Physics, Modeling and Design Implications of RF Correlated Noise in SiGe HBTs
by
Ziyan Xu
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
May 5, 2013
Keywords: SiGe HBT, RF noise, compact modeling, noise extraction, LNA
Copyright 2013 by Ziyan Xu
Approved by
Guofu Niu, Chair, Alumni Professor of Electrical and Computer Engineering
Fa Foster Dai, Professor of Electrical and Computer Engineering
Stuart Wentworth, Associate Professor of Electrical and Computer Engineering
Bogdan Wilamowski, Professor of Electrical and Computer Engineering
Abstract
Accurate noise compact modeling and efficient noise extraction techniques are required for
RF circuit design. Understanding the impact of noise sources and the noise propagation is also
necessary for device and circuit noise optimization. In this work, we discuss RF noise physics,
modeling, extraction and circuit design implications. The related compact model parameters are
determined based on the experimental results of various SiGe HBTs.
After a review of previous noise model and their implementation in compact models, we de-
velop a much improved physics-based compact noise model for use with any existing compact
models. We investigate the impact of CB SCR transit time effect on noise parameters of bipolar
transistors, together with the noise transport in the neutral base. A model suitable for compact
model implementation is developed. The resulting frequency dependence and correlation of termi-
nal current noises can be generated from independent white noise sources, which is important as
the current standard simulators are unable to handle correlated noise sources.
We present a new compact modeling approach to extraction of intrinsic transistor terminal
current noises and terminal resistance thermal noise. The extraction method are based on the
transfer function which can be calculated using ac small signal simulation results. Thus this method
is independent of specific compact model and avoids tedious element by element de-embedding
procedure.
The relevant importance of noise sources are evaluated in various noise representations and
proved to be varying from one representation to another. It is even unfair to claim the noise source?s
dominance within one single representation as it depends on each noise power spectre density and
noise parameter. The base resistances are identified to be the most important elements of the ex-
trinsic network that determine the intrinsic terminal noise current propagation towards the external
ii
terminals. Such propagation increases the NFmin due to intrinsic noise currents considerably, mak-
ing it much higher than theNFmin due toRb?s thermal noise. Rb?s role as an impedance element can
be much more important than its role as a thermal noise source for practical SiGe HBTs. Analytical
expression are derived to demonstrate that the well known effect of noise correlation is shown to
be highly dependent on Rb. Consequently, Rb reduction should continue for NFmin improvement
despite a nearly negligible NFmin due to Rb?s thermal noise.
Impact of correlated RF noise on LNA design is also examined. The useful simultaneous
noise and impedance matching conditionally holds based on the simplified analytical derivation
results. Simulation results show that noise matching requires a considerably larger transistor and
power consumption in the presence of intrinsic terminal current noise correlation. The actual noise
figure of LNA designed using SPICE model are found to be overall comparable to that of the
correlated model designed LNA, which is due to the small noise conductance tolerating significant
noise mismatch.
iii
Acknowledgments
My deepest gratitude goes first and foremost to my major professor, Dr. Guofu Niu, for his
encouragement and guidance all along the way. Dr. Niu is not only an inspiring mentor but also an
outstanding role model.
I would like to thank the other members of my committee, Dr. Fa Dai, Dr. Stuart Wentworth
and Dr. Bogdan Wilamowski. They never turned down my request and alway cooperate in a timely
manner. Their constant help is an important factor to my academic growth. I would like also to
thank my university reader, Dr. Xiao Qin. His participation helped to make sure my PhD study
had a smooth ending.
Many thanks also go to Kejun Xia and Pei Shen. Collaborations with them on the noise study
were pleasant and memorable.
I would like to thank my parents and families for their self-less support, understanding and
love, which make me feel secure and confident. At last, I would like to thank my best friend and
husband. To meet him is the best thing that ever happened to me; to marry him is the most correct
decision that I have ever made.
iv
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 RF Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Terminal Current Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Noise Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Noise Representations of Two Port Network . . . . . . . . . . . . . . . . . 6
1.2.2 Noise figure and Noise Parameter . . . . . . . . . . . . . . . . . . . . . . 11
1.3 SiGe HBTs Noise Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Noise Compact Modeling and Implementation . . . . . . . . . . . . . . . . . . . . . . 19
2.1 van Vliet?s model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 CB SCR Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 A survey of previous noise compact modeling methods . . . . . . . . . . . . . . . 24
2.3.1 Noise compact model of Mextram group . . . . . . . . . . . . . . . . . . 24
2.3.2 Noise compact model of HICUM group . . . . . . . . . . . . . . . . . . . 26
2.3.3 Noise transit time modeling approach . . . . . . . . . . . . . . . . . . . . 27
2.3.4 New method with RC delayed noise current . . . . . . . . . . . . . . . . . 28
2.4 Noise source model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.1 Consider CB SCR transit time alone . . . . . . . . . . . . . . . . . . . . . 31
2.4.2 Intrinsic Base and CB SCR . . . . . . . . . . . . . . . . . . . . . . . . . . 38
v
2.4.3 General Compact Modeling Implementation . . . . . . . . . . . . . . . . . 42
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Compact Model Based Noise Extraction . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1 Extraction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.1 Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.2 Extraction procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Verification using synthesized data . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Experimental Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3.1 Model parameter determination . . . . . . . . . . . . . . . . . . . . . . . 59
3.3.2 Frequency dependent extraction results . . . . . . . . . . . . . . . . . . . 61
3.3.3 Bias dependent extraction results . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 Noise Source Importance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1 Equivalent circuit simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 Analytical derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.1 Z-Noise Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.2 Y-Noise representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.3 Chain Noise Representation . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3 Noise Parameter Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.1 Analytical Models of Noise Parameters . . . . . . . . . . . . . . . . . . . 79
4.3.2 Model comparison with measurement . . . . . . . . . . . . . . . . . . . . 88
4.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5 Impact on Low Noise Amplifier Design . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.1 Analytical derivation of LNA noise figure . . . . . . . . . . . . . . . . . . . . . . 94
5.2 Simultaneous noise and impedance matching . . . . . . . . . . . . . . . . . . . . 97
5.3 Simulation Results based on a Cascade LNA . . . . . . . . . . . . . . . . . . . . . 98
vi
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A Noise Representation Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 108
B Derivation of Noise Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
C Matlab Code for Intrinsic Noise Extraction . . . . . . . . . . . . . . . . . . . . . . . . 114
D Verilog-A Code for Compact Noise Model Implementation . . . . . . . . . . . . . . . 121
E Derivation of relation between TintY and TY . . . . . . . . . . . . . . . . . . . . . . . . 122
vii
List of Figures
1.1 Energy band diagram of a graded-base SiGe HBT [1] . . . . . . . . . . . . . . . . . . 2
1.2 Summary of SiGe BiCMOS technology from IBM. This chart is copied from [2]. . . . 3
1.3 Shockley?s impedance field method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 RF noise sources of a transistor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 A simplified example of transistor as a noisy two port. . . . . . . . . . . . . . . . . . 7
1.6 Y representation of a linear noisy two-port network. . . . . . . . . . . . . . . . . . . . 7
1.7 Chain representation of a linear noisy two-port network. . . . . . . . . . . . . . . . . 8
1.8 Z representation of a linear noisy two-port network. . . . . . . . . . . . . . . . . . . . 9
1.9 H representation of a linear noisy two-port network. . . . . . . . . . . . . . . . . . . . 10
1.10 Illustration of the definition of Noise figure for an amplifier. . . . . . . . . . . . . . . 12
1.11 Asymmetric paraboloid of noise figure in the three-dimensional co-ordinate system. . . 14
1.12 Illustration of noise parameters of a two-port network. . . . . . . . . . . . . . . . . . 15
1.13 NFmin versus frequency for SiGe HBTs from four SiGe BiCMOS technologies, in-
cluding three high-performance variants at the 0.5-, 0.18-, and 0.13-?m nodes as well
as a slightly higher breakdown voltage variant at the 0.18- ?m node. This figure is
copied from [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
viii
1.14 NFmin versus Jc at 10 GHz for 250 GHz SiGe HBT with AE = 0.094 ? 9.62?m2.
Measured data are shown by symbols. This figure is copied from [4]. . . . . . . . . . 17
1.15 NFmin versus Jc at 25 GHz for 250 GHz SiGe HBT with AE = 0.094 ? 9.62?m2.
Measured data are shown by symbols. This figure is copied from [4]. . . . . . . . . . 18
2.1 Comparison of noise parameters simulated for an IBM SiGe HBT using design kit
with measurement from 2 to 26 GHz. Inside the design kit, HICUM model is used. . . 20
2.2 Y-parameter modeling result using equivalent circuit with NQS input. ?0 = 270cm2/(V ?s),
electron life time= 1.54x10?7 s, T = 300 K, dB = 45 nm, VBE = 0.8 V. fT = 184
GHz for ? = 5. fT = 83 GHz for ? = 0 (copied from [5]). . . . . . . . . . . . . . . . 23
2.3 Network representation of the charge partitioning model. . . . . . . . . . . . . . . . . 25
2.4 Illustration the terminal noise currents. ib0 is from emitter hole velocity fluctuation. ib1
and ic1 are from base electron velocity fluctuation. ic1 becomes ic2 through CB SCR
transport, which also produces ib2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Calculated noise current spectral density normalized by 2qIC vs frequency. AE =
0.8?20?m2. IC = 3.55 mA. fT = 15 GHz. ?c is set to 0.7?f, ?b is set to 0.2?f and
? = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6 Calculated correlation normalized by 2qIC vs frequency. AE = 0.8?20?m2. IC =
3.55 mA. fT = 15 GHz. ?c is set to 0.7?f, ?b is set to 0.2?f and ? = 3. . . . . . . . . . 31
2.7 Equivalent circuit of proposed compact noise modeling implementation. . . . . . . . . 34
2.8 Verilog-A sample code of proposed noise model. . . . . . . . . . . . . . . . . . . . . 35
2.9 Comparison of measured and simulated noise parameters vs frequency. . . . . . . . . 37
2.10 Comparison of measured and simulated noise parameters vs IC at 5 GHz. . . . . . . . 38
ix
2.11 Comparison of measured and simulated noise parameters vs IC at 10 GHz. . . . . . . 39
2.12 Calculation results of PSDs of intrinsic ic, ib and their correlation with different ?b, ?c
and ? combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.13 Normalized correlation with different ?b, ?c and ? combinations. . . . . . . . . . . . . 42
2.14 Equivalent circuit for Verilog-A implementation of the completely physics based ver-
sion of the proposed model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.15 Comparison of measured and simulated noise parameters vs frequency. . . . . . . . . 46
2.16 Comparison of measured and simulated noise parameters vs IC at 5 GHz. . . . . . . . 46
2.17 Comparison of measured and simulated noise parameters vs IC at 10 GHz. . . . . . . 47
2.18 Comparison of measurement and simulated noise parameters by making ?b =?f. . . . 48
3.1 Illustration of noise extraction method based on small-signal equivalent circuit. . . . . 50
3.2 Illustration of noise extraction method based on lumped four-port network. . . . . . . 50
3.3 Illustration of transfer of internal terminal noise currents ibi and ici, various resistance
thermal noise irk to external terminal noise currents ibx and icx with ac shorted base-
emitter and collector-emitter voltages. . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 Sample Verilog-A code of nodes setup in VBIC model and symbols with external
nodes displayed in schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5 Illustration of simulating transfer functions NTicx,ci and NTibx,ci by ac small signal
analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.6 Screen shot of ADS simulation schematic. . . . . . . . . . . . . . . . . . . . . . . . . 56
x
3.7 Intrinsic noise extraction flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.8 Comparison of Sici?c, Sibi?b and Sici?b as a function of frequency. . . . . . . . . . . . . . . 58
3.9 Comparison of the extractedSicii?ci,Sibii?b
i
andSibii?b
i
with their input values as a function
of frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.10 Comparison of Y-parameters from simulation and measurement as a function of fre-
quency at IC = 3.97 mA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.11 Comparison of noise parameters as a function of frequency from measurement and
simulations with thermal noises plus correlated intrinsic current noises and thermal
noises plus uncorrelated intrinsic current noises. . . . . . . . . . . . . . . . . . . . . . 61
3.12 Comparison of Sicxi?cx, Sibxi?bx and Sicxi?bx as a function of frequency from measurement
and simulations with thermal noises plus correlated intrinsic noises, the correlated
intrinsic noises alone and the thermal noises alone. . . . . . . . . . . . . . . . . . . . 62
3.13 Comparison of extracted Sicii?ci, Sibii?bi and Sicii?bi as a function of frequency from mea-
surement with different values of base resistance and simulations with and without
correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.14 Comparison of extracted Sicii?ci, Sibii?bi and Sicii?bi as a function of frequency from mea-
surement with two different values of effect base resistances and simulations with and
without correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.15 Comparison of extracted Sicii?ci, Sibii?bi and Sicii?bi as a function of IC from measurement
with different values of base resistance and simulations with and without correlation. . 65
4.1 NFmin obtained from measurement andNFmin obtained from simulations with different
noise sources turned on. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
xi
4.2 Sici?c obtained from measurement and Sici?c obtained from simulations with different
noise sources turned on. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Illustration of the full model and simplified model of the device. . . . . . . . . . . . . 69
4.4 Simulated Y11 and Y21 using the simplified model for intrinsic transistor plus Rb and
using the complete HICUM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.5 Simulated noise transfer functions as a function of frequency at IC = 3.97 mA with
no extrinsic network element, with all extrinsic network elements, with only Rb and
with only Re. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.6 The elements of Cicm,ibm obtained with [NTibc] simulated using complete extrinsic net-
work compared with the elements of Cicm,ibm obtained with [NTibc] simulated using
only a lumped Rb in the extrinsic network. . . . . . . . . . . . . . . . . . . . . . . . 72
4.7 Illustration of Z presentation in the cases of the intrinsic device alone and the intrinsic
device plus Rb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.8 The correlation matrix of Z-noise representation with Rb = 0, noiseless Rb and ther-
mal noisy Rb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.9 Illustration of Y presentation in the cases of the intrinsic device alone and the intrinsic
device plus Rb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.10 Correlation matrix of Y representation from calculations. . . . . . . . . . . . . . . . . 78
4.11 Correlation matrix of chain noise representation from calculation with Rb = 0?, ther-
mal noiseless Rb and thermal noisy Rb. . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.12 Equivalent small-signal circuit with source noise vs, device thermal noise vb and cur-
rent noise ib and ic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
xii
4.13 Equivalent small-signal circuit with source noise vs, device thermal noise vb and cur-
rent noise ib and ic. Emitter resistance is neglected. . . . . . . . . . . . . . . . . . . . 82
4.14 Comparison of NFmin calculated from analytical equations and measurement. . . . . . 88
4.15 Comparison of NFmin from 50 GHz SiGe HBT as a function of frequency from sim-
ulations with correlated intrinsic terminal current noises due to CB SCR plus thermal
noises, correlated intrinsic terminal current noises alone and thermal noises alone. . . . 90
4.16 Simulated noise parameters using zero Rb, noiseless Rb and noisy Rb as a function of
frequency at 3.97 mA; both uncorrelated SPICE model and new correlated model are
included for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.17 Simulated noise parameters using zero Rb, noiseless Rb and noisy Rb as a function of
IC at 10 GHz; both uncorrelated SPICE model and new correlated model are included
for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.1 Simplified equivalent circuit for a transistor with an emitter inductor Le and a base
inductor Lb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2 Simplified equivalent circuit for a "noisy" transistor with an emitter inductor Le and a
base inductorLb. The transistor has noise sources including the terminal noise current
ic and ib and the thermal noise vb of base resistance Rb. Power Source has a noise
source vs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 Modeled and measured noise parameters versus JC at 5 GHz. . . . . . . . . . . . . . . 99
5.4 Schematic of the SiGe HBT cascode LNA used. . . . . . . . . . . . . . . . . . . . . . 100
5.5 The noise matching source impedance of design using the correlated noise model ver-
sus Jc at 5 GHz and 10 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
xiii
5.6 Emitter length (LE) for Ropt=50 ? and IC versus JC at 5 GHz. . . . . . . . . . . . . . 101
5.7 NFLNA and NFmin,LNA of designs using SPICE and new noise models versus JC at 5
GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
xiv
List of Tables
1.1 Transformation matrices to calculate noise representations . . . . . . . . . . . . . . . 11
2.1 Noise currents of different mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 Noise Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
xv
Chapter 1
Introduction
Noise is a term describing any existing interference in a certain environment. It is therefore
most of the time distressing and complicated. In electrical circuit and communication system,
noise is anything but the desired signal. We can easily cite examples of this kind of noise from
daily life. It could be the snow flake in the image when we watch TV; it could be the cacophony
when we listen to the radio or make a cell phone call; or it could be the humming sound from the
transformer device in your house.
Physically, noise presents in the form of the spontaneous fluctuations in current, voltage and
even temperature in the electronic device, circuits and system. It sets the lower limits to the mea-
surement accuracy and the strength of signal that can be processed correctly. By effective shielding
methods, noise impact from outside environment and interconnection parasitics may be reduced.
The main noise contribution then comes from the device employed in the system. This noise cur-
rent or voltage detected at device terminals is the sum of the propagation of carrier current density
fluctuations due to velocity fluctuations or carrier number fluctuations during carrier thermal mo-
tion. Carrier number fluctuations may be reduced by reducing introduced traps and defects with
more mature process. However, the velocity fluctuations inherent in carrier random thermal motion
can not be reduced even with perfect fabrication. Therefore, low-noise devices are always sought
for low-noise RF application with higher frequency request.
A good candidate for low-noise operation is Silicon-Germanium (SiGe) heterojunction bipo-
lar transistor(HBT). After being studied and developed for several decades, SiGe technology
has become practical reality. The heart of SiGe technology is the SiGe HBT, which is the first
practical bandgap engineering device realized in silicon and can be integrated with the modern
1
CMOS technology. By seamlessly introducing the graded Ge layer into the base of bipolar tran-
sistor (BJT), SiGe HBT technology exceeds the conventional Si BJT in both DC, RF and noise
performance[6][7]. Fig. 1.1 shows that the graded-Ge induces an extra drift field in the neutral
base. The smaller base bandgap increases the electron injection at emitter-base junction, and
therefore the collector current density and current gain. The induced drift field also accelerates
minority carrier transportation, yielding a decreased base transit time and higher cut-off frequency.
This, together with its low noise feature, is why SiGe HBTs have been widely used in commercial
and military wireless communication application. Fig. 1.2 is a SiGe BiCMOS chart of IBM show-
ing the evolution of performance and minimum lithographic feature size over years from 0.5 ?m
technology to 0.13 ?m technology [2]. The DOTFIVE project, uniting eleven of the best academic
and institutes partners in Europe, has also demonstrated the realization of SiGe HBTs operating at
a maximum frequency close to 0.5 THz (500 GHz) at room temperature [8].
Figure 1.1: Energy band diagram of a graded-base SiGe HBT [1]
.
In this chapter, We first briefly introduce the main RF noise sources in SiGe HBTs and def-
initions of different noise representations as well as noise parameters. We will focus on the basic
idea and method of noise study and give a general overview of status of the reported the RF noise
performance of SiGe HBTs.
2
Figure 1.2: Summary of SiGe BiCMOS technology from IBM. This chart is copied from [2].
1.1 RF Noise
The operation of semiconductor devices is based on the carrier transportation. Under the
action of external forces and the interaction with the lattice perturbations or other carriers, electrons
and holes undergo a kind of Brownian motion whereby the velocity of each carrier exhibits large
fluctuations [9]. The single particle fluctuation is large, while the collective fluctuations are small.
From a device and circuit standpoint, the small collective fluctuation, however, propagates to the
external device terminal and produces spontaneous fluctuations in current and voltage.
Fig. 1.3 illustrates Shockley?s impedance field method [10]. Current density fluctuation
?In/p (r) caused by velocity fluctuation at origin r. Current density fluctuation propagates towards
the contact rcontact through the impedance factor Zn/p (r,rcontact) during thermal motion and results
in noise voltage fluctuation V (rcontact) at the contact.
From the equivalent circuit and compact modeling stand point, the velocity fluctuation caused
by majority carrier thermal motion can be equivalently expressed by the thermal noises of resis-
tors, while the velocity fluctuation caused by minority carrier thermal motion can be equivalent
3
Figure 1.3: Shockley?s impedance field method.
expressed by the intrinsic terminal current noises, which are the most important two kinds of
noises in RF range. Fig. 1.4 illustrates the thermal noise sources of the base resistance, emitter
resistance and collector resistance respectively as well as the terminal current noises of base and
collector current.
Figure 1.4: RF noise sources of a transistor.
1.1.1 Thermal Noise
The thermal noise of resistances describe by the Nyquist theorem is caused by random mo-
tion of the majority carriers and observed in thermal equilibrium, meaning regardless of applied
4
external bias [11]. The thermal noise can be regarded as a diffusion noise or velocity fluctuation
noise [12]. The power spectral density (PSD) of thermal noise voltage is usually given by
Svr,vr? = 4KTR, (1.1)
and that of thermal noise current is
Sir,ir? = 4KTR . (1.2)
K is the Boltzmann constant and T is temperature in Kelvin. R represents the thermal resistance.
Strictly speaking, the resistances are bias temperature dependant. The thermal noise is nearly
constant at RF range, but not necessarily "white" due to carrier thermal motion nature [13] [14].
1.1.2 Terminal Current Noise
Traditionally, base and collector current noises are treated "shot" like, which means the carri-
ers overcoming the junction potential barrier flow in a completely independent manner. The PSDs
of shot noise are
Sib,ib? = 2qIB, (1.3)
and
Sic,ic? = 2qIC. (1.4)
IB and IC are base and collector DC current. Specifically, in a bipolar transistor, the base current
shot noise 2qIB results from the flow of base majority holes across the emitter-base junction po-
tential barrier. The reason that IB appears in the base shot noise is that the amount of hole current
overcoming the EB junction barrier is determined by the minority hole current in the emitter, IB.
Similarly, the collector current shot noise results from the flow of electrons over the collector-base
junction potential barrier, and has a spectral density of 2qIC. However, it is generally believed
that the emitter-base junction is the origin of both base and collector noise [15]. Historically, the
concept of shot noise originated from the random noise in a vacuum thermionic diode and the shot
noise of transistors is of diffusion noise type. The transition of carriers across the CB junction,
which is usually reverse-biased for low-noise amplification, however, is a drift process. Therefore,
5
a dc current passing through such a junction alone does not have intrinsic shot noise. The collector
current shows shot noise only because the electron current is transported from the EB junction and
injected into the CB junction.
The above two views may lead to the same collector current noise, but different base current
noise and base collector current noise correlation at higher frequency [16]. Practically, the uncor-
related "shot" like 2qI noise model cannot successfully predict the device noise behavior at high
frequency. Increasing interest to higher frequency application urges that the correlation of base
and collector noise current has to be considered. The concept of noise transport is well accepted
and the noise transit time as a signature of noise correlation is implemented in different approach
[16] [17] . We will focus on the base and collector terminal current noise modeling in Chapter 2.
1.2 Noise Characterization
One important question is how to examine and evaluate the noise in the device. A useful
way to analysis device noise is to use the two port network theory. The transistor can be classified
as a twoport. As long as the device noise is much smaller in magnitude compared to the device
external biases, the device noise can be treated linearly [18]. Therefore, we can treat the device
noise problem as a linear noisy twoport problem.
