ABSTRACT
Title of Dissertation: MATERIALS CHARACTERIZATION FOR
SUB-MICRON SUPERCONDUCTING
INTERCONNECTS IN RECIPROCAL
QUANTUM LOGIC CIRCUITS
Cougar Alessandro Tomas Garcia
Doctor of Philosophy, 2022
Dissertation Directed by: Professor Steven M. Anlage
Department of Materials Science and Engineering
Today?s server based computing consumes a considerable amount of energy. Reciprocal
Quantum Logic (RQL) is a classical logic family within superconducting electronics, and is a
candidate for energy efficient computing technologies. Similar to the current complimentary
metal-oxide semiconductor technologies, RQL interconnects are responsible for dissipating the
majority of the energy. The energy dissipated in RQL interconnects comes from finite resistive
losses in the superconducting wires and embedded dielectrics at radio frequencies. Therefore,
material properties, processing, and performance are critical to understanding the mechanisms of
loss and mitigation of power dissipation in RQL interconnects.
This dissertation presents work on three aspects of materials characterization of RQL
interconnects: implementing a method to deconvolve superconducting and dielectric losses, evaluating
losses in three generations of RQL fabrication, and understanding the microscopic physics that
determines the performance of RQL interconnects in a temperature and frequency range from
1.5-6 K and 3-12 GHz, respectively.
A novel method to accurately deconvolve superconducting and dielectric losses by exploiting
their frequency dependence is described. Furthermore, a finite element modeler is used to accurately
extract the losses. This method is termed Dispersive Loss Deconvolution. The designed microstrip
transmission line resonators are fabricated in a 0.25 ?m RQL fabrication process composed of
Nb wires embedded in Tetraethyl orthosilicate (TEOS) dielectric. The Nb and TEOS losses as a
function of microstrip width down to 0.25 ?m are modeled and measured.
The electrical and physical material properties for 3 RQL processes over 5 wafers are
evaluated. The electrical properties were evaluated by characterizing resonators in cryogenic
dip probes and a dry system with ?10 mK temperature control. The physical properties were
evaluated using Transmission Electron Microscopy and Energy-Dispersive Spectroscopy. Two of
the processes use chemical mechanical polishing (CMP) to planarize the Nb wires, and the other
using reactive ion etching (RIE) to define Nb wires.
At 4.2 K, the Nb loss in the 0.25 ?m resonators between the 3 processes were surprisingly
distinct. The two CMP processes yield Nb losses up to 2 times higher relative to the RIE process,
and have a discernible increase in loss by as much as 20% going from 4 to 0.25 ?m microstrip
widths. For the RIE process, there is no detectable upturn in Nb losses for microstrip widths
down to 0.25 ?m. Most notably, the RIE process produced 0.25 ?m Nb wires with loss reaching
the theoretical lower limit of intrinsic surface resistance Rs = 17 ?? at 4.2 K and 10 GHz.
The superior RIE process may be linked to the incorporation of thin metal passivation layers
protecting the Nb, which prevented Nb oxide from participating in additional loss mechanisms.
For all 3 processes and microstrip widths from 0.25-4 ?m, the TEOS losses had negligible width
dependence and varied by as much as ?20%.
From the electrical characterization at 4.2 K, it was found that the Nb wires are the limiting
loss mechanisms in RQL interconnects. As temperature is decreased below 4.2 K, it is well
known that Nb loss will exponentially decrease and amorphous dielectrics like TEOS can have
loss with a non-monotonic temperature dependence depending on the input power. This offered
the opportunity to explore a possible optimum operating temperature to minimize power dissipation
by the RQL interconnects. At relatively low input powers, TEOS became the limiting loss
mechanism for temperatures below 3 K, and I conclude this can be attributed to losses coming
from two-level system tunneling relaxation and resonant absorption processes.
The work in this dissertation describes the development of methods to aid in characterization,
design, and fabrication of RQL interconnects, and can be extended to potentially other Single
Flux Quantum and Quantum Computing technologies.
MATERIALS CHARACTERIZATION FOR
SUB-MICRON SUPERCONDUCTING
INTERCONNECTS IN RECIPROCAL
QUANTUM LOGIC CIRCUITS
by
Cougar Alessandro Tomas Garcia
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2022
Advisory Committee:
Professor Steven M. Anlage, Chair/Advisor
Professor Ichiro Takeuchi
Professor Christopher J. Lobb
Professor John Cumings
Professor Christopher J. K. Richardson
Industry Mentor:
Dr. Vladimir V. Talanov, Northrop Grumman
?c Copyright by
Cougar Alessandro Tomas Garcia
2022
Dedication
TO MY WIFE, MARLANEA
MY FATHER, MANUEL GARCIA
MY MOTHER, DARA GARCIA
MY GRANDPA & GRANDMA, ORLAN & DIAMANTE HENDRICKSON
AND MY FATHER-IN-LAW, VINCE M. FRANK
ii
Acknowledgments
?If I have seen further it is by
standing on the shoulders of
Giants?.
Sir Isaac Newton stated in a letter
to Robert Hooke in 1675, the same
decade of his pioneering work in
optics and gravity.
First and foremost, I would like to thank my thesis advisor Professor Steven Anlage for
his guidance in my journey and his willingness to enlighten my understanding and appreciation
of the physical world. In the same light, I thank Dr. Vladimir Talanov, my co-advisor and
industry mentor at Northrop Grumman for his his foundational work this thesis was built upon.
Additionally, I would like to thank Dr. Nicholas Rizzo for his life and technical mentorship from
day one of my scientific career. Thanks also goes to Dr. Elie Track, an influence to many around
the world and has played a motivational role on my journey. In like manner, thanks goes to Dr.
Joshua Strong, Dr. Anna Herr, Dr. Quentin Herr for their creation of new technologies and their
willingness to have me on their team, as it ultimately inaugurated my dissertation work. Special
thanks goes to Dr. Daniel Queen for his guidance and fruitful conversations that ultimately helped
me understand how low temperature amorphous dielectric properties contributed in Chapters 4
iii
and 5. Thanks also goes to Kathy Gray, John Fusco, Dr. Aaron Martin, Andrea Yeiser, Jay
Fetterman, Sean McLaughlin, and Dr. Marc Sherwin for programatically supporting my work.
Thanks and appreciation also goes to Dr. Henry Luo, Dr. Ryan Dowdy, Dr. Nathan Siwak,
Dr. Eric Jones, Chris Pinion, Dr. Chris Kirby, Andrew Brownfield, Nancyjane Bailey, Dr. Steven
Sendelbach, Dr. Alex Sirota, Dr. Ed Kurek, Kelsey McCusker, Angelica Dayhoff, Dr. Anton
Sidorov, Dr. Melissa Loving, Dr. Thomas Ambrose, Dr. Eric Seabron, Dr. Eric Goodwin,
Dr. John X. Przybyz, Dr. Aaron Lee, Dr. Laura Lenard, Will Callis, Dillon Merenich, Zach
McDonald, Liam Downey, Keith Crider, Dr. Phillip Fischer, Justin Goodman, Dr. Flavio Griggio,
and Dr. Timothy Kohler.
Furthermore, I would like to thank my father for his love and support. I would also like
to give special thanks to my wife for her unconditional love, support, and unrelenting patience
during this journey.
Lastly, thank God!
iv
Table of Contents
Dedication ii
Acknowledgements iii
Table of Contents v
List of Tables viii
List of Figures ix
List of Abbreviations xi
List of Variables xiv
Chapter 1: Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Technological Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Chapter 2: Materials for Superconducting Interconnects 11
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Superconductor Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 A Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Complex Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.3 Surface Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.4 Temperature Dependence of Surface Impedance . . . . . . . . . . . . . . 27
2.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Amorphous Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.1 The Dielectric Loss Tangent tan ? . . . . . . . . . . . . . . . . . . . . . 30
2.3.2 Dielectric Loss Mechanisms and Phonon Scattering at T < 10K . . . . . 32
2.3.3 Temperature Dependence of tan ? . . . . . . . . . . . . . . . . . . . . . 37
2.3.4 Frequency Dependence of tan ? . . . . . . . . . . . . . . . . . . . . . . 41
2.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Fabrication Processes for RQL Interconnects . . . . . . . . . . . . . . . . . . . . 45
2.4.1 Brief Background of CMOS Technologies . . . . . . . . . . . . . . . . . 45
2.4.2 SCE Fabrication Technologies . . . . . . . . . . . . . . . . . . . . . . . 47
2.4.3 Chemical Mechanical Planarization (CMP) . . . . . . . . . . . . . . . . 48
v
2.4.4 Cloisonne? Process (Dielectric CMP) . . . . . . . . . . . . . . . . . . . . 50
2.4.5 Damascene Process (Metal CMP) . . . . . . . . . . . . . . . . . . . . . 50
2.4.6 Thermal Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Chapter 3: Modeling Lossy 3D Superconductor-Dielectric 2-Port Networks 54
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 Materials Characterization of Interconnect Morphology . . . . . . . . . . . . . . 56
3.2.1 Scanning Transmission Electron Microscopy (STEM) . . . . . . . . . . . 57
3.2.2 Energy-Dispersive Spectroscopy (EDS) . . . . . . . . . . . . . . . . . . 61
3.3 Theory and Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3.1 RLGC Telegrapher?s equations . . . . . . . . . . . . . . . . . . . . . . 66
3.3.2 Geometric Factors in Inhomogeneous Transmission Line . . . . . . . . . 67
3.3.3 Geometric Factors in Homogeneous Transmission Line . . . . . . . . . . 69
3.3.4 RLGC and Q-Factors From Impedance Matrix [Z] . . . . . . . . . . . . 72
3.4 HFSS Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.4.1 Parallel Plate Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4.2 Defining Solve-Inside Superconductor . . . . . . . . . . . . . . . . . . . 82
3.4.3 Solve-Inside Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.4.4 Microstrip Transmission Line (MTL) . . . . . . . . . . . . . . . . . . . 85
3.5 MTL Geometric Factor ? Extraction . . . . . . . . . . . . . . . . . . . . . . . . 89
3.6 MTL RLGC and ? Sensitivity to Fill . . . . . . . . . . . . . . . . . . . . . . . 92
3.7 MTL ? Sensitivity to RQL Fabrication . . . . . . . . . . . . . . . . . . . . . . . 97
3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Chapter 4: Characterization of Microwave Loss in RQL Interconnects at 4.2 K 102
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2 Resonator Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.2.1 Resonant Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.2.2 Critical Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4 Method Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.4.1 Non-Dispersive tan ? Model . . . . . . . . . . . . . . . . . . . . . . . . 117
4.4.2 Homogeneous Superconductor, Dispersive tan ? Model . . . . . . . . . . 124
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Chapter 5: Temperature Dependent Characterization of Microwave Loss in RQL Interconnects 137
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2 Temperature Dependent Resonator Measurements . . . . . . . . . . . . . . . . . 139
5.3 Multiplexed Microstrip Transmission Line (MUX-MTL) Resonators . . . . . . . 150
5.4 Using the Dispersive Deconvolution Method to Simultaneously Measure tan ?(T )
and Rs(T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5.5 TLS Absorption and Relaxation Microwave Loss in TEOS Above 1K . . . . . . 158
5.6 Dielectric Loss Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
vi
5.7 Temperature Dependence of the Power Dissipation in Nb and TEOS . . . . . . . 174
5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
Chapter 6: Summary and Conclusions 180
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Appendix A: FIB Cross Sections of Process A and B 186
Appendix B: Estimation of Radiation Q-Factor 189
Appendix C: Temperature Dependent Resonator Measurements 190
C.1 Measurements System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
Appendix D: Relevant Equations for Parallel Plate Transmission Line Resonator 192
Bibliography 196
vii
List of Tables
3.1 STEM Results Summary Table . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2 HFSS Results Table of Fill Effect Process A . . . . . . . . . . . . . . . . . . . . 96
3.3 Nominal Fabrication and Material Properties . . . . . . . . . . . . . . . . . . . . 97
4.1 Rs and tan ? Results Summary Table - Process A, B, C . . . . . . . . . . . . . . 105
5.1 Measured TLS-Phonon Dipole Coupling in Various Materials Summary Table . . 164
5.2 MTL Resonator Temperature Dependent Results Summary Table . . . . . . . . . 171
A.1 FIB Results Summary Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
viii
List of Figures
1.1 RQL Interconnects - Materials Paradigm Reference Frame . . . . . . . . . . . . 4
1.2 Energy versus Delay for CMOS and RQL Interconnects . . . . . . . . . . . . . . 8
2.1 Wave Propagation into Superconductor . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Swihart TM wave in Parallel Plate Transmission Line . . . . . . . . . . . . . . . 21
2.3 Acoustic and Electronic Dielectric loss Measurement Agreement . . . . . . . . . 34
2.4 Diagram of Temperature Regimes of Energy Dissipation in Amorphous Dielectrics 39
2.5 Estimate of loss tangent tan ? plateau at 10 GHz for ??-SiO2 . . . . . . . . . . . 41
2.6 CMOS versus RQL Damascene Process . . . . . . . . . . . . . . . . . . . . . . 46
2.7 Superconducting Electronics Fabrication Process Diagram . . . . . . . . . . . . 49
3.1 STEM Images Processes Side-by-Side 0.25 ?m . . . . . . . . . . . . . . . . . . 59
3.2 STEM Images Processes Side-by-Side 1 ?m . . . . . . . . . . . . . . . . . . . . 60
3.3 STEM Images Morphology Features . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 EDS Measured Ar - Damascene Process A and B . . . . . . . . . . . . . . . . . 63
