 683?
699, Mar. 2003.
244
[38] J.C. Guo and Y.M. Lin, ?A new lossy substrate deembedding method for sub100 nm
RF CMOS noise extraction and modeling,? IEEE Transactions on Electron Devices, vol.
53, no. 2, pp. 339?347, Feb. 2006.
[39] C. Enz, ?An CMOS transistor model for RF IC design valid in all regions of operation,?
IEEE Transactions on Microwave Theory and Techniques, vol. 50, no. 1, pp. 342?359, Jan.
2002.
[40] Thomas H. Lee, The Design of CMOS RadioFrequency Integrated Circuits, Cambridge
University Press, Dec. 2003.
[41] Tajinder Manku, ?Microwave CMOS?device physics and design,? IEEE Journal of Solid
State Circuits, vol. 34, no. 3, pp. 277?285, Mar. 1999.
[42] M. Reisch, HighFrequency Bipolar Transistors, Springer, 2003.
[43] H. Hillbrand and P. Russer, ?An e cient method for computer aided noise analysis of
linear amplifier networks,? IEEE Transactions on Circuits and Systems, vol. 23, no. 4, pp.
235?238, Apr. 1976.
[44] TAURUS, 2D Device Simulator, Synopsys.
[45] DESSIS, 2D Device Simulator, version 9.0, Synopsys.
[46] W.M.C. Sansen and R.G. Meyer, ?Characterization and measurement of the base and
emitter resistances of bipolar transistors,? IEEE Journal of SolidState Circuits, vol. 7,
pp. 492?498, Dec. 1972.
[47] G.F. Niu, W.E. Ansley, S. Zhang, J.D. Cressler, C.S. Webster, and R.A. Groves, ?Noise
parameter optimization of UHV/CVD SiGe HBT?s for RF and microwave applications,?
IEEE Transactions on Electron Devices, vol. 46, no. 8, pp. 1589?1598, Aug. 1999.
[48] W. Shockley, J.A. Copeland, and R.P. James, ?The impedance field method of noise calcu
lation in active semiconductor devices,? in Quantum theory of atoms, molecules, and the
solidstate. 1966, pp. 537?563, Academic Press.
[49] D.L. Harame, J.H. Comfort, J.D. Cressler, E.F. Crabb?e, J.Y.C. Sun, B.S. Meyerson, and
T. Tice, ?Si/SiGe epitaxialbase transistors Part II: process integration and analog applica
tions,? IEEE Transactions on Electron Devices, vol. 42, no. 3, pp. 469?482, Mar. 1995.
[50] F. Bonani, G. Ghione, M.R. Pinto, and R.K. Smith, ?An e cient approach to noise analysis
through multidimensional physicsbased models,? IEEE Transactions on Electron Devices,
vol. 45, no. 1, pp. 261?269, Jan. 1998.
[51] Yan Cui, Guofu Niu, Yun Shi, and D.L. Harame, ?Spatial distribution of microscopic
noise contributions in SiGe HBT,? IEEE Topical Meeting on Silicon Monolithic Integrated
Circuits in RF Systems, pp. 170?173, Apr. 2003.
245
[52] Christoph Jungemann, Burkhard Neinhus, S. Decker, and Bernd Meinerzhagen, ?Hierar
chical 2D DD and HD Noise Simulations of Si and SiGe Devices?Part II: Results,? IEEE
Transactions on Electron Devices, vol. 49, no. 7, pp. 1258?1264, July 2002.
[53] Guofu Niu, Shiming Zhang, J.D. Cressler, A.J. Joseph, J.S. Fairbanks, L.E. Larson, C.S.
Webster, W.E. Ansley, and D.L. Harame, ?SiGe profile design tradeo s for RF circuit
applications,? Technical Digest of International Electron Devices Meeting, vol. 49, pp.
573?576, Dec. 1999.
[54] Guofu Niu, Shiming Zhang, J.D. Cressler, A.J. Joseph, J.S. Fairbanks, L.E. Larson, C.S.
Webster, W.E. Ansley, and D.L. Harame, ?Noise modeling and SiGe profile design trade
o s for RF applications [HBTs],? IEEE Transactions on Electron Devices, vol. 47, no. 11,
pp. 2037?2044, Nov. 2000.
[55] Guofu Niu, W.E. Ansley, S. Zhang, J.D. Cressler, C.S. Webster, and R.A. Groves, ?Noise
parameter optimization of UHV/CVD SiGe HBT?s for RF and microwave applications,?