1.2.1 Noise Representations of Two Port Network
A noisy two port network, can be equivalently described by a noiseless two port with two
equivalent noise sources, regardless of the complicacy of the network topology. There are families
of two-port representations, including impedance representations (Z representation), admittance
representation (Y representations), Hybrid representation (H representation) and chain represen-
tation (ABCD representation). Each representation is corresponding to one noise representation.
The expressions of each noise representation are related to the internal topology. The different two
noise sources combinations and transfer function between different noise representations all lead
to the different relative importance of noise sources.
6
Y representation
Fig. 1.5 shows an simplified example of transistor as a noisy two port. The upper case letters
I and V indicate the Fourier transforms of the current and voltage, which are frequency depen-
dent [18]. In this noisy two port, there are three original noise sources, the thermal noise voltage
source vb of base resistance rb, the intrinsic terminal current noise sources ibi and ici. This noisy
Figure 1.5: A simplified example of transistor as a noisy two port.
two port can be described by a noiseless two port with an equivalent current noise source i1 at the
input and an equivalent current noise source i2 at the output port, as shown in Fig. 1.6. This is Y
noise representation. i1 and i2 therefore consist of noise contribution from ibi, ici and rb. From the
Figure 1.6: Y representation of a linear noisy two-port network.
intrinsic node, we have input current I1 ?i1 and output current I2 ?i2. Y representation I?V
7
relations including equivalent noise sources for Y representation are
?
?I1?i1
I2?i2
?
?=
?
?Y11 Y12
Y21 Y22
?
??
?
?V1
V2.
?
? (1.5)
The PSD?s of i1 and i2 and their correlation are Si2,i2?, Si1,i1? and Si2,i1?. The chain noise matrix is
defined as
CY =
?
?Si1,i1? Si1,i2?
Si2,i1? Si2,i2?.
?
? (1.6)
The Y noise representation are often used for modeling terminal current noise. (1.6) is also written
as
CY =
?
?Sibxi?bx Sibxi?cx
Sicxi?bx Sicxi?cx.
?
? (1.7)
ibx and icx here are the equivalent noise current sources instead of intrinsic noise origin.
ABCD representation
The chain noise representation, also referred as the general-circuit-parameter representation,
describe the noise of two port network using an equivalent voltage noise sourceva and an equivalent
current noise source ia both at the input port, as shown in Fig. 1.7. The chain representation are
most often used to calculate noise parameter. I?V relations including equivalent noise sources
Figure 1.7: Chain representation of a linear noisy two-port network.
8
for the chain representation are
?
?I1?ia
I2
?
?=
?
?Y11 Y12
Y21 Y22
?
??
?
?V1?va
V2.
?
? (1.8)
The PSDs of va and ia and their correlation are Sva,va?, Sia,ia? and Sia,va? (Sva,ia?). The chain noise
matrix is defined as
Ca =
?
?Sia,ia? Sia,va?
Sva,ia? Sva,va?.
?
? (1.9)
Z representation
The Z noise representation describes the noise of two port network using an equivalent voltage
noise source v1 at the input and an equivalent voltage noise source v2 at the output port, as shown
in Fig. 1.8. The Z noise representation is best used for de-embedding the noise of series parasitics.
I?V relations including equivalent noise sources for Z representation are
Figure 1.8: Z representation of a linear noisy two-port network.
?
?I1
I2
?
?=
?
?Y11 Y12
Y21 Y22
?
??
?
?V1?v1
V2?v2.
?
? (1.10)
9
The PSD?s of v1 and v2 and their correlation are Sv2,v2?, Sv1,v1? and Sv2,v1?. The chain noise matrix
is defined as
CZ =
?
?Sv1,v1? Sv1,v2?
Sv2,v1? Sv2,v2?.
?
? (1.11)
H representations
The H noise representation describes the noise of two port network using an equivalent voltage
noise source vh at the input and an equivalent current noise source ih at the output port, as shown
in Fig. 1.9. I?V relations including equivalent noise sources for H representation are
Figure 1.9: H representation of a linear noisy two-port network.
?
? I1
I2?ih
?
?=
?
?Y11 Y12
Y21 Y22
?
??
?
?V1?vh
V2.
?
? (1.12)
The PSD?s of vh and ih and their correlation are Svh,vh?, Sih,ih? and Svh,ih?. The chain noise matrix is
defined as
CH =
?
?Svh,vh? Svh,ih?
Sih,vh? Sih,ih?.
?
? (1.13)
Transformation between different noise representations
The chain noise representation, Y noise representation, Z noise representation, and H noise
representations can be transformed from one to the others. An example of transformation between
10
Table 1.1: Transformation matrices to calculate noise representations
CA CY CZ CH
CprimeA
?
?1 0
0 1
?
?
?
?0 A12
1 A22
?
?
?
?1 ?A11
0 ?A21
?
?
?
?1 A12
0 A22
?
?
CprimeY
?
??Y11 1
?Y21 0
?
?
?
?1 0
0 1
?
?
?
?Y11 Y12
Y21 Y22
?
?
?
??Y11 0
?Y21 1
?
?
CprimeZ
?
?1 ?Z11
0 ?Z21
?
?
?
?Z11 Z12
Z21 Z22
?
?
?
?1 0
0 1
?
?
?
?1 ?Z12
0 ?Z22
?
?
CprimeH
?
?1 ?H11
0 ?H21
?
?
?
??H11 0
?H21 1
?
?
?
?1 ?H12
0 ?H22
?
?
?
?1 0
0 1
?
?
ABCD representation and other representations is shown in Appendix A. More generally, the
transformation can be realized by the matrix operation
Cprime =T?C?T?, (1.14)
where C is the original noise correlation matrix and Cprime is the resulting noise correlation matrix. T
is the transformation matrix given in Table 1.1, and T? is the transpose conjugate of T . The used
two-port network parameters, Y-, Z-, H-, ABCD-, are summarized as in Table 1.1.
1.2.2 Noise figure and Noise Parameter
The figure of merit that describes the level of excess noise present in the system is noise factor,
F. It is defined to be the signal-to-noise ratio (SNR) at the input divided by the the SNR at the
output,
F = Si/NiS
o/No
, (1.15)
with Si being the input signal power, So being the output signal power, Ni being the input noise
power and No being the output noise power. Noise figure is the noise factor in decibels,
NF = 10?log (F). (1.16)
11
The concept of noise figure is easy to understand when we consider an amplifier, as shown in
Fig. 1.10. Both the signal and noise at the input are amplified by the gainG of the amplifier. If the
Figure 1.10: Illustration of the definition of Noise figure for an amplifier.
amplifier is perfect, the output noise is also equal to the input noise amplified by the amplifier?s
gain, resulting in the same SNR at both the input and output. However, the noise inherent the
amplifier contributes additional noise at the output, so SNR at the output is smaller than that at the
input, resulting inF being greater than one, orNF being greater than 0 dB. (1.15) can be rewritten
as
F = SiS
o
?NoN
i
= 1G?NoN
i
. (1.17)
If the amplifier inherent noise is represented by equivalent noise sources at the input, as in the
chain representation,
F = 1 + Namp,iN
i
. (1.18)
If the amplifier inherent noise is represented by equivalent noise sources at the output,
F = 1 + Namp,oGN
i
. (1.19)
12
Both Namp,i and Namp,o are dependent on the source impedance ZS or source admittance YS.
The noise figure of a circuit is determined by the source termination and the noise properties of
the circuit. The noise properties are usually expressed by noise parameters, including the minimum
noise figure NFmin, the noise resistance Rn and the optimum source admittance Yopt. Yopt = Gopt +
jBopt. F related to noise parameters by
F =Fmin + RnG
S
vextendsinglevextendsingleY
S?Yopt
vextendsinglevextendsingle2 , (1.20)
where YS =GS +jBS is the admittance of source. NFmin = 10 log(Fmin). Or
F =Fmin + GnR
S
vextendsinglevextendsingleZ
S?Zopt
vextendsinglevextendsingle2 , (1.21)
where ZS =GS +jBS = 1/YS and Zopt =Ropt +jXopt = 1/Yopt.
Clearly, noise figure reaches it minimum, NFmin, when YS = Yopt (ZS = Zopt), i.e., at "noise
matching", as shown in Fig. 1.11. Each set of four noise parameters define a noise surface which
is an asymmetric paraboloid. Each NF, and the corresponding YS must be located on this noise
surface. NFmin is the vertex of the paraboloid. Rn, determined by the curvature of the paraboloid,
represents the sensitivity of NF deviation from NFmin to noise mismatch [19].
From linear noisy two-port theory, the noise parameter can be related to the noise correlation
matrices [18]. The chain noise representation is most often used, as illustrated in Fig. 1.12
The relation between noise parameters and PSDs of chain representations are
Rn = Sva,va?4kT , (1.22)
Gopt =
radicaltpradicalvertex
radicalvertexradicalbtSi
a,ia?
Sva,va??
bracketleftBiggIfracturparenleftbigS
ia,va?
parenrightbig
Sva,va?
bracketrightBigg2
, (1.23)
Bopt =?Ifractur
parenleftbigS
va,va?
parenrightbig
Sva,va? , (1.24)
Rn = Sva,va?4kT , (1.25)
13
Figure 1.11: Asymmetric paraboloid of noise figure in the three-dimensional co-ordinate system.
Fmin = 1 +
radicalBig
Sva,va?Sia,ia??bracketleftbigIfracturparenleftbigSia,va?parenrightbigbracketrightbig2 +RfracturparenleftbigSia,va?parenrightbig
2kT
= 1 + 2Rn
parenleftBigg
Gopt +Rfractur
parenleftbigS
ia,va?
parenrightbig
Sva,va?
parenrightBigg
, (1.26)
NFmin = 10 log10 Fmin.
T is noise temperature in Kelvin [20] and is conventionally taken to be room temperature, 290 K.
The chain noise representation correlation matrix can also be expressed using the noise pa-
rameters. As
Sva,va? = 4KTRn, (1.27)
Sia,ia? = 4KTRnvextendsinglevextendsingleYoptvextendsinglevextendsingle2 , (1.28)
Sia,va? = 2KT (Fmin?1)?4kKTRnYopt, (1.29)
14
Figure 1.12: Illustration of noise parameters of a two-port network.
or
CA = 4KT
?
? Rn
Fmin?1
2 ?RnY
?
opt
Fmin?1
2 ?RnYopt Rn
vextendsinglevextendsingleY
opt
vextendsinglevextendsingle2
?
? (1.30)
The detailed derivation procedure can be found in Appendix B.
1.3 SiGe HBTs Noise Performance
When we consider the single transistor as a two-port network, the noise parameters and noise
representation matrices describe the noise characteristics of the transistor itself. They are indepen-
dent of the source and load termination. Among the noise parameters, NFmin is the most important
one. Small NFmin is most desirable for device optimization when noise is regarded.
In [7] and [15], the analytical expression ofNFmin is related to transistor transconductancegm,
current gain ?, base resistance rb and cut-off frequency fT
NFmin = 1 + 1? +
radicalbig
2gmrb
radicalBigg
1
? +
parenleftbiggf
fT
parenrightbigg2
. (1.31)
A few simplifications are employed in derivation. First, intrinsic base and collector terminal cur-
rent noise are "shot" like, 2qI, and assumed to be uncorrelated. Second, base resistance are lumped
equivalent resistance providing thermal noise but does not impact on intrinsic terminal noise cur-
rent propagation. Third, the circuit application requires gmrb >> 1/2. In the following chapters,
15
we will develop more accurate model with correlation and evaluate the base resistance impact on
intrinsic noise propagation.
The simplicity of (1.31) here helps us to understand how the device noise performance is
related to DC/AC characteristics. At a fixed bias, NFmin increases with frequency increasing.
Increasing ?, fT and decreasing rb will help to reduce NFmin. Thus the higher fT, smaller rb
and comparable current gain make SiGe HBTs have a better noise feature when compared to
the traditional Si BJTs, while the continued development in lithography and other innovations
in advanced technology bring a persistent improvement in transistor noise performance. Below
is shown a few examples of the reported NFmin of SiGe HBTs over the last decade. Fig. 1.13
shows NFmin versus frequency for a sampling of IBM high-performance HBTs at the 0.5-, 0.18-,
and 0.13- ?m nodes as well as a slightly higher breakdown voltage variant at the 0.18- ?m node
[3]. NFmin < 1dB is realized in SiGe HBT of 0.13 ?m technology below 20 GHz. Fig. 1.14
and Fig. 1.15 shows NFmin versus JC at 10 GHz and 25 GHz of SiGe HBT from 0.13 ?m SiGe-
BiCMOS process featuring fT = 250 GHz [21]. NFmin below 1 dB is achieved at 10 GHz.
16
Figure 1.13: NFmin versus frequency for SiGe HBTs from four SiGe BiCMOS technologies,
including three high-performance variants at the 0.5-, 0.18-, and 0.13- ?m nodes as well as a
slightly higher breakdown voltage variant at the 0.18- ?m node. This figure is copied from [3] .
Figure 1.14: NFmin versus Jc at 10 GHz for 250 GHz SiGe HBT with AE = 0.094?9.62?m2.
Measured data are shown by symbols. This figure is copied from [4].
17
Figure 1.15: NFmin versus Jc at 25 GHz for 250 GHz SiGe HBT with AE = 0.094?9.62?m2.
Measured data are shown by symbols. This figure is copied from [4].
18
Chapter 2
Noise Compact Modeling and Implementation
Successful circuit design requires accurate compact transistor models that can faithfully and
efficiently describe transistor electrical characteristics across a wide frequency, biasing and temper-
ature range. Accurate noise compact model is in particular important for mixed-signal analog and
RF circuit design, e.g., low noise amplifier (LNA) design. Currently the noise model in industry
standard compact models is essentially the same as what was in early SPICE Gummel-Poon model.
All terminal resistances have the usual 4kTR thermal noise. The base and collector currents show
uncorrelated 2qI "shot" noises which are frequency independent.
Semiconductor technology development is driven by aiming at realizing larger systems and
lower cost with scaling, specifically higher density, lower power and higher speed. Higher opera-
tion frequency is especially requested for RF application and overall challenges the conventional
model accuracy at high frequency. Fig. 2.1 compares noise parameters simulated using IBM? s de-
sign kit which employs the HICUM transistor model with measurement for a SiGe HBT featuring
peak fT = 36 GHz from IBM 0.5/0.35 ?m technology [22]. De-embedding was done by on-chip
open and short structures. The model parameters have been extracted to reproduce well fitted DC
current and S-parameters. The compact model, however, does not reproduce the noise parameters
well. NFmin is severely overestimated above 5 GHz. The real part of Yopt is poorly modeled as
well. The imaginary part of Yopt is well modeled. Rn is reasonably modeled, but increases with
frequency and becomes higher than measurement above 10 GHz.
2.1 van Vliet?s model
Many efforts have been made to improve high frequency compact noise modeling in HBTs.
One popular approach is to develop the noise compact model based on van Vliet?s pioneering
19
0 10 20 30
0
2
4
6
8
frequency (GHz)
NF
min
(dB)
0 10 20 30
0
5
10
15
20
frequency (GHz)
R
n
(
?
)
0 10 20 30
0
0.02
0.04
0.06
frequency (GHz)
Real(Y
opt
) (
?
-1
)
0 10 20 30
-0.06
-0.04
-0.02
0
frequency (GHz)
Imag(Y
opt
) (
?
-1
)
Measurement
Design kit
Figure 2.1: Comparison of noise parameters simulated for an IBM SiGe HBT using design kit with
measurement from 2 to 26 GHz. Inside the design kit, HICUM model is used.
work. Van Vliet?s model was derived about 4 decades ago [23] [12]. Device noise comes from
the fluctuations of the number of particles due to generation and recombination processes and
fluctuations due to thermal motion of carriers (thermal noise or diffusion noise). Impact of these
fluctuations can be taken into account by adding stochastic noise to the continuity and current
equations resulting in the Langevin equations of transport theory [11]. Van Vliet?s model solves the
Langevin equation of base minority carrier velocity fluctuation (and hence current density noise)
propagation towards the two boundaries of the neutral base. In the 1-D condition, the Langevin
equation based on Drift-Diffusion model for the base electron diffusion noise is
Dn ?
2
?x2trianglen+?nE
?
?xtrianglen?
trianglen
?c ?j?trianglen+? (?) = 0. (2.1)
E is the built-in electric field. ?n is the electron high field mobility
?n = ?n0
?
radicalbigg
1 +
parenleftBig
?n0E
?sat
parenrightBig?, (2.2)
20
where ?sat is saturation velocity, ?n0 is low field mobility and ? is a constant specifying how
abruptly the velocity goes into saturation. The noise source is
? (?) =? (?) + 1q ??x? (?), (2.3)
where ? (?) is GR noise source and ? (?) is diffusion noise source
? (?) = 4q2DnN (x). (2.4)
N (x) is the DC electron concentration. GR noise can be negligible at RF and only diffusion noise
is considered. The boundary condition is
trianglen|x=0 = 0,trianglen|x=dB = 0. (2.5)
We can use the Green function method to solve the Langevin equation [9]. van Vliet solved the 3-D
Langevin equation for base electron noise using Green?s function method. A unique feature of van
Vliet?s model is that the base and collector current noises, as well as their correlation, are explicitly
expressed as a function of the intrinsic Y-parameters due to base minority carrier. In the derivation,
the Y-parameters are expressed by Green?s functions in linear fashion using the extended Green
theorem. The PSDs of noise are initially quadratic in Green?s functions. In order to make the
connection between noise PSD and the Y-parameter, it is convenient to transform the noise PSD
into a result whose main part is linear in the Green?s functions. Finally the base electron noises are
related to the Y-parameters of the base region [23] [5]. The original results are in common-base
configuration but can be transformed into common-emitter configuration as
SBib,ib? = 4kTRfracturparenleftbigYB11parenrightbig?2qIBB, (2.6)
SBic,ic? = 2qIC + 4kTRfracturparenleftbigYB22parenrightbig, (2.7)
SBic,ib? = 2kTparenleftbigYB21 +YB?12 parenrightbig?2qIC. (2.8)
The superscriptB refers to base. IBB is the DC neutral base recombination current, which is negligi-
ble in modern transistors. We emphasize thatYB in van Vliet?s derivation refers to the Y-parameters
21
of the intrinsic base region only and requires rigorous frequency domain solution, meaning the
frequency dependence of SBib,ib?, SBic,ic?, and their correlation, are taken into account through the
frequency dependence of the Y-parameters of base YB due to the base electron transport.
The frequency dependence ofYB for the base was first examined by [24] using a 1D transistor
structure.
YB11 = ICV
T
j ??
TB
bracketleftbigg
1?j ??
TB
braceleftbigg?coth ?
2 + 1 +?
??1 + exp (??) ?
3?2
2 (??1 + exp (??))2
bracerightbigg
+...
bracketrightbigg
(2.9)
1
YB21 =
VT
IC
bracketleftbigg
1 +j ??
TB
braceleftbigg?coth ?
2 + 1 +?
??1 + exp (??)
bracerightbigg
+...
bracketrightbigg
(2.10)
where VT =KT/q, ? is a measure for electrical field (assuming ? =triangleEG/VT for SiGe HBT) and
?TB = 1/?b with ?b being base transit time. This result was given for drift transistor when it was
developed. We are using it here for SiGe HBT as there is Ge induced electrical field in the base
and we are using it for constant doping and linear Ge profile. This results can be approximately
regarded as that the base minority carrier charge responds to the base emitter voltage with an
input delay time, after which the collector current at the end of base region responds to the stored
base minority carrier charge with output delay time. The input delay time represents the input
non-quasistatic (NQS) effect and the output delay time represents the output excess phase delay.
Results of YB are shown in Fig. 2.2[5]. With a larger ?, YB has a stronger frequency dependence.
The input NQS effect becomes stronger at a given frequency for a larger Ge gradient device, as
shown byRfractur(Y11). In [25], van Vliet model is extended to include emitter minority carrier induced
base noise. The PSDs of SEBib,ib?, SEBic,ic? and their correlation can be obtained as
SEBib,ib? = 4kTRfracturparenleftbigYEB11 parenrightbig?2qIB, (2.11)
SEBic,ic? = 2qIC + 4kTRfracturparenleftbigYEB22 parenrightbig, (2.12)
SEBic,ib? = 2kTparenleftbigYB21 +YEB?12 parenrightbig?2qIC. (2.13)
22
Figure 2.2: Y-parameter modeling result using equivalent circuit with NQS input. ?0 =
270cm2/(V ?s), electron life time= 1.54x10?7 s,T = 300 K,dB = 45 nm,VBE = 0.8 V.fT = 184
GHz for ? = 5. fT = 83 GHz for ? = 0 (copied from [5]).
where YEB represents Y-parameters of base and emitter region. IB is mainly due to is the hole
current at the emitter injection point, essentially the mount of base current due to the injection of
base holes.
The intrinsic Y-parameters in a compact model do not have all the necessary frequency depen-
dence required for van Vliet?s model, as detailed in Section 2.3. Therefore directly implementing
van Vliet?s model into compact models may cause some issues.
2.2 CB SCR Effect
van Vliet model considers only the minority carrier transport in base but does not consider the
transport effect in collector-base junction space charge region (CB SCR), which becomes impor-
tant and even dominant in the aggressively scaled transistor. The collector current 2qIC noise from
the van Vliet model was derived as the minority electron current noise (for NPN) at the end or col-
lector side of the neutral base. This electron noise current travels through the CB SCR, producing
extra correlated terminal current noises [26], while Y-parameters have considerable contributions
23
from CB SCR transport. From a Y-parameter standpoint, CB SCR electron transport modifies
transistor Y-parameter, and to first order, causes an extra diffusion capacitance component.
Y11 =YB11 + (1??)YB21,Y21 =?YB21, (2.14)
where
?= 1?exp(?2j??c)2j??
c
, (2.15)
?c the well known collector transit time corresponding to CB SCR transport. The drift of electrons
across the CB SCR induces additional base hole current. The noise in the electron current exiting
the neutral base gets transported across the CB SCR, and in the process, directly creates additional
base hole current noise. Depending on HBT design, at higher frequencies this extra base hole
current noise can be comparable to, or even dominate over the "regular? base current noise that is
described by the van Vliet model. The van Vliet?s model has been recently extended for modern
SiGe HBTs, and included CB SCR effect [25]. However, like van Vliet model, the model requires
accurate intrinsic transistor Y-parameters [25] [27]. As a result, empirical modifications have to be
made to produce improved noise modeling result in [28] and [29].
2.3 A survey of previous noise compact modeling methods
Most popular noise compact modeling approach is implementing van Vliet?s model into mod-
ern compact models. This approach is being explored in research versions of two industry standard
compact models, namely Mextram and HICUM [28] [29]. Correlation between base and collector
current noise are added. Note that in the official versions of HICUM and Mextram that are cur-
rently implemented in commercial simulators, including Cadence Spectre and Agilent ADS, the
respective correlated noise implementations reported in [28] and [29] are not yet available.