3.5 EDS Maps - Damascene Process B . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.6 EDS Plot NbOx Layer Damascene Process A and B . . . . . . . . . . . . . . . . 64
3.7 EDS Maps and Line Scan Process C . . . . . . . . . . . . . . . . . . . . . . . . 65
3.8 Block Diagram of 2-Port Network Parameters . . . . . . . . . . . . . . . . . . . 74
3.9 HFSS Simulated Model Types . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.10 HFSS Model of Parallel Plate Waveguide . . . . . . . . . . . . . . . . . . . . . 81
3.11 HFSS PPW Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.12 HFSS MTL Model Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.13 HFSS MTL Magnetic Fringing Fields . . . . . . . . . . . . . . . . . . . . . . . 88
3.14 HFSS Results of Geometric Factor vs MTL Width . . . . . . . . . . . . . . . . . 91
3.15 HFSS STEM of Fill Process A . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.16 HFSS Simulation of Fill Effect Process A . . . . . . . . . . . . . . . . . . . . . 94
3.17 HFSS Parametric Corner Analysis on MTL . . . . . . . . . . . . . . . . . . . . 98
3.18 HFSS Geometric Factor Variation of MTL . . . . . . . . . . . . . . . . . . . . . 100
4.1 MTL Physical Chip Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2 MTL Measured Qi For Varied Coupling . . . . . . . . . . . . . . . . . . . . . . 109
4.3 Wafer Map Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.4 Measured S21 and S?21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.5 Measured Qi vs ? vs w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.6 Geometric Factors ? from HFSS versus Parallel Plate and Nominal . . . . . . . . 115
ix
4.7 Geometric Factors ? from HFSS versus Parallel Plate and Nominal . . . . . . . . 118
4.8 Rs versus MTL width, Parallel plate and Inhomogeneous Superconductor Model 120
4.9 Results (Homogeneous Superconductor, non-dispersive tan ? Model) . . . . . . . 123
4.10 Dispersive tan ? estimates by Conductor Loss Subtraction . . . . . . . . . . . . . 125
4.11 Width Dependence of Dispersive tan ? estimates by Conductor Loss Subtraction . 126
4.12 Qi versus Dispersive tan ? Model Using 3-Parameter Fit . . . . . . . . . . . . . 128
4.13 MTL Width Dependence of the Dispersive tan ? 3-Parameter . . . . . . . . . . . 129
4.14 MTL Width Dependence of Rs and tan ? Using Dispersive tan ? 3-Parameter Fits 130
5.1 MTL, Process B, 4 ?m, versus Temperature . . . . . . . . . . . . . . . . . . . . 140
5.2 MTL, Power Conservation Reference Node . . . . . . . . . . . . . . . . . . . . 142
5.3 MTL Q?,Ti for -20 and -35 dBm Input Power . . . . . . . . . . . . . . . . . . . . 145
5.4 MTL Peak Current from T=1.7-4.5 K for -20 and -35 dBm Input Power . . . . . 146
5.5 Q?1i (?n) for MTL widths 0.25 - 4 ?m at 1.7 K . . . . . . . . . . . . . . . . . . . 147
5.6 Calculated Peak Currents for MTL widths 0.25 - 4 ?m at 1.7 and 4.2 K . . . . . . 149
5.7 MUX-MTL Physical Chip Layout . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.8 MUX-MTL Temperature dependent S21versusFrequency . . . . . . . . . . . . 152
5.9 MUX-MTL Temperature dependent Qi versus Frequency . . . . . . . . . . . . . 153
5.10 Dispersive Loss Deconvolution Method Applied to Temperature DependentQi(T )
Using MUX-MTLs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.11 Non-Monotonic Temperature Dependence of TEOS Dielectric Constant in MUX-
MTL Resonators at T < 3K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.12 Frequency Dependence of TLS Dipole Coupling in TEOS . . . . . . . . . . . . . 166
5.13 Measurement of TLS Relaxation and Absorption From TEOS Loss Tangent Using
MUX-MTL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.14 TLS Relaxation and Absorption Fit to Q?1i Data . . . . . . . . . . . . . . . . . . 170
5.15 Stored Power and Peak Currents in MUX-MTL Resonators Below 3 K . . . . . . 173
5.16 Microwave Loss Spectroscopy of TEOS Amorphous Dielectric at T = 1.5? 3K 175
5.17 Measurements of Power Dissipation in RQL interconnects using MUX-MTLs . . 177
A.1 Process A FIB Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
A.2 Process B FIB Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
C.1 Temperature Dependent RF Measurements System . . . . . . . . . . . . . . . . 191
x
List of Abbreviations
ABF Annular Bright Field
AIST Advanced Industrial Science and Technology
AQFP Adiabatic Quantum Flux Parametron
BCS Bardeen, Cooper, and Schrieffer theory
BF Bright Field
CD Critical Dimension
CS Conducting Strip
CVD Chemical Vapor Deposition
CMOS Complimentary Metal-Oxide-Semiconductor
CMP Chemical Mechanical Polishing (or Planarization)
dBm Decibel-Milliwatts
DC Direct Current
DLD Dispersive Loss Deconvolution
EDS Energy-Dispersive Spectroscopy
eSFQ Energy Efficient Single Flux Quantum
FEM Finite Element Modeler
FIB Focused Ion Beam
xi
GP Ground Plane
HFSS High Frequency Structure Simulator
HTS High Temperature Superconductor
IBM International Business Machines Corporation
IL Insertion Loss
IRDS International Roadmap for Devices and Systems
JJ Josephson Junction
LL MIT Lincoln Laboratory
MTL Microstrip Transmission Line
MUX-MTL Multiplexed Microstrip Transmission Line
PEC Perfect Electrical Conductor
PECVD Plasma-Enhanced Chemical Vapor Deposition
PPW Parallel Plate Waveguide
PTL Passive Transmission Line
PVD Physical Vapor Deposition
QFP Quantum Flux Parametron
RF Radio Frequency
RIE Reactive-Ion Etching
RQL Reciprocal Quantum Logic
RSFQ Rapid Single Flux Quantum
xii
SEM Scanning Electron Microscopy
SFQ Single Flux Quantum
SIBC Surface Impedance Boundary Condition
STEM Scanning Transmission Electron Microscopy
SCE Superconducting Electronics
TE Transmitted Electron
TEM Transmission Electron Microscopy
TEOS Tetraethyl Orthosilicate
TLS Two Level System
VNA Vector Network Analyzer
ZC Z-Contrast
ZOR Zeroth-Order Resonator
xiii
List of Critical Symbols
? = 8.85? 10?12 vacuum permittivity (F m?10 ), Eq. 2.18?
?wave = ?0/?0 = 120? intrinsic wave impedance (?), Eq. 2.26
?0 = 4? ? 10?7 vacuum permeability (N A?2), Eq. 2.3
c = 3? 108 vacuum speed of light (m s?1), Eq. 2.36
h? = 1.06? 10?34 reduced Planck?s constant (J s), Eq. 2.6
k = 1.38? 10?23B Boltzmann constant (J K?1), Eq. 2.4
? The transmission line attenuation constant (m?1), Eq. 3.19
? ?1db ultrasonic attenuation (m or dB)
? The transmission line phase constant (m?1), Eq. 3.19
?cm Transmission line partial geometric factor associated with
losses in the m-th conductor (?/sq), Eq. 3.6
?c0 Net conductor geometric factor
at reference frequency ?0, Eq. 3.13
?c0m Transmission line partial geometric factor at that frequency
associated with loss in the mth conductor, Eq. 3.11
?d Transmission line partial geometric factor associated with
losses in that dielectric layer or tube, Eq. 3.41
?dn Transmission line partial geometric factor associated with
loss in the nth dielectric layer or tube, Eq. 3.6
? The transmission complex propagation constant
xiv
of a transmission line (m?1), Eq. 2.12
? BCS energy gap of a superconductor (J), Eq. 2.4
??abs The change in dielectric constant
due to TLS resonant absorption, Eq. 5.11
?r relative dielectric constant, Eq. 2.26
??rel The change in dielectric constant
due to TLS relaxation, Eq. 5.10
? magnetic penetration depth (m), Eq. 2.2
?eff effective magnetic penetration depth (m), Eq. 2.25
?EM the wavelength of the propagating wave (m), Eq. 2.17
?L London penetration depth (m), Eq. 2.2
n?2 The permanent dipole coupling of TLSs
having a density of states n, Eq. 5.12
n?? 2 The induced dipole coupling of TLSs
having a density of states n, Eq. 5.11
?0 microscopic coherence length
of a bulk superconductor (m), Eq. 2.8
? = ?1 ? i?2 complex conductivity (??1m?1), Eq. 2.5
?n normal metal conductivity (??1m?1), Eq. 2.5
?rel characteristic relaxation time (s), Eq. 2.30
Xeff effective surface reactance (?/sq), Eq. 2.23
? angular frequency (rad s?1), Eq. 2.5
a The TLS material constant, Eq. 5.15
xv
C Transmission line capacitance per unit length (F m?1), Eq. 3.5
d superconductor thickness (m), Eq. 2.22
fn Resonant frequency of the n-th mode
for a microstrip transmission line resonator (Hz), Eq. 4.1
G Transmission line conductance per unit length (S m?1), Eq. 3.4
g Coupling factor of a Capacitively coupled
microstrip, Eq. 4.2
Ipeak The magnitude of the peak RF current in the resonator (A) Eq. 5.5
ITLSc critical intensity for TLS in weak-strong electric
field interaction (W m?2), Eq. 2.36
IL Insertion Loss (dB), Eq. 4.2
J 2c is the critical energy intensity for the saturation of TLSs (W/m ) Eq. 5.19
L Transmission line inductance per unit length (H m?1), Eq. 3.3
l Transmission line length (m), Eq. 3.18
`mfp normal electron mean free path (m), Eq. 2.8
`ph phonon mean free path (m), Eq. 2.33
`tun,res TLS resonant tunneling contribution
to phonon mean free path (m), Eq. 2.34
`tun,rel TLS tunneling relaxation contribution
to phonon mean free path (m), Eq. 2.34
`class,rel classical relaxation contribution
to phonon mean free path (m), Eq. 2.34
xvi
M? The average TLS coupling energy (eV) Eq. 5.16, Eq. 5.16
ne total electron density (m?3), Eq. 2.1
ns superfluid electron density (m?3), Eq. 2.1
Pin The incident power at the input of the resonator
after cable attenuation (W ), Eq. 5.1
Pr The reflected power to the VNA (W ) 5.1
Pst The circulating (stored) power in the resonator (W ), Eq. 5.4
Pt The transmitted power (W ), Eq. 5.1
Pd The power dissipated by the resonator (W ), Eq. 5.1
Qc Partial Q-factor associated with the conductor loss, Eq. 3.1
Qd Partial Q-factor associated with the dielectric loss, Eq. 3.1
Qi Internal or intrinsic Q-factor, Eq. 3.1
R Transmission line resistance per unit length (?m?1), Eq. 3.3
Reff effective surface resistance (?/sq), Eq. 2.24
Rs intrinsic surface resistance (?/sq), Eq. 2.19
Rs0 Intrinsic surface resistance at reference frequency ?0 (?/sq), Eq. 3.13
Rs0m Transmission line intrinsic resistance
at a reference frequency ?0 of the m-th conductor (?/sq), Eq. 3.11
Rsm Transmission line effective intrinsic resistance
of the m-th conductor (?/sq), Eq. 3.6
|S21| The transmission coefficient magnitude (dB) Eq. 4.2
T temperature (K), Eq. 2.6
xvii
T1 TLS relaxation time (s), Eq. 2.36
T2 TLS dephasing time (s), Eq. 2.36
Tc superconducting critical temperature (K), Eq. 2.1
tan ? dielectric loss tangent, Eq. 2.26
tan ?0n Transmission line loss tangent of the n-th dielectric layer or tube, Eq. 3.11
tan ?n Transmission line effective loss tangent of the n-th dielectric layer or tube, Eq. 3.6
tan ?class,rel classical relaxation contribution to the dielectric loss tangent, Eq. 2.35
tan ?tun,rel TLS tunneling relaxation contribution to the dielectric loss tangent, Eq. 2.35
tan ?tun,res TLS resonant tunneling contribution to the dielectric loss tangent, Eq. 2.35
Vpeak The magnitude of the peak RF voltage in the resonator (V) Eq. 5.6
w microstrip width (m), Eq. D.3-D.6
Xs intrinsic surface reactance (?/sq), Eq. 2.19
Y shunt admittance (??1 m?1), Eq. 2.11
Z series impedance (?m?1), Eq. 2.10
Zeff effective surface impedance (?/sq), Eq. 2.21
Z0 Characteristic impedance of the microstrip transmission line resonator (?), Eq. 3.20
Zfeed Characteristic impedance of the feed line (?), Eq. 4.4
[Zij] Impedance matrix (?), Eq. 3.22
Zin Input impedance (?), Eq. 4.4
Zs intrinsic surface impedance (?/sq), Eq. 2.20
xviii
Zsub dielectric substrate bulk impedance (?/sq), Eq. 2.24
z(0) transformed impedance at film interface normalized to Zs, Eq. 2.22
z(d) transformed impedance up to the film interface normalized to Zs, Eq. 2.22
xix
Chapter 1: Introduction
1.1 Overview
Since the birth of the information age, there has been a relentless need for more computing
power. In the 1950s-1970s mainframe computers were only accessible through terminals in close
proximity. Today, high performance data centers can be accessed remotely from anywhere around
the world. Data centers are now essential to almost every part of the world economy - universities,
government institutions, large companies, small companies, and even individuals. Server-based
computing is now becoming the primary user for complimentary metal-oxide semiconductor
(CMOS) technologies, amounting to considerable energy consumption. Additionally, Moore?s
Law is expected to flatten by 2025, preventing CMOS from doubling in performance every
2 years [1]. If energy demands continue to evolve across the world as a consequence of the
growing usage of available computing power and Moore?s law comes to an end, the total energy
consumption of data centers across the world is projected to reach 65 GW per year in 2030
(? 70% increase from today) [2], approximately equivalent to the output of 65 power plants, 31
Hoover Dams, or 6.5 billion LED Bulbs.
The power consumption and performance challenges of CMOS need to be addressed, and
are being studied by the International Roadmap for Devices and Systems (IRDS) [3]. Logic
families within superconducting electronics (SCE) are candidates to address power dissipation.
1
Reciprocal Quantum Logic (RQL), an SCE digital logic technology, has been reported to be 300
times more energy efficient than CMOS [4], and has demonstrated the largest working circuit
to date with ? 105 logic units on a single chip [5]. Additionally, RQL has a significant speed
advantage over CMOS, where the upper frequency set by the plasma frequency of the Josephson
junctions on the order of fp ? 250 GHz [4]. However, RQL is still a young technology as
compared to CMOS having consistently ? 1010 logic units per circuit. Several engineering
challenges are limiting large scale RQL-based computation.
In particular, superconducting interconnects, which serve as RQL?s clocking, power delivery
system, and serial data links. Moreover, unlike CMOS where the ground planes only serve as a
medium for data propagation and electrical isolation between signals, RQL ground planes are
engineered to mitigate magnetic flux being trapped in circuit areas that can reduce operating
margins and even catastrophic circuit failure [6, 7]. Material quality and defectivity affect both
power dissipation and magnetic flux trapping in RQL circuits, and the focus of this dissertation
is the former.