IEEE Transactions on Electron Devices, vol. 46, no. 8, pp. 1589?1598, Aug. 1999.
[56] S.J. Jeng, B. Jagannathan, J.S. Rieh, J. Johnson, K.T. Schonenberg, D. Greenberg,
A. Stricker, H. Chen, M. Khater, D. Ahlgren, G. Freeman, K. Stein, and S. Subbanna, ?A
210GHz fT SiGe HBT with a nonselfaligned structure,? IEEE Electron Device Letters,
vol. 22, pp. 542?544, Nov. 2001.
[57] T.F. Meister and et. al., ?SiGe bipolar technology with 3.9 ps gate delay,? Proceeding of
IEEE Bipolar/BiCMOS Circuit and Technology, pp. 103?106, Sept. 2003.
[58] Y. Cui, G. Niu, Y. Shi, C. Zhu, L. Najafizadeh, J. D. Cressler, and A. Joseph, ?SiGe
Profile Optimization for Improved Cryogenic Operation at High Injection,? IEEE Bipo
lar/BiCMOS Circuits and Technology Meeting, Oct. 2006.
[59] C. Jungemann, B. Neinh?us, C. D. Nguyen, B. Meinerzhagen, R. W. Dutton, J. Scholten,
and L. F. Tiemeijer, ?Hydrodynamic Modeling of RF Noise in CMOS Devices,? Technical
Digest of IEEE International Electron Devices Meeting, pp. 803?873, Dec. 2003.
[60] The National Technology Roadmap for Semiconductors, Semiconductor Industry Associa
tion, 2001.
[61] JungSuk Goo, W. Liu, ChangHoon Choi, K.R. Green, Zhiping Yu, T.H. Lee, and R.W.
Dutton, ?The equivalence of van der Ziel and BSIM4 models in modeling the induced
gate noise of MOSFETs,? Technical Digest of International Electron Devices Meeting, pp.
811?814, Dec. 2000.
[62] JungSuk Goo, High Frequency Noise in CMOS Low Noise Amplifiers, Ph.D. Thesis,
Stanford University, 2001.
246
[63] Guofu Niu, Yan Cui, and S. S. Taylor, ?Microscopic RF noise simulation and noise source
modeling in 50 nm le CMOS,? Tech. Dig. of IEEE Topical Meeting on Silicon Monolithic
Integrated Circuits in RF Systems, pp. 123?126, 2004.
[64] Xiaodong Jin, JiaJiunn Ou, ChihHung Chen, Weidong Liu, M.J. Deen, P.R. Gray, and
Chenming Hu, ?An e ective gate resistance model for CMOS RF and noise modeling,?
Technical Digest of International Electron Devices Meeting, pp. 961?964, Dec. 1998.
[65] Yuhua Cheng and M. Matloubian, ?High frequency characterization of gate resistance in
RF MOSFETs,? IEEE Electron Device Letters, vol. 22, no. 2, pp. 98?100, Feb. 2001.
[66] T.A. Fjeldly, T. Ytterdal, and Y. Cheng, ,? May 2003.
[67] Mansun Chan, Kelvin Hui, R. Ne , Chenming Hu, and Ping Keung Ko, ?A relaxation time
approach to model the nonquasistatic transient e ects in MOSFETs,? Technical Digest
of International Electron Devices Meeting, pp. 169?172, Dec. 1994.
[68] J. Tao, A. Rezvani, and P. Findley, ?RF CMOS gate resistance and noise characterization,?
IEEE International Conference on SolidState and IntegratedCircuit Technology, vol. 1,
pp. 159?162, Oct. 2004.
[69] Q. Q. Liang, J. D. Cressler, G. Niu, Y. Lu, G. Freeman, D. C. Ahlgren, R. M. Malladi,
K. Newton, and D. L. Harame, ?A simple fourport parasitic deembedding methodology
for highfrequency scattering parameter and noise characterization of SiGe HBTs,? IEEE
Transactions on Microwave Theory and Techniques, vol. 51, no. 11, pp. 2165?2174, Nov.
2003.
[70] C.Y. Wong, J.Y.C. Sun, Y. Taur, C.S. Oh, R. Angelucci, and B. Davari, ?Doping of n+ and
p+ polysilicon in a dualgate CMOS process,? Technical Digest of International Electron
Devices Meeting, pp. 238?241, 1988.
[71] R. Rios and N.D. Arora, ?Determination of ultrathin gate oxide thickness for CMOS struc
tures using quantum e ects,? Technical Digest of International Electron Devices Meeting,
pp. 613?616, 1994.