2.3.1 Noise compact model of Mextram group
This model formulation is developed based on charge partitioning. In charge partitioning, a
part of the total diffusion charge is partitioned between the emitter and the collector. Collector and
24
emitter currents are given by [30] [31]
IE =IDC + (1??cp) ddtQtot, (2.16)
IC =IDC??cp ddtQtot, (2.17)
WhereIDC is the DC emitter to collector transport current. The net build-up of charge isIE?IC =
dQtot/dt. ?cp is the charge partitioning factor and has a value between 0 and 1. 1??cp means
the fraction of charge reclaimed by the emitter, as illustrated in Fig. 2.3. The PSDs of base and
Figure 2.3: Network representation of the charge partitioning model.
collector intrinsic noise and their correlation are given as [28]:
Sib,ib? = 2qIB, (2.18)
Sic,ic? = 2qIC, (2.19)
Sic,ib? = 2qj??cpQtot. (2.20)
In this model formulation, van Vliet?s model is brutally used, together with highly simplifying
and inconsistent use of quasi-static Y-parameters where non-quasistatic Y-parameters are required.
The step-by-step "derivation" of (2.18),(2.19) and (2.20) from van Vliet model, (2.6), (2.7) and
(2.8) is given below. According to Fig. 2.3,
Y11 = IBV
T
+j?Ctot, (2.21)
25
where
Ctot = dQtotdV
BE
. (2.22)
Both forward and reverse Early effect are neglected in IB, IC and Qtot.
Y21 = ICV
T
?j??CPCtot. (2.23)
Sib,ib? = 4KTRfractur(Y11)?2qIB with neglecting Early effect in IB and exponential IB?VBE relation.
Rfractur(Y11) = dIBdVBE = IBVT = qIBKT , which is alway giving frequency independent Sib,ib? = 2qIB. Y21 =
dIC
dVBE ?j??cpCtot =
IC
VT ?j??cpCtot =
qIC
KT ?j??cpCtot gives
Sic,ib? = 2KTY?21?2qIC =j??cpCtot?2KT =j??cp2qVTCtot?. (2.24)
Again Ctot = dQtotdVBE and Qtot = Qtot,s exp
parenleftBig
VBE
VT
parenrightBig
, so CtotVT = Qtot and Sic,ib? = j??cpQtot If normal-
ized correlation C = Sic,ib? negationslashradicalbigSic,ic?Si,ib? is calculated, one may find that |C| > 1 beyond some
frequency point. For purpose of mathematical consistent in the model implementation, an extra
term 2q(??cpQtot)2/IC is added to Sib,ib?. This extra term does not have a physical basis in this
model, but we will show in the later section that "the extra term" becomes important at higher fre-
quency and is essentially the key to improve the Mextram model fitting results. Again, Mextram
noise model does not attempt to include CB SCR effect. In reality, Qtot has all charges and ?cp is
the equivalent to the ratio factor of ?c negationslash ?f. Therefore Mextram model is equivalent to CB SCR
alone model detailed in Section 2.4.1, but physical basis is wrong.
2.3.2 Noise compact model of HICUM group
The PSDs of base and collector of HICUM model are
Sib,ib? = 2qIB
bracketleftBig
1 + 2?qfBfparenleftbig??fparenrightbig2
bracketrightBig
, (2.25)
Sic,ic? = 2qIC, (2.26)
Sic,ib? =?2qICj??f?it. (2.27)
26
?f is transit time, ?qf and ?it are the ratio of the minority charge and transfer current time delay
with respect to the transit time. The model implementation is based on system theory of transfer
of the noise signal in the linear network. The correlation is realized in the compact model using
uncorrelated noise sources and dummy sources.
In this HICUM solution, a frequency dependent Rfractur(Y11) is produced by adding non-quasi-
static effect. That, however, has little effect on the overall transistor Y-parameters, as such fre-
quency dependence is masked by the extra delays caused by parasitics, and is nearly impossible to
extract from measured Y-parameters. In this model, the inherent correlation of base and collector
current noise is still derived from noise transport in the neutral base.
2.3.3 Noise transit time modeling approach
Another recently reported approach is to develop an approximate compact model implementa-
tion of the noise transit time model [32], a small signal model that previously showed good results
in some HBTs [17] [16] [33] [15].
The "noise transit time" ?c in the original small signal models of [17] and [16] had a vague
physical meaning, but was later found to be close to but not identical to the collector transit time?c
in 2-D microscopic noise simulation of a SiGe HBT [33]. This is not a coincidence, and is instead
due to the similarity between the "noise transit" concept and physical process of correlated noise
generation induced by collector-base space charge region (CB SCR) transport [26] [25] [15] [34].
The PSDs are
Sib,ib? = 2qparenleftbigIb +|?ej??|2ICparenrightbig, (2.28)
Sic,ic? = 2qIC, (2.29)
Sic,ib? = 2qICparenleftbigej???1parenrightbig. (2.30)
The compact model implementation of the noise transit time model in [32] has several prob-
lems. It relies on a specific large signal equivalent circuit formulation, and cannot be applied
generally to an existing large signal compact model. In addition, the noise transit time, which
27
is responsible for generating noise correlation and frequency dependence, is set by a transcapac-
itance, and thus will affect both transistor y-parameters and noise parameters. This is undesired
from a general compact modeling standpoint, as one typically would like to have the ability to
separately adjust noise transit time to fit noise parameters without affecting y-parameters.
2.3.4 New method with RC delayed noise current
A new approach to implementing correlated terminal noises by placing an RC delayed noise
current between the base and collector nodes has been recently developed [35]. The frequency
dependence of the intrinsic noise sources due to the CB SCR transport effect has been modeled with
an accuracy value up to the second order of ?. Another advantage of this model is its capability
of implementing both the real and imaginary part of correlation. Overall, it gives the best fit to
measurement data compared with the CB SCR model and SPICE model. A major drawback of
this model is less flexibility for traditional transistor whose noise transport is still dominated by
base transit time instead of CB SCR transit time.
2.4 Noise source model
Based on analysis of prior works detailed above, we would like to avoid all the unphysical
results from trying to implement the van Vliet model in a large signal compact transistor model
using the simplified intrinsic y-parameters available in the compact model. It is also clear that we
need to account for the CB SCR transport effect and eventually keep the general applicability for
those transistor less dominated by ?c .
Fig. 2.4 illustrates the proposed model for including both minority carrier velocity fluctuation
induced terminal current noises and additional noises from the CB SCR transport effect. ib0 is the
base current noise resulting from velocity fluctuation of minority holes in the emitter. ib0 has a
PSD of 2qIB at low frequency. Velocity fluctuations of base minority electrons produce a base
current noise ib1, and an electron current noise at the end of the neutral base, ic1. ib1 and ic1 are
correlated as they both result from base electron velocity fluctuations. Here bi, ci, and ei are the
intrinsic base, collector and emitter nodes in the "parent" compact models, i.e., HICUM, Mextram,
28
Figure 2.4: Illustration the terminal noise currents. ib0 is from emitter hole velocity fluctuation.
ib1 and ic1 are from base electron velocity fluctuation. ic1 becomes ic2 through CB SCR transport,
which also produces ib2 .
or VBIC [36]. Table. 2.1 summarize the specific models related to different noise currents. "Shot"
like model, which is implemented in the standard SPICE simulators, takes frequency independent
and uncorrelated noise sources ib0 and ic1 only into account. Intrinsic base model includes ib0, ib1
and ic1. Correlation comes from ib1 and ic1. The research version of Mextram and HICUM models
are good examples of noise compact model employing intrinsic base transport effect. CB SCR
model considers the CB SCR transport effect besides the "Shot" like noise sources, therefore ic1
is "modified" by CB SCR region and becomes ic2 at the exit of CB SCR. Noise correlation comes
fromic2 and the inducedib2. The "consolidated" model includes the noise transport in both intrinsic
base region and CB SCR. Base and collector noise current correlation comes from ic2 and the sum
of ib1 and ib2. Note that ib1 and ib2 are also correlated.
We bring forward and show the calculation results of different PSDs normalized by 2qIC
based on a 36 GHz SiGe HBTs in Fig. 2.5. The model equations used will be developed and
fully detailed in the following Section 2.4.1 and Section 2.4.2. Device has AE = 0.8?20?m2
29
Table 2.1: Noise currents of different mechanisms
Effect ib ic Correlation
"Shot" like ib0 ic1 None
Intrinsic Base ib0,ib1 ic1 ib1ic1
CB SCR ib0,ib2 ic2 ib2ic2
Intrinsic Base+CB SCR ib0,ib1,ib2 ic2 (ib1 +ib2)ic2
0 5 10 15 20 25 30
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
freq(GHz)
noise current spectral density normalized by 2qI
c
Sib
0
ib
0
*
Sib
1
ib
1
*
Sib
2
ib
2
*
S(ib
1
+ib
2
)
imag(Sic
1
ib
1
*
)
imag(Sic
2
ib
2
*
)
imag(Sic
2
(ib
1
+ib
2
)
*
)
Figure 2.5: Calculated noise current spectral density normalized by 2qIC vs frequency. AE =
0.8?20?m2. IC = 3.55 mA. fT = 15 GHz. ?c is set to 0.7?f, ?b is set to 0.2?f and ? = 3.
and is biased at IC = 3.55 mA with fT = 15 GHz. ?c is set to 0.7?f, ?b is set to 0.2?f and ? = 3.
Intuitively, the base noise current contribution for this device is less than the collector noise current
and their correlation. Also, Sibi?b is dominated by the contribution of Sib2i?b
2
. The imaginary part
ofSic2i?b
12
is dominated by the imaginary part of Sic2i?b
2
. CB SCR transport effect is more important
than the noise transport in the neutral base. Fig. 2.6 shows that the real part of correlation is much
smaller than the imaginary part.
30
0 5 10 15 20 25 30
-0.5
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
freq(GHz)
correlation normalized by 2qI
c
imag(Sic
1
ib
1
*
)
imag(Sic
2
ib
2
*
)
imag(Sic
2
(ib
1
+ib
2
)
*
)
real(Sic
1
ib
1
*
)
real(Sic
2
ib
2
*
)
real(Sic
2
(ib
1
+ib
2
)
*
)
Figure 2.6: Calculated correlation normalized by 2qIC vs frequency. AE = 0.8?20?m2. IC = 3.55
mA. fT = 15 GHz. ?c is set to 0.7?f, ?b is set to 0.2?f and ? = 3.
2.4.1 Consider CB SCR transit time alone
With scaling and the introduction of graded SiGe HBTs, neutral base transit time has been
decreasing constantly. In the same time, the collector-base junction space charge region (CB SCR)
transit time does not decrease as much and becomes significant, which was not accounted for by the
noise and Y-parameter expressions of van Vliet?s. Recently, we have reexamined the microscopic
noise transport in modern SiGe HBTs using van Vliet?s approach, and extended that work by
including the CB SCR transport effect [25] [26]. The resulting model, like van Vliet?s, requires
the use of strictly physically self consistent description of the frequency dependent y-parameters,
and cannot be implemented in compact models required for IC design.
For now, we "neglect" the frequency dependence and correlation of noises resulting from base
electrons, and include only the CB SCR transport effect. This at first was based on the consideration
that the base cutoff frequency 1/2pi?b is much higher thanfT of modern device, and thus for typical
RF applications operating below 20 GHz, the ?low? frequency approximation should hold for van
31
Vliet?s model, at least at low injection, , as ib1 is least important compared in ib0, ib1 and ib2. Both
ib1 and ib0 are much less important than ib2 for the modern device. As shown below, this allows
considerable simplification, and yields a set of noise equations suitable for all existing compact
models with only a single CB SCR transit time parameter ?c. In other words, any deviation of the
base and collector current noises from the traditional independent shot noise model is a result of
CB SCR electron transport effect, rather than frequency dependent base electron noise generation
described by the van Vliet model.
Model Derivation and Description
First, let us consider the base and collector noise currents without CB SCR effect. One might
understand them either from the viewpoint of shot noise producing passage of majority carriers
through emitter-base junction potential barrier or the viewpoint of microscopic minority carrier
velocity fluctuation and transport, which produce the traditional description:
Sib0ib0? =?ib1ib1??/trianglef = 2qIB, (2.31)
Sic1ic1? =?ic1ic1??/trianglef = 2qIC, (2.32)
Sic1ib0? = 0. (2.33)
ib0 is the base noise current associated with hole injection into emitter (or emitter hole velocity
fluctuation in microscopic view), and ic1 is the collector noise current associated with electron in-
jection into base (or base electron velocity fluctuation). Due to the ?low? frequency approximation
we make, they are both frequency independent, and uncorrelated.
Transport of the ic1 noise current across the CB SCR induces an extra base current noise ib2,
which can be calculated from the difference between electron noise current at the exit of CB SCR,
ic2, and ic1, as illustrated in Fig. 2.4. Following [37],
ic2 =ic1 1?exp(?2j??c)2j??
c
. (2.34)
where ?c =Wc/(2?sat), the collector transit time.
32
To obtain expressions suitable for compact modeling, we then assume ?c? lessmuch 1, which is
satisfied for typical applications. The exponential term in (2.34) can then be approximated with a
Taylor series of order 2:
ex?1 +x+ 12x2, (2.35)
(2.34) can then be rewritten as
ic2 ?ic1 (1?j??c). (2.36)
We then have
Sic2ic2? ?2qIC(1 + (??c)2) ?2qIC. (2.37)
ib2 then becomes
ib2 =ic1 ?ic2 =ic1 (j??c), (2.38)
and
Sib2ib2? = 2qIC?2?c2. (2.39)
The correlation term is
Sic2ib2? =?j2qIC??c. (2.40)
We have neglected the real part of the correlation as it is proportional to (??c)2, a second order
term. The normalized correlation C = Sic2ib2?/radicalbigSib2ib2?Sic2ic2? is then negative imaginary unity,
which allows a straightforward implementation into compact models.
Compact Modeling Implementation
Conventional noise computation implemented in all simulators are only able to handle uncor-
related noise sources. Therefore we need to implement the correlations from uncorrelated sources.
Our implementation is similar to that of MOSFET correlated drain and gate noises in MOSFET
recently presented in [38]. Instead of expressing everything in terms of the normalized correla-
tion [38], here we directly implement the correlation between ib2 and ic2. The equivalent circuit is
33
Figure 2.7: Equivalent circuit of proposed compact noise modeling implementation.
shown in Fig. 2.7. Note that ib0 is uncorrelated with ib2 and ic2, and we no longer need ic1 in the
equivalent circuit.
To generate the correlation between ib2 and ic2, we make both current sources controlled by
the voltage of the same dummy node, na. The dummy node is connected to ground by a 1? noise
free resistor. A unity noise source current ina is placed in parallel to the resistor to produce the
dummy noise voltage.
ib2 and ic2 are generated from V (na) as follows
ib2 =gn1ddt(V (na)), (2.41)
ic2 =gn2V (na), (2.42)
34
X01X02X03X02X04X05X06X07X04X08X03X01X01X09X08X0A
X01X0BX06X08X09X04X0CX0DX09X08X0EX01X01X0BX0FX09X08X0A
X01X05X10X11X12X01X13X01X14X15X16X17X18X19X0FX1AX0A
X01X1BX1CX1DX14X01X13X01X05X10X11X12X17X1CX1DX14X0A
X01X1BX1CX1EX1FX01X13X01X05X10X11X12X17X1CX1EX1FX0A
X01X1CX0DX0BX0FX09X08X0EX01X01X20X21X01X01X10X0CX07X05X02X0FX09X11X07X22X02X0DX1FX0EX0A
X01X1CX0DX0BX0FX09X08X0EX01X01X20X21X01X01X23X0DX0BX0FX09X08X0EX0A
X01X24X09X1FX01X13X01X22X12X06X05X0DX01X1BX1CX1DX14X0EX17X25X04X0A
X01X24X09X14X01X13X01X22X12X06X05X0DX1BX1CX1DX14X0EX0A
X01X1CX0DX0BX26X02X0EX01X20X21X01X10X0CX07X05X02X0FX09X11X07X22X02X0DX1BX1CX1EX1FX26X01X27X22X0CX11X05X28X0EX01X21X01X24X09X1FX17X29X29X05X0DX23X0DX0BX0FX09X08X0EX0EX0A
X01X1CX0DX04X26X02X0EX01X20X21X01X24X09X14X17X0DX23X0DX0BX0FX09X08X0EX0EX0AX01X01
Figure 2.8: Verilog-A sample code of proposed noise model.
where gn1 and gn2 are coefficients that can be determined to reproduce (2.39), (2.37) and (2.40).
The results are
gn1 =
radicalbig
2qIC?c, (2.43)
gn2 =
radicalbig
2qIC. (2.44)
In frequency domain, the ddt operator becomes j?.
Sib2ib2? = (gn1?)2?SVna = 2qIC?2?c2, (2.45)
Sic2ic2? =gn22?SVna = 2qIC, (2.46)
Sic2ib2? =gn2?(?j?gn1) =?j2qIC??c. (2.47)
This reproduces (2.39), (2.37) and (2.40). Fig. 2.8 shows a sample Verilog-A code.
The next question is how to determine ?c. In typical compact models, the total transit time ?f
as a function of biasing current and voltage is available. Here we assume that ?c is proportional to
?f, ?c = fg1?f, with fg1 being a proportionality constant determined from fitting noise parameter
versus frequency data. Strictly speaking, one should separately model ?c, preferably from fitting
noise data, which will necessitate additional model development.
35
Experimental Results
The proposed noise model is implemented using Verilog-A, and compared with noise mea-
surements on a SiGe HBT technology optimized for wireless power applications [39]. A 0.8?20?3
?m2 standard device with a 36 GHz peakfT was measured. The HICUM model is used here, while
the model can be implemented with any other compact models. Model parameters are extracted
using standard procedures, including DC and Y-parameter fitting. Base resistance related model
parameters (Rbx and Rbi0) need to be finely tuned against Rn at low frequency. Noise model pa-
rameter fg1 is extracted during optimization of noise parameter fitting at high frequency.
At RF, in addition to the transistor terminal current noise, there is terminal resistance thermal
noise. To examine their relevant importance, we also run simulations with only resistance thermal
noise turned on. Fig. 3.11 shows minimum noise figure NFmin, noise resistance Rn, real part of
optimum generator admittance Rfractur(Yopt), and imaginary part Ifractur(Yopt) as a function of frequency.
VCE=3.3 V and IC=3.55 mA. The traditional independent shot noise model, denoted as "SPICE
noise model", is compared with the proposed model. In both models, the terminal resistance
thermal noise is included.
For NFmin, the resistance noise contribution is negligibly small. However, it is a significant
contributor to noise resistance Rn, primarily from the base resistances. Its impact on Yopt is much
smaller than the base and collector current noises.
NFmin increases monotonically with frequency, as expected, and Rn is almost independent of
frequency. Compared with the traditional model, the new model significantly improves the overall
noise parameter modeling accuracy for all of the four noise parameters. This also indicates the
importance of CB SCR transit time effect on noise parameters. An exception isIfractur(Yopt), which is
slightly worse. The extracted value of fg1 is 0.757, which indicates that ?c dominates ?f in this
technology, which is also expected given the collector design for high breakdown voltage.
Using the same set of model parameters, noise parameters are simulated as a function of IC
at 5 and 10 GHz. VCE=3.3 V. The results are shown in Fig. 2.10 and Fig. 4.17 for 5 and 10 GHz,
respectively. The cutoff frequency fT vs IC is added in Fig. 2.10, and peak fT happens when IC
36
0 10 20 30
0
2
4
6
8
freq (GHz)
NF
min
(dB)
0 10 20 30
0
5
10
15
20
freq (GHz)
R
n
(
?
)
0 10 20 30
0
0.02
0.04
0.06
freq (GHz)
Real(Y
opt
)
0 10 20 30
-0.06
-0.04
-0.02
0
freq (GHz)
Imag(Y
opt
)
Measurement
SPICE Noise Model
New Noise Model
Terminal Resistance Noise Only
V
CE
= 3.3 V
I
C
= 3.55 mA
A
E
= 0.8x20x3?m
2
Figure 2.9: Comparison of measured and simulated noise parameters vs frequency.
37
0 0.01 0.02 0.03
0
2
4
6
8
I
C
(A)
NF
min
(dB)
0 0.01 0.02 0.03
0
5
10
15
20
I
C
(A)
R
n
(
?
)
0 0.01 0.02 0.03
-0.06
-0.04
-0.02
0
I
C
(A)
Imag(Y
opt
)
0 0.01 0.02 0.03
0
0.02
0.04
0.06
I
C
(A)
Real(Y
opt
)
0
40
f
T
(GHz)
Measurement
SPICE Noise Model
New Noise Model
freq = 5 GHz
O
f
T
Figure 2.10: Comparison of measured and simulated noise parameters vs IC at 5 GHz.
= 0.020 A. The fg1 determined from low injection is used as is for all biases. The discrepancy
between model and data becomes larger at higher injection. This could be caused by the deviation
of S?ic1ic1 from 2qIC at high injection and/or the change of fg1.
2.4.2 Intrinsic Base and CB SCR
Solving the microscopic transport equation in intrinsic base will obtain the current noise re-
lated to intrinsic Y-parameters [23]
Sib1ib1? = 4kTRe{Y11,i}, (2.48)
Sic1ic1? = 2qIC + 4kTRe{Y22,i}?2qIC, (2.49)
Sic1ib1? = 2kT (Y21,i +Y12,i?gm). (2.50)
38
0 0.01 0.02 0.03
0
2
4
6
8
I
C
(A)
NF
min
(dB)
0 0.01 0.02 0.03
0
0.02
0.04
0.06
Real(Y
opt
)
0 0.01 0.02 0.03
0
5
10
15
20
I
C
(A)
R
n
(
?
)
0 0.01 0.02 0.03
-0.06
-0.04
-0.02
0
I
C
(A)
Imag(Y
opt
)
Measurement
SPICE Noise Model
New Noise Model
freq = 10 GHz
I
C
(A)
Figure 2.11: Comparison of measured and simulated noise parameters vs IC at 10 GHz.
Approximation to first and second order expansion of Y-parameters under constant doping
and uniform bandgap in base region in frequency domain will get
Sib1ib1? = 2qIC 13 (??b)2 , (2.51)
Sic1ib1? =?j2qIC 13??b. (2.52)
The expressions are easy for compact modeling implementation.
For SiGe HBTs, Ge ramp induces electric field and (2.51) and (2.52) will have to be modified
by ?, which is a measure for the electric field. For the constant doping and electrical field, ? =
39
triangleEg/VT [24], andtriangleEg is the bandgap difference across the neutral base,
Sib1i?b
1
= 2qIC?n (??b)2 , (2.53)
Sic1i?c1 = 2qIC, (2.54)
Sic1i?b
1
=?j2qIC?n??b. (2.55)
?n = 2A (2.56)
?n =A+B (2.57)
A= 3?
2
2 (??1 + exp (??))2 ?
?+ 3
??1 + exp (??) (2.58)
B = ?coth
parenleftbig1
2?
parenrightbig+ 1 +?
??1 + exp (??) ?
3?2
2 (??1 + exp (??))2 (2.59)
We already have Sic2i?c2 (2.37) and we find Sib12i?b
12
and Sic2i?b
12
:
Sib12i?b
12
= 2qICbracketleftbig?n (??b)2 + (??c)2bracketrightbig+ 2qICparenleftbig2?n?2?b?cparenrightbig (2.60)
Sic2i?b
12
=?j2qIC?(?c +?n?b)?2qIC?2?c
parenleftbigg1
3?c +?n?b
parenrightbigg
(2.61)
Note thatib1 andib2 are correlated by the minority carrier transport in the base region , so we denote
ib12 =ib1 +ib2 .