RQL interconnects are composed of superconducting wires down to submicron dimensions
embedded in a dielectric. The RQL interconnect performance is governed by the power dissipation
in these materials due to radio frequency (RF) resistive losses. The power dissipation in RQL
interconnects is orders of magnitude lower than CMOS interconnects, but can dwarf that of the
power in RQL logic, and so the superconductor and dielectric RF losses should be minimized.
In CMOS circuits, static and dynamic energy dissipation have become the main limiting
factors for further device scaling [8]. Specifically, at maximum scaling and density of CMOS
circuits, the clock power becomes the dominant contributor to dynamic power dissipation relative
to the logic. Resonator-based clocks in CMOS have been shown to reduce power dissipation
2
[9]. The 300x higher energy efficiency of RQL relative to CMOS was calculated for only the
net power dissipated by the logic alone. Recently, it was found that an RQL circuit with a
metamaterial resonator clock yields an energy efficiency of ? = P ?1RQL(PRQL + PRes) ? 0.3,
where PRQL is the total dynamic power burned by the logic and PRes is the total static power
burned by the resonator clock [10]. Therefore, the energy efficiency of RQL relative to CMOS
drops down to 90x, significantly reducing its competitive advantage.
Materials science and engineering plays a primary role in creating state-of-the-art computing
systems and optimizing their performance. As capabilities increase, so does their complexity,
making the analysis and improvement at the single component level more demanding. Grand
challenges, such as the ending of Moore?s law and the world wide need for more computing
power, can be addressed via the materials science tetrahedron [11]. Specifically, RQL interconnect
power effeciency can be optimized by examining the materials processing (Fig. 1.1a), structure
(Fig. 1.1b), properties (Fig. 1.1c), and performance (Fig. 1.1d). From this perspective, power
dissipation in RQL interconnects may be alleviated with a study of fabrication processes, material
morphology, electrical properties, and their affect on RQL performance. The information gleaned
from such a study can be incorporated into the emerging systems materials engineering triangle
[12], enabling system-level planning to facilitate maturation of superconducting electronics (SCE)
technologies (e.g. RQL).
This dissertation is focused on characterizing RQL superconductor-dielectric interconnect
losses (see Fig. 1.1). This is accomplished by: (i) characterization of the interconnects electrical
properties by developing a novel methodology to deconvolve intrinsic RF losses, (ii) evaluating
the processing of interconnects using complimentary characterization techniques of the material
morphology, and (iii) optimizing the performance by reducing the RF losses in superconducting
3
Figure 1.1: Addressing power dissipation with the materials science paradigm (tetrahedron). The
submicron micronstrip transmission line (MTL) resonator lies at the center of the tetrahedron as
the characterization ?vehicle?. a) Three chemcial mechanical planarization (CMP) processes of
RQL MTLs were examined in this work. b) An example of studying the microstructure and
morphology an MTL (see Chapter 3) c) MTL resonator measurements to extract the RF resistive
loss propoerties of the superconudctor and dielectric materials (see Chapters 4 and 5). d) With
new predictive models, improvements in the performance can be applied to increasing the energy
efficiency of RQL meta material clock networks (see Chapters 4 and 5)
4
interconnects.
The dissertation is organized into six chapters. In the remainder of this first chapter, I review
the technological challenges of RQL interconnects. In Chapter 2, I discuss the electrodynamics of
superconductors and amorphous dielectrics, and RQL fabrication processes. Chapter 3 introduces
the non-idealities of RQL interconnects caused by fabrication processes, describes a method to
characterize the superconductor and dielectric RF losses independently, and presents a procedure
to model superconductors in HFSS in order to extract the RF losses in the superconductor.
In Chapter 4, I present a resonator design and measurements of RF losses at 4.2 K in RQL
interconnects down to 0.25 ?m dimensions. In this chapter, the measured intrinsic losses of the
superconductor and dielectric, Rs and tan ?, will be related back to the morphology analyzed in
Chapter 3. In Chapter 5, I explore the temperature dependence of RF losses in Nb and TEOS
from 1.5-4.5 K to better understand if temperature plays a role in RQL interconnect performance.
Finally, in Chapter 6, I provide a summary of my main results, followed by recommendations for
future research.
1.2 Technological Challenges
It was first theorized by Brian Josephson in 1962 that supercurrent could tunnel between
two superconducting electrodes if the barrier to tunneling was thin enough [13]. Soon after,
Anderson and Rowell measured this tunneling effect [14], followed by John Rowell patenting the
use of the Josephson junction (JJ) as a cryogenic logic device [15]. Thus, the birth of Josephson
electronics and IBM?s thrust to demonstrate a Josephson based computer in the 1960s-1980s.
IBMs effort, using Pb based JJs, ended in 1983 due to only a 2.5 times speed improvement
5
relative to CMOS. The higher reliability of Nb based JJs developed by Gurvitch at Bell Labs [16]
allowed for larger scale circuits and complexity, renewing SCE in early 1987. This is around
the same time high-temperature superconductivity was discovered. Voltage-state gates were used
during this time, where logical 0 (1) is represented by a voltage level of zero (2.5 mV), but
requires a large footprint with significant logic power dissipation due to the incorporation of
resistors [17]. Miraculously, Josephson microprocessors were functional and demonstrated by
Fujitsu Laboratories in Japan [18]. These processors were built one-for-one with CMOS using
niobium junctions having a few thousand gates and memory of a few kilobits. Although, memory,
processing margins, footprint, and power consumption were the limiting factors preventing from
competing with CMOS.
Single flux quantum (SFQ) based logic devices, relying on the current (or phase) state to
do computation, can have (i) 1-2 orders of magnitude reduction in power dissipation, (ii) 2-4
times smaller footprint compared to voltage-state Josephson logic [19], and (iii) potential for up
to 50 times higher clock rates compared to CMOS [20]. SFQ based logic families are rapid single
flux quantum (RSFQ) logic [21], [22], quantum flux parametron (QFP) [23], reciprocal quantum
logic (RQL) [4], energy efficient single flux quantum (eSFQ) [24], and adiabatic quantum flux
parametron (AQFP) [25]. The biggest advantage for SFQ technologies is computation speed,
where greater than 100 GHz data rate has been demonstrated [26, 27]. Additionally, SFQ can
be used as the control processor in quantum computing systems [28, 29, 30], and other niche?
application specific circuits [31]. Recently, scalability and energy efficiency have been reviewed
in comparison to CMOS [32, 33]. Without sacrificing computation speed, RQL logic currently
has a big advantage in energy efficiency, with the largest JJ count per chip to date (a measure of
the scalability) [5]. Superconducting interconnects need to be considered further.
6
Superconducting microwave transmission line (MTL) are the most common interconnects,
supporting low bandwidth signals for clocks and power distribution [34, 35], as well as high
bandwidth signals to propagate SFQ pulses that carry bits of information [36]. The latter is
called a passive transmission line (PTL). Compared to SiO2 and Cu at room temperature, PTLs
composed of SiO2 dielectric and Nb at 4.2 K are near lossless. For example, the energy dissipation
in CMOS interconnects limits the clock frequency of processors to 4 GHz and forces the amount
of transistors that need to be powered off at any given time to prevent the chip temperature from
exceeding thermal limits [32].
Fig. 1.2 shows the energy versus delay for CMOS and RQL interconnects, adopted from
the International Roadmap For Devices and Systems (IRDS) 2020 edition. Here we are not
considering AQFP technology, since it does not use PTL interconnects.
Accounting for the cooling energy to keep circuits at 4 K with > 10W cooling power, PTL
power dissipation is 1? 10?16 J (0.3 ?W at 3 GHz) for a 1 mm long PTL (?RQL interconnect?
arrow in Fig. 1.2). Therefore, RQL MTL interconnects can have 2-4 orders magnitude lower
dissipation relative to CMOS.
As mentioned above, to further scale RQL technology, it is desired to have global clock
and power distribution with spatially equal magnitude and phase. Recently, a metamaterial clock
resonator has been designed and demonstrated to deliver power to 48k JJs operating at 3.5 GHz
supporting uniform power distribution across 3? 3mm2 area at 4.2 K [10]. This clock design is
compatible with AC-powered SFQ logic families (e.g. RQL and QFP). Unfortunately, this comes
at an energy cost. For the above circuit size, the resonator clock has a best case energy efficiency
estimated to be ? = P ?1RQL(PRQL + PRes) ? 0.3, where PRes ? 18 ?W is the power burned
by the resonator clock, and PRQL ? 40 ?W is the power burned by the logic when all logic units
7
Figure 1.2: Adopted from the International Roadmap For Devices and Systems 2020 edition
(IRDS) [37]. (Yellow Box) Arrow pointing to the energy dissipated by a RQL 1 mm long passive
transmission line (PTL) interconnects. (Red squares) CMOS interconnects energy dissipation
and delay for comparison.
8
are saturated (producing all 1s every clock cycle). This means the resonator clock is now limiting
the energy efficiency of RQL, as PRQL is much lower under normal operation of the circuit with
only a small percentage of logic gates drawing power from the clock at any given time.
RQL resonator clock interconnects are made of microstrip transmission line (MTL) networks
[10]. For a fixed circuit size and damping ratio of the junctions, resonator clock efficiency is
related to the intrinsic Q-factor of the resonator by ? ? (1 + Q?1i )?1 with nominal Q?1i =
Q?1 +Q?1 ? 4? 10?3 at 10 GHz, where Q?1 and Q?1c d c d are proportional to the power dissipated
and therefore RF losses in the Nb wires and SiO2 dielectric, respectively. In RQL fabrication, the
SiO2 is typically made using a chemical vapor deposition process using a tetraethyl Orthosilicate
(TEOS) precursor, and so SiO2 will be referred by its common precursor TEOS throughout
this work. The power dissipated in TEOS and Nb are notable and comparable at 4.2 K, which
contribute to the low Qi in RQL clocks. The loss mechanisms and fabrication limitations will be
discussed further in Chapter 2.
As a consequence, reducing RF losses in superconductor-dielectric resonant systems is
critical for the scaling of RQL and AC-powered SFQ technologies. Furthermore, it is also
important to fully characterize power dissipation as a function of frequency (MHz-GHz) and
temperature (mK-K) to assure optimum operating points, as non-monotonic functioning are
possible. The technological challenge this work will aim to address through materials science
is power dissipation in RQL clock interconnects by modeling and characterizing microstrip
transmission line resonators.
9
1.3 Summary
In summary, there is an opportunity to increase the power efficiency of RQL resonator-
based interconnects to increase the technology?s competitive advantage. This dissertation intends
to remedy this state of affairs by providing novel approaches to characterize the microwave losses
of Nb and TEOS. I will present techniques that serve as tools to understand both the intrinsic
properties and extrinsic effects due to fabrication. These approaches include finite element
modeling (e.g. HFSS), physical characterization of the material microstructure, a method to
deconvolve the Nb and TEOS loss in a single measurement, and temperature dependent loss
spectroscopy. The methods established can be used to predict the power efficiency of RQL
interconnects for a given material system, and provide a path for future work.
10
Chapter 2: Materials for Superconducting Interconnects
2.1 Overview
In this chapter, I will introduce the intrinsic material properties responsible for power
dissipation in RQL interconnects. Each subsection will briefly review the history of superconductors
and dielectrics. The superconductor intrinsic resistance Rs and dielectric loss tangent tan ?
will be defined. Their respective temperature dependence and frequency dependence will also
described, supported by models and measurements. The assumptions, concepts, and equations
will be used throughout Chapters 3-5.
2.2 Superconductor Electrodynamics
2.2.1 A Brief History
Superconductivity was first discovered in 1911 by Heike Kamerlingh Onnes, the pioneer
of helium liquification [38], when he measured a zero resistance on a mercury sample at 4.2 K
[39]. In 1934, Gorter and Casimir developed the thermodynamic two-fluid model [40], where
the total electron density ne in a superconductor is split into two parallel channels: a normal
electron density nn and a frictionless electron density ns. The non-dissipative electron channel
ns is now commonly referred to as the superfluid electron density, coined from the discovery of
11
superfluidity in 4He (zero viscosity) by Kapitsa [41] and independently by Allen and Misener
[42] in 1937. At absolute zero temperature T = 0 the superfluid electron density equals the total
electron density ns/ne = 1. At a thermodynamic critical temperature Tc of a superconductor
(e.g. Nb bulk or thin film), the superfluid electron density is equal to zero ns/ne = 0, having a
model temperature dependent form
( )
ns ? T
4
= 1 (2.1)
ne Tc
In 1935, Fritz and Heinz London proposed the two constitutive relations describing the
electrodynamics response of a superconductor in an electromagnetic field [43]. The second
London equation paired with Ampere?s law gives
?2 1B = B (2.2)
?2L
where B is the vector magnetic field inside the superconductor, and ?L is the London penetration
depth
?
mc2
?L = (2.3)
4?n 2ee
where ?0 is the vacuum magnetic permeability, c is the speed of light in vacuum, and m and
e are the mass and charge of an electron, respectively. Note, the London penetration depth
is traditionally defined in terms of the total electron density, and is not temperature dependent
as shown in Eq. 2.1. The temperature dependence of ?L(T ) will be reviewed in Section 2.2.2.
Eq. 2.2 suggests an applied external magnetic fieldH0 decays exponentially inside a superconductor
12
to a magnetic field of H = H0/e at a distance into the surface equal to the London penetration
depth ?L ? 40 nm for bulk Nb, where e is the first power exponential. In normal metals, the
decay length is associated with the commonly known skin effect and the corresponding skin depth
of Cu is ? 700 nm at ?/2? = 10 GHz [44]. Also note, the skin depth of Cu is infinite at ? = 0
while ?L is finite.
Jumping ahead, in 1957 the microscopic theory for superconductivity was established
by Bardeen, Cooper, and Schrieffer (BCS) theory [45]. In accordance with BCS theory, the
superfluid electrons have a bound (e.g. lower energy) state mediated by phonon lattice vibrations.
The superfluid state consists of Cooper pairs. Using the simple electrodynamic model description,
locally constricting the lattice around one paired electron in a single Cooper pair allows the other
paired electron to later travel in the opposite direction with stronger lattice Coulomb interactions.
The lowest-energy excitations of the superconduction states requires energy 2?, where ? is the
BCS energy gap. At zero temperature T = 0, the Nb BCS energy gap, a strongly coupled
superconductor, can be estimated as [46]
? ? 1.97kBTc (2.4)
where kB = 86 ?eV/K is the Boltzmann constant. A strongly coupled superconductor means
there is a strong electron-phonon coupling between the Cooper pairs and the phonon lattice
energy h??ln, where h? is the Planck constant. For the measurements in Chapters 4 and 5 of Nb thin
films, assuming Tc ? 9.2K, the stimulus frequencies f ? 10GHz are much lower than the gap
frequency f  2?/h ? 750GHz. Estimates of the Nb energy gap from temperature dependent
measurements of microstrip transmission line resonators will be presented in Chapter 5 using the
13
dispersive loss deconvolution (DLD) method.