[72] A. Scholten, Private communication.
247
APPENDICES
248
APPENDIX A
MATLAB PROGRAMMING FOR OPENSHORT DEEMBEDDING IN CHAPTER 2
correction = 1; % correction = 0: matrix operation; 1: correction
k=1.38e23;
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
% Sparameters 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 sparameter to Y parameters
temp=50*((1+s(1,1))*(1+s(2,2))s(1,2)*s(2,1));
y(1,1)=((1s(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))*(1s(2,2))+s(1,2)*s(2,1))/temp;
% Sparameters 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 sparameter to Y parameters
temp=50*((1+s(1,1))*(1+s(2,2))s(1,2)*s(2,1));
y_open(1,1)=((1s(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))*(1s(2,2))+s(1,2)*s(2,1))/temp;
% Sparameters 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 sparameter to Y parameters
temp=50*((1+s(1,1))*(1+s(2,2))s(1,2)*s(2,1));
y_short(1,1)=((1s(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))*(1s(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)=(NFmin1)/2Rn*conj(Yopt);
Ca_dut(2,1)=(NFmin1)/2Rn*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=yy_open;
yi_short=y_shorty_open;
% 7. deembed Cy_DUT from the parallel parasitic
Cyi_dut=Cy_dutCy_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_dutZi_short;
%12. Deembed Czi_dut from series parasitics to get
% the correlation matrix Cz of the intrinsic transistor
Cz=Czi_dutCzi_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 openshort 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((50Zopt_new)/(50+Zopt_new));
ang_Gama_new(i)=angle((50Zopt_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[(4.446e1 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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 = 5.30793553232469E05
$!Varset ImY11 = (omega*2.40555589212422E14 )
$!Varset ReY12 = 1.14676737756815E06
$!Varset ImY12 = (2.13716228513946E15*omega)
$!Varset ReY21 = 4.08080901277576E03
273
$!Varset ImY21 = (3.39993575257787E14*omega)
$!Varset ReY22 = 4.04802562859283E06
$!Varset ImY22 = ( 3.90143182411877E15*omega)
#create 1.plt for Sv1
$!Newlayout
$!READDATASET ?"ise:lay" "ise:lc" "MFBD1/msh10_msh.grd"
"MFBD/acfself_bjt_1_00num_acgf_des.dat.gz"?
DATASETREADER = ?DFISE Loader?
$!ALTERDATA
EQUATION = ?{tLNVSD} = {LNVSD}?
$!ALTERDATA
EQUATION = ?{Sv1} = {pLNVSD}?
$!WRITEDATASET "MFBD/1.dat"
INCLUDEGEOM = NO
INCLUDECUSTOMLABELS = NO
VARPOSITIONLIST = [12,29]
BINARY = No
USEPOINTFORMAT = Yes
PRECISION = 9
#create 2.plt for Sv2
$!Newlayout
$!READDATASET ?"ise:lay" "ise:lc" "MFBD1/msh10_msh.grd"
"MFBD/acfself_bjt_2_00num_acgf_des.dat.gz"?
DATASETREADER = ?DFISE Loader?
$!ALTERDATA
EQUATION = ?{tLNVSD} = {LNVSD}?
$!ALTERDATA
EQUATION = ?{Sv2} = {pLNVSD}?
$!WRITEDATASET "MFBD/2.dat"
INCLUDEGEOM = NO
INCLUDECUSTOMLABELS = NO
VARPOSITIONLIST = [12,29]
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/acfself_bjt_1_2_00num_acgf_des.dat.gz"?
DATASETREADER = ?DFISE Loader?
$!ALTERDATA
EQUATION = ?{RetLNVXVSD} = {ReLNVXVSD}?
$!ALTERDATA
EQUATION = ?{ImtLNVXVSD} = {ImLNVXVSD}?
$!ALTERDATA
EQUATION = ?{ReSv12} = {RepLNVXVSD}?
$!ALTERDATA
EQUATION = ?{ImSv12} = {ImpLNVXVSD}?