This model is well physics based and can cover:
1. Classic devices have ?b effect only and ? = 0 with uniform base doping. This is van Vliet
model.
2. Classic devices have ?b effect only and ?> 0.
3. Devices have CB SCR effect only, i.e. ?c only.
4. Devices have both intrinsic base and CB SCR effect. ?b > 0 and ?c > 0.
Fig. 2.13 shows model calculation results of (2.37) (2.60) and (2.61)with different ?b, ?c and
? combinations. IC = 3.55 mA, IB = 32?A. fT = 1negationslash (2pi (?b +?c)). ?F = 3 ps.
1. ?b =?F, ?c = 0 and ? = 0.
40
2. ?b =?F, ?c = 0 and ? = 4.
3. ?b = 0, ?c =?F and ? = 0.
4. ?b =?c = 0.5?F and ? = 0.
5. ?b =?c = 0.5?F and ? = 4.
Sici?c does not change. ? = 4 leads to a stronger frequency dependence ofSibi?b and Sici?b. ?c impact
Sibi?b and Sici?b more effectively than ?b.
0 0.2 0.4 0.6 0.8 1
1.114
1.114
1.114
1.114
1.114
1.114
x 10
-21
S
icic
*
(A
2
/Hz)
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
x 10
-21
S
ibib
*
(A
2
/Hz)
0 0.2 0.4 0.6 0.8 1
-0.8
-0.6
-0.4
-0.2
0
x 10
-21
freq/f
T
?
(S
icib
*
) (A
2
/Hz)
0 0.2 0.4 0.6 0.8 1
-1.5
-1
-0.5
0
x 10
-21
freq/f
T
?
(S
icib
*
) (A
2
/Hz)
?
b
= ?
F
, ?
c
=0, ? = 0
?
b
= ?
F
, ?
c
=0, ? = 4
?
b
= 0, ?
c
=?
F
, ? = 0
?
b
= ?
c
= 0.5?
f
, ? = 0
?
b
= ?
c
= 0.5?
f
, ? = 4
Figure 2.12: Calculation results of PSDs of intrinsic ic, ib and their correlation with different ?b, ?c
and ? combinations.
Fig. 2.13 shows normalized correlation results.
1. ?b = ?F, ?c = 0 and ? = 0 leads to normalized correlation to the classical van Vliet model?s
value, C =?jnegationslash?3
2. ? = 4 increases normalized correlation for both cases ?=0 and ?c = 0.5?F.
3. ?c =?F, ?b = 0 and ? = 0 leads to normalized correlation around 1.
41
0 0.2 0.4 0.6 0.8 1
0.5
1
1.5
freq/f
T
Normalized Correlation
?
b
= ?
F
, ?
c
=0, ? = 0
?
b
= ?
F
, ?
c
=0, ? = 4
?
b
= 0, ?
c
=?
F
, ? = 0
?
b
= ?
c
= 0.5?
f
, ? = 0
?
b
= ?
c
= 0.5?
f
, ? = 4
Figure 2.13: Normalized correlation with different ?b, ?c and ? combinations.
2.4.3 General Compact Modeling Implementation
Next task is to generate the correlated ib12 and ic2 noises from uncorrelated white noise
sources.
A close inspection of (2.37), (2.60) and (2.61) shows that:
1. Sic2i?c2 is dominated by a constant 2qIC term, so long as ?2?2c lessmuch 1, typically true. The
negative sign of the second term is difficult to produce without involving additional low-
pass filtering of basic white noise sources. From a modeling standpoint, if we ?change?
the second term to a positive number still proportional to ?2?2c, it can be then produced
in Verilog-A with a ddt function. As long as the coefficient is small, which is the case in
the proposed implementation below, it has negligible effect on final result. This is indeed
verified to be the case. Having addt term in ic2 helps generating a real part of the correlation
that is proportional to ?2.
42
Figure 2.14: Equivalent circuit for Verilog-A implementation of the completely physics based
version of the proposed model.
One way to model ic2 is to use a unity voltage v1 source as illustrated in Fig. 2.14. ddt
becomes j? in noise analysis:
ic2 =g1v1 +g2j?v1, (2.62)
Sic2i?c2 =g21 +g22?2, (2.63)
g1 is then set toradicalbig2qIC.
2. Rfractur
parenleftBig
Sic2i?b
12
parenrightBig
??2, whileIfractur
parenleftBig
Sic2i?b
12
parenrightBig
??. This can be achieved if we have a g3j?v1 term in
ib12 to create the correlation:
ib12 =g3j?v1 +g4j?v2, (2.64)
43
where the other term, g4v2, generates the part of ib12 uncorrelated with ic2, from v2, another
unity voltage noise source uncorrelated with v1. We then have:
Ifractur
parenleftBig
Sic2i?b
12
parenrightBig
=?g1g3?. (2.65)
Rfractur
parenleftBig
Sic2i?b
12
parenrightBig
=g2g3?2, (2.66)
Sib12i?b
12
=g23?2 +g4?2, (2.67)
3. We can therefore determine g3 fromIfractur
parenleftBig
Sic2i?b
12
parenrightBig
and previously determined g1 according to
(2.65).
4. We then determine g2 fromRfractur
parenleftBig
Sic2i?b
12
parenrightBig
and previously determined g3 according to (2.66).
5. We accept the g2 determined above so long as g22?2 lessmuch g21, that is, g22?2 lessmuch 2qIC. This
indeed turns out to be the case. Even in the extreme case of CB SCR effect dominating, the
finalg22?2 is 2qIC??2?2c/9, which is a better approximation ofSic2i?c2 than previous compact
model implementations. For instance, in [32], the approximation is 2qIC(1 +?2?2c ).
6. We then determine g4 from Sib12i?b
12
and previously determined g3.
Overall we have now completely reproduced Sic2i?c2 , Sib12i?b
12
, Rfractur
parenleftBig
Sic2i?b
12
parenrightBig
, Ifractur
parenleftBig
Sic2i?b
12
parenrightBig
using
two uncorrelated unity voltage sources, with a negligible small error in Sic2i?c2 .
The ib0 base current noise is uncorrelated with any others, and can be easily added as shot
noise between bi and ei.
We are now able to evaluate the relative importance of each noise current by calculating
their frequency dependence according the analytical expressions above. We assume ?c and ?b
are proportional to ?f here, ?c and ?b have different VCB dependence though. fg1 and fg2 are
proportionality constants. ?c = fg1?f and ?b = fg2?f. ? is again the Ge induced electric field
constant. ?c = 0.7?f, ?b = 0.2?f and ? = 3, which are estimated from the same 36 GHz SiGe HBT
having total transit time ?f = 3.48ps. Note that for different device from different technology and
application, the ratios of ?c/?f and ?b/?f should be varied and carefully chosen.
44
Table 2.2: Noise Model Parameters
Model fg1 fg2 ?
Spice-like 0 0 0
Intrinsic Base 0 0.2 3
CB SCR 0.7 0 0
Intrinsic Base+CB SCR 0.7 0.2 3
We next examine the results of noise parameters. we are able to obtain the different models
related to different physics effects from the same piece of Verilog-A code, by turning on or off the
three parameter?s value, The HICUM model will be used, while the model can be implemented
with any other compact models. Four cases corresponding to Table 2.1 are present in Table 2.2.
1) Typical Spice model, fg1 = 0, fg2 = 0 and ? = 0 ; 2) Only intrinsic base effect is considered,
fg1 = 0, fg2 = 0.2 and ? = 3 ; 3) Only CB SCR effect is considered,fg1 = 0.7, fg2 = 0 and ? = 0
; 4) both intrinsic base and CB SCR effect are considered, fg1 = 0.7, fg2 = 0.2 and ? = 3. Physics
wise, for a given base, varying ? changes ?b?s value. However, for modeling, one may use ? as an
additional parameter to adjust correlation.
All the DC and AC model parameters are extracted using standard procedures. As we know,
RB affect noise characteristics significantly and is extracted from DC characteristics for this case.
Fig. 2.15 shows minimum noise figure NFmin, noise resistance Rn, real part of optimum gen-
erator admittanceRfractur(Yopt), and imaginary partIfractur(Yopt) as a function of frequency. VCE=3.3 V and
IC=3.55 mA. Using the same set of model parameters, noise parameters are also simulated as a
function of IC at 5 and 10 GHz. VCE=3.3 V. The results are shown in Fig. 2.16 and Fig. 2.17.
The CB SCR models and the model including both CB SCR and intrinsic base transport
effect give similar simulation results and are closer to the measurement data than the first two
models which mainly exclude the CB SCR effect. This means that CB SCR dominates the total
transit time for this technology and enough to be responsible for the noise modeling. As for other
technology, such as a transistor of high performance, the results could be different.
A quick experiment by turning off the CB SCR effect, making ?b = ?f and ? = 3 against
measurement in Fig. 2.18 shows an acceptable fitting results, which is obviously wrong as ?c
45
0 10 20 30
0
2
4
6
8
freq (GHz)
NFmin (dB)
0 10 20 30
0
5
10
15
20
freq (GHz)
Rn (
?
)
0 10 20 30
0
0.02
0.04
0.06
freq (GHz)
Real(Yopt) (
?
-1
)
0 10 20 30
-0.06
-0.04
-0.02
0
freq (GHz)
Imag(Yopt) (
?
-1
)
Measurement
Spice Model
Intrinsic Base
CB SCR
Intrinsic Base+CB SCR
Figure 2.15: Comparison of measured and simulated noise parameters vs frequency.
0 10 20 30
0
2
4
6
8
I
C
(mA)
NFmin (dB)
0 10 20 30
0
5
10
15
20
I
C
(mA)
Rn (
?
)
0 10 20 30
0
0.02
0.04
0.06
I
C
(mA)
Real(Yopt) (
?
-1
)
0 10 20 30
-0.06
-0.04
-0.02
0
I
C
(mA)
Imag(Yopt) (
?
-1
)
0
40
f
T
(GHz)
Measurement
Spice Model
Intrinsic Base
CB SCR
Intrinsic Base + CB SCR
@ 5GHz
f
T
O
Figure 2.16: Comparison of measured and simulated noise parameters vs IC at 5 GHz.
46
0 10 20 30
0
2
4
6
8
I
C
(mA)
NFmin (dB)
0 10 20 30
0
5
10
15
20
I
C
(mA)
Rn (
?
)
0 10 20 30
0
0.02
0.04
0.06
I
C
(mA)
Real(Yopt) (
?
-1
)
0 10 20 30
-0.06
-0.04
-0.02
0
I
C
(mA)
Imag(Yopt) (
?
-1
)
Measurement
Spice Model
Intrinsic Base
CB SCR
Intrinsic Base + CB SCR
@10 GHz
Figure 2.17: Comparison of measured and simulated noise parameters vs IC at 10 GHz.
dominates?f in such power device. Thus the extraction of the noise model parametersfg1, fg2 and
? cannot totally rely on optimizer which may give multiple unreasonable solutions.
2.5 Conclusion
We have reviewed various noise models and developed a physics-based compact noise model
for use with any existing compact transistor model, primarily HICUM, Mextram, and VBIC for
SiGe HBTs, corresponding Verilog-A implementation compatible with all major circuit simulators.
This model is able to be reduced to all extreme cases and therefore cover both conventional discrete
transistors and modern advanced transistors. Comparison with noise measurement on a SiGe HBT
technology shows that overall the noise modeling results are much improved.
47
0 10 20 30
0
2
4
6
8
freq (GHz)
NFmin (dB)
0 10 20 30
0
5
10
15
20
freq (GHz)
Rn (
?
)
0 10 20 30
0
0.02
0.04
0.06
freq (GHz)
Real(Yopt) (
?
-1
)
0 10 20 30
-0.06
-0.04
-0.02
0
freq (GHz)
Imag(Yopt) (
?
-1
)
Measurement
Intrinsic Base ?
b
= ?
f
Figure 2.18: Comparison of measurement and simulated noise parameters by making ?b =?f.
48
Chapter 3
Compact Model Based Noise Extraction
Of critical importance to developing better models for RF noise in terminal noise currents
is to be able to experimentally extract them. There are two kinds of existing extraction methods.
One is based on element by element de-embedding of passive parasitics through a series of two-
port network operations on a simplified small signal equivalent circuit [40], as shown in Fig. 3.1.
Enclosed in the dash box is the intrinsic transistor. Noise sources include thermal noise sources
of rbi, rbx, re and rc, and the intrinsic noise sources of terminal current ib and ic. The other one is
based on a lumped 4-port network modeling of parasitic elements [41][42], as shown in Fig. 3.2.
iN,e1, iN,e2, iN,i1 and iN,i2 are noise current sources at port 1-4, respectively. iN,int1 and iN,int2 are
the noise current sources of the intrinsic two-port system. The 4-port noise current sources and the
correlation matrix can be written as
SY4 =
?
?SYn,ee SYn,ei
SYn,ie SYn,ii
?
?= 4KTReal
?
?Yee Yei
Yie Yii
?
?. (3.1)
Yee, Yei, Yie and Yii are four 2?2 matrices, which can be obtained from 4-port I-V relations. One
can calculate the intrinsic noise correlation matrix as
SYn,int =YT?1 (SYn,total?SYn,ee)Y?T?1?SYn,ii +YT?1SYn,ei +SYn,eiY?T?1, (3.2)
where YT =YeiparenleftbigYINT +Yiiparenrightbig?1. This method requires additional de-embedding structures.
These methods, however, are difficult to implement for extracting noise sources from mea-
sured noise parameters and a transistor compact model. Even if one goes through the lengthy
process of analytically formulating all of the parasitic elements from linearization of the large
signal equivalent circuit, these approaches are highly inefficient and error prone, due to the large
49
Figure 3.1: Illustration of noise extraction method based on small-signal equivalent circuit.
Figure 3.2: Illustration of noise extraction method based on lumped four-port network.
50
number of circuit elements involved and much more complex small signal elements. These ex-
traction methods cannot possibly handle complex controlled sources, which certainly exists in all
compact models, for instance, in the delicate epi-layer models. Furthermore, extraction procedure
is model specific due to its analytical nature.
In this chapter, we present a general purpose method of extracting RF noise in SiGe HBT base
and collector currents using the very same compact models used for RFIC design. It is shown that
the fittings of Y-parameter, noise parameters and the external terminal noise currents are all needed
to meaningfully extract base resistances and intrinsic terminal noise currents experimentally.
3.1 Extraction Method
3.1.1 Basic Idea
Figure 3.3: Illustration of transfer of internal terminal noise currents ibi and ici, various resistance
thermal noise irk to external terminal noise currents ibx and icx with ac shorted base-emitter and
collector-emitter voltages.
Our basic idea is to utilize ac small signal analysis in circuit simulation to find out the response
currents under short circuit boundary conditions placed at the external transistor terminals when
51
we place an ac current excitation at all noise sources, including the intrinsic base and collector
terminals and the terminals across each resistance, as illustrated in Fig. 3.3.
1. ici and ibi are the intrinsic terminal noise currents. The positive directions are from ci to ei
and bi to ei.
2. icx and ibx are the response noise currents, which are the equivalent input and output noise
currents using admittance representation for the whole transistor [43] [7]. The positive di-
rections are from e to c and from e to b. Short circuit are imposed at two ports.
3. each resistor rk has a thermal noise current excitation irk , Sirki?rk = 4kT/rk with k = bi
(intrinsic base), bx (extrinsic base), e (emitter) and c (collector).
Only a few selected parasitic elements are shown in the extrinsic network for illustration.
However, we emphasize that complete intrinsic and extrinsic networks as defined by the compact
model are used in actual extraction.
In matrix form, we can write
?
?icx
ibx
?
? as sum of
?
?icm
ibm
?
?, contribution from ici and ibi, and
contribution from irk, k =bx,bi,e and c as follows.
?
?icx
ibx
?
?=
?
?icm
ibm
?
?+summationdisplay
k
[NTrk]irk, (3.3)
where ?
?icm
ibm
?
?= [NTibc]?
?
?ici
ibi
?
?, (3.4)
[NTibc] is a 2 ? 2 transfer function matrix describing transfer of ici and ibi towards icx and ibx.
[NTrk] is 2?1 transfer function matrix for propagation of irk towards icx and ibx. Observe that all
of the irk are uncorrelated with each other, and uncorrelated with ibi and ici. According to standard
two port noise admittance representation [7][18][43], we denote the correlation matrix of two noise
52
currents i1 and i2 as
Ci1,i2 =
?
?Si1i?1 Si1i?2
Si2i?1 Si2i?2
?
?=
?
?i1
i2
?
?
parenleftBig
i1? i2?
parenrightBig
negationslash trianglef, (3.5)
where Sixi?y = ixi?y/trianglef. Cicx,ibx denotes correlation matrix of icx and ibx; Cici,ibi denotes correlation
matrix of ici and ibi; Cicm,ibm denotes correlation matrix of icm and ibm. Using (3.3)(3.4)(3.5), Cicx,ibx
can be written as
Cicx,ibx =Cicm,ibm +
summationdisplay
k
[NTrk]?Sirki?rk ?[NTrk]?, (3.6)
Cicm,icm = [NTibc]?Cici,ibi?[NTibc]?. (3.7)
summationtext
k[NTrk]?Sirki?rk?[NTrk]
? represents the contributions of all resistance thermal noises toCi
cx,ibx.
Cici,ibi is then related to Cicm,ibm by:
Cici,ibi = [NTibc]?1?Cicm,ibm?([NTibc]?)?1, (3.8)
Cicm,ibm =Cicx,ibx?
summationdisplay
k
[NTrk]?Sirki?rk ?[NTrk]?. (3.9)
3.1.2 Extraction procedure
To extract Cici,ibi from noise parameters, we need to:
1. calculate Cicx,ibx from minimum noise figure NFmin, noise resistance Rn, optimum generator
admittance Yopt and Y-parameters [18] [40] [43]. Correlation matrix is first obtained from
noise parameters , and Y representation matrix is then obtained using matrix transformation
as shown in Chapter 1.
Cicx,ibx =TA2Y
?
?
?4kT
?
? Rn
Fmin?1
2 ?RnY
?
opt
Fmin?1
2 ?RnYopt Rn
vextendsinglevextendsingleY
opt
vextendsinglevextendsingle2
?
?
?
??T?A2Y. (3.10)
53
TA2Y =
?
??Y21 0
?Y11 1
?
?;Fmin = 10NFmin10 .
T here is noise temperature 290 K, which is not same as the T in 4kTrk.
2. simulatesummationtextk[NTrk]?Sirki?rk ?[NTrk]?, the total contribution of terminal resistance thermal
noise to Cicx,ibx. Summation can be simply achieved in a single simulation by turning on all
thermal noise sources (irk) and turning offici and ibi.
3. simulate [NTibc] by placing ac current excitations at the current noise locations and observ-
ing icx and ibx. All the nodes of interest in the large signal equivalent circuit can be made
externally visible and accessible in circuit simulation by using Verilog-A [44]. Fig. 3.4
shows a sample code of nodes setup in VBIC model [36] Verilog-A code and symbols with
external nodes displayed in schematic.
Figure 3.4: Sample Verilog-A code of nodes setup in VBIC model and symbols with external
nodes displayed in schematic.
54
Fig. 3.6 shows an example of simulating NTibx,ci and NTicx,ci. An ac current excitation ici,ac
is placed between ci and ei. cx and bx are both ac shorted to ex. Two elements of [NTibc],
NTibx,ci and NTicx,ci are then evaluated according to their definitions:
NTicx,cidefinesicx,aci
ci,ac
vextendsinglevextendsingle
vextendsingle
ibi,ac=0
,NTibx,cidefinesibx,aci
ci,ac
vextendsinglevextendsingle
vextendsingle
ibi,ac=0
, (3.11)
Similarly, the other two (NTibx,bi and NTicx,bi ) can be obtained by placing an excitation
current source between the intrinsic bi and ei nodes. A screen shot of ADS simulation
schematic is also shown in Fig. 3.6.
The extraction flow is summarized in Fig. 3.7.
Figure 3.5: Illustration of simulating transfer functions NTicx,ci and NTibx,ci by ac small signal
analysis.
3.2 Verification using synthesized data
To verify the proposed intrinsic noise source extraction method, we take our CB SCR noise
model [34] and implement it in the VBIC compact model using Verilog-A [44]. We then run ADS
simulation to generate noise parameters, and use them in place of measured noise parameters to
55
Figure 3.6: Screen shot of ADS simulation schematic.
extract ici and ibi, which are represented by their correlation matrix consisting of Sibii?bi, Sicii?ci and
Sicii?bi.
A comparison with the model equations of ici and ibi implemented in our Verilog-A code
immediately tells us if the extraction method works or not. Since everything is done in simulation,
resistance noise is precisely known, a working extraction should produce the same ici and ibi used
in the model.
The extraction steps are:
1. Calculate correlation matrix of icx and ibx, Cicx,ibx, Sibxi?bx, Sicxi?cx, Sicxi?bx and Sibxi?cx from noise
parameters NFmin, Rn and Yopt. The matrix is obtained from ac analysis using simulator, e.g.
Agilent ADS [45].
2. Calculate Sirki?rk = 4KT negationslash rk from the compact model equation for thermal noise for all
resistances.
3. Calculate (3.9) by subtracting summationtext
k=bx,bi,e,c,s
[NTrk]irk from Cicx,ibx .
4. Simulate the transfer function matrix [NTibc] and [NTrk] by placing an ac current excitation
at the noise source location and observing the responses in icx and ibx.
56
Figure 3.7: Intrinsic noise extraction flow.
Below we show simulations for a SiGe HBT from a 0.18?m BiCMOS technology [46]. The
device has an emitter size of 0.48?20?1 ?m2 and peak fT of 55 GHz. Most of design kit model
parameters are kept as it is, except for Rbx and Rbi, which are finely tuned to fit experimental noise
data, as detailed below. For this device, the extracted fg1 is 0.51, Rbx is 9.2 ? and Rbi is 22.5 ?.
Next section details how to determine their values.
A set of frequency swept data at IC = 3.97 mA is chosen as an example. Fig. 3.8 shows
Sici?c, Sici?b and Sibi?b from different extraction steps. The PSD values of Sibmi?bm, Sicmi?cm and Sicmi?bm are
much smaller thanSibxi?bx, Sicxi?cx andSicxi?bx for the technology used. Sibx,ri?bx,r, Sicx,ri?cx,r andSicx,ri?bx,r are
57
thermal noise contributions, which are dominated by base resistance thermal noise contributions
Sibx,Rbi?bx,Rb, Sicx,Rbi?cx,Rb and Sicx,Rbi?bx,Rb. Furthermore, for the SiGe HBT technology used, base resis-
tance thermal noise contributes significantly toicx and ibx, making intrinsic noise source extraction
more difficult, as detailed below.
0 10 20 30
0
0.2
0.4
0.6
0.8
1
x 10
-20
Frequency (GHz)
S
icic
* (A
2
/Hz)
0 5 10 15 20 25 30
0
2
4
6
x 10
-22
Frequency (GHz)
S
ibib
* (A
2
/Hz)
0 10 20 30
-10
-5
0
x 10
-22
Frequency (GHz)
?