2.2.2 Complex Conductivity
In 1940, Heinz London measured resistive loss in superconducting Sn by means of calorimetry
at high frequencies, and proposed it was due to a two-fluid model (interpenetrating mixture of
normal and superconducting electrons) [47]. Later in 1947, Sir Brian Pippard measured the
surface impedance of superconducting Sn and Hg at 1.2 GHz using a quarter-lambda resonator
[48]. I will come back to surface impedance measurements in Section 2.2.3. The above experiments
and many more led to the Mattis and Bardeen theory in 1958 that unified the anomalous skin
effect for normal and superconducting metals [49].
In weak magnetic fields and in the low frequency limit h??  ?, the Mattis and Bardeen
expressions for complex conductivity ? = ?1 ? i?2 of a superconductor can be approximated by
[50, 51]
?1 = 2?(T ) e
?(T )/kBT
?(T )/k T 2 ln
?(T ) (2.5)
?n kBT (1+e B ) kBT
?2 = ??(T ) tanh ?(T ) (2.6)
?n h?? 2kBT
where ? is the angular frequency, ?(T ) is the temperature dependent BCS superconducting
gap, ?n is the normal metal conductivity, and ?1 and ?2 are the real and imaginary components
of the complex conductivity at temperature T . At absolute zero temperature T = 0, the real
conductivity is effectively zero ?1 = 0. Eq. 2.5 was approximated by Kautz [50] and Eq. 2.6 by
Tinkham [51] for all temperatures below T < Tc.
Similarly, in weak magnetic fields and in the low frequency limit h??  ?, the two-fluid
14
model yields a complex conductivity of a superconductor
? = ?1 ? i? ? ? (n ?1 ?1 ?22 n n/ne)? i?0 ? ? (2.7)
where nn/ne is the normal electron density fraction, ? = 2?f is the angular frequency, and ?n
is the normal metal conductivity proportional to the normal electron mean free path ?n ? `mfp
[49]. Here, ?2 is London?s conductivity and ? is the resulting magnetic penetration depth of the
superconductor that can deviate from the bulk London penetration depth.
Using Pippard?s modification to the London theory [52], and assuming the dirty limit where
the microscopic (bulk) coherence length ?0 is of the same order as the electron mean free path
`mfp, the magnetic penetration depth can be approximated as
? ? ?L(?0/`mfp)1/2 (2.8)
where ?0 is the characteristic coherence length of the superconductor.
Eq. 2.8 suggests the screening length will effectively increase relative to the microscopic
London penetration depth as a metal becomes more impure ? > ?L. Hereby the magnetic
penetration depth is sensitive to material defects (e.g. impurities, grain boundaries, dislocations),
where higher concentrations of defects increase scattering and therefore decreasing the `mfp. For
example, bulk Nb polycrystalline material have ?L ? ?0 ? 39 nm with `mfp  ?0, whereas
nanocrystalline Nb thin films have `mfp ? 7 nm due to increased defect scattering mechanisms
(e.g. grain boundary scattering), and results in a magnetic penetration depth of approximately
? ? 90 nm. This is a good approximation for Nb thin films [53] and patterned submicron wires
15
[54] described in Chapters 3-5.
Before the now accepted BCS theory, the two-fluid model developed by Gorter and Casimir
[40] described the temperature dependent behavior of superconductors by assuming the density
fraction of superconducting electrons ns/ne would drop to zero at the superconducting critical
temperature Tc with dependence
?(T ) = ?(0) [1? (T/T )?]?1/2c (2.9)
where ?(0) is the real (measured) magnetic penetration depth at temperature T = 0, and ? is an
empirical fitting parameter. For low Tc superconductors ? ? 4 [51] (the approximation used in
Eq. 2.1) and for high-temperature superconductors ? ? 2 [55].
Comparing Eqs. 2.5-2.6 and Eq. 2.7 show a relationship between the superconducting
gap ? and the penentration depth ?. Abiding by the assumption that h??  ?, it is assumed
here that the penetration depth ? is frequency-independent. Additionally, since all measurement
temperatures are T  Tc and well below the gap frequency for Nb, it is assumed here that
?1  ?2. The complex conductivity is more accurately expressed by the BCS theory but
is beyond the scope of this work, and can be found in Ref. [45] and is nicely summarized in
Ref. [17]. I will refer back to the complex conductivity in the next section and use it to express
the surface impedance of the superconductor. Using the complex conductivity, Nb microstrip
transmission lines are modeled in Chapter 3 and simulated parameters are used to deconvolve the
intrinsic surface resistance Rs from measurements in Chapters 4 and 5.
16
2.2.3 Surface Impedance
The ultimate conclusion from this section will relate the complex conductivity ? into an
intrinsic surface impedance Zs to obtain the mechanism of resistive power dissipation coming
from the superconductor in RQL interconnect, i.e. intrinsic resistance Rs. Furthermore, the
superconducting parallel plate transmission line will be introduced. It will be shown that as the
magnetic penetration depth ? becomes comparable to the finite thickness and width dimensions
of a superconducting microstrip transmission line, a method is needed to accurately extract Rs
to compare RQL interconnects of different dimensions on equal footing and analyze extrinsic
effects due to fabrication.
At the end of the 19th century, engineers primarily relied on circuit theory to simplify
complex circuit analysis and design without the aid of computers. Discrete components (e.g.
transmission line) could be ?lumped? and connected into a network only using resistors, inductors,
and capacitors. With an an electromagnetic wave applied to a combined circuit or discrete
component at radio frequencies (RF), the alternating currents and voltages can be related to the
circuit by an intrinsic impedance Zs introduced by Oliver Lodge in 1889 [56].
First, consider a wave propagating from vacuum to a media as shown in Fig. 2.1 with a
relative dielectric constant ?r or a conductivity ? (real or complex). Fig. 2.1a is an analogy to
a transverse electromagnetic (TEM) plane wave propagating along the x-direction into a perfect
electrical conductor transmission line with media having a relative dielectric constant ? > 1.
Using the coordinate system in Fig. 2.2, the complex voltage V and current I for a wave propagating
17
along the x-axis of a transmission line can be described by the following three equations [57]
dV/dx = ?ZI (2.10)
dI/dx = ?Y V (2.11)
?
? = ZY (2.12)
where Z and Y are the distributed series impedance and the distributed shunt admittance of the
line, respectively, and ? is the propagation constant of the line. This will be revisited in Chapter 3.
The propagation of plane electromagnetic waves along the x-direction in a lossy medium
shown in Fig. 2.1b can be described also by the following three equations
dE/dx = ?i??H (2.13)
dH/dx = ?(? + i??)H (2.14)
?
? = i??(? + i??) (2.15)
where E is the electric field vector pointing along the y-direction, H is the magnetic field vector
pointing along z-direction, ? is the wave angular frequency, ? = ?0?r is the permittivity of the
medium, ? = ?0?r is the permeability of the medium, ?r is the relative dielectric constant of the
medium, ?r is the relative permeability of the medium, ? is the medium propagation constant,
and ? is the conductivity of the medium.
Thus, the impedance term from transmission line theory using V and I could be extended
to electromagnetic wave theory E and H, and the intrinsic impedance takes the general form and
18
Figure 2.1: Diagram showing wave propagation from vacuum into media with a relative
dielectric constant ?r or conductivity ? (real or complex). a) Analogy to transmission line with
media having a relative dielectric constant ? > 1 showing the Complex voltage V and current
I as the wave propagates into media described by Eqs. 2.10-2.12. b) The case for a normal
incidence propagating plane wave into media (see Eqs. 2.13-2.15). c) The case for a glancing
incidence propagating plane wave for media with ?r > 1 with refracted angle of ? > 0. In
general, the refracted wave is an in-homogeneous wave. This is a wave for which the planes of
constant phase are not parallel to the planes of constant amplitude [58]. d) The case for glancing
incidence propagating plane wave for media with ?r  1 with refracted angle of ? ? 0. Cases
c) and d) have been adopted from [58] using Snell?s law of refraction in conductive media (see
Eq. 2.17).
is related to the media conductivity by
?
Ey i?? ?
Zs = = = (2.16)
Hz ? + i?? ? + i??
19
Fig. 2.1c and Fig. 2.1d are two cases for a wave propagating at glancing incidence relative
to the vacuum-media interface. Using Snell?s law of refraction in conductive media [58],
?
2?? ?0
sin? ? sin ? ? k0? 2 sin ? ? ?/?EM (2.17)
?
where ? is the incident angle for wave propagating in vacuum, ? is the refraction angle for
wave propagating in the media, ? is the angular frequency, ?0 is the vacuum permittivity, ?0 is
the vacuum permeability, ? is the conductivity of the medium the wave is refracted into, k0 =
?
? ?0?0 is the vacuum wave number, ?EM is the wavelength of the propagating electromagnetic
wave. The left hand side approximation in Eq. 2.17 can be used for case Fig. 2.1c, where
the media has an absolute relative dielectric constant greater than 1, |?r| > 1. The middle
approximation in Eq. 2.17 can be used for case Fig. 2.1d for a superconductor where the relative
dielectric constant is negative and its absolute value is much greater than 1 (see Section 3.4.2
where a superconductor is defined as a dielectric with negative permittivity in HFSS). The middle
approximation in Eq. 2.17 assumes the conductivity of the superconductor is ? ? ?2 from Eq. 2.7
assuming ?2  ?1. Assuming a Nb magnetic penetration depth of 40 nm, ? ? 1.2? 10?5 rad.
Consequently, Eq. 2.17 and Fig. 2.1d exemplify that for a normal or glancing incident wave
propagating into a superconductor where |?r|  1 the refracted wave will propagate normal to
the vacuum-superconductor interface. This is the basis for why a superconducting parallel plate
transmission line supports a transverse magnetic (TM) propagation wave and is termed a Swihart
wave [59]. In Fig. 2.2, the wave is propagating along the y-direction which is glancing incidence
to the two superconductor-dielectric interfaces. At x = 0, Ey = 0 and at the superconductor
20
Figure 2.2: Diagram showing a superconducting parallel plate transmission line with a Swihart
transverse magnetic (TM) wave [59] propagating along the y-axis, where looking into and out
of the page the superconducting plates are infinitely long. The dielectric material between the
plates has a thickness s, and each superconducting plate has a thickness d. The propagating wave
is transverse magnetic because there is a small, finite y-component to the electric field at the
superconductor-dielectric boundaries. The y-component of the electric field is responsible for
the curvature of the electric field lines near the boundary (blue).
dielectric interfaces (x = ?s/2), the Ey has a finite magnitude. It can also be seen that Fig. 2.1a
and Fig. 2.2 are equivalent and Eqs. 2.10-2.12 can be used. As the dielectric thickness becomes
small, and comparable to the magnetic penetration depth ?, Ey becomes significant and can
greatly affect the phase velocity vp (see Appendix Eq. D.7, where fn ? vp).
Assuming the superconducting plate thicknesses d and dielectric thickness s in Fig. 2.1
are much larger than the magnetic penetration depth d  ? and s  ?, respectively, the Ey
magnitude is negligible to Ex magnitude, and therefore a surface impedance boundary condition
(SIBC) can be enforced.
The Leontovich-Shchukin SIBC approximation is derived by using the following assumptions
but in the context of a good superconductor. The wavelength and magnetic penetration depth ?
traveling in the media must be small relative to the material dimensions. As a consequence, the
21
material must be sufficiently thick (bulk) where reflected fields from the back surface/interface
are neglected. Furthermore, the EM fields vary slowly on the surface of the medium relative to
the wavelength. This constitutes a flat surface, or its smallest radius of curvature must be large
relative to the magnetic penetration depth. Assuming the above, and referring to the coordinate
system defined in Fig. 2.2, for a wave traveling in the y-direction along a plane surface, i.e.
the plane between superconductor plates and dielectric in Fig. 2.2, the complex intrinsic wave
impedance is in units of ohms per square and proportional to the ratio of the electric and the
magnetic fields by
? ? ? ?
Ey ?
Z ? i?? ?? i??wave = ? = ? (2.18)Hz plane wave ? + i?? ???? ?
where ? = ?0?r is the permeability of the medium, ? = ?0?r is the permittivity of the medium,
Ex is the magnitude of the electric field pointing out-of-plane, Hy is the magnitude of the
magnetic field pointing in-plane, and ? is the wave source frequency. The right approximation in
Eq. 2.18 holds for good, nonmagnetic conductive media, where the relative permeability ?r and
permittivity ?r are equal to unity ?r = ?r = 1, and so the permeability and permittivity are that
of the vacuum ? = ?0 and ? = ?0, respectively. As it can be seen, Eq. 2.18 is not dependent on
coordinates and assumes the wave impedance is constant over the plane conducting surface. In
other words, the intrinsic impedance Zwave connects the electromagnetic response of conducting
media with its conductivity function dependent on material properties.
Eq. 2.18 can be used as a surface impedance boundary condition (SIBC), and can greatly
simplify complex problems. Today, SIBC are used in finite-element modelers (FEM) to simplify
geometries and numerically solve the electromagnetic fields faster. Sergey Yuferev and Nathan
22
Ida give a good overview of the different SIBCs, their history, and applications [60]. An example
of using a SIBC to validate HFSS simulations can be found in Section 3.4.1.
In 1938, the impedance concept was later generalized and applied to engineering applications
(e.g. power engineering, transmission lines) by Sergei Schelkunoff [61] who first coined the
skin effect term. In this work, assuming a superconducting medium with complex conductivity
? = ?1 ? i?2 is nonmagnetic with ? = ?0 and a good conductor with negligible displacement
current ?  ??, the intrinsic impedance simplifies to
?
i?0?
Zs = Rs + iXs = (2.19)
?1 ? i?2
where Rs is the intrinsic surface resistance and Xs is the intrinsic surface reactance of the
superconducting media. Taking the approximation for the complex conductivity in Eq. 2.7
assuming, T  Tc, ?1  ?2, and h??  ? it can be shown that the two-fluid superconductor
intrinsic impedance can be written as
1
Zs = Rs + iX ? ?2 2 3s
2 0
? ? ?1 + i?0?? (2.20)
where ?0 is the vacuum magnetic permeability, ? is the stimulus angular frequency, ? is the
magnetic penetration depth defined in Eq. 2.8, and ?1 is the conductivity of the normal electrons.