$!WRITEDATASET "MFBD/1_2.dat"
INCLUDEGEOM = NO
INCLUDECUSTOMLABELS = NO
VARPOSITIONLIST = [12,1516]
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 = [16]
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*ReY22ImY11*ImY22ReY12*ReY21+ImY12*ImY21)
$!Varset Imdelta0 = (ReY11*ImY22+ReY22*ImY11ReY12*ImY21ReY21*ImY12)
$!Varset abs2delta0 = (Redelta0*Redelta0 + Imdelta0*Imdelta0)
$!Varset x = (Redelta0*ReY21+Imdelta0*ImY21)
$!Varset Redelta1 = (x/abs2Y21)
$!Varset x = (Imdelta0*ReY21Redelta0*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*ReY21ReY22*ImY21)
$!Varset Imdelta2 = (x/abs2Y21)
$!Varset abs2delta2 = (Redelta2*Redelta2+Imdelta2*Imdelta2)
$!Varset x = (Redelta0*ReY22+Imdelta0*ImY22)
$!Varset Redelta3 = (x/abs2Y21)
$!Varset x = (Imdelta0*ReY22Redelta0*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*ReY12ReY11*ImY12)
$!Varset Rey = (ReY21*ReY22+ImY21*ImY22)
$!Varset Imy = (ImY21*ReY22ReY21*ImY22)
$!Varset Rez = (ReY21*ReY11+ImY21*ImY11)
$!Varset Imz = (ImY21*ReY11ReY21*ImY11)
$!Varset Rew = (ReY22*ReY12+ImY22*ImY12)
$!Varset Imw = (ImY22*ReY12ReY22*ImY12)
$!Varset Reu = (ReY22*ReY11+ImY22*ImY11)
$!Varset Imu = (ImY22*ReY11ReY22*ImY11)
$!Varset Rev = (ReY21*ReY12+ImY21*ImY12)
$!Varset Imv = (ImY21*ReY12ReY21*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/finalfselfpnum.dat"
INCLUDEGEOM = NO
INCLUDECUSTOMLABELS = NO
VARPOSITIONLIST = [114]
BINARY = no
USEPOINTFORMAT = yes
PRECISION = 9
$!FIELDLAYERS SHOWMESH = NO
$!Fieldlayers showcontour = Yes
$!TWODAXIS YDETAIL{ISREVERSED = YES}
$!GLOBALCONTOUR LEGEND{SHOW = YES}
$!FIELD [118] CONTOUR{CONTOURTYPE = FLOOD}
$!ADDONCOMMAND
ADDONID = ?ISE TCAD ADDon?
COMMAND = ?ORTHOSLICE X 1.75 Frame 001?
$!WRITEDATASET "MFBD/1dcutfselfpnum.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 1.1610737e195)]
)
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 9.3631897e224), (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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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=1e3 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 = 1.6e19;
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_freq5):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 = zzb; 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=1.38066e023;
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*gia1bc1^2);
bso1 = bc1;
yopt1 = gso1+j*bso1;
gammaopt1 = (1yopt1*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=1e1 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=1e1 MaxStep=1
Goal {Parameter=vg.dc Voltage=0.2}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg0.2acqmhd")
Quasistationary (
initialstep = 0.5 MinStep=1e1 MaxStep=1
Goal {Parameter=vg.dc Voltage=0.3}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg0.3acqmhd")
Quasistationary (
320
initialstep = 0.5 MinStep=1e1 MaxStep=1
Goal {Parameter=vg.dc Voltage=0.4}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg0.4acqmhd")
Quasistationary (
initialstep = 0.5 MinStep=1e1 MaxStep=1
Goal {Parameter=vg.dc Voltage=0.5}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg0.5acqmhd")
Quasistationary (
initialstep = 0.5 MinStep=1e1 MaxStep=1
Goal {Parameter=vg.dc Voltage=0.6}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg0.6acqmhd")
Quasistationary (
initialstep = 0.5 MinStep=1e1 MaxStep=1
Goal {Parameter=vg.dc Voltage=0.7}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg0.7acqmhd")
Quasistationary (
initialstep = 0.5 MinStep=1e1 MaxStep=1
Goal {Parameter=vg.dc Voltage=0.8}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg0.8acqmhd")
Quasistationary (
initialstep = 0.5 MinStep=1e1 MaxStep=1
Goal {Parameter=vg.dc Voltage=0.9}
){
Coupled {poisson Electron ElectronTemperature}
}
save(fileprefix = "vd0vg0.9acqmhd")
Quasistationary (
321
initialstep = 0.5 MinStep=1e1 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=1e2 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=1e2 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=1e2 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=1e2 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=1e2 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=1e2 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=1e2 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=1e2 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=1e2 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=1e2 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}
}
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}
C.3 MATLAB Programming for Simulation Results
C.3.1 Main file
close all; clear all; clc;
q = 1.6e19;
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));
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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