(S
icib
*)(A
2
/Hz)
0 10 20 30
-10
-5
0
x 10
-22
Frequency (GHz)
?
(S
icib
*) (A
2
/Hz)
i
cx
i
bx
*
i
cx,r
i
bx,r
*
i
cx,rb
i
bx,rb
*
i
cm
i
bm
*
i
cx
i
cx
*
i
cx,r
i
cx,r
*
i
cx,rb
i
cx,rb
i
cm
i
cm
i
bx
i
bx
*
i
bx,r
i
bx,r
*
i
bx,rb
i
bx,rb
*
i
bm
i
bm
*
Figure 3.8: Comparison of Sici?c, Sibi?b and Sici?b as a function of frequency.
Fig. 3.9 shows the extracted intrinsic noise from simulation compared with their "input" val-
ues calculated using (2.37), (2.39) and (2.40). The uncorrelated 2qIC and 2qIB are also included
as reference. A perfect reproduction of ici and ibi is observed, proving the correctness of the ex-
traction method.
3.3 Experimental Extraction
We now extract the intrinsic terminal current noises from noise measurements on the same
SiGe HBT. VBIC compact model [36] is used. Ideally, we expect the extraction method can be
directly applied to experimental data after obtaining good dc/ac fitting. However, experimental
extraction is much more complicated as shown below.
58
0 10 20 30
1
1.1
1.2
1.3
1.4
x 10
-21
Frequency (GHz)
S
ic
i
ic
i
*(A
2
/Hz)
0 10 20 30
0
1
2
3
4
x 10
-23
Frequency (GHz)
S
ib
i
ib
i
*(A
2
/Hz)
0 10 20 30
-1
0
1
2
3
x 10
-29
Frequency (GHz)
?
(S
ic
i
ib
i
*)(A
2
/Hz)
0 10 20 30
-2
-1.5
-1
-0.5
0
x 10
-22
Frequency (GHz)
?
(S
ic
i
ib
i
*)(A
2
/Hz)
Extracted
Input
I
C
= 3.97 mA
2qI
B
2qI
C
Figure 3.9: Comparison of the extracted Sicii?ci, Sibii?b
i
and Sibii?b
i
with their input values as a function
of frequency.
3.3.1 Model parameter determination
We start from compact model parameter extraction by fitting dcI?V data and ac Y-parameters.
Using Rb from fitting dc I?V data and ac Y-parameters alone, however, often results in unsat-
isfactory noise parameter fitting as well as unphysical Cici,ibi, because noise parameters and Cici,ibi
extraction results were found to be much more sensitive to Rb than Y-parameters. The extrinsic
base resistance Rbx and the intrinsic Rbi can be approximately lumped into a single Rb for the
device used, as shown below. Fig. 3.10 shows the simulation results of Y-parameters at IC = 3.97
mA, using Rb = 29 ? and 31 ?. They both give good Y-parameter fitting. In fact, other Rb values
close to 30 ? will also give similarly good Y-parameter fitting.
We clearly do not want to apply the whole noise extraction procedure with everyRb value and
then determine which Rb is correct for noise extraction. Therefore we extract Rb by fitting noise
parameters and the noise correlation matrix of icx and ibx together using a correlated noise model.
The simplified version of correlated noise model that neglects the neutral base transit time effect
as detailed in Chapter 2 is used here for its acceptable accuracy for the device used, meaning only
frequency independent base plus CB SCR transport effect is considered.
59
0 10 20 30
0
0.02
0.04
?
(Y
11
) (S)
Frequency(GHz)
0 10 20 30
0
0.02
0.04
?
(Y
11
) (S)
Frequency(GHz)
0 10 20 30
-3
-2
-1
0
x 10
-3
?
(Y
12
) (S)
Frequency(GHz)
0 10 20 30
-3
-2
-1
0
x 10
-3
?
(Y
12
) (S)
Frequency(GHz)
0 10 20 30
0
0.05
0.1
?
(Y
21
) (S)
Frequency(GHz)
0 10 20 30
-0.1
-0.05
0
?
(Y
21
) (S)
Frequency(GHz)
0 10 20 30
0
5
x 10
-3
?
(Y
22
) (S)
Frequency(GHz)
0 10 20 30
0
5
x 10
-3
?
(Y
22
) (S)
Frequency(GHz)
Measurement
Simulation R
b
= 31?
Simulation R
b
= 29 ?
I
C
= 3.97 mA
Figure 3.10: Comparison of Y-parameters from simulation and measurement as a function of fre-
quency at IC = 3.97 mA.
Fig. 3.11 compares noise parameter measurement with simulations obtained using two dif-
ferent intrinsic terminal noise current models, the uncorrelated SPICE model and CB SCR model.
Base resistance values and noise model parameters ?c using CB SCR model are both optimized to
get the best fitting. Simulated noise parameters are clearly better using CB SCR model than SPICE
model.
Fig. 3.12 shows Sicxi?cx, Sibxi?b
x
and Sicxi?b
x
, including those calculated from measured noise
parameters using (3.10) and those optimized simulation results using CB SCR model. Sicxi?cx,
Sibxi?b
x
and Sicxi?b
x
from thermal noise alone are also included. Sicxi?cx and Sibxi?b
x
are dominated by
thermal noise. This, however, does not mean NFmin is dominated by thermal noise, as we will
show below. Together with the noise parameters fitting, Rb is extracted to be 31 ?.
60
0 10 20 30
0
2
4
6
Frequency (GHz)
NF
min
(dB)
0 10 20 30
0
10
20
30
40
50
Frequency (GHz)
R
n
(
?
)
0 10 20 30
0
0.01
0.02
0.03
Frequency (GHz)
?
(Y
opt
) (S)
0 10 20 30
-0.03
-0.02
-0.01
0
Frequency (GHz)
?
(Y
opt
) (S)
Measurement
CB SCR model
SPICE model
I
C
= 3.97 mA
Figure 3.11: Comparison of noise parameters as a function of frequency from measurement and
simulations with thermal noises plus correlated intrinsic current noises and thermal noises plus
uncorrelated intrinsic current noises.
3.3.2 Frequency dependent extraction results
Fig. 3.14 compares the extracted Sicii?ci, Sibii?b
i
and Sicii?b
i
as a function of frequency using Rb =
29 and 31 ?. Modeling results using uncorrelated SPICE model and CB SCR model are both
included for reference. Below 15 GHz, Sicii?ci extracted using Rb = 31 ? is close to 2qIC. The
real part of Siciibi? extracted using Rb = 29 ? has a large positive value, which is contradictory to
known physics that Siciibi? is dominated by the imaginary part [23][12][25]. The imaginary part of
Siciibi? is negative and decreases linearly with frequency for both Rb values. Overall, the extracted
Sicii?ci, Sibii?bi and Sicii?bi are clearly different using two Rb values. Rb = 31 ? gives more reasonable
extraction results than Rb = 29 ? below 15 GHz. Therefore, noise extraction is more sensitive to
Rb than Y-parameters. The extraction ofRb needs to be included as part of the noise extraction. For
61
0 10 20 30
0
0.2
0.4
0.6
0.8
1
x 10
-20
Frequency (GHz)
S
ic
x
ic
x
* (A
2
/Hz)
0 10 20 30
-5
0
x 10
-22
Frequency (GHz)
?
(S
ic
x
ib
x
*) (A
2
/Hz)
0 10 20 30
0
2
4
6
x 10
-22
Frequency (GHz)
S
ib
x
ib
x
* (A
2
/Hz)
0 10 20 30
-1
-0.5
0
x 10
-21
Frequency (GHz)
?
(S
ic
x
ib
x
*) (A
2
/Hz)
Measurement
Correlated+Thermal
Thermal alone
I
C
= 3.97 mA
Figure 3.12: Comparison of Sicxi?cx, Sibxi?bx and Sicxi?bx as a function of frequency from measurement
and simulations with thermal noises plus correlated intrinsic noises, the correlated intrinsic noises
alone and the thermal noises alone.
similar reasons, we use a correlated intrinsic current noise model to help obtain more meaningful
extraction results.
Fig. 3.14 shows the extracted Sicii?ci, Sibii?bi and Sicii?bi as a function of frequency, together with
the correlated model and simple shot noise model for ibi and ici. Below 15 GHz, Sicii?ci is close
to 2qIC. The imaginary part of Sicii?bi is negative and decreases linearly with frequency. The real
part of Sicii?bi is positive and opposite to our model, (2.40). However, it is not important because its
value is much less than the imaginary part of the correlation. Sibii?bi increases with frequency and is
higher than 2qIB.
Above 15 GHz, the noise extraction results are noisy, in part because of the noise in the
original noise parameter measurement, as illustrated by the zoom-in insert in Fig. 3.11.
62
Note that extraction was made with two sets ofRbi and Rbx. The sum of Rbi+Rbx is the same.
At this low IC, the final extraction result is the practically the same, indicating that only the total
value matters.
0 10 20 30
-1
0
1
2
x 10
-21
Frequency (GHz)
S
ic
i
ic
i
*(A
2
/Hz)
0 10 20 30
-1
0
1
2
3
x 10
-22
Frequency (GHz)
S
ib
i
ib
i
* (A
2
/Hz)
0 10 20 30
-5
0
5
10
15
x 10
-23
Frequency (GHz)
?
(S
ic
i
ib
i
*(A
2
/Hz)
0 10 20 30
-4
-2
0
2
4
x 10
-22
Frequency (GHz)
?
(S
ic
i
ib
i
*) (A
2
/Hz)
Meas R
bx
= 9.2 ?R
bi
= 22.5?
Meas R
bx
= 12.2 ?R
bi
= 18.5?
Correlated
Shot
I
C
= 3.97 mA
Figure 3.13: Comparison of extracted Sicii?ci, Sibii?bi and Sicii?bi as a function of frequency from mea-
surement with different values of base resistance and simulations with and without correlation.
Fig. 3.13 shows the extracted Sicii?ci, Sibii?bi and Sicii?bi as a function of frequency, with two
different effectiveRb values. Rb = 31? is optimized with noise parameters after dcI?V and ac Y-
parameter fitting. Rb = 29? is determined only by dc I?V and ac Y-parameter fitting. Rb = 29?
leads to an unphysical extraction results even at low frequency, therefore Rb optimization with
noise parameters are necessary for intrinsic noise extraction.
3.3.3 Bias dependent extraction results
Fig. 3.15 show the extracted Sicii?ci, Sibii?bi and Sicii?bi as a function of IC at 5 GHz. Sibii?bi and
Sicii?bi are well fitted with the Rbi = 22.5? and Rbx = 9.2? except the highest bias point. Above
10 mA, the extracted Sicii?ci is clearly higher than model, i.e. 2qIC. One possible reason is 2qIC
does not describe the current dependence very well at higher current. This result is consistent with
the result obtained from small-signal equivalent circuit based noise extraction in [40]. Another
possible reason is that the current dependence of Rbeff is not well modeled in VBIC.
63
0 10 20 30
-2
0
2
4
x 10
-21
Frequency (GHz)
S
ic
i
ic
i
*(A
2
/Hz)
0 10 20 30
-5
0
5
10
x 10
-22
Frequency (GHz)
S
ib
i
ib
i
* (A
2
/Hz)
0 10 20 30
-5
0
5
10
x 10
-22
Frequency (GHz)
?
(S
ic
i
ib
i
*)(A
2
/Hz)
0 10 20 30
-10
-5
0
5
x 10
-22
Frequency (GHz)
?
(S
ic
i
ib
i
*) (A
2
/Hz)
SPICE model
CB SCR model
ExtractedR
b
= 29?
Extracted R
b
= 31 ?
I
C
= 3.97 mA
Figure 3.14: Comparison of extracted Sicii?ci, Sibii?bi and Sicii?bi as a function of frequency from mea-
surement with two different values of effect base resistances and simulations with and without
correlation.
At higher IC, the two sets of Rbx and Rbi lead to clearly different intrinsic noises, because
Rbx +Rbi/qB are different. At lower IC, qB reduces to unity. Given that in this technology Sicxi?cx
is dominated by thermal noise, the current dependence of Rbeff needs to be further investigated,
possibly with a different model for its current dependence.
The same extraction method and procedure can be applied to any device. We have success-
fully done this on the 36 GHz SiGe HBT used in Chapter 2 as well.
3.4 Conclusion
We have developed a general purpose method of experimentally extracting base resistances
and correlated RF noises in the intrinsic base and collector currents using the same compact model
64
0 5 10 15 20
0
0.5
1
1.5
2
x 10
-20
I
C
(mA)
S
ic
i
ic
i
*(A
2
/Hz)
0 5 10 15 20
0
0.5
1
1.5
2
2.5
x 10
-22
I
C
(mA)
S
ib
i
ib
i
* (A
2
/Hz)
0 5 10 15 20
-2
0
2
4
6
8
x 10
-22
I
C
(mA)
?
(S
ic
i
ib
i
*)(A
2
/Hz)
0 5 10 15 20
-20
-15
-10
-5
0
5
x 10
-22
I
C
(mA)
?
(S
ic
i
ib
i
*)(A
2
/Hz)
Meas R
bx
= 9.2 ?R
bi
= 22.5?
Meas R
bx
= 12.2 ?R
bi
= 18.5?
Correlated
Shot
Frequency = 5GHz
Figure 3.15: Comparison of extracted Sicii?ci, Sibii?bi and Sicii?bi as a function of IC from measurement
with different values of base resistance and simulations with and without correlation.
used for RFIC design. The method is verified with synthetic data, and then applied to measure-
ment data. Practical issues associated with removing thermal noise contributions are discussed,
together with a method for compact noise model parameter extraction. Most meaningful extrac-
tion results are obtained with simultaneous fitting of Y-parameters, noise parameters and external
terminal noise currents using a correlated noise model. The results also show that compact noise
modeling accuracy and noise source extraction from noise parameter measurement both become
more difficult at higher frequency or higher current.
65
Chapter 4
Noise Source Importance Evaluation
As we pointed out in the previous chapters, there are mainly two kinds of RF noise sources in
bipolar transistors, i.e., the terminal resistance thermal noise and the correlated intrinsic terminal
current noises. A logical question is which noise source is more important and in general how to
evaluate their importance, as well as the importance of noise correlation.
Fig. 4.1 shows measured and simulated minimum noise figure (NFmin) versus frequency from
2 to 26 GHz of the same 0.35 ?m technology SiGe HBT used for model development in Chapter 2
at IC = 3.55 mA [22]. The emitter area is 0.8?20?3 ?m2. Simulation is made using a modified
version of the HICUM model that implements the collector base space charge region (CB SCR)
transport noise model described in Chapter 2. NFmin is simulated with all the noise sources turned
on, with only intrinsic current noise sources turned on, and with only thermal noise sources turned
on to evaluate the relative importances of the two types of noise.
The NFmin due to thermal noises alone is very small and below 1 dB in the whole fre-
quency range. The NFmin due to intrinsic current noises is dominant. The NFmin due to intrin-
sic noise from popular uncorrelated 2qI model is larger than that from correlated noise model,
particularly at higher frequencies. It is generally believed that the intrinsic noise correlation
is the key factor to reduce NFmin and many efforts have been made to model noise correlation
[17][16][28][47][48][29][4][35]. However, we will show in the following that the intrinsic noise
correlation itself is not able to reduce NFmin.
One may attempt to conclude from examiningNFmin that intrinsic current noise is much more
important than thermal noise. However, if we examine the power spectral density (PSD) of the
external collector current noise under ac short circuit condition at both the base and collector for a
common emitter configuration, Sici?c, shown in Fig. 4.2 for the same device and biasing condition,
66
0 5 10 15 20 25 30
0
1
2
3
4
5
6
Frequency (GHz)
NF
min
(dB)
V
CE
= 3.3 V
I
C
= 3.55 mA
A
E
= 0.8x20x3?m
2
Measurement
Correlated Current Noise+Thermal Noise
Correlated Current Noise alone
Thermal Noise alone
Uncorrelated Current Noise alone
Figure 4.1: NFmin obtained from measurement andNFmin obtained from simulations with different
noise sources turned on.
we come to a different conclusion. Note that here ic refers to the external collector noise current
obtained (i.e. icx in Fig. 3.3) instead of the intrinsic collector noise current. Thermal noise con-
tribution dominates Sici?c, which has the side effect of making extraction of intrinsic noise current
from noise measurement difficult [49].
A logical and important question is which conclusion is correct, and how the relative im-
portance of different noise source should be evaluated, which we address in this chapter. Instead
of evaluating only NFmin or Sici?c, we examine all noise parameters and all of the elements of the
noise correlation matrices for all noise representations introduced in Chapter 1, including the Y
representation, the Z representation and the chain (ABCD) representation.
To have a complete picture of the relevant importance of individual noise sources, we compare
all elements of the noise correlation matrices for all representation calculated from turning on
individual noise sources. Analytical expressions are derived to obtained better insight.
67
0 5 10 15 20 25 30
0
0.5
1
1.5
2
2.5
3
3.5
x 10
-21
freq (GHz)
S
icic
* A
2
/Hz
Measurement
Simulation
Intrinsic Current Noise alone
Thermal Noise alone
Figure 4.2: Sici?c obtained from measurement and Sici?c obtained from simulations with different
noise sources turned on.
4.1 Equivalent circuit simplification
Our goal is to derive analytical expressions of noise correlation matrices in different represen-
tation, which can then be used to quantify relative importance of each noise source explicitly. The
method is to relate the external correlation matrix of the noise representation to the intrinsic noise
current correlation matrix that is made of Sibii?b
i
, Sicii?ci and Sicii?bi, PSDs of intrinsic noise current ici,
ibi and their correlation. This can be achieved through analytically combining linear noisy two-port
networks, starting from the intrinsic transistor network.
Terminal resistances, the base resistance Rb in particular, have two roles in determining
transistor noise parameters and the equivalent noise sources for any representation. They are
impedance elements, and thermal noise sources at the same time. Fig. 4.3(a) illustrates a typi-
cal equivalent circuit for bipolar transistor. A few selected parasitic elements of extrinsic network
are shown including the resistances which produce thermal noise. The real extrinsic circuit may
be even more complex depending on the complexity of a specified compact model. Therefore, it is
difficult and tedious to have all the extrinsic parasitics included for analytical derivation. We have
68
Figure 4.3: Illustration of the full model and simplified model of the device.
chosen to keep a single lumped Rb in the extrinsic network, as shown in Fig. 4.3(b). The justifica-
tion is that collector resistances are less important and emitter resistance is of very small value for
SiGe HBT. The capacitances are noiseless and do not have distinct impact on Y parameters except
at very high frequencies. Based on this simplified two-port network, we derive below expressions
of noise correlation matrices of the whole transistor for Z, Y and chain representations. To sim-
plify derivation, the intrinsic base current noise source is placed between intrinsic base and emitter
terminals [7].
69
0 20 40 60
0
0.05
0.1
0.15
?
(Y
11
) (S)
0 20 40 60
0
0.05
0.1
0.15
0.2
?
(Y
21
) (S)
Frequency (GHz)
0 20 40 60
-0.08
-0.06
-0.04
-0.02
0
Frequency (GHz)
?
(Y
21
) (S)
0 20 40 60
0
0.05
0.1
0.15
0.2
?
(Y
11
) (S)
intrinsic + R
b
Full model
I
C
= 3.5mA
Figure 4.4: Simulated Y11 and Y21 using the simplified model for intrinsic transistor plus Rb and
using the complete HICUM model .
Fig. 4.4 compares the Y11 and Y21 simulated using the full model, i.e. complete HICUM
model, and the simplified model, i.e. only a lumped Rb is used in the extrinsic network. That is,
all of the parasitics in extrinsic network are removed in the Verilog-A code of HICUM, except for
Rb. The two simulation results are very close. Y11 and Y21 from the simplified model capture the
frequency dependence of those from the full model very well, which justifies the model simplifi-
cation for the HBT used. Y12 and Y22 of the device used are not shown here due to their extremely
small magnitude.
Another evidence to prove that dominance ofRb among the extrinsic elements is to utilize the
noise extraction method proposed in last chapter. To find out if there is one extrinsic element (or
more) that dominates, we can keep extrinsic network elements individually and compare results
70
0 10 20 30
-1
0
1
Frequency (GHz)
?
(NT
ibx,bi
)
0 10 20 30
-0.5
0
0.5
Frequency (GHz)
?
(NT
ibx,ci
)
0 10 20 30
-4
-2
0
Frequency (GHz)
?
(NT
icx,bi
)
0 10 20 30
0
2
4
Frequency (GHz)
?
(NT
icx,bi
)
0 10 20 30
0
1
2
Frequency (GHz)
?
(NT
icx,ci
)
0 10 20 30
-1
0
1
Frequency (GHz)
?
(NT
icx,ci
)
0 10 20 30
-0.5
0
0.5
Frequency (GHz)
?
(NT
ibx,ci
)
0 10 20 30
-0.5
0
0.5
1
1.5
Frequency (GHz)
?
(NT
ibx,bi
)
whole transistor
intrinsic transistor + R
b
intrinsic transistor
intrinsic transistor +R
e
Figure 4.5: Simulated noise transfer functions as a function of frequency at IC = 3.97 mA with no
extrinsic network element, with all extrinsic network elements, with only Rb and with only Re.
with keeping all of them. To insure it is completely clean intrinsic transistor before adding any
parasitics, we can simulate its Cicx,ibx, the correlation matrix of PSDs of icx and ibx, and see if it
is identical to its Cici,ibi, the correlation matrix of ici and ibi from model calculation. The transfer
functions in [NTibc], defined in Chapter 3, are shown in Fig. 4.5. Examples with keeping Re alone
and Rb alone in the extrinsic network are included. With only a lumped Rb = Rbx +Rbi in the
extrinsic network, the transfer functions are nearly identical to those simulated with keeping all
of the extrinsic network elements. The transfer functions simulated with only Re in the extrinsic
network are approximately same as the transfer functions with no extrinsic element.
To further confirm the dominance of Rb in determining the transfer function matrix [NTibc],
we compare Cicm,ibm, the correlation matrix of icx and ibx due to ici and ibi, obtained using (3.7),
with [NTibc] simulated using complete extrinsic network and with [NTibc] simulated using only
a lumped Rb in the extrinsic network, as shown in Fig. 4.6. The two Cicm,ibm obtained are ap-
proximately identical. Therefore, Rb is the most dominant element in the extrinsic network in
determining [NTibc].
71
0 10 20 30
-1
0
1
2
x 10
-21
Frequency (GHz)
S
ic
m
ic
m
*
(A
2
/Hz)
0 10 20 30
-1
0
1
2
3
x 10
-22
Frequency (GHz)
S
ib
m
ib
m
*
(A
2
/Hz)
0 10 20 30
-1
-0.5
0
0.5
1
1.5
x 10
-22
Frequency (GHz)
?
(S
ic
m
ib
m
*
) (A
2
/Hz)
0 10 20 30
-4
-2
0
2
4
x 10
-22
Frequency (GHz)
?
(S
ic
m
ib
m
*
) (A
2
/Hz)
C
ibm,icm
C
ibm,icm
Rb only
Figure 4.6: The elements of Cicm,ibm obtained with [NTibc] simulated using complete extrinsic net-
work compared with the elements of Cicm,ibm obtained with [NTibc] simulated using only a lumped
Rb in the extrinsic network.