Eq. 2.20 above models a bulk superconductor where the two-fluid model can be assumed. In this
work we are concerned with measuring the intrinsic resistance Rs as it determines the power lost
due to the superconductor in RQL interconnects. In this context, the intrinsic resistance is that of
the effective bulk property of the superconductor and its relation to finite thickness effects will be
23
discussed later in this section. In normal metals, where ?1  ?2, the real and imaginary parts of
?
the intrinsic resistance are equal, and frequency dependence Rs = Xs ? ?. For Cu at 10 GHz,
the intrinsic resistance isRs ? 26m? and? 2m? at temperatures T = 300 and 10 K, respectively.
For a superconductor like Nb, assuming a good isotropic superconductor with ?1  ?2, Nb has
a quadratic frequency dependence of the intrinsic resistance R 2s ? ? [62, 63]. For clean bulk
niobium, Rs ? 20 ? 40 ?? at 4.2 K and 10 GHz. Furthermore, for thin film Nb, the BCS Rs
can be as low as 17 ?? at 4.2 K and 10 GHz when ?1 is at a theoretical optimum for the case
?0/lmfp ? 4?, where ?0 is the microscopic (bulk) coherence length [64]. This means intrinsic
resistance can be lower for Nb thin films compared to bulk. To achieve minimum Rs ? 17??
in Nb, the optimal mean free path is lmfp ? 10 nm for ?0 ? 40 nm. This is conveniently close
to typical Nb lmfp values for thin films having thicknesses d ? 100 ? 200 nm [53], which is
the typical wiring layer thickness in RQL fabrication (see section 2.4). Using Eq. 2.8, assuming
?0 ? ?L = 40 nm, and lmfp = 10 nm yields a magnetic penetration depth of ? = 80 nm. This is
within 12% of the measured value of ? = 90 nm for submicron Nb wires determined from self
inductance measurements at T=4.2 K with exceptional hardware-to-model correlation [54, 65].
It is important to note that Eq. 2.20 above describes the impedance of a bulk (infinitely
thick) superconductor. Here we deal with finite thin film superconductors that have been patterned
into narrow strips, making the above equation incomplete. It is of vital importance in this work
to ?extract? the intrinsic bulk impedance from an ?effective? impedance so we analyze extrinsic
effects changing Zs as the transmission line dimensions shrink. A significant amount of work has
been done to modify the impedance definition to account for non-ideal or ?non-bulk? situations,
and they will be further reviewed and compared to the intrinisic impedance Zs.
Starting in the 1930s, radio technology facilitated theory development of electromagnetic
24
wave propagation from a radiating antenna over the Earth?s surface. Solutions to the wave
propagation can be very complex as the waves can travel in and reflect off of many media
including the earth?s surface, embedded surfaces, and air over long distances where the curvature
of the Earth needs to be considered. To simplify the problem, Leontovich [66, 67] and Shchukin
[68] independently proposed an approach where the air region only needs to be considered by
making the Earth?s surface a surface impedance boundary condition (SIBC).
For the case where the magnetic penetration depth becomes comparable to the material
or film-thickness ? ? 2d, modifications to the intrinsic impedance Zs need to be made. The
modifications need to be made because unlike the Leontovich-Shchukin SIBC approximation,
reflected fields across the bottom surface of the finite thickness superconductor cannot be neglected
due to the condition ? ? 2d. Furthermore, thin-film superconductors are typically deposited
on dielectric substrates, so the electromagnetic fields at the dielectric-superconducting interface
as well as into the dielectric need to also be accounted for. In the 1980s high temperature
superconductors (HTSs) were discovered. Klein et al. came up with the theory for the effective
surface impedance Zeff in 1990 to account for finite thickness effects [69] since HTSs have
very large magnetic penetration depths relative to the typical thin-film thicknesses ? ? 2d. First,
assuming an infinitely thick dielectric substrate below the superconducting thin film, the effective
surface impedance was calculated by using an impedance transformation method having the form
Zeff = z(d)Zs (2.21)
z(0) + tanh(?d)
z(d) = (2.22)
1 + z(0) tanh(?d)
where d is the film thickness, Zs is the intrinsic surface impedance given by Eq. 2.20, z(0) is the
25
impedance at the film-substrate interface, z(d) is the transformed impedance up to the film surface
d, z(0) and z(d) are normalized to Zs of the film, and ? = ? + i? is the complex propagation
constant of the superconductor.
From this and accounting for the dielectric substrate, the effective surface impedance Reff
of a superconducting plate can be approximated by [69, 70]
Zeff = Reff + iXeff (2.23)
[ ] ( )
2
Reff = R coth (d/?) +
d/? + Xs d/?s 2 2 (2.24)sinh (d/?) Zsub sinh (d/?)
Xeff = ??0?eff (2.25)
( )
Zsub =
?0 1 + i tan ?1/2 (2.26)
? 2r
where Reff is the effective surface resistance, Xeff is the effective surface reactance, ?eff =
? coth (d/?) is the effective penetration depth, ?0 is the permeability of free space, Zsub is the
bulk impedance of the dielectric substrate, ?r and tan?? are the relative dielectric constant and
loss tangent of the substrate, respectively, and ?0 ? ?0/?0 = 120? ? is the intrinsic wave
impedance.
It can be inferred from Eqs. 2.23-2.25 that the primary contribution to the enhancement of
Zs is due to the ratio of the film thickness relative to the magnetic penetration depth d/?. As the
film thickness becomes small relative to the magnetic penetration depth d/?  1, the effective
impedance Zeff dramatically increases as a consequence of the current density increasing inside
of the superconducting film by approximately a factor coth (d/?). Conversely, as d/?  1,
Eq. 2.23 reduces to Zs of a bulk superconductor in Eq. 2.20. The second term of Reff is due
26
to power transmission into the dielectric substrate, and decreases in effect as the impedance
mismatch between the dielectric substrate and and superconducting film increases Xs/|Zsub| 
1 and is independent of the dielectric substrate loss tangent tan ?, assuming Xs  Rs and
transmitted power is not reflected back to the film.
Notice in Eq. 2.24, neglecting the substrate contribution, Reff is the intrinsic resistance
multiplied by an effective geometric factor in brackets. If one wanted to compare two samples
with different film thicknesses, one would need to use Eq. 2.24 to extract the intrinsic (bulk)
resistance. The same concept holds for extracting the intrinsic reactance Xs from Eq. 2.25.
2.2.4 Temperature Dependence of Surface Impedance
The temperature dependence of the superconductor bulk intrinsic reactance Xs (magnetic
penetration depth ?) from Eq. 2.20 can be approximated by assuming two-fluid model found
in Eq. 2.9. The temperature dependence of the intrinsic resistance Rs is better approximated
from the temperature dependence of the BCS gap. At temperatures approximately T ? Tc/2 the
intrinsic resistance scales exponentially on the ratio of the BCS gap and temperature by Rs(T ) ?
e??(T )/kBT . In this temperature range, the magnetic penetration depth is assumed to be constant,
having the value ? found in Eq. 2.9. Below a certain temperature the intrinsic resistance becomes
temperature independent having the value Rres, and is Tres ? 1.8 K for thin film Nb [71]. With
this, the temperature dependence of the intrinsic impedance Zs0 at a reference frequency ?0 at
T ? Tc/2 can be approximated by
??(T )
Rs0(T ) = R (0)
1 e kBT +R A ?B(T/Tc)s0 res0 ? e + C for T ? Tc/2 (2.27)T T
Xs0(T ) = ?0?0?(T ) coth (d/?(T )) (2.28)
27
?(T ) = ?(0) [1? (T/T )?]?1/2c (2.29)
where Rres0 is the residual intrinsic resistance at a reference frequency ?0 below T ? 1.8 K for
thin film Nb, Rs0(0) is the BCS zero temperature intrinsic resistance, A is an empirical fitting
parameter, B/Tc = ?(0)/kBTc is the gap ratio fitting parameter, C = Rres0 is the residual
intrinsic resistance fitting parameter, d is the superconductor thin film thickness, and ?(0) is the
magnetic penetration depth at zero temperature.
Valente-Feliciano et al. found B/Tc = ?(0)/kBTc ? 1.89 for Nb thin films [71], within
5% agreement to approximation of ?(0)/kBTc ? 1.97 made in Eq. 2.4. The approximation on
the right in Eq. 2.27 will be used in Chapter 5 to fit temperature dependent microstrip transmission
line (MTL) data having fitted values of ?/kBTc ? 2.1.
2.2.5 Summary
In summary, to meaningfully compare results for RQL interconnects having different microstrip
transmission line (MTL) widths requires the determination of the intrinsic resistance Rs shown
in Eq. 2.20. Thin-film superconductors screen electromagnetic fields on the length scale of
the magnetic penetration depth ?. Increased defectivity in the material results in an increase
in ?, causing the magnetic field penetration to drive further inside of the thin-film or MTL.
Furthermore, an increase in the magnetic penetration depth means the RF currents sample defects
buried deeper in the wire, and can induce an increase in Rs. The surface impedance boundary
condition (SIBC) first introduced by Leontovich and Shchukin simplifies the problem making
it easier to account for non-idealities relative to bulk solutions. Klein et al. approximated the
SIBC condition for an infinitely wide thin film superconductor on a dielectric substrate where
28
? is comparable or much larger than the thin film thickness ? ? 2d. Although, an additional
treatment is needed to account for lossy MTLs where ? also becomes comparable to the dielectric
separation thickness s between the conducting strip and the groundplane as well as the MTL
width w. With the advancement and maturation of finite element modelers, it is possible to
numerically solve the electromagnetic fields and extract the intrinsic impedance Zs, and the
method will be carried out in Chapter 3 to extract Rs from measured data in Chapter 4 and
5.
2.3 Amorphous Dielectrics
Silicon dioxide SiO2 fabricated via chemical vapor deposition using a tetraethyl orthosilicate
(TEOS) precursor is the primary dielectric material used in the microstrip transmission line
resonators measured in chapters 3-5. Silicon oxide in this context will be referred to by its
precursor TEOS throughout this dissertation. TEOS is deposited on Si wafers and is limited to
temperatures T ? 150?C (see section 2.4.6). Using common plasma-enhanced chemical vapor
deposition processes similar to references [72, 73], the kinetics are such that surface diffusion
is limited, preventing the growth of a crystalline lattice. The TEOS amorphous structure is
confirmed by the presence of a diffuse ring in electron diffraction image similar to reference [74].
The combination of the T ? 150?C thermal budget and amorphous structure of TEOS limits the
lower bound of the loss tangent to approximately tan ? ? 1?10?3 at 4.2 K. Comparatively, this is
a factor of 3-4 of magnitude higher than thermal oxide grown SiO2 [75]. Therefore, this gives rise
to non-ideal high power dissipation coming from the TEOS dielectric in RQL interconnects and
is a motivation of this work to review the fundamental material properties and loss mechanisms.
29
The typical operating temperatures for classical superconducting logic and quantum computing
circuits are T ? 4.2 K and T  1 K, respectively. At these temperatures, the power loss
from the amorphous (glassy) dielectrics depends strongly on the stimulus energy h??, where ? is
the applied electric field frequency. Therefore, it is important to briefly review the history and
explicitly state the temperature and frequency regime carried out in measurements performed
in chapters 4 and 5. First, I will introduce the fundamental dielectric loss tangent tan ? that
describes how energy is dissipated in a dielectric material due to relaxation processes. Then, I
will briefly review the history of low temperature measurements of tan ?. Here, I will focus the
review on the difference between crystalline and amorphous dielectrics. I will also incorporate
the relative dielectric constant ?r, as its temperature and frequency dependence are governed by
the same mechanisms as tan ?. For a proper introduction, the two-level system model will be
included in the discussion. To conclude this section, I will summarize the historical work that
has been done by showing the universality of the tan ? temperature dependence and its respective
mechanisms, where varying ? more or less shifts the intersection of temperature regimes. Finally,
I will explicitly state the mechanism dominating tan ? in different temperature and frequency
regimes used in the measurements carried out in Chapters 4 and 5.
2.3.1 The Dielectric Loss Tangent tan ?
A simple RC circuit analogy can be used to describe dielectric relaxation [76]. In a linear
system a dielectric material can be modeled having a resistance R due to some Joule losses in the
dielectric, and capacitance C capable of holding a chargeQ. Here, we assume a seriesRC circuit
with time-independent R and C. For a step-function voltage applied to the dielectric, where the
30
voltage maximum is V = V0 at time t = 0 and zero V = 0 for time t > 0, the time-dependent
discharging can be described as
Q = CV0e
?t/?rel (2.30)
where Q reaches a fraction 1/e after a time ?rel, which is the characteristic relaxation time, or
RC time constant equal to ? i?trel = RC. For a sinusoidal voltage stimulus V (t) = V0e at angular
frequency ?, the charge has both in-phase and quadrature responses, and can be considered
complex, and takes the form
( )
1 ? ??relQ(?, t) = CV0 i ei?t (2.31)
1 + ?2? 2rel 1 + ?
2? 2rel
Eq. 2.31 shows a time lag between the charge response Q and applied voltage V . This phase lag
can be represented by the tangent of an angle ? as tan ?. Assuming tan ? << 1 implies ? ? ??rel,
where ?rel is the characteristic time lag between Q and V . The time lag induces a dissipation of
energy. In other words, if the sinusoidal voltage was turned off, the dielectric charge will ?relax?
to a lower energy state, and that time lag, or characteristic relaxation time ?rel will dissipate some
amount of energy. It can be shown that the energy dissipated W and ? are related by W ? sin ?
[76]. For small angles ?, the energy dissipated in the dielectric is directly proportional to ?, hence
the commonly used term loss tangent tan? and loss angle ?. For a parallel RC circuit, which is
more analogous to our circuits here, the loss tangent is related to dielectric relaxation time
tan ? = 1/??rel (2.32)
31
Both the phonon (quantized lattice vibrations) and polarization (charge) can be described with the
above RC circuit model. For an acoustic frequency (sound wave) stimulus, the acoustic energy
loss is caused by the lattice viscosity, or resistance of the vibrating atoms to displacement. Here,
the resistance R is proportional to the viscosity and C is proportional to the stored energy in
lattice vibrations. The time lag of lattice displacements induces energy loss.
If a dielectric material can be electrically polarized by a stimulus due to some displacement
of charge (e.g. doped dielectric crystals, ionic crystals, amorphous dielectrics), then charge
displacement relative to its lattice vibrations under electromagnetic stimuli will induce power
dissipation. Hence, phonon vibrational modes and electronic polarization are coupled and dissipate
power in dielectric materials.
2.3.2 Dielectric Loss Mechanisms and Phonon Scattering at T < 10K
As mentioned at the beginning of section 2.3, the TEOS dielectric measured chapters 3-5
is considered amorphous. Therefore, in the context of this work, only amorphous dielectrics will
be considered, but it is important to briefly summarize the differentiation between the material
properties of glassy and crystalline dielectrics at cryogenic temperatures approximately T <
100 K. The surprising difference was seen in heat capacity measurements of vitreous, meaning
glassy, silica and crystalline ??quartz by Zeller and Pohl in 1971 [77]. At 0.1 K, the specific heat
capacity CD of vitreous silica was 2 orders of magnitude higher than quartz. Crystalline quartz
follows the typical Debye dependence CD ? T 3 below its Debye temperature approximately
T  TD ? 360 K [78]. At these temperatures, the phonons, or lattice vibrations, can only
occupy long wavelength vibrational modes and can be treated as an elastic continuum, and so the
32
heat capacity is proportional to the thermal conductivity by
1
?D = CDvD`ph (2.33)
3
where CD is the Debye specific heat capacity per volume of the heat carrying phonons, vD is
the phonon velocity, and `ph is the phonon mean free path due to scattering by lattice defects.
Referring back to the Zeller and Pohl measurements, it was found that vitreous silica, and
universally for amorphous dielectric materials for T  TD, have a temperature dependent
specific heat and thermal conductivity proportional to ? T and ? T 2, respectively. Early
phenomenological models tried to modify the specific heat capacity alone to account for the
? T dependence, but were not consistent and are further reviewed in [79]. This led to further
experimental evidence that the anomalous thermal properties arise from additional excitations
inducing a change in the specific heat and suggests the phonon mean free path has a temperature
dependence `ph(T ).
Similarly to the thermal properties, there is a close relationship between the acoustic and
dielectric attenuation/loss properties in amorphous dielectrics at temperatures approximately below
100 K [80]. For example, for amorphous dielectrics there is a quintessential maximum in loss/attenuation
at a temperature Tmax. Anderson and Bommel confirmed this by plotting the temperature of this
maximum loss peak as a function of stimulus frequency ?/2? and found exceptional agreement
between acoustic and electrical measurements (see Fig. 2.3). An Arrhenius relationship suggests
a thermally activated process and will be mentioned briefly later. Notice, Fig. 2.3 shows that
Tmax will increase to higher temperatures as the measurement frequency is increased.
In 1972 Anderson [82] and Phillips [83] proposed a two-level system (TLS) model to
33
Figure 2.3: This figure is adopted from reference [81] and discussed in reference [80] Fig.
6.2-6.3.
34
describe the thermal properties of amorphous dielectrics at temperatures below the crystalline
Debye temperature TD. In the TLS model, atoms (or electrons, no one knows) occupy one of two
minima capable of tunneling between the two states quantum mechanically. Consequently, the
manifestation of tunneling between two energy states scatters phonons, prompting an additional
mechanism of dissipated energy (i.e. TLS loss). With this, the TLS model universally explains
the temperature dependent specific heat ? T and thermal conductivity ? T 2 of amorphous
dielectrics at temperatures T < 1 K. Effectively, and to support the context of this dissertation,
the anomalously low thermal conductivity for amorphous dielectrics at temperatures T  10 K
arrises from low phonon scattering lengths `ph as a result of strong resonant absorption of thermal
phonons by the TLSs [80].
Referring back to Fig 2.3, the dissipated loss coming from TLSs have been measured by
acoustic attenuation and the dielectric loss tangent, both of which are related. The measured
acoustic attenuation ?db is proportional to the inverse phonon mean free path ? ? `?1db ph . Therefore,
at temperatures below T < Tmax and at a given reference frequency ?0, the temperature dependent
dielectric loss tangent and phonon mean free path are approximately proportional tan ?(T, ?0) ?
`?1ph (T, ?0) [84]. The frequency dependent proportionality between dieletric loss tangent and
ultrasonic absorption is tan ? ? ?db/?. Note, the frequency dependence of ?db depends on
which temperature below T < Tmax the measurement is taking place, and will be discussed
further.
Since it is established that energy dissipation measured by the loss tangent tan ? is coupled
to phonon scattering at temperatures T < 10 K, I will focus on the mechanisms causing the
temperature and frequency dependence of `ph(T, ?). The TLS loss can be split up into two
phonon scattering mechanisms, TLS tunneling due to resonant absorption `tun,res and to relaxation
35
`tun,rel [79]. Resonant absorption occurs when TLSs are resonant with the applied electric or
acoustic fields at particular energies and frequencies. Relaxation of TLSs occur when non-
resonant TLSs having a dipole moment are driven out of non-equilibrium in an applied field
and scatter phonons as the TLSs ?relax? in a decaying oscillatory fashion during redistribution
between the energy levels. Schickfus et al. was the first to measure the energy dissipation in
vitreous silica due to both TLS loss mechanisms below T  1 K [84] using the 2 cavity test
setup [85]. Loss due to resonant absorption manifests in measurements at low applied electric
fields, and at higher applied fields the TLSs are saturated and loss is dominated by TLS relaxation.
At higher temperatures in the approximate range 10 K < Tmax ? 100 K, ?classical?
relaxation dominate the phonon scattering process. Contrary to TLS relaxation via tunneling
between two energy states, classical relaxation scatters phonons by the dissipation of energy
from physical charge displacement (see Eq. 2.32) [76, 86]. The phonon mean free path due to
classical relaxation is `class,rel. Temperatures higher T > 10 K will not be considered since it
is beyond what is measured in chapter 3 and 4, but are mentioned for completeness. Therefore,
the three mechanisms contributing to the total phonon free path lph(T, ?) at temperatures below
T < 10K with temperature and frequency dependence are
l (T, ?)?1 = `?1ph tun,res(T, ?) + `
?1
tun,rel(T, ?) + `
?1
class,rel(T, ?) (2.34)
where the phonon scattering mean free path due to TLS resonant tunnelling, TLS tunneling
relation, and classical charge (dipole) relaxation are `tunn,res, `tunn,rel, and `class,rel, respectively.
Mentioned previously, the dielectric relaxation is proportional to the ultrasonic attenuation
tan ?(T, ?0) ? ?db(T, ?0) [84]. The ultrasonic attenuation is inversely proportional to the phonon
36
mean free path by ?db(T, ? ?10) ? lph (T, ?0) (see [79]). Accordingly, the total dielectric loss
tangent tan ? can be split up into the three loss mechanisms from Eq. 2.34 in the following way
tan ?(T, ?) = tan ?tun,res(T, ?) + tan ?tun,rel(T, ?) + tan ?class,rel(T, ?) (2.35)
where the dissipative loss associated TLS resonant tunnelling, TLS tunneling relaxation, and
classical charge (dipole) relaxation are tan ?tunn,res, tan ?tunn,rel, and tan ?class,rel, respectively,
and are all temperature and frequency dependent.
2.3.3 Temperature Dependence of tan ?
Referring to Fig. 2.4 starting from the lowest temperature, depending on the applied electric
field strength E, the dielectric loss is dominated by either TLS resonant tunneling (absorption)
or TLS relaxation processes. The response of TLSs to applied electric fields is dependent on the
coherence between their two eigenfunctions formally treated for solving the problem developed
for optical saturation [87] and magnetic resonance [88]. These methods have been applied to
TLSs in glasses [89] and a straighforward summary can be found in a review by Phillips et al.
[79]. The treatment for TLS field strength dependence in the acoustic case can be translated to the
electric case and the equations from Schickfus et al. for tan ? dependence on field strength will
be presented in Chapter 5. Essentially, at very low temperatures T  1 K, the critical intensity
ITLSc (in Wm
?2) is defined as the cross-over point from TLS resonant absorption (weak field) to
TLS relaxation (strong field) and can be written as [79]
2
TLS 3h? ?r?0cIc = (2.36)2p20T1T2
37
where ?r is the relative dielectric constant, p0 is the electric dipole moment (inCm), c is the speed
of light, and T1 and T2 are the TLS relaxation and dephasing times, respectively. For electric field
intensities greater than the critical intensity ITLS  ITLSc , TLS resonant absorption is saturated
and relaxation processes will dominate. For electric field intensities less than the critical intensity
ITLS  ITLSc , TLS resonant absorption processes will dominate.
Referring again to Fig. 2.4, below T? and for weak electric fields, tan ? is dominated by
TLS resonant absorption processes and is proportional to the inverse of the phonon mean free
path with temperature dependence tan ? ?1res,tun(?0) ? `res,tun(?0) ? tan h??0/kBT . Below T? ,
for strong electric fields, and for long relaxation minimum TLS relaxation times ??min  1,
tan ? is dominated by TLS relaxation and so called ?one-phonon? processes having temperature
dependence tan ?rel,tun(?0) ? `?1rel,tun(?0) ? T 3 [86]. Above T? , where relaxation times begin
to decrease due to thermal excitation ??min  1, TLS relaxation still dominates but a ?plateau?
is reached for a long range of temperatures yielding tan ?rel,tun(?0) ? `?1rel,tun(?0) = const.
Here, it is important to note that I am not reviewing or incorporating the soft potential model
first proposed by Karpov et al. and thoroughly reviewed in references [90, 91]. Essentially, the
soft potential model can explain the ?plateau?? and other features in the data for measurements at
higher frequency (? THz) by incorporating the existence of soft localized modes accounting for
anharmonicity. For the measurements carried out in chapters 3 and 4, the temperature ranges are
below T? in the frequency range ?/2? = 1 ? 20 GHz. Above T ?, classical relaxation processes
dominate where charge ?hopping? dominates with temperature dependence tan ?rel,class(?0) ?
`?1rel,class(?0) ? T [76, 79, 86]. At Tmax, the thermal energy is equal to the ?hopping? barrier
height and thermal vibrations now dominate the system above this temperature where static spin
polarizations are negligible [86].
38
Figure 2.4: Diagram depicting the regimes of mechanisms dominating the temperature
dependent tan ?. The general diagram was adopted from reference [86] for dependence of `ph
and modified to include the TLS absorption regime at the lowest temperatures and lowest applied
E fields, which is the most widely used and understood regime today. The loss tangent and
inverse mean free path at a reference frequency ?0 have temperature dependent proportionality
tan ? ? `?1ph The majority of the notations are adopted from references [90, 91, 92] that
thoroughly review all of the mechanisms relating to energy dissipation, but the soft potential
model is not included or reviewed here. The y-axis is plotted as the inverse quality factor or
internal friction Q?1 from acoustic measurements, which is proportional to the loss tangent
tan ?, the ultrasonic attenuation ?db typically measured in dB/cm, and the inverse phonon mean
free path `ph by Q?1 ? tan ? ? ?db ? `?1ph . The critical temperatures T? , T ?, and Tmax are
the transition temperature regimes are dominated by TLS resonant tunneling, TLS relaxation,
classical relaxation processes as temperature is increased. Note, this entire plot can shift to the
left in or regions can squeeze/widen in temperature depending on the relative energy scales of the
various processes to the thermal energy kBT , applied frequency ?.
39
Note that the general trend in Fig 2.4 can shift left/right, or regions can squeeze/widen,
in temperature depending on the relative energy scales of the various processes unique to the
material such as (i) the thermal energy kBT , and (ii) applied frequency ?. As an example,
internal friction data was taken from reference [93] and re-plotted to estimate T? in Fig. 2.5. Note,
the inverse acoustic internal friction Q-factor is proportional to the loss tangent Q?1 ? tan ?.
These plots clearly show a frequency dependence of T? . This is analogous to the frequency
dependence of Tmax mentioned and confirmed by Anderson and Bommel shown in Fig. 2.3. It
can be estimated that T? ? 22K at a measurement (stimulus) frequency of 10 GHz for ??SiO2.
This serves as an initial estimate for measurements performed here at temperatures below 10 K
and applied frequencies of 10 GHz suggesting the dominant mechanism contributing to loss is
the TLS relaxation processes in Chapters 4 and 5.
It is interesting to note that below T? , the temperature power law dependence of tan ? is
T 2.3, which is less than T 3 for the 43 MHz measurement in Fig. 2.5. Schickfus and Hunklinger
measured a tan ? temperature dependence of T 2.4 from 2-8 K in the TLS relaxation regime at
10 GHz [84]. Furthermore, Strom et al. measured a tan ? ? T 2.4 dependence in the relaxation
regime for Na ?-alumina at 11.5 GHz with the transition temperature between TLS absorption
and relaxation at 5 K [94]. Therefore, deviations from tan ? ? T 3 dependence in the TLS
relaxation regime is not uncommon. This is most likely due to the measurements taking place at
temperatures near T? or near the TLS resonant absorption regimes.
Here I assume TLS relaxation processes dominate dielectric loss tan ? in measurements
carried out in this work. I will not treat losses due to TLS absorption processes since measurements
here are carried out above T > 1K and all TLSs are saturated above this temperature. Although,
the temperature dependent data in chapter 4 suggests we see the onset of TLS absorption regime.
40
Figure 2.5: The (left) plot is data re-plotted from reference [93] Fig. 3. In this plot, i fit the low
temperature regime (TLS relaxation) and the higher temperature regime (classical relaxation) of
the 43 MHz data (yellow). The (right) plot is estimated T? temperatures taken at the arrows shown
in (left) plot. At 10 GHz, the extrapolated temperature is T? ? 22K.
With that, the model used here for the tan ? temperature dependence is
tan ?(T, ?) ? tan ? 3tun,rel(T, ?) ? T (2.37)
2.3.4 Frequency Dependence of tan ?
Considering only loss due to TLS relaxation with ??min  1 and T < T? , the phonon
scattering length `ph ? `tun,rel is constant as a function of frequency (see Eq. 2.34). In this
temperature regime, the loss tangent has a frequency dependence
tan ? ? ??1, T < T? , strong E field (2.38)
41
Although, specifically in the amorphous dielectric TLS community, the frequency dependence of
tan ? in Eq. 2.38 is rarely measured or has possibly never been explicitly confirmed experimen-
tally.
For comparison, the frequency dependence of tan ? in the temperature regime where TLS
absorption dominates, the loss tangent is frequency independent by
tan ?(?) = constant, T  T? , weak E field (2.39)
In parallel communities the frequency dependence of dielectrics with finite electrical conductivity
have been measured suggesting universality due to many body interactions. The electrical conductivity
is related to the dielectric loss tangent by [95]
??? + ?
tan ? = ? (2.40)?
where ? is the angular frequency, ? and ?? are the real and imaginary part of the complex
dielectric constant, respectively, and ? is the finite electrical conductivity in the dielectric.