4.2 Analytical derivations
In all derivations, we include Rb?s contribution in two steps. In step 1, Rb only acts as an
element of extrinsic network which modifies the propagation of ibi and ici, and changes two-port
parameters. In step 2, the thermal noise Rb is added.
4.2.1 Z-Noise Representation
Fig. 4.7(a) illustrates Z-noise representation. v1i and v2i are the equivalent input and output
voltage noise sources for the intrinsic network. Fig. 4.7(b) and Fig. 4.7(c) show that adding Rb
between external input terminal and v1i is equivalent to adding Rb between external v1i and the
intrinsic base terminal. Therefore, v1i is equal to v1, v1 =v1i and v2 =v2i. Adding the noiseless Rb
does not affect v1 and v2 at all.
The two-port parameters, Z parameters, however, are affected. Here we denote the Z pa-
rameter matrix as TZ rather than Z as it is going to be used to transform CY to CZ, according to
72
Figure 4.7: Illustration of Z presentation in the cases of the intrinsic device alone and the intrinsic
device plus Rb.
Table 1.1. TZ is different from TintZ due to Rb:
TZ =
?
?Z
int
11 +Rb Z
int
12
Zint21 Zint22
?
?, (4.1)
and
TintZ =
?
?Z
int
11 Z
int
12
Zint21 Zint22
?
?, (4.2)
which is an excellent demonstration of Rb?s impact as an impedance element.
73
0 20 40 60
0.5
1
1.5
2
x 10
-19
freq (GHz)
S
v1v1*
(V
2
/Hz)
0 20 40 60
2
4
6
8
x 10
-12
freq (GHz)
S
v2v2*
(v
2
/Hz)
0 20 40 60
-3
-2.5
-2
x 10
-16
freq (GHz)
?
(S
v1v2*
) (V
2
/Hz)
0 20 40 60
0
2
4
6
x 10
-17
freq (GHz)
?
(S
v1v2*
) (V
2
/Hz)
Noisy R
b
Noiseless R
b
Rb = 0 ?
Figure 4.8: The correlation matrix of Z-noise representation withRb = 0, noiselessRb and thermal
noisy Rb.
In general from two-port theory detailed in Chapter 1,
CZ =
?
?v1
v2
?
?
parenleftBig
v1 v2
parenrightBig?
negationslash trianglef
=TintZ
?
?i1
i2
?
?
parenleftBig
i1 i2
parenrightBig?
negationslash trianglefTintZ ? (4.3)
The Z representation noise correlation matrix can be represented as
CZ =CintZ =TintZ CY,iTintZ ?, (4.4)
where
CYi =
?
?Sibiib?i Sibiic?i
Siciib?i Siciic?i
?
?, (4.5)
74
and
TintZ =
?
?
Yint22
triangleYint ?
Yint12
triangleYint
? Yint21triangleYint Yint22triangleYint
?
?,triangleYint =Yint11 Yint22 ?Yint12 Yint21 . (4.6)
This is simply result of Z =Y?1. CZ can be related to CYi and the intrinsic Yint as
Sv1v?1 = 1|triangleYint|2
parenleftBigvextendsinglevextendsingle
Yint22 vextendsinglevextendsingle2Sibii?bi?2RfracturparenleftbigYint12 Yint22i?Sicii?biparenrightbig+vextendsinglevextendsingleYint12 vextendsinglevextendsingle2 Sicii?ci
parenrightBig
, (4.7)
Sv1v?2 = 1|triangleYint|2 parenleftbig?Yint22 Yint21 ?Sibii?bi +Yint12 Yint21 ?Sicii?bi +Yint22 Yint11 ?Sibii?ci?Yint12 Yint11 ?Sicii?ciparenrightbig,
Sv2v?2 = 1|triangleYint|2
parenleftBigvextendsinglevextendsingle
Yint21 vextendsinglevextendsingle2 Sibii?bi?2RfracturparenleftbigYint11 Yint21 ?Sicii?biparenrightbig+vextendsinglevextendsingleYint11 vextendsinglevextendsingle2 Sicii?ci
parenrightBig
.
The noiselessRb gives the sameCZ as the intrinsic network only. Although thermal noiseless
Rb has no impact on the correlation matrix of Z-noise representation, it still changes noise param-
eters through Y parameters, as detailed below. Thermal noise ofRb will only add a 4kTRb term to
Sv1v1?, as shown in Fig. 4.8. Therefore
CZ(noisy Rb) =CZ(noiseless Rb) +CRbZ =
?
?Sv1v?1 Sv1v?2
Sv2v?1 Sv2v?2
?
?+
?
?4kTRb 0
0 0
?
?. (4.8)
We then compare calculation results using (4.8) from three scenarios: 1. Rb = 0 ?; 2.
a noiseless Rb = 6.8 ?, its experimentally extracted value; 3. a noisy Rb = 6.8 ?. Fig. 4.8
shows results of comparison up to 50 GHz. The results with Rb = 0 ? and noiseless Rb show no
difference. Overall Sv1v?1 is much smaller than Sv2v?2 and Sv2v?1 , which are dominated by intrinsic
current noise.
4.2.2 Y-Noise representation
We first examine how ici and ibi propagate to external node by a thermal noiseless Rb under
ac short circuit conditions at both the input and the output. The resulting currents at the collector
and base are denoted as icm and ibm. The correlation matrix is defined by
CY (noiseless Rb) =
?
?Sibmi?bm Sibmi?cm
Sicmi?bm Sicmi?cm
?
?. (4.9)
75
Figure 4.9: Illustration of Y presentation in the cases of the intrinsic device alone and the intrinsic
device plus Rb
As shown in Fig. 4.9, noises of two port network in Z representation can be equivalently
expressed by Y representation. The intrinsic Y representation noise correlation matrix CY,i can be
transformed from CZ,i multiplied by an transformation matrix TintY , which is equal to the intrinsic
Y parameter matrix.
CY,i =
?
?i1
i2
?
?
parenleftBig
i1 i2
parenrightBig?
negationslash trianglef =TintY
?
?v1
v2
?
?
parenleftBig
v1 v2
parenrightBig?
negationslash trianglefTintY ? =TintY CZ,iTintY ?,
where
TintY =
?
?Y
int
11 Y
int
12
Yint21 Yint22
?
?. (4.10)
With the noiseless Rb,
CY (noiseless Rb) =TYCZ(noiseless Rb)TY?, (4.11)
where
TY =
?
?Y11 Y12
Y21 Y22
?
? (4.12)
= 11 +Yint
11 Rb
?
?Y
int
11 Y
int
12
Yint21 Yint22 +RbtriangleYint
?
?. (4.13)
The detailed derivation of relation betweenTintY andTY can be found in Appendix E. AsCZ(noiseless Rb) =
CZ,i, we can relate CY (noiseless Rb) to CY,i by matrix operation. Recall from Subsection 4.2.1,
76
CintZ =TintZ CY,iTintZ ?, we then have
CY (noiseless Rb) =TYTintY ?1CY,iTintY ??1TY?. (4.14)
We can now obtain the PSDs of external noise currents as
Sibmi?bm = Sibii
?
bivextendsingle
vextendsingle1 +Yint11 Rbvextendsinglevextendsingle2,
Sicmi?cm =Sicii?ci + R
2
b
vextendsinglevextendsingleYint
21
vextendsinglevextendsingle2 S
ibii?bivextendsingle
vextendsingle1 +Yint11 Rbvextendsinglevextendsingle2 ?
2RfracturparenleftbigRbYint21 ?Siciibi?parenrightbig
parenleftbig1 +R
bYint11
?parenrightbig , (4.15)
Sicmi?bm =? RbY
int
21 Sibii?bivextendsingle
vextendsingle1 +Yint11 Rbvextendsinglevextendsingle2 +
Sicii?bi
parenleftbig1 +R
bYint11
?parenrightbig.
With an extremely small Rb, the external PSDs reduce to the intrinsic ones.
The contribution of Rb?s thermal noise to ib and ic can be considered as ib,Rb = 4kTRbY11 and
ic,Rb = 4kTRbY21 under ac short conditions. Y11 and Y21 can be related to Yint11 and Yint21 by (4.13).
Therefore the contribution of Rb?s thermal noise to total Sicic?, Sibib? and Sici?b are
SibRbi?b
Rb
= 4kTRb
vextendsinglevextendsingle
vextendsinglevextendsingle Y
int
11
1 +Yint11 Rb
vextendsinglevextendsingle
vextendsinglevextendsingle
2
,
SicRbi?cRb = 4kTRb
vextendsinglevextendsingle
vextendsinglevextendsingle Y
int
21
1 +Yint11 Rb
vextendsinglevextendsingle
vextendsinglevextendsingle
2
, (4.16)
SicRbi?b
Rb
= 4kTRb Y
int
21 Y
int
11
?
vextendsinglevextendsingle1 +Yint
11 Rb
vextendsinglevextendsingle2.
The values of SibRbi?b
Rb
, SicRbi?cRb and SicRbi?cRb increase with frequency and are related to f2. As
shown in [49], base resistance thermal noise dominates over the total thermal noise contributions
and contributes significantly to ic and ib. Overall,
CY (noisy Rb) =CY (noiseless Rb) +CRbY =
?
?Sibmi?bm Sibmi?cm
Sicmi?bm Sicmi?cm
?
?+
?
?Sib,Rbi?b,Rb Sib,Rbi?c,Rb
Sic,Rbi?b,Rb Sic,Rbi?c,Rb
?
?. (4.17)
77
Fig. 4.10 shows the calculation results using (4.15)-(4.17). Sici?c, Sibi?b and the imaginary part of
Sici?b are dominated by thermal noise. The real part of Sici?b is less important because the value of
correlation is dominated by its imaginary part.
0 20 40 60
0
1
2
3
4
x 10
-21
freq (GHz)
S
icic*
(A
2
/Hz)
0 20 40 60
0
0.5
1
1.5
2
2.5
x 10
-21
freq (GHz)
S
ibib*
(A
2
/Hz)
0 20 40 60
-1
0
1
2
3
x 10
-22
freq (GHz)
?
(S
icib*
) (A
2
/Hz)
0 20 40 60
-2
-1.5
-1
-0.5
0
x 10
-21
freq (GHz)
?
(S
icib*
) (A
2
/Hz)
R
b
= 0 ?
Noiseless R
b
Noisy R
b
R
b
thermal noise alone
Figure 4.10: Correlation matrix of Y representation from calculations.
4.2.3 Chain Noise Representation
We now derive the chain noise representation matrix expressions, which directly relate to
noise parameters. Recall from Chapter 1, the chain noise representation correlation matrix CA can
be transformed from CY multiplied by an transformation matrix TA,
CA =TACYTA?, (4.18)
where
TA =
?
?0 A12
1 A22
?
?=
?
?0
?1
Y21
1 ?Y11Y21
?
?=
?
??0 ?
1+Yint11 Rb
Yint21
1 ?Yint11Yint
21
?
??. (4.19)
78
The expressions of Siai?a, Svav?a and their correlation can be obtained from (4.15)-(4.18). Without
thermal noise,
Siai?a =Sibii?bi +
vextendsinglevextendsingle
vextendsinglevextendsingleY
int
11
Yint21
vextendsinglevextendsingle
vextendsinglevextendsingle
2
Sicii?ci?2Rfractur
parenleftbiggYint
11
Yint21 Sicii
?
bi
parenrightbigg
, (4.20)
Svav?a = Sicii
?
civextendsingle
vextendsingleYint21 vextendsinglevextendsingle2 +Sibii
?
biR
2
b?2Rfractur
parenleftbiggR
bSicii?bi
Yint21
parenrightbigg
+
parenleftBigvextendsinglevextendsingle
Yint11 Rbvextendsinglevextendsingle2 + 2RfracturparenleftbigYint11 Rbparenrightbig
parenrightBig S?i
ciicivextendsingle
vextendsingleYint21 vextendsinglevextendsingle2
?2Rfractur
parenleftBigg
R2bYint11 ?
Yint21 Sicii
?
bi
parenrightBigg
,
Siav?a = Y
int
11vextendsingle
vextendsingleYint21 vextendsinglevextendsingle2Sicii
?
ci?
Sibii?ci
Yint21 ? +RbSibii
?
bi?2Rfractur
parenleftbiggR
bYint11
Yint21 Sicii
?
bi
parenrightbigg
+ Rb
vextendsinglevextendsingleYint
11
vextendsinglevextendsingle2
vextendsinglevextendsingleYint
21
vextendsinglevextendsingle2 Sicii?ci.
Observe from (4.20) that a noiseless Rb changes not only the Y parameters but also introduces
many terms to Svav?a and Siav?a, although noiseless Rb has no impact on Siai?a. The thermal noise of
Rb adds only a 4kTRb term to Svav?a. That is
CA(noisy Rb) =CA(noiseless Rb) +CRbA =
?
?Svav?a Svai?a
Siav?a Siai?a
?
?+
?
?4kTRb 0
0 0
?
?. (4.21)
Fig. 4.11 shows the simulation results for all three scenarios for chain noise representation. It is
clear that Rb has only changed Svav?a and Rfractur(Svai?a), and thermal noise contribution shows up only
in Svav?a. Intrinsic current noise dominates Siai?a andIfractur(Svai?a)
4.3 Noise Parameter Implications
4.3.1 Analytical Models of Noise Parameters
To obtain additional insight into device design and optimization for device noise performance,
analytical expressions of noise parameters are desirable. The noise figure and noise parameters
can always be related to the noise representations and small-signal parameters of the device. In
this section, we derive the expressions of noise parameters based on the simplified small-signal
equivalent circuit. We assume device is at room temperature and equal to noise temperature 290
K.
79
0 20 40 60
0
1
2
3
x 10
-19
freq (GHz)
S
vava*
(V
2
/Hz)
0 20 40 60
0
0.5
1
1.5
x 10
-21
freq (GHz)
S
iaia*
(A
2
/Hz)
0 20 40 60
0
0.5
1
x 10
-20
freq (GHz)
?
(S
iava*
) (W/Hz)
0 20 40 60
0
2
4
6
8
x 10
-21
freq (GHz)
?
(S
iava*
) (W/Hz)
Noisy R
b
Noiseless R
b
R
b
= 0
Figure 4.11: Correlation matrix of chain noise representation from calculation with Rb = 0?,
thermal noiseless Rb and thermal noisy Rb.
The simplified equivalent circuit is shown in Fig. 4.12. The intrinsic Y parameter can be
obtained as,
YINT11 =j?Cbe;
YINT21 =gm; (4.22)
YINT12 =YINT22 = 0.
Other assumption includes:
fT = gm2piC
be
. (4.23)
The noise figure of the device can be defined as
F = 1 + Noise output due to deviceNoise output due to source. (4.24)
80
Figure 4.12: Equivalent small-signal circuit with source noise vs, device thermal noise vb and
current noise ib and ic.
The transistor has noise sources including the terminal noise currentic andib and the thermal noise
vb of base resistance rb. Power Source has a noise source vs. The source impedance is ZS. Noise
current ic and ib are correlated to each other, and thermal noise vb is independent to ic and ib.
Therefore, NF can be rewritten as
NF = 1 + < (iout,ic +iout,ib),(iout,ic +iout,ib)
? >+
. (4.25)
iout,ic, iout,ib,iout,rb and iout,Rs are output noise current respectively due to ic, ib, rb and Rs (the real
part of ZS). They can be calculated by removing all the other noise sources.
iout,ic
vbe?j?Cbe?(ZS +rb) +vbe + (vbe?j?Cbe +gmvbe +ic)re = 0 (4.26)
As re of device is small in the modern technology, we neglect re for now. The equivalent circuit is
even simplified as in Fig. 4.13. Then
vbe = 0 (4.27)
and
iout,ic =ic (4.28)
81
Figure 4.13: Equivalent small-signal circuit with source noise vs, device thermal noise vb and
current noise ib and ic. Emitter resistance is neglected.
iout,ib
(vbe?j?Cbe +ib) (ZS +rb) +vbe = 0
vbe =? ib (ZS +rb)j?C
be (ZS +rb) + 1
(4.29)
and
iout,ib =? gmib (ZS +rb)j?C
be (ZS +rb) + 1
. (4.30)
< (iout,ic +iout,ib),(iout,ic +iout,ib)? >
The output noise power produced by ic and ib is
(iout,ic +iout,ib),(iout,ic +iout,ib)?
=
parenleftbigg
ic? gmib (ZS +rb)j?C
be (ZS +rb) + 1
parenrightbiggparenleftbigg
ic? gmib (ZS +rb)j?C
be (ZS +rb) + 1
parenrightbigg?
(4.31)
As ZS =RS +jXS, (4.31) can be written as
< (iout,ic +iout,ib),(iout,ic +iout,ib)? >=+ g2
m|ZS +rb|
2
1 +?2C2be|ZS +rb|2?2?CbeXS
?gm (ZS +rb)?+gm (ZS +rb) +gmj?Cbe|ZS +rb|2 (?)
1 +?2C2be|ZS +rb|2?2?CbeXS
82
Let us define = (Gu +jBu)trianglef and = (Gu?jBu)trianglef,
< (iout,ic +iout,ib),(iout,ic +iout,ib)? >
=+ g2
m|ZS +rb|
2
1 +?2C2be|ZS +rb|2?2?CbeXS
? 2Gutrianglefgm (RS +rb) + 2ButrianglefgmXS?2Butrianglefgm?Cbe|ZS +rb|
2
1 +?2C2be|ZS +rb|2?2?CbeXS
(4.32)
iout,rb
vbe?j?Cbe (ZS +rb) +vbe +vb = 0
vbe =? vbj?C
be (ZS +rb) + 1
(4.33)
and
iout,rb =? gmvbj?C
be (ZS +rb) + 1
(4.34)
The output noise power produced by rb is
=
parenleftbigg
? gmvbj?C
be (ZS +rb) + 1
parenrightbiggparenleftbigg
? gmvbj?C
be (ZS +rb) + 1
parenrightbigg?
= g
2
m
1 +?2C2be|ZS +rb|2?2?CbeXS (4.35)
iout,RS
vbe?j?Cbe (ZS +rb) +vbe?vS = 0
vbe = vSj?C
be (ZS +rb) + 1
(4.36)
and
iout,RS = gmvSj?C
be (ZS +rb) + 1
(4.37)
83
The output noise power produced by RS is
=
parenleftbigg
? gmvSj?C
be (ZS +rb) + 1
parenrightbiggparenleftbigg
? gmvSj?C
be (ZS +rb) + 1
parenrightbigg?
= g
2
m
1 +?2C2be|ZS +rb|2?2?CbeXS (4.38)
Noise figure
Substituting (4.32), (4.35) and (4.38) into (4.25) , we get
NF = 1 + rbR
S
+ Sici
?c
parenleftbig1 +?2C2
be
parenleftbig(R
S +rb)2 +X2S
parenrightbig?2?C
beXS
parenrightbig
g2m4KTRS
+ Sibi
?
b
parenleftbig(R
S +rb)2 +X2S
parenrightbig
4KTRS ?
2Gu (RS +rb)
gm4KTRS ?
2BuXS
gm4KTRS +
2Bu?Cbeparenleftbig(RS +rb)2 +X2Sparenrightbig
gm4KTRS
(4.39)
Therefore, the noise figure NF is related to the source impedance, the device small-signal equiva-
lent circuit elements and the noise spectrum densities of internal terminal noise currents.
With rb = 0
To find out the optimum XS, Xopt to minimize the noise figure, we have
?F
?XS = 0 (4.40)
XS
parenleftbigg
Sici?c +Sibi?b
parenleftBig?T
?
parenrightBig2
+ 2Bu?T?
parenrightbigg
? 1?C
be
parenleftBig
Sici?c +Bu?T?
parenrightBig
(4.41)
where ?T =gm/Cbe. Therefore, we obtain the expression of Xopt,
Xopt =
1
?Cbe
parenleftbigS
ici?c +Bu?T?
parenrightbig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
. (4.42)
To find out the optimum RS, similar to obtain Xopt, we have
?F
?RS = 0 (4.43)
84
Sici?c
parenleftBigg
R2S?
parenleftbigg
XS? 1?C
be
parenrightbigg2parenrightBigg
+Sibi?bparenleftbigR2S?X2Sparenrightbig
parenleftBig?T
?
parenrightBig2
+ 2Bu?T?
parenleftbigg
R2S?X2S + XS?C
be
parenrightbigg
= 0
(4.44)
Substituting XS with Xopt of (4.42), we obtain Ropt,
R2opt =
Sici?c
parenleftBig
1
?Cbe
parenrightBig2
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
?
parenleftBigg 1
?Cbe
parenleftbigS
ici?c +Bu?T?
parenrightbig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
parenrightBigg2
(4.45)
Ropt =
radicaltpradicalvertex
radicalvertexradicalvertex
radicalbt Sici
?c
parenleftBig
1
?Cbe
parenrightBig2
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
?
parenleftBigg 1
?Cbe
parenleftbigS
ici?c +Bu?T?
parenrightbig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
parenrightBigg2
(4.46)
Substituting (4.42) and (4.46) into (4.39) leads to an expression of NFmin:
NFmin = 1? 2Gug
m4KT
+
radicalBig
Sici?cSibi?b ?B2u
2KTgm . (4.47)
As|Gu|<<|Bu|, we assume Gu = 0 here,
NFmin = 1 +
radicalBig
Sici?cSibi?b ?B2u
2KTgm . (4.48)
With CB SCR model in Chapter 2,
Sici?c = 2qIC, (4.49)
Sibi?b = 2qIB + 2qIC(??c)2, (4.50)
Gu = 0,Bu =?j2qIC??c. (4.51)
(4.47) is rewritten as
NFmin = 1 + q
radicalbigI
CIB
KTgm . (4.52)
It is clear to see that NFmin is independent of correlation between base and collector noise current
and even independent of frequency. This is against existing understanding that noise correlation in
general decreases NFmin, which also implies that the reduction of NFmin due to noise correlation
85
depends on rb. With further simplification, gm =qIC/KT and ? =IC/IB,
NFmin = 1 +
radicalBigg
1
?. (4.53)
With thermal noiseless rb
Xopt does not change compared to rb = 0,
Xopt =
1
?Cbe
parenleftbigS
ici?c +Bu?T?
parenrightbig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
. (4.54)
Ropt
R2opt =r2b +
parenleftBig
1
?Cbe
parenrightBig2parenleftbig
Sici?c ?2Gurbgmparenrightbig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
?
parenleftBig
1
?Cbe
parenrightBig2parenleftbig
Sici?c +Bu?T?parenrightbig2
parenleftBig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Buparenleftbig?T?parenrightbig
parenrightBig2 (4.55)
Ropt =
radicaltpradicalvertex
radicalvertexradicalvertex
radicalvertexradicalbtr2
b +
parenleftBig
1
?Cbe
parenrightBig2parenleftbig
Sici?c ?2Gurbgmparenrightbig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
?
parenleftBig
1
?Cbe
parenrightBig2parenleftbig
Sici?c +Bu?T?parenrightbig2
parenleftBig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Buparenleftbig?T?parenrightbig
parenrightBig2 (4.56)
Substituting (4.54) and (4.56) into (4.39) and assuming Gu = 0, NFmin becomes
Fmin = 1 + 12KT
parenleftbigg?
?T
parenrightbigg2parenleftbig
Ropt +rbparenrightbig?
parenleftbigg
Sici?c +Sibi?b
parenleftBig?T
?
parenrightBig2
+ 2Bu?T?
parenrightbigg
. (4.57)
Using the CB SCR transport model equations of (4.49),(4.50) and (4.51),
Fmin = 1 +
?
??
radicaltpradicalvertex
radicalvertexradicalbt
g2mr2b
parenleftBiggparenleftbigg
?
?T
parenrightbigg2
(1??T?c)2 + 1?
parenrightBigg2
+ 1? +gmrb
parenleftBiggparenleftbigg
?
?T
parenrightbigg2
(1??T?c)2 + 1?
parenrightBigg?
??.
(4.58)
Fmin = 1 +
radicalBigg
g2mr2b
parenleftbigg
(1??T?c)2 + 1?
parenrightbigg2
+ 1? +gmrb
parenleftBiggparenleftBiggparenleftbigg
?
?T
parenrightbigg2
(1??T?c)2 + 1?
parenrightBiggparenrightBigg
. (4.59)
86
With rb thermal noise
rb does not impact on Xopt
Xopt =
1
?Cbe
parenleftbigS
ici?c +Bu?T?
parenrightbig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
. (4.60)
The thermal noise of rb results in an additional term in R2opt
R2opt = 4KTrb
parenleftbig?T
?
parenrightbig2
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
+r2b +
parenleftBig
1
?Cbe
parenrightBig2parenleftbig
Sici?c ?2Gurbgmparenrightbig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
(4.61)
?
parenleftBig
1
?Cbe
parenrightBig2parenleftbig
Sici?c +Bu?T?parenrightbig2
parenleftBig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Buparenleftbig?T?parenrightbig
parenrightBig2
Ropt =
?A
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
(4.62)
where
A=r2b
parenleftbigg
Sici?c +Sibi?b
parenleftBig?T
?
parenrightBig2
+ 2Bu?T?
parenrightbigg2
+
parenleftbigg
4KTrb
parenleftBig?T
?
parenrightBig2
+Sici?c 1g2
m
parenleftBig?T
?
parenrightBig2parenrightbiggparenleftbigg
Sici?c +Sibi?b
parenleftBig?T
?
parenrightBig2
+ 2Bu?T?
parenrightbigg
? 1g2
m
parenleftBig?T
?
parenrightBig2parenleftBig
Sici?c +Bu?T?
parenrightBig2
(4.63)
Substituting (4.60) and (4.62) into (4.39) and assuming Gu = 0, the expression of NFmin is
Fmin = 1 + rbR
opt
+ 12KT
parenleftbigg?
?T
parenrightbigg2parenleftbig
Ropt +rbparenrightbig?
parenleftbigg
Sici?c +Sibi?b
parenleftBig?T
?
parenrightBig2
+ 2Bu?T?
parenrightbigg
. (4.64)
Fmin = 1 +
rb
parenleftBig
Sici?c +Sibi?b
parenleftBig
?T
?
2
parenrightBig
+ 2Bu?T?
parenrightBig
?A
+ 12KT
parenleftbigg?
?T
parenrightbigg2parenleftbigg
rb
parenleftbigg
Sici?c +Sibi?b
parenleftBig?T
?
parenrightBig2
+ 2Bu?T?
parenrightbigg
+
radicalbig
A
parenrightbigg
(4.65)
87
Using the same transport model,
A=B?(2qIC)2 (4.66)
where
B =
parenleftbigg 1
gm
parenrightbigg2parenleftBig?
T
?
parenrightBig4 1
?
+rb
parenleftbigg
(1??T?c)2 + 1?
parenleftBig?T
?
parenrightBig2parenrightbigg
?
parenleftbigg 2
gm
parenleftBig?T
?
parenrightBig2
+rb
parenleftbigg
(1??T?c)2 + 1?
parenleftBig?T
?
parenrightBig2parenrightbiggparenrightbigg
(4.67)
Fmin = 1 +
(1??T?c)2 + 1?parenleftbig?T?parenrightbig2
?B
?
?rb +gm
parenleftbigg?
?T
parenrightbigg2?
?rbradicalbigB+ B
(1??T?c)2 + 1?parenleftbig?T?parenrightbig2
?
?
?
?
(4.68)
4.3.2 Model comparison with measurement
0 5 10 15 20 25 30
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Freq (GHz)
NF
min
(dB)
Measurement
Analytical Modeling
IBM 0.35um Tech
IBM 0.13um Tech
Figure 4.14: Comparison of NFmin calculated from analytical equations and measurement.
Fig. 4.14 compares analytical NFmin with measurement for two different technologies. The
device from IBM 0.35 ?m technology has a peak fT = 35 GHz, AE = 0.8?20?3?m2 and total
base resistance equal to 7 ?. Device is biased at IC = 3.55 mA and fT = 15 GHz. ?c = 0.75?f.
The device from IBM 0.13?m technology has a peakfT = 160 GHz,AE = 0.12?12?m2 and total
88
base resistance equal to 11 ?. Device is biased at IC = 2.10 mA and fT = 80 GHz. ?c = 0.4?f.
For the device from IBM 0.35 ?m technology, the calculated NFmin based on analytical model is
at an average 0.3 dB higher than the measured NFmin; for the device from 0.13 ?m technology, the
calculated NFmin is at an average 0.15 dB higher than the measured NFmin. For both devices, the
analytical model has captured the frequency dependence of NFmin.
We have established that the base resistances are the most important extrinsic circuit elements
in determining the noise transfer functions for the propagation of the intrinsic terminal noise cur-
rents ici and ibi towards the external terminals, and they can be approximately lumped into a single
Rb for the device used. The best known role that Rb plays in the context of transistor noise discus-
sion, however, is acting as a thermal noise source. One previously in general thinks of the intrinsic
noise currents (ici and ibi) and the resistor thermal noise current (irb) as independent noise sources.
Comparison of (4.48) (4.57) and (4.65) shows that the correlation of ic and ib highly depends on
the existence of Rb.
4.3.3 Simulation Results
A popular technique of comparing the relevant importance of noise sources is to turn on them
individually and compare the noise parameters obtained with turning on them together. Modern
simulators all support such comparison.
Fig. 4.15 shows the simulated NFmin at IC = 3.97 mA in the situations of: 1) turning on
thermal noise sources only (black), 2) turning on terminal current noise sources using CB SCR
transport model only while keeping all the resistances noiseless (blue) and 3) turning on both
thermal noise sources and terminal current noise sources (red). NFmin obtained with thermal noise
only is clearly the smallest. The NFmin due to thermal noises is less than the NFmin due to intrinsic
terminal current noises. In other words, Rb?s impact on NFmin as thermal noise source is weak,
although thermal noise contribution dominates Sicxi?cx and Sibxi?b
x
as shown in Fig 3.12. This is
possible as NFmin is more directly related to the functions in the noise chain representation than in
the admittance representation [43] [7][18].
89
0 5 10 15 20 25 30
0
0.5
1
1.5
2
2.5
3
3.5
4
Frequency (GHz)
NF
min
(dB)
Simulation: Current Noise + Thermal Noise
Simulation: Current Noise
Simulation: Thermal Noise
I
C
= 3.97 mA
Figure 4.15: Comparison of NFmin from 50 GHz SiGe HBT as a function of frequency from
simulations with correlated intrinsic terminal current noises due to CB SCR plus thermal noises,
correlated intrinsic terminal current noises alone and thermal noises alone.
Next we examineRb?s impact as the dominant extrinsic circuit element onNFmin. We compare
the simulation results using SPICE model and a more generic new model, which includes the
noise transport in both intrinsic base and CB SCR and accurate to higher frequency than CB SCR
transport model [35]. For both models, simulations are made with:
1. zero base resistances;
2. the base resistances as extracted experimentally, but with thermal noise turned off;
3. the base resistances as extracted experimentally, with thermal noise turned on.
The purpose is to distinguish Rb?s two roles, as a thermal noise source, which is best known, and
as the dominant extrinsic network element in determining propagation ofici and ibi towards icx and
ibx.
Fig. 4.16 shows the noise parameters simulated at IC = 3.97 mA from 2 to 50 GHz. For
NFmin, the important findings include:
1. without Rb, NFmin from both uncorrelated and correlated model are much lower than with
Rb. The difference is larger at higher frequency. Without Rb, the difference between us-
ing uncorrelated SPICE model and correlated new model can be barely observed at even
90
0 20 40 60
0
5
10
NF
min
(dB)
0 20 40 60
-20
0
20
40
60
80
Rn (
?
)
0 20 40 60
0
0.02
0.04
0.06
Frequency (GHz)
?
(Y
opt
) (S)
0 20 40 60
-0.2
-0.15
-0.1
-0.05
0
?
(Y
opt
) (S)
Frequency (GHz)
SPICE model; R
b
= 0 ?
New model; R
b
= 0 ?
SPICE model; noiseless R
b
New model; noiseless R
b
SPICE model; noisy R
b
New model; noisy R
b
Figure 4.16: Simulated noise parameters using zero Rb, noiseless Rb and noisy Rb as a function of
frequency at 3.97 mA; both uncorrelated SPICE model and new correlated model are included for
comparison.
high frequency. So noise correlation does not fundamentally decrease NFmin, which was
previously believed to be true [17][16] [28][47][48][34][35].
2. it is however true that noise correlation reduces NFmin, for the Rb value found in the actual
device, whetherRb is noisy or noiseless. This means reduction ofNFmin by correlation isRb
dependent, and thus will vary with technology.
3. NFmin increases considerably with noiseless Rb, for both uncorrelated and correlated ici and
ibi, although Rb as a noise source itself leads to small NFmin.
4. thermal noise of Rb further increases NFmin.
Therefore reduction of Rb should continue to be a priority in future technology development, not
only for higher fmax and power gain, but also for lower NFmin, despite the fact that the NFmin due
to Rb?s thermal noise is already negligible.
91
Rn is very small without Rb and almost frequency independent with either model and thermal
noise of Rb further increases Rn. Without Rb, the magnitude of Ifractur(Yopt) is higher than Rfractur(Yopt) at
high frequency. With Rb, the magnitudes ofIfractur(Yopt) from both two models decrease and become
close to each other. Both magnitudes of Rfractur(Yopt) and Ifractur(Yopt) decrease after adding the thermal
noise of Rb.
0 10 20 30
0
2
4
6
NF
min
(dB)
0 10 20 30
-10
0
10
20
30
40
R
n
(
?
)
0 10 20 30
0
0.05
0.1
I
C
(mA)
?
(Y
opt
) (S)
0 10 20 30
-0.1
-0.05
0
?
(Y
opt
) (S)
I
C
(mA)
SPICE model; R
b
= 0 ?
New model; R
b
= 0 ?
SPICE model; noiseless R
b
New model; noiselessR
b
SPICE model; noisy R
b
New model; noisy Rb
Figure 4.17: Simulated noise parameters using zero Rb, noiseless Rb and noisy Rb as a function
of IC at 10 GHz; both uncorrelated SPICE model and new correlated model are included for
comparison.
Fig. 4.17 shows bias dependent noise parameter simulation results at 10 GHz. The major
difference between different models emerges after adding noiselessRb and the difference increases
with increasing IC. Noisy Rb further adds approximately 1 dB NFmin. Both magnitudes ofRfractur(Yopt)
and Ifractur(Yopt) are higher without Rb, and the magnitudes of Rfractur(Yopt) and Ifractur(Yopt) from uncorrelated
model are lower than from correlated model. Opposite to NFmin and Rn, Yopt from two models are
close to each other with Rb. With Rb, the magnitudes ofRfractur(Yopt) from both two models decrease,
as well asIfractur(Yopt). Noisy Rb further decreases the magnitudes ofRfractur(Yopt) andIfractur(Yopt).
92
4.4 Conclusions
Analytical expressions of noise correlation matrices for various noise representations are de-
rived and used to successfully explain simulation and measurement results. Relative importance
of individual noise sources is examined for all elements of the noise correlation matrices of the Y,
Z and chain representations. The base resistances are identified to be the dominant extrinsic tran-
sistor equivalent circuit elements that determine the propagation of the intrinsic terminal current
noises towards the extrinsic terminals. Further insight is then obtained into how correlated intrinsic
current noises and base resistances affect transistor noise parameters. For the first time, we show
that the reduction of NFmin by noise correlation is a strong function of Rb, and for zero Rb, noise
correlation would have no impact on NFmin. As far as the impact on transistor NFmin is concerned,
base resistance thermal noise as a noise source is actually not that important. Instead, Rb as a
circuit element, modifies the transfer of intrinsic noise currents towards extrinsic terminals, and
such modification causes significant increase of NFmin, particularly at higher frequency, regardless
of whether the intrinsic terminal noise currents are correlated or not. The reduction of transistor
NFmin by correlation is shown to be a strong function of Rb.
93
Chapter 5
Impact on Low Noise Amplifier Design
Low noise amplifier (LNA), usually the first stage of the receiver, must be able to amplify
low noise as low as -100 dBm, while maintaining sufficient signal-to-noise-ratio and adding suf-
ficient low noise to the circuit. The main contributor to the LNA noise figure is the RF noise of
the transistor. In the previous chapter,we have discussed the noise physics and presented noise
model on the transistor level. Adding correlation of terminal noise base and collector current has
successfully improved the accuracy of the noise modeling results. On the circuit level, however,
additional question remains.
An important RF property of bipolar transistor derived in [50] is that simultaneous noise and
impedance match can be achieved through transistor sizing and the use of two inductors placed
the emitter and base. At a given JC, real part of noise matching is achieved by adjusting transistor
size such that Ropt = 50 ?, in a 50 ? system. An emitter inductor Le provides an input resistance
Rin =50 ?. A base inductorLb cancels out the input reactance and at the same time transforms the
source noise matching reactance of the LNA to 0 ?. This important property that highly simplifies
RF LNA design, however, was derived using the SPICE noise model that does not consider the
frequency dependent correlation, together with additional approximations. A logical question is
how frequency dependent correlation affects simultaneous noise and (input) impedance matching.
5.1 Analytical derivation of LNA noise figure
Assuming that the RF operating frequency is far above f? (where the ac ? begins to decrease
mainly due to Cbe), and that Miller effect is negligible, the equivalent circuit of an simple LNA
consisting one single transistor can be shown as in Fig. 5.1.
94
Figure 5.1: Simplified equivalent circuit for a transistor with an emitter inductor Le and a base
inductor Lb.
Next we consider the noise of the LNA. As illustrated in Fig. 5.2, the transistor has noise
sources including the terminal noise current ic and ib and the thermal noise vb of base resistance
Rb. Power Source has a noise source vs. We now derive the noise figure of the transistor with Le
and Lb. The noise figure of LNA is defined as
NF = 1 + Noise output due to LNANoise output due to source. (5.1)
Using the very same method of single transistor noise figure derivation in Chapter 4, the analytical
expression of LNA?s noise parameters can be obtained as,
Xopt,LNA =
1
?Cbe
parenleftbigS
ici?c +Bu?T?
parenrightbig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
+?(Le +Lb). (5.2)
95
Figure 5.2: Simplified equivalent circuit for a "noisy" transistor with an emitter inductor Le and
a base inductor Lb. The transistor has noise sources including the terminal noise current ic and ib
and the thermal noise vb of base resistance Rb. Power Source has a noise source vs.
R2opt,LNA = 4KTRb
parenleftbig?T
?
parenrightbig2
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
+R2b +
parenleftBig
1
?Cbe
parenrightBig2parenleftbig
Sici?c ?2GuRbgmparenrightbig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
(5.3)
?
parenleftBig
1
?Cbe
parenrightBig2parenleftbig
Sici?c +Bu?T?parenrightbig2
parenleftBig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Buparenleftbig?T?parenrightbig
parenrightBig2
Fmin,LNA = 1 + RbR
opt
+ 12KT
parenleftbigg?
?T
parenrightbigg2parenleftbig
Ropt +Rbparenrightbig?
parenleftbigg
Sici?c +Sibi?b
parenleftBig?T
?
parenrightBig2
+ 2Bu?T?
parenrightbigg
. (5.4)
Compared with (4.60) (4.62) and (4.64), the noise parameters of LNA can be related to the noise
parameters of the transistor alone as
Xopt,LNA =Xopt +?(Le +Lb) (5.5)
Ropt,LNA =Ropt (5.6)
NFmin,LNA =NFmin (5.7)
96
Adding Le and Lb does not impact on Ropt and NFmin but impacts on Xopt. The above conclusion
was previous obtained based on derivation with uncorrelated intrinsic terminal current noise, and
adding the correlation of ib and ic does not vitiate it.
5.2 Simultaneous noise and impedance matching
Next we inspect whether the simultaneous noise and impedance matching is still valid. As
shown in Fig. 5.1, the input impedance looking into the circuit before Lb is given by
Zin = vini
in
, (5.8)
where
iin =vbe?j?Cbe, (5.9)
and
vin = (j?Lb +Rb)?iin +vbe +j?Le (iin +gmvbe). (5.10)
Substituting (5.9) and (5.10) into (5.8),
Zin =j?Lb +j?Le +Rb + 1j?C
be
+?TLe, (5.11)
as
?T = gmC
be
. (5.12)
The ac ? is also defined as
?RF = gmj?C
be
= ?Tj?. (5.13)
A resistive component ?TLe is produced by using the emitter inductor. The value of Le and Lb is
needed to match the RF source impedance, meaning
Rfractur(Zin) =Rb +?TLe =RS, (5.14)
and
Ifractur(Zin) =?Lb +?Le? 1?C
be
= 0. (5.15)
97
For simultaneous impedance and noise matching, the values of Le and Lb required for the source
impedance matching should be identical to the values required forRopt =RS andXopt = 0?. Thus
Xopt,LNA =
1
?Cbe
parenleftbigS
ici?c +Bu?T?
parenrightbig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
+?(Le +Lb) = 0. (5.16)
R2opt,LNA =R2S = 4KTRb
parenleftbig?T
?
parenrightbig2
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
+R2b +
parenleftBig
1
?Cbe
parenrightBig2parenleftbig
Sici?c ?2GuRbgmparenrightbig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
(5.17)
?
parenleftBig
1
?Cbe
parenrightBig2parenleftbig
Sici?c +Bu?T?parenrightbig2
parenleftBig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Buparenleftbig?T?parenrightbig
parenrightBig2
The value of Ropt,LNA can be obtained though sizing the transistor. Xopt = 0 requires
1
?Cbe
parenleftbigS
ici?c +Bu?T?
parenrightbig
Sici?c +Sibi?bparenleftbig?T?parenrightbig2 + 2Bu?T?
= 1?C
be
. (5.18)
This condition can be automatically satisfied when we have
f =
radicalBigg
fT
??c, (5.19)
? is the DC current gain. ?c is proportional to ?f. Even we assuming ?c and ? is constant before
high injection, strict simultaneous noise and impedance matching can be only achieved by carefully
choosing frequency at a certain bias, as fT is bias dependent.
5.3 Simulation Results based on a Cascade LNA
Simulations are based on the SiGe HBT with 36 GHz peak fT. The design kit provided
by IBM uses the HICUM model for HBTs. The CB SCR noise model is used. The collector
transit time parameter was extracted by fitting measured noise parameters. Other compact model
parameters were extracted by fitting dc I-V curves and y-parameters.
Fig. 5.3 shows the modeled and measured noise parameters versus current density (JC) at
VCE = 3.3 V and f = 5 GHz, including the minimum noise figure (NFmin), noise resistance (Rn),
98
0 0.5 1 1.5
0
5
10
15
J
C
(mA/um
2
)
NF
min
(dB)
0 0.5 1 1.5
0
1
2
3
J
C
(mA/um
2
)
R
N
/50
0 0.5 1 1.5
0
20
40
60
J
C
(mA/um
2
)
R
opt
(
?
)
0 0.5 1 1.5
0
20
40
60
80
J
C
(mA/um
2
)
X
opt
(
?
)
Measurement
SPICE Model
New Model
A
E
= 3x20x0.8 um
2
Freq: 5 GHz
Figure 5.3: Modeled and measured noise parameters versus JC at 5 GHz.
real and imaginary part of noise matching source impedance (Ropt and Xopt). The SiGe HBT used
has an emitter area of 0.8?20?3 ?m2 and a peak fT of 36 GHz. Noise measurements were made
using an ATN system, and noises of the probing pads and interconnects leading from the pads to
HBT terminals are de-embedded with open-short method. The CB SCR model produces much
more accurate noise parameters than the SPICE noise model as expected. Observe that the noise
matching source resistanceRopt from measurement and new model are higher than from the SPICE
model. Consequently, at a given current densityJC (or VBE), for noise matching through transistor
sizing, one needs to use a larger size than given by the SPICE model, as detailed below.
A cascode topology shown in Fig. 5.4 is chosen for its better reverse isolation and excellent
frequency stability [51]. Cb is for DC blocking, and Lbias is for AC blocking. Here we choose
to use the same size for Q1 and Q2. Size adjustment is made by changing emitter length (LE).
We also optimize the output matching network for output impedance matching. Rc is fixed during
optimization but can be optimized as well.
Fig. 5.5 shows the noise matching source impedance designs using new correlated model
versus JC at 5 GHz and 10 GHz. Real part noise matching is achieved using the size calculated
from Ropt per emitter length (LE), without optimization. Xopt,LNA is generally close to 0 ?. At
99
Figure 5.4: Schematic of the SiGe HBT cascode LNA used.
both 5 GHz and 10 GHz,Xopt,LNA decreases withJC increasing. Xopt,LNA = 0? happens at smaller
JC at 5 GHz than at 10 GHz. This can be explained by (5.19), as fT increases with JC before high
injection and a smaller fT is required at 5 GHz than at 10 GHz and thus smaller JC to achieve
simultaneous noise and impedance matching.
Fig. 5.6 shows LE and IC of cascode LNAs designed by using SPICE and new noise model
versus JC at 5 GHz. For the JC, a larger emitter length is required for source resistance noise
match, i.e. Ropt = 50?, in all the new correlated noise model LNA designs. This directly translates
into a higher IC and thus higher power consumption at simultaneous noise and input impedance
matching. The root cause of this can be found from transistorRopt difference between two models.
for the same transistor size, the new model Ropt is higher than SPICE model Ropt. Compared to
measurements and the new model, the SPICE noise model underestimates Ropt, and consequently
requires a smaller emitter length for adjusting Ropt to 50 ?.
Fig. 5.7 showsNFLNA andNFmin,LNA, gain and IIP3 of versusJC at 5 GHz. For designs made
using both models, NFLNA is nearly identical to NFmin,LNA. Given that noise matching is done for
100
0.05 0.15 0.25 0.35 0.45 0.5
44
45
46
47
48
49
50
51
52
J
C
(mA/um
2
)
R
opt, LNA
(
?
)
0.05 0.15 0.25 0.35 0.45 0.5
-8
-6
-4
-2
0
2
4
6
J
C
(mA/um
2
)
X
opt, LNA
(
?
)
10G
5G
Figure 5.5: The noise matching source impedance of design using the correlated noise model
versus Jc at 5 GHz and 10 GHz.
0.05 0.1 0.15 0.2 0.25 0.3 0.35
20
40
60
80
J
C
(mA/um
2
)
Emitter length L
E
(um)
0
5
10
15
I
C
(mA)
L
E
, new model design
L
E
, SPICE model design
I
C
, new model design
I
C
, SPICE model design
I
C
L
E f= 5GHz, V
CC
= 3V
Figure 5.6: Emitter length (LE) for Ropt=50 ? and IC versus JC at 5 GHz.