In 1961, Pollak et al. observed a frequency dependence of the conductivity ? in n-type Si
at temperatures T ? 4.2K having the form [96]
? = As?
b (2.41)
where As and b are empirical fitting parameters. Pollak et al. found that b ? 0.8 fit for a range of
temperatures below 10 K. The model described by Pollak et al. explaining the frequency power
dependence requires low concentrations of localized conduction states allowing electron (charge)
42
to ?hop? between states giving rise to an increase in conductivity. With the application of a steady
state alternating current, charge will be displaced by the ionized group-3 and group-5 impurities
systematically incorporated into the Si by Pollak et al. experiment. The ?hopping? time of the
charge is analogous to a relaxation time having some phase lag relative to the applied electric
field (refer back to Eq. 2.31 and discussion). Therefore, energy can be dissipated through this
mechanism and witnessed by measurements of the loss tangent tan ?.
A host of studies found a similar frequency dependence in materials with low conductivity,
predominantly dielectrics, that satisfy the presence of non-interacting dipoles free to rotate or
displace in a medium. In 1977 Jonscher [97], and later updated by Ngai et al. in 1979 [98],
presented the universality of dielectric frequency dependent response supported by half a century
worth of data. The universal power law proposed have the forms [97]
tan ? ? ?b?1, 0 < b < 1 (2.42)
? ? ?b, 0 < b < 1 (2.43)
Ngai et al. refers specifically to TLS as a possible electron-phonon mechanism adhering to the
power dependent universality [98]. Although, the TLS and the universal dielectric power law
were not explicitly connected.
Kaiser et al. argues that measurements using lumped element resonators at 4.2 K of
amorphous thin film dielectrics, TLS relaxation processes dominate the measured tan ? [99].
Assuming non-interacting dipoles with equal relaxation times ?rel at GHz frequencies, Kaiser et
al. makes the power exponent a free fitting parameter b to fit to Eq. 2.43 [97, 98]. Kaiser et
al. suggests the parameter bd depends on the coupling of TLSs to each other, where bd = 0 is
43
non-interacting (Debye-like), bd = 0.5 nearest neighbor interactions, and bd > 0.6 is attributed to
many-body interactions [98]. Although outside of the scope of this work, it is still worth noting
that amorphous and crystalline metals can exhibit similar exponents with bd ? 0.8 in Pollak?s
experiments of n-type Si as an example mentioned earlier. This is worth exploring further in
experimentation building off the work presented in Chapters 3-5.
Considering both the (i) proposed universal frequency dependence in Eq. 2.43, and (ii) TLS
relaxation frequency dependence in Eq. 2.38 assuming ??min  1 and T < T? , the empirical
frequency dependence of the dielectric loss tangent can be fit to
tan ? = B ?b?1d , 0 < b < 1 (2.44)
where Bd and b are power law free fitting parameters. For b = 0, the frequency dependence
simplifies to that expected for TLS relaxation. For 0 < b < 1 the frequency dependence follows
universal scaling supported by measurements at 4.2 K [99].
2.3.5 Summary
In this work, the dielectric studied is tetraethyl orthosilicate (TEOS). The thin film TEOS
material fabricated (see Section 2.4 and 3.2) and measured (see Chapters 4 and 5) is considered
to be an amorphous (glassy) dielectric.
The resonator measurements carried out here (see Chapters 4 and 5) were done in the
temperature and frequency range 1.5-10 K, 1-15 GHz, respectively, in an electric field regime
between strong and weak. In this range, I assume two-level system (TLS) relaxation processes
are dominating tan ?.
44
In Chapter 4, for the dielectric loss tangent tan ?, we will assume both non-dispersive
(tan ? = constant) [100] and dispersive (tan ? ? ?b?1) [79, 97, 98, 99, 101] loss tangent models
over a frequency range ?/2? ? 1 ? 12 GHz at T = 4.2 K. In Chapter 5, I will attempt to fit
the frequency dependence of tan ? as a function of temperature, and compare tan ? temperature
dependence to the expected? T 3 for TLS relaxation processes [84]. Furthermore, I will claim the
non-monotonic temperature dependence of tan ? is due to TLS relaxation and resonant absorption
loss and leads to TEOS being the dominate loss mechanism in RQL interconnects below 4 K.
2.4 Fabrication Processes for RQL Interconnects
2.4.1 Brief Background of CMOS Technologies
Both CMOS and superconducting electronics (SCE) technologies have similar interconnect
fabrication challenges, one of them being planarization. In CMOS, to achieve higher performance
and lower cost per chip for successive technology generations, there is an approximate linear
increase in number of devices leading to a quadratic increase in the number of interconnects
required [102]. This demand is met by reducing the pitch between wires or increasing the total
number of interconnect layers. This increase in density has a negative effect on performance, and
has been solved by qualitative improvements of the fabrication processes. Planarization of each
successive layer is paramount when increasing density, as the minimum feature sizes and pitch
are set by the depth of focus in the lithography process [103].
To fabricate interconnects, current CMOS technologies use some form of a damascene Cu
electroplating process for on-chip metalization [104]. The damascene process will be explained
below. Electroplating is the ideal process to conformally fill high-aspect ratio dielectric trenches
45
Figure 2.6: a) SEM image adopted from reference [105] as an example of high aspect ratio
trench filling using Cu electroplating in a CMOS damascene process (before metal planarization).
b) RQL damascene process after planarization showing a Nb interconnect deposited using
magnetron sputting at T ? 150? as an example of low aspect ratio trench filling. More
characterization on this in Chapter 3.
with Cu. SCE technologies are currently limited to trench aspect ratios of < 1 and resort to
less conformal sputtering processes. Fig. 2.6 exemplifies the low aspect ratio filling of RQL
interconnects relative to common CMOS electroplating technologies [105, 106, 107, 108, 109].
In CMOS, the push for higher density of submicron interconnects increases the overall
resistivity of the Cu. This increase in resistivity induces an increase in energy dissipation and
Joule heating. RQL interconnects can provide advantage over this, and was discussed in Section 1.2.
46
2.4.2 SCE Fabrication Technologies
Currently, the following companies have advanced fabrication processes producing SCE
logic circuits: SEEQC [110, 111, 112], National Institute of Advanced Industrial Science and
Technology (AIST) in Japan [113, 114]; D-Wave Systems Inc. using Cypress semiconductor
foundry in Minnesota (proprietary process unpublished, reviewed here [115]); and MIT Lincoln
Laboratory (LL) in Massachusetts [115]. Parallel to semiconductor microelectronic scaling,
interconnect wire width, pitch, and total number of layers determine the scalability of SFQ
circuits. The LL process currently has eight wiring layers with minimum wire width and spacing
(pitch) of 0.35 and 0.5 ?m, respectively [115]. Unlike CMOS where minimum dimensions
are defined by the transistors, the interconnects in SCE have comparable or larger minimum
dimensions relative to the logic units (Josephson junctions), therefore determining current scaling
of the technology. Niobium is the most commonly used metal in SCE, as it has a relatively short
penetration depth ? and long coherence length [53, 54], relatively low RF losses [71], and can be
consistently fabricated having high critical current densities [54]. Silicon dioxide SiO2 is the most
common interlayer dielectric used in SCE and is adopted from semiconductor technologies [116].
These dielectrics provide isolation between metal layers, and the dielectric synthesis and quality
can have an impact on the metal layer performance (inductance, RF loss) as well as determining
the speed and dispersion of data propagation [36], governed by the relative dielectric constant ?r
and loss tangent tan ?, respectively.
47
2.4.3 Chemical Mechanical Planarization (CMP)
For both CMOS and SFQ based planarized processes (e.g. [115]), interconnect fabrication
cannot be made without chemical-mechanical planarization (CMP) [117, 118, 119, 120]. Pioneered
at IBM, CMP is a technique to remove material with undesired topography by abrasive action in
a chemical etching environment. The physical abrasion is done using a rotating polishing pad in
contact with the wafer mediated by a chemical alumina slurry. There is also a pad conditioner to
remove residual contaminants to maintain constant abrasion on the wafer (i.e. ?sandpaperness?).
Furthermore, there are different types of endpoint detection methods to determine when the
metal/dielectric has been planarized. The two most common in-situ endpoint detection methods
are: ISRMTM (interferometry based) [121, 122] developed by AMAT [123], and eddy current
based techniques [124]. There are also many other techniques to perform endpoint detection
[125, 126]. This process can achieve a planar surfaces to sub nm levels over a 200 mm wafer,
but due to the coupled physical-chemical nature of this technique, induced material defects are
of primary concern. The two primary planarized processes are the cloisonne? and damascene
process. In regard to availability, these planarization processes enable both high metal densities
per layer (approximately 50-95%) and many layers to be used as interconnects. Although, for
superconducting interconnects, as wire dimensions decrease so does the electron mean free path
giving rise to higher surface resistanceRs and power dissipation (see Eq. 2.20 and Sections 2.2.2-
2.2.3).
48
Figure 2.7: Superconducting Electronics (SCE) fabrication processes showing the generalized a)
cloisonne? process involving a metal etch followed by a dielectric CMP, and b) the damscene pro-
cesses involving a dielectric etch followed by a metal fill and CMP. Here, the dielectric Tetraethyl
orthosilicate (TEOS) and metal niobium (Nb) were used to fabricate (MTL) interconnects with
both the cloisonne? and damascene process. Samples from Process A and B were fabricated using a
damascene process. Samples from Process C were fabricated using a cloisonne? process. Physical
characterization of the MTLs can be found in Section 3.2. Electrical characterization of the MTLs
can be found in Chapter 4 and Chapter 5.
49
2.4.4 Cloisonne? Process (Dielectric CMP)
Arguably, the first metal-dielectric fabrication process was Cloisonne?, which is the ancient
art originating from Egypt of decorating objects with intricate metal designs encapsulated in
insulating enamel to give a colorful finish. The resulting surface of the object is rough after the
firing step to adhere the enamel onto the metal and is sanded down to give it a smooth finish. In
microelectronics fabrication, the Cloisonne? process is started by depositing the interconnect metal
(Cu, Nb) with physical vapor deposition (PVD). After the dense metal (e.g. Cu, Nb) interconnect
design has been defined by photo-lithography and patterned using reactive-ion etching (RIE), the
metal is embedded with insulating dielectric (e.g. SiO2, SiNx) using chemical-vapor-deposition
(CVD) processes [127]. Since CVD deposits conformally over the metal, the resulting surface
is planarized to make the surface planar and smooth. See Fig. 2.7. PVD and RIE processes
are relatively inexpensive and produce excellent metal thickness uniformity (between 1-10%).
Additionally, RIE is more scalable than the liftoff process [118]. Although, the cloisonne? process
is unable to produce higher density (small wire pitch) interconnects. The RIE etch selectivity at
corners of the photoresist and high metal density areas, which ultimately limit the metal thickness
and induced metal sidewall angle. Typically in practice, CMOS resorts to a damascene (metal
CMP) process to achieve the highest metal densities. Fortunately, there are many ?knobs? that
can be turned turned in CMOS/SCE to get the cloisonne? process to meet desired requirements.
2.4.5 Damascene Process (Metal CMP)
In constrast to cloisonne?, the damascene process [119, 120] starts with a uniform dielectric
deposited on a planarized wafer followed by patterning the dielectric ?trenches? using photolithography
50
and oxide RIE. Metal is then deposited to fill the troughs, over filled (over burdened), and then
metal is CMPed down to the dielectric surface, leaving behind only recessed metal wire. See
Fig. 2.7. This process is the primary planarization process producing the highest yield and
largest processing window. A large processing window means the performance of the device
or product being fabricated is not sensitive to small changes in the fabrication process (e.g. small
changes in temperature during metal deposition). In general, dielectrics are amiable to the process
(lithography, RIE). As a consequence, this technique suffers from the poor trench filling qualities
of refractory metals. Magnetron Sputtering techniques, although optimal in many ways (e.g. high
deposition rates, process control, thickness uniformity), do not uniformly fill trenches because
the flux concentration of atoms towards the recessed areas is hard to control [128]. Additionally,
uniform coverage inside the trench facilitated by surface diffusion of adsorbed atoms is almost is
extremely difficult due to magnetron sputter energy scales [129]. Because of this, trench filling
at corners is especially difficult and commonly causes voiding inside of the trench due to metal
deposited on the upper edge surface eventually blocking incoming flux. At the same time, this
blocks atoms/plasma from leaving the trench. As one can expect, this process can induce a
plethora of defects on the metal (contaminants, residual stress, voids, etc) due to trench filling
and CMP.
Both cloisonne? and damascene processes used in CMOS/SCE technologies give a planarized
interconnect balanced by performance trade-offs. Changes in the process parameters have been
shown to mitigate short comings of each process [102, 120]. For example, depsition parameters
(e.g. deposition pressure, RF/DC bias, substrate temperature) have been tuned to achieve optimal
adatom energies for uniform filling of trenches. Thin protective layers (e.g. Ti, Al, Ta) have
been used as diffusion barriers. Trade-offs can be further mitigated to enhance performance by
51
advances in the fundamental understanding of the materials and their response to fabrication
changes.
2.4.6 Thermal Budget
Somewhat analogous to atomic mobility through self-diffusion in Al based wiring in CMOS
due to fabrication temperatures reaching 40% of the Al melting temperature [102], the thermal
budget in SFQ technologies is primarily set by the activation temperature of oxygen migration
into Nb interconnects and intermixing at Josephson junction oxide interfaces. The latter could
be alleviated by having the Jospehson junction integration at the end of the fabrication process.
Therefore Nb interconnects are the thermal budget limiting factor where fabrication temperatures
do not typically exceed T ? 150?C. As a reference, 0.1 and 1 at. % of oxygen in Nb reduces
the superconducting critical temperature by approximately 0.2 and 1 K, respectively [130]. In
addition, the normal state resistivity at 4.2 K has been shown to increase by a factor of 30 for
0.5 at. % oxygen in Nb. An increase in normal state resistivity will decrease the electron mean
free path ` and increase the magnetic penetration depth ? (see Eq. 2.8). With this, low deposition
temperatures make it difficult to fabricate high quality SiO2 and Nb thin film material. For CMOS
fabrication, the typical deposition temperature is T ? 350?C for SiO2 using a TEOS precursor,
which is approximately 1/3 the melting temperature of Cu. An increased thermal budget would
allow for higher substrate (wafer) deposition temperatures, facilitating ideal growth kinetics of
metals/dielectrics for high quality materials.