101
0.05 0.1 0.15 0.2 0.25 0.3 0.35
1
1.5
2
2.5
3
J
C
(mA/um
2
)
NF
LNA
(dB)
0.05 0.1 0.15 0.2 0.25 0.3 0.35
1.5
2
2.5
J
C
(mA/um
2
)
NF
min, LNA
(dB)
SPICE model
New correlated model
freq= 5 GHz V
CC
=3V
Figure 5.7: NFLNA and NFmin,LNA of designs using SPICE and new noise models versus JC at 5
GHz.
both model designs, NFLNA follows NFmin, which is higher for the SPICE model, as we expect
from transistor level modeling shown earlier in Fig. 5.3.
However, experiment of re-simulating the SPICE model designed LNAs using our new cor-
related noise model shows that the SPICE model designs is fairly close to the noise figure of the
new model designs, despite the clear inaccuracy of the SPICE noise model used in the design [52].
The reason is that the noise conductance Gn remains small enough such that the resulting F?Fmin
is small compared to Fmin. As a result, NFLNA is still dominated by NFmin,LNA.
5.4 Conclusion
We have examined the impact of high frequency noise correlation on LNA design and per-
formance. Analytical derivation shows that simultaneous noise and input impedance matching
conditionally holds in presence of high frequency noise correlation. Noise matching, however,
requires a considerably larger transistor and power consumption.
102
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106
Appendices
107
Appendix A
Noise Representation Transformation
The transformation equations are copied from [53].
108
109
110
111
Appendix B
Derivation of Noise Parameters
Zi denotes the input impedance of the two port. ZS denotes the source impedance (ZS =
1/YS = 1/(GS +jBS)) . The noise current delivered by the source to the noise free two port is
in =?is ZsZ
i +ZS
, (B.1)
and
Ni =Rfractur(Zi) = 4KTGs |ZS|
2
|Zi +ZS|2Rfractur(Zi)trianglef. (B.2)
The noise current delivered to the noise free two port by the correlated noise voltage va and noise
current ia is
iprimen =?va 1Z
i +ZS
?ia ZSZ
i +ZS
, (B.3)
and
Nprimei =Rfractur(Zi) =parenleftbigSvav?a +Siai?a|Z+S|2 + 2RfracturparenleftbigSiav?aZSparenrightbigparenrightbig 1|Z
i +ZS|2
Rfractur(Zi)trianglef. (B.4)
F = 1 + Svav
?a|YS|2 +Siai?a + 2Rfractur
parenleftbigS
iav?aY?S
parenrightbig
4KTGS (B.5)
To find out Bopt, ?F?BS = 0, we then get
Bopt =?Ifractur
parenleftbigS
iav?a
parenrightbig
Svav?a . (B.6)
112
To find out Gopt, ?F?GS = 0. Substituting BS =Bopt,
Gopt =
radicaltpradicalvertex
radicalvertexradicalbtSi
ai?a
Svav?a ?
IfracturparenleftbigSiav?aparenrightbig2
Svav?a2 . (B.7)
Then we obtain Fmin,
Fmin = 1 +
radicalBig
Svav?aSiai?a ?RfracturparenleftbigSvav?aparenrightbig2 +RfracturparenleftbigSvav?aparenrightbig
2KT . (B.8)
113
Appendix C
Matlab Code for Intrinsic Noise Extraction
function NoisePara_Extraction_5PAE
color = {?k?,?b?,?g?,?y?,?m?,?r?,?r--?,?r-o?};
%path = ?H:\Noise5PAE\Meas_XiaoyunSimu?;
%myPath = ?Y:\public\ziyan\Noise7WL\Data\7WL_HPp48x20x1\?;
myPath = ?D:\Noise7WL\Data\7WL_HPp48x20x1\?;
FigNum=1;
Ltext={?open-short?};
AE = 0.48*20*1;
fopen1 = ?NpHp_p48x20_Open3?;
fshort = ?NpHp_p48x20_Short4?;
[freq_OS,Sopen,Topen]=read_SP_Fswp([myPath,fopen1],?21?,?MA?,?GHz?);
[ftemp,Sshort,Tshort]=read_SP_Fswp([myPath,fshort],?21?,?MA?,?GHz?);
% ==================================================================
if 1% frequency sweep de-embedding
NPBiasx = {?Ic3p97_FDEL?,?Ic7p68_FDEL?};
SPBiasx = {?Ic3p97_SP?,?Ic7p68_SP?};
F_name = ?NpHp_p48x20_Vc2_?;
for BiaNo = 1;
File_F_NP{BiaNo} = [F_name,NPBiasx{BiaNo}];
File_F_SP{BiaNo} = [F_name,SPBiasx{BiaNo}];
switch BiaNo
case 1, SBias_Fswp = [0.841,2,32.0850e-6,3.9745e-3];
case 2, SBias_Fswp = [0.859,2,64.7950e-6,7.6410e-3];
end
[fdut_Fswp,Sdut_Fswp,TdutSP_Fswp]=..
read_SP_Fswp([myPath,File_F_SP{BiaNo}],?21?,?MA?,?GHz?);
[NPdut_Fswp,TdutNP_Fswp]=read_NP_Fswp([myPath,File_F_NP{BiaNo}],?MA?,?GHz?);
%------------% NFmin before deem
114
freq_swp = NPdut_Fswp(:,1)*1e-9;
NFmin_b4os = NPdut_Fswp(:,2);
Fmin_b4os=10.^(NFmin_b4os/10);
Xtext = ?frequency (GHz)?;
X = fdut_Fswp*1e-9;
% In case, open short frequency setup is different from dut
Fswp_index = find_list(freq_OS,fdut_Fswp);
Sopen_Fswp = Sopen(Fswp_index,:);
Sshort_Fswp = Sshort(Fswp_index,:);
Freq = fdut_Fswp;
[YINT_Fswp,CY_INT_Fswp,PNoise_INT_Fswp]= ...
DeemNP_OS_ziyan(fdut_Fswp,Sdut_Fswp,Sopen_Fswp,Sshort_Fswp,NPdut_Fswp);
%%%%%%%%%%%%%%Ypara%%%%%%%%%%%%%%%%%%%%
figure(9);%?21?
subplot(4,2,1);hold on;
plot(Freq*1e-9,real(YINT_Fswp(:,1)),?b-*?);
ylabel(?real(Y11) (S)?);
subplot(4,2,2);hold on;
plot(Freq*1e-9,imag(YINT_Fswp(:,1)),?b-*?);
ylabel(?imag(Y11) (S)?);
subplot(4,2,3);hold on;
plot(Freq*1e-9,real(YINT_Fswp(:,3)),?b-*?);
ylabel(?real(Y12) (S)?);
subplot(4,2,4);hold on;
plot(Freq*1e-9,imag(YINT_Fswp(:,3)),?b-*?);
ylabel(?imag(Y12) (S)?);
subplot(4,2,5);hold on;
plot(Freq*1e-9,real(YINT_Fswp(:,2)),?b-*?);
ylabel(?real(Y21) (S)?);
subplot(4,2,6);hold on;
plot(Freq*1e-9,imag(YINT_Fswp(:,2)),?b-*?);
ylabel(?imag(Y21) (S)?);
subplot(4,2,7);hold on;
plot(Freq*1e-9,real(YINT_Fswp(:,4)),?b-*?);
ylabel(?real(Y22) (S)?);xlabel(?Freq(GHz)?);
subplot(4,2,8);hold on;
plot(Freq*1e-9,imag(YINT_Fswp(:,4)),?b-*?);
ylabel(?imag(Y22) (S)?);xlabel(?Freq(GHz)?);
%********************% NFmin after deem
NFmin_os = PNoise_INT_Fswp(:,2)
115
Gopt_os = PNoise_INT_Fswp(:,3);
RN_os = PNoise_INT_Fswp(:,4)
Yopt_os=PNoise_INT_Fswp(:,5)
Fmin_os=10.^(NFmin_os/10);
Si1 = CY_INT_Fswp(:,1); %CY_os11
Si21 = CY_INT_Fswp(:,2); %CY_os21
Si12 = CY_INT_Fswp(:,3); %CY_os12
Si2 = CY_INT_Fswp(:,4); %CY_os22
mfigure(11);
subplot(4,2,7);hold on;
plot(freq_swp,real(Si1),?b-*?);
xlabel(?freq (GHz)?);ylabel(?real Sibib* (A^2/Hz)?);
subplot(4,2,8);hold on;
plot(freq_swp,imag(Si1),?b-*?);
xlabel(?freq (GHz)?);ylabel(?imag Sibib* (A^2/Hz)?);
subplot(4,2,1);hold on;
plot(freq_swp,real(Si2),?b-*?);
xlabel(?freq (GHz)?);ylabel(?real Sicic*(A^2/Hz)?);
subplot(4,2,2);hold on;
plot(freq_swp,imag(Si2),?b-*?);
xlabel(?freq (GHz)?);ylabel(?imag Sicic*(A^2/Hz)?);
subplot(4,2,5);hold on;
plot(freq_swp,real(Si12),?b-*?);
xlabel(?freq (GHz)?);ylabel(?real Sibic*(A^2/Hz)?);
subplot(4,2,6);hold on;
plot(freq_swp,imag(Si12),?b-*?);
xlabel(?freq (GHz)?);ylabel(?imag Sibic*(A^2/Hz)?);
subplot(4,2,3);hold on;
plot(freq_swp,real(Si21),?b-*?);
xlabel(?freq (GHz)?);ylabel(?real Sicib*(A^2/Hz)?);
subplot(4,2,4);hold on;
plot(freq_swp,imag(Si21),?b-*?);
xlabel(?freq (GHz)?);ylabel(?imag Sicib*(A^2/Hz)?);
datapath = ?D:\Noise7WL\Ads7WL_p48x20_g2_prj\Trans0621\?;
NT_ibi = sprintf(?%sNT_ibi_Vc2Ic3p97.txt?, datapath);
NT_ici = sprintf(?%sNT_ici_Vc2Ic3p97.txt?, datapath);
[RNT_Ibx_ibi] = textread(NT_ibi,??,25,...
?delimiter?,? ?,?headerlines?,1);
freq = RNT_Ibx_ibi(:,1)*1e-9;
116
RNT_Ibx_Ibi = RNT_Ibx_ibi(:,2);
[INT_Ibx_ibi] = textread(NT_ibi,??,25,...
?delimiter?,? ?,?headerlines?,29);
INT_Ibx_Ibi = INT_Ibx_ibi(:,2);
[RNT_Icx_ibi] = textread(NT_ibi,??,25,...
?delimiter?,? ?,?headerlines?,57);
RNT_Icx_Ibi = RNT_Icx_ibi(:,2);
[INT_Icx_ibi] = textread(NT_ibi,??,25,...
?delimiter?,? ?,?headerlines?,85);
INT_Icx_Ibi = INT_Icx_ibi(:,2);
[RNT_Ibx_ici] = textread(NT_ici,??,25,...
?delimiter?,? ?,?headerlines?,1);
freq = RNT_Ibx_ibi(:,1)*1e-9;
RNT_Ibx_Ici = RNT_Ibx_ici(:,2);
[INT_Ibx_ici] = textread(NT_ici,??,25,...
?delimiter?,? ?,?headerlines?,29);
INT_Ibx_Ici = INT_Ibx_ici(:,2);
[RNT_Icx_ici] = textread(NT_ici,??,25,...
?delimiter?,? ?,?headerlines?,57);
RNT_Icx_Ici = RNT_Icx_ici(:,2);
[INT_Icx_ici] = textread(NT_ici,??,25,...
?delimiter?,? ?,?headerlines?,85);
INT_Icx_Ici = INT_Icx_ici(:,2);
format long;
NTIcx_Ibi = RNT_Icx_Ibi+INT_Icx_Ibi*j
NTIcx_Ici = RNT_Icx_Ici+INT_Icx_Ici*j
NTIbx_Ibi = RNT_Ibx_Ibi+INT_Ibx_Ibi*j
NTIbx_Ici = RNT_Ibx_Ici+INT_Ibx_Ici*j
NTI = [NTIcx_Ici,NTIcx_Ibi,NTIbx_Ici,NTIbx_Ibi];
NTIcx_Ibi_conj = RNT_Icx_Ibi-INT_Icx_Ibi*j;
NTIcx_Ici_conj = RNT_Icx_Ici-INT_Icx_Ici*j;
NTIbx_Ibi_conj = RNT_Ibx_Ibi-INT_Ibx_Ibi*j;
NTIbx_Ici_conj = RNT_Ibx_Ici-INT_Ibx_Ici*j;
NTI_conj =[NTIcx_Ici_conj,NTIbx_Ici_conj,NTIcx_Ibi_conj,NTIbx_Ibi_conj];
det_NTI = 1./(NTIcx_Ici.*NTIbx_Ibi-NTIcx_Ibi.*NTIbx_Ici);
NTI_inv = [NTIbx_Ibi.*det_NTI,...
-NTIcx_Ibi.*det_NTI, -NTIbx_Ici.*det_NTI, NTIcx_Ici.*det_NTI];
NTI_conj_inv11 = real(NTI_inv(:,1))-j*imag(NTI_inv(:,1));
NTI_conj_inv12 = real(NTI_inv(:,3))-j*imag(NTI_inv(:,3));
NTI_conj_inv21 = real(NTI_inv(:,2))-j*imag(NTI_inv(:,2));
117
NTI_conj_inv22 = real(NTI_inv(:,4))-j*imag(NTI_inv(:,4));
NTI_conj_inv = [NTI_conj_inv11,NTI_conj_inv12,NTI_conj_inv21,NTI_conj_inv22];
%datapathR = ?D:\NoiseTransfer\Ndata_new\?;
%SNTR = sprintf(?%sSNTR_new_correction.txt?, datapathR);
SNTR = sprintf(?%sSNTR_Vc2Ic3p97_ex.txt?, datapath);
[NTRI] = textread(SNTR,??,?delimiter?,? ?,?headerlines?,0);
NTRI1 = NTRI(:,1)+j*NTRI(:,2);
NTRI2 = NTRI(:,3)+j*NTRI(:,4);
NTRI3 = NTRI(:,5)+j*NTRI(:,6);
NTRI4 = NTRI(:,7)+j*NTRI(:,8);
IX_os = [Si2-NTRI1,Si21-NTRI2,Si12-NTRI3,Si1-NTRI4]; %Ic, Icib, Ibic,Ib
%Si_int= multiple(NTI_inv,IX_os,NTI_conj_inv) %ic, icb, ibc, ic;
Si_int_pre1 = NTI_inv(:,1).*IX_os(:,1)+ NTI_inv(:,2).*IX_os(:,3);
Si_int_pre2 = NTI_inv(:,1).*IX_os(:,2)+ NTI_inv(:,2).*IX_os(:,4);
Si_int_pre3 = NTI_inv(:,3).*IX_os(:,1)+ NTI_inv(:,4).*IX_os(:,3);
Si_int_pre4 = NTI_inv(:,3).*IX_os(:,2)+ NTI_inv(:,4).*IX_os(:,4);
Si_int1 = Si_int_pre1.*NTI_conj_inv(:,1)+ Si_int_pre2.*NTI_conj_inv(:,3)
Si_int2 = Si_int_pre1.*NTI_conj_inv(:,2)+ Si_int_pre2.*NTI_conj_inv(:,4);
Si_int3 = Si_int_pre3.*NTI_conj_inv(:,1)+ Si_int_pre4.*NTI_conj_inv(:,3);
Si_int4 = Si_int_pre3.*NTI_conj_inv(:,2)+ Si_int_pre4.*NTI_conj_inv(:,4);
mfigure(2);
subplot(4,2,1);hold on;
%plot(freq_swp,real(Si_int(:,1)),?b?);
plot(freq_swp,real(Si_int1),?k-o?);
xlabel(?freq (GHz)?);ylabel(?real Sicic*?);
subplot(4,2,2);hold on;
%plot(freq_swp,imag(Si_int(:,1)),?b?);
plot(freq_swp,imag(Si_int1),?k-o?);
xlabel(?freq (GHz)?);ylabel(?imag Sicic* ?);
subplot(4,2,3);hold on;
%plot(freq_swp,real(Si_int(:,2)),?b?);
plot(freq_swp,real(Si_int2),?k-o?);
xlabel(?freq (GHz)?);ylabel(?real Sicib*?);
subplot(4,2,4);hold on;
%plot(freq_swp,imag(Si_int(:,2)),?b?);
plot(freq_swp,imag(Si_int2),?k-o?);
xlabel(?freq (GHz)?);ylabel(?imag Sicib*?);
subplot(4,2,5);hold on;
%plot(freq_swp,real(Si_int(:,3)),?b?);
plot(freq_swp,real(Si_int3),?k-o?);
118
xlabel(?freq (GHz)?);ylabel(?real Sibic*?);
subplot(4,2,6);hold on;
%plot(freq_swp,imag(Si_int(:,3)),?b?);
plot(freq_swp,imag(Si_int3),?k-o?);
xlabel(?freq (GHz)?);ylabel(?imag Sibic*?);
subplot(4,2,7);hold on;
%plot(freq_swp,real(Si_int(:,4)),?b?);
plot(freq_swp,real(Si_int4),?k-o?);
xlabel(?freq (GHz)?);ylabel(?real Sibib*?);
subplot(4,2,8);hold on;
%plot(freq_swp,imag(Si_int(:,4)),?b?);
plot(freq_swp,imag(Si_int4),?k-o?);
xlabel(?freq (GHz)?);ylabel(?imag Sibib*?);
if 0
Si_out_pre1 = NTI(:,1).*Si_int1+NTI(:,2).*Si_int3;
Si_out_pre2 = NTI(:,1).*Si_int2+NTI(:,2).*Si_int4;
Si_out_pre3 = NTI(:,3).*Si_int1+NTI(:,4).*Si_int3;
Si_out_pre4 = NTI(:,3).*Si_int2+NTI(:,4).*Si_int4;
Si_out1 = Si_out_pre1.*NTI_conj(:,1)+Si_out_pre2.*NTI_conj(:,3)+NTRI1;
Si_out2 = Si_out_pre1.*NTI_conj(:,2)+Si_out_pre2.*NTI_conj(:,4)+NTRI2;
Si_out3 = Si_out_pre3.*NTI_conj(:,1)+Si_out_pre4.*NTI_conj(:,3)+NTRI3;
Si_out4 = Si_out_pre3.*NTI_conj(:,2)+Si_out_pre4.*NTI_conj(:,4)+NTRI4;
mfigure(10006);
subplot(4,2,1);hold on;
plot(freq_swp,real(Si2),?b?);
plot(freq_swp,real(Si_out1),?r?);
xlabel(?freq (GHz)?);ylabel(?real Sicic*?);
subplot(4,2,2);hold on;
plot(freq_swp,imag(Si2),?b?);
plot(freq_swp,imag(Si_out1),?r?);
xlabel(?freq (GHz)?);ylabel(?imag Sicic* ?);
subplot(4,2,3);hold on;
plot(freq_swp,real(Si21),?b?);
plot(freq_swp,real(Si_out2),?r?);
xlabel(?freq (GHz)?);ylabel(?real Sicib*?);
subplot(4,2,4);hold on;
plot(freq_swp,imag(Si21),?b?);
plot(freq_swp,imag(Si_out2),?r?);
xlabel(?freq (GHz)?);ylabel(?imag Sicib*?);
subplot(4,2,5);hold on;
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plot(freq_swp,real(Si12),?b?);
plot(freq_swp,real(Si_out3),?r?);
xlabel(?freq (GHz)?);ylabel(?real Sibic*?);
subplot(4,2,6);hold on;
plot(freq_swp,imag(Si12),?b?);
plot(freq_swp,imag(Si_out3),?r?);
xlabel(?freq (GHz)?);ylabel(?imag Sibic*?);
subplot(4,2,7);hold on;
plot(freq_swp,real(Si1),?b?);
plot(freq_swp,real(Si_out4),?r?);
xlabel(?freq (GHz)?);ylabel(?real Sibib*?);
subplot(4,2,8);hold on;
plot(freq_swp,imag(Si1),?b?);
plot(freq_swp,imag(Si_out4),?r?);
xlabel(?freq (GHz)?);ylabel(?imag Sibib*?);
end
end
end
% =========================================================
return
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Appendix D
Verilog-A Code for Compact Noise Model Implementation
electrical na, nb; //added
branch (na) b_na;
branch (nb) b_nb;
real sic, sib;
real gn1, gn2, gn3, gn4; //added
twoq = 2.0 * ?P_Q;
sic = twoq*abs(it);
sib = twoq*abs(ibei);
I(b_na) <+ white_noise(1);
I(b_nb) <+ white_noise(1);
I(b_na) <+ V(b_na);
I(b_nb) <+ V(b_nb);
gn1 = sqrt(sic);
gn2 = -sqrt(sic)*Tf*fg1*((1.00/3.00)*Tf*fg1
+ (1.00/3.0+ eta*(1.00/9.00))*fg2*Tf)/((Tf*fg1
+ (1.00/3.0+ eta*(1.00/9.00))*fg2*Tf));
gn3 = sqrt(sic)*(Tf*fg1 + (1.00/3.0+ eta*(1.00/9.00))*fg2*Tf); //optional
gn4 = sqrt(sic)*fg2*Tf*sqrt((1.00/3.00+eta*(4.00/45.00))
-pow((1.00/3.00+eta*(1.00/9.00)),2));
I(bi,ei) <+ white_noise(sib, "shot")+gn3*ddt(V(b_na)) +gn4*ddt(V(b_nb));
I(ci,ei) <+ gn1*(V(b_na))+ gn2*ddt(V(b_na));
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Appendix E
Derivation of relation between TintY and TY
I?V relation of TY is
I1 =Y11V1 +Y12V2, (E.1)
I2 =Y21V1 +Y22V2. (E.2)
I?V relation of TintY is
I1 =Yint11 (V1?I1rb) +Yint12 V2, (E.3)
I2 =Yint21 (V1?I1rb) +Yint22 V2. (E.4)
Then we have
Y11V1 +Y12V2 =Yint11 (V1?I1rb) +Yint12 V2, (E.5)
Y21V1 +Y22V2 =Yint21 (V1?I1rb) +Yint22 V2. (E.6)
Therefore
Y11V1 +Y12V2 =Yint11 (V1?I1rb) +Yint12 V2, (E.7)
Y21V1 +Y22V2 =Yint21 (V1?I1rb) +Yint22 V2. (E.8)
Replace I1 in (E.8) with (E.2),
Y11V1 +Y12V2 =Yint11 (V1?(Y11V1 +Y12V2)rb) +Yint12 V2, (E.9)
Y21V1 +Y22V2 =Yint21 (V1?(Y11V1 +Y12V2)rb) +Yint22 V2. (E.10)
(E.10) should be independed of V1 and V2. Therefore
Y11V1 =Yint11 V1?Yint11 Y11rbV1, (E.11)
Y12V2 =Yint12 V2?Yint11 Y12rbV2, (E.12)
Y21V1 =Yint21 V1?Yint11 Y21rbV1, (E.13)
Y22V2 =Yint22 V2?Yint21 Y12rbV1, (E.14)
We are then able to obtain the relation between TY and TintY .
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