52
2.5 Summary
In summary, the fabrication process has a significant impact on RQL interconnect morphology
of Nb wires and the low temperature thermal budget invokes the TEOS dielectric to be amorphous
in character. Consequently, the intrinsic microwave losses need to be monitored for changes in the
process and the mechanisms causing extrinsic loss should be mitigated. Models of the microwave
losses will be presented in Chapter 3 and demonstrated in Chapter 4 at 4.2 K. The morphology
will be further explored in Chapter 3, and then compared to RF characterization in Chapter 4.
Temperature dependent models will be presented and demonstrated in Chapter 5.
53
Chapter 3: Modeling Lossy 3D Superconductor-Dielectric 2-Port Networks
3.1 Introduction
A resonator remains the only practical way to accurately measure intrinsic resistance Rs
[62, 131] and loss tangent tan ? [100, 132] at microwave frequencies. The resonator internal Q-
factor is related to the partial Q-factors associated with the conductor loss Qc and the dielectric
loss Qd by Q?1i = Q?1c +Q
?1
d . There will be more discussion on the measurement techniques in
Chapter 4.
For an MTL embedded in a single dielectric material, the loss tangent is related to the partial
dielectric Q-factor by tan ? = Q?1d and is independent of the MTL geometry. To extract the
intrinsic resistance from measurements of Qc, one needs to accurately solve the geometric factor
?, where Rs ? ?/Qc. The geometric factor can be solved analytically for simple geometries or
numerically for more complex geometries.
The analytical modeling work to account for Zs (and ?) contribution on transmission line
characteristics are accurate only when the current distributions are known and/or geometries are
such that the conducting strip width w is greater than the dielectric thickness s: w/s > 1 (see
Table 3.1) [133, 134, 135, 136, 137]. Sheen et al. describes the limitations in detail, presents a
new method accounting for current distributions, and numerically solves for the lossy stripline
parameters (e.g. ?) using a proprietary Finite Element Modeler (FEM) [138].
54
Recently, commercially available FEM products have been used to simulate superconducting
transmission lines. Analytical work applied to a variety of lossless transmission line geometries
(microstrip, stripline, etc.) has been compared to a commercially available software [65], and is in
good agreement with SQUID based inductance measurements [54]. Other FEM implementations
have been used to simulate MTLs with lossless superconductor properties by using plane sheets
with user-defined surface impedance, but is only accurate for MTLs with w  ? [139], [140].
Piccinini et al. simulated lossy normal metal CPWs for CMOS Quantum Computing systems
using HFSS similar to our method for superconducting MTLs but did not solve inside the normal
metal due to its negligible skin depth at microwave frequencies [141, 142]. Additionally, the
superconducting microwave losses and current distributions of coplanar waveguides with strip
widths down to 50 nm have been simulated and compared with measured data [143, 144]. In the
Quantum Computing community, Comsol 2D simulations are used to calculate the participation
ratios of varying thin dielectrics contributing to two-level system (TLS) loss and are generally
not concerned with simulating the negligible superconductor loss at mK temperatures [145, 146].
In addition to the need for numerical solutions when one is dealing with nonuniform current
distributions, no analytical theory can account for the common fabrication effects impacting
the MTL - sidewall trench tilt, rounded edges of wires, in-homogeneous superconductor or
dielectric materials due to intermixing at material interfaces, cross-sectional thickness miss-
targeting, etc. Furthermore, to maintain a desired metal density ? 50% for each layer in the
damascene planarization process, small superconducting fill strips on the order of 0.5?3?m need
to be included above, below and beside MTLs running perpendicular or parallel to conducting
strip, which again is almost impossible to account for in the analytical calculations.
The expected nonuniform current distribution cannot be solved analytically for our 0.25-
55
micron fabrication process (similar to [115]), where the dielectric separation thickness s, superconductor
thickness d, conducting strip cross-section width w, and magnetic penetration depth are all
comparable: s ? d ? w ? ? ? 0.25 ?m. Furthermore, to extract an accurate Rs, the
exact cross-sectional geometry should be measured by Scanning Electron Microscopy (SEM)
or Transmission Electron Microscopy (TEM). Therefore, in this chapter, to extract Rs from
measurements, we will numerically solve for ? using HFSS coupled with TEM measurements.
3.2 Materials Characterization of Interconnect Morphology
The damascene fabrication process is the desired planarization process for interconnects to
scale a technology. As mentioned in Section 2.4, it is a challenge to fill dielectric trenches with
refractory metals (like Nb) using magnetron sputtering [128]. This creates non-ideal morphology
in the metal interconnect. In this section, the morphology of damascened Nb-TEOS fabrication
processes will be examined with scanning transmission electron microscopy (STEM) coupled
with energy-dispersive x-ray spectroscopy (EDS). The sample preparation and measurements
will be briefly presented with the focus on the primary results that can affect RF losses in the
Nb and TEOS. This analysis will be referred to in Chapters 4 and 5 to help explain physical
mechanisms of RF loss measured by RF electrical characterization.
Reviews on STEM can be found in the following references [147, 148, 149, 150, 151]
D?Alfonso et al., later confirmed by Chu et al., were the first to do spatial atomic mapping using
EDS with a STEM [152, 153]. A review on this technique can be found in reference [147, 154].
56
3.2.1 Scanning Transmission Electron Microscopy (STEM)
To accurately calculate the ? from HFSS, the cross sectional geometry of an MTL needs
to be accurately known. With ?, the intrinsic resistance Rs can be extracted from measurements
of Qc (see Eq. 3.13). Focused Ion Beam Microscopy (FIB) (see Appendix A), and Scanning
Transmission Electron Microscopy (STEM) were used to measure the MTL down to sub-nm
accuracy.
The STEM imaging was done as a service by EAG Laboratories [155]. First a carbon
protective coating was applied prior to Focused Ion Beam (FIB) milling of the structure. A cross-
section was prepared using an FEI Helios 660 Dual Beam FIB/SEM and thinned to electron
transparency. Reviews on specimen preparation for TEM using FIB can be found in the following
references [156, 157] Samples were imaged using Hitachi 2700 Aberration Corrected Scanning
Tunnelling Electron Microscope (ACSTEM) with an accelerating voltage of 200 kV and a nominal
probe size of ? 0.1 ? 1nm. The focused probe is scanned across the sample surface, and
transmitted electrons are imaged. Transmitted Electron (TE) STEM images form from electrons
transmitted through the samples that are un-scattered or scattered to low angles and are similar
to and commonly referred to as Bright Field (BF) images (Fig. 3.15a,c). Transmitted electron
(TE) is the first and standard imaging method. Annular bright field (ABF) imaging is analagous
to TE, but with enhanced contrast from weakly scattered electrons from light elements by using
a ?preventing disk? (see Fig 1 in [158] and reference [149]). Atomic Contrast (or Z-contrast)
STEM images are formed by electrons transmitted through the sample that have scattered out to
higher angles (Fig. 3.15b). These are acquired using annular dark-field (ADF) detectors. If the
electrons scatter to high enough angles the image intensity is roughly proportional to the square of
57
the atomic number (Z2), hence these are termed Z-contrast (ZC) images [159, 160]. In summary,
TE images are most useful in this section to analyze the dielectric material (e.g. density, porosity)
while ZC images are useful to analyze the morphology of the metal wires (see Figs. 3.1 and 3.3).
Below, in Table 3.1 summarizes the measured cross sectional critical dimensions. The
measurements were taken from the STEM images found from Fig. 3.1 and Fig. 3.2.
Table 3.1: STEM Results Summary table for all three processes measured in this work. The
?mean width? column reports the minimum/max width, pertaining to the top and bottom width
of the wire (see Fig. 3.1 and Fig. 3.2). All measurements were done at EAG Laboratories [155]
In Fig. 3.1 and 3.2, the bottom of all images are closest to the wafer substrate. The top
of all images are closest to dielectric-air interface. For process A and C, the ground plane is
below the wire, or conducting strip (CS). For process B, the ground plane is above the CS. The
CS (wire) in Process A and B have an inverted trapazoid shape from the damascene process
described in Section 2.4. In process C, the CS has a trapazoid shape from the cloisonne? process
(see Section 2.4). The bottom metal in Fig. 3.2(a) and top metal in Fig. 3.2(b) are 6x0.4 ?m fill
metal strips with 0.3 ?m spacing, and will be discussed in Section 3.6.
Aside from geometries, information about the morphologies can be analyzed from STEM
images. Fig. 3.3 highlights the primary features that can have an impact on the Nb and TEOS
58
Figure 3.1: STEM images taken from representative samples of 0.25 ?m MTLs fabricated by
Process a) A, b) B, and c) C. The top row of images (a-c) are ZC, where bright is high Z and
dark is low Z. The bottom row are TE images. The bottom of all images are closest to the wafer
substrate. The top of the images are closest to dielectric-air interface. Process A and C have the
ground plane (GP) below the wire (or conducting strip CS). Process B has the GP above the CS.
All measurements were done at EAG Laboratories [155].
RF losses. Namely, damascene processes A and B show distinct lower metal density lines going
diagonally inward from from the bottom edges of the CS. This is primarily caused during trench
filling process when two growth directions (trench bottom, side wall) meet. This also gives rise
to the voids, or very low density metal, found along the intersection lines (see features 1 (yellow)
and 2 (orange) in Fig. 3.3). For the cloisonne? process C, the Nb CS has a relatively continuous
columnar grain structure, typical for sputtered Nb. The missing metal at the top and left edges of
the 0.25-um CS is from the ?undercut? during RIE (see Fig. 3.3(b) feature 6).
From the ZC images, a 8-10 nm NbOx layer can be seen at the top of the planarized
trench in damascene processes A and B. For process B only, there are lower density regions in
the dielectric and can be see in Fig. 3.3(d). Both of these oxide features will be discussed further
59
Figure 3.2: STEM images taken from representative samples of 1 ?m MTLs fabricated by
Process a) A and b) B. Both images are TE images. The bottom of all images are closest to the
wafer substrate. The top of the images are closest to dielectric-air interface. Process A and C
have the ground plane (GP) below the wire (or conducting strip CS). Process B has the GP above
the CS. The bottom metal in a) and top metal in b) are 6x0.4 ?m fill metal strips with 0.3 ?m
spacing, and will be discussed in Section 3.6. All measurements were done at EAG Laboratories
[155].
in Section 3.2.2.
At 0.25-micron dimensions, the electronmagnetic fringing fields become a significant fraction
in cross section [134]. Additionally, the current distribution will be nonuniform taking up a
significant fraction of the cross sectional area. Furthermore, the location of defects relative to
the current density concentrations in cross section can have a significant impact on the Nb and
TEOS RF losses, and will be discussed in detail in Section 4.5 Hence, the motivation to model
the electromagnetic fields around and inside the Nb layers, as well as numerically solving ? (see
Section 3.4).
60
Figure 3.3: Highlighted Morphology features from STEM images for Processes A, B and C.
Note, image contrast has been enhanced to highlight features. a) Feature 1 (orange) highlights the
intersection between two grain orientations. Feature 2 (yellow) highlights voids or areas of low
metal concentration. b) Feature 3 (purple) highlights a continuous Nb columnar grain. Feature 6
(pink) highlights the ?undercut? of the Nb during RIE. c) Feature 4 (green) highlights the NbOx
layer found in Process A and B. d) Feature 5 (blue) highlights the lower density areas (bubbling
and streaks) found in Process B only. All measurements were done at EAG Laboratories [155].
3.2.2 Energy-Dispersive Spectroscopy (EDS)
Energy Dispersive X-Ray Spectroscopy (EDS) will be used to survey the spatial concentration
of elements with a detection limit of > 1at% (for elements Li-Pb). Although, the measurements
here were standardless, and therefore accuracy down to 1at% cannot be claimed. Rather, the
analysis will only qualitatively compare between samples and processes. This tool will help
61
understand the oxide compositions, and possible contamination of elements in the Nb layers.
EDS mapping was done immediately after STEM, as it only requires a different detector
inside the system. A 0.1-1 nm diameter electron beam is scanned across the sample surface as
X-rays are detected.
Fig. 3.4 shows a detectable amount of Ar content (? 1at%) in damascene process A and B.
The larger amount of noise for Process A signal in Fig. 3.4(d) is due to an overall lower number
of counts relative to Process B measurement. Additionally, the Process A sample is most likely
more electron transparent (thinner). Hence the reason why Process A and B are plotted together
normalized to their respective Nb signal. The sputtering pressure during trench filling is typically
increased for more ?conformal? coverage of the trench [161]. The mechanism is an increased
Ar pressures induces higher net columnated flux of metal ions depositing into the trench. As a
result, there is also higher concentration of Ar plasma able to get trapped inside the trench/Nb.
Ar content was not detected in EDS measurements of Process C.
Fig. 3.5 shows the EDS maps spatial maps of Nb, Si, and O for damascene Process B.
The top down line scan in Fig. 3.5(a) starts with the TEOS layer at position 0 nm and suggests
a stoichiometry close to SiO. This insert of Fig. 3.5(a) zooms into the TEOS-Nb oxide interface.
There is a plateau of the Nb signal from 23-27 nm and then a gradualy increase (decrease) of the
Nb (O) signal from 27-33 nm. The overall NbOx 10 nm (greater than a typical niobium native
oxide layer), and there is a stable oxide layer of 5 nm with a stoichiometry somewhere between
NbO ? NbO2.
Fig. 3.6(a) shows the O/Nb ratio at the niobium oxide interface for both damscene Process
A and B. The abscence of a plateau suggests Process B does not have a stable oxide layer. Oxygen
content decreases to zero over the same exact length scale as Process A of 10 nm. Fig. 3.6(b)
62
Figure 3.4: EDS measurement of Ar content in damascene process A and B. EDS map of Ar
signal superimposed on the STEM image for damascene process a) A and b) B. The white dashed
lines signify the EDS line scans plotted in c) and d). c) The semi-quantitative atomic percent of
Ar in Process A, showing there is a detectable amount of Ar. d) The Ar signal normalized to the
Nb to compare process A and B. The background signal is there for reference. All measurements
were done at EAG Laboratories [155]
is the Tc dependence of O/Nb ratio for bulk cast alloys of niobium and oxygen adopted from
Hulm et al. [162]. It can be inferred that a bulk alloy of niobium and 60% oxygen would have
a Tc ? 7 K. A thin film of Nb, especially at 5 nm, will have a significantly suppressed Tc,
suggesting the 5 nm, 60% rich niobium oxide may be a normal conductor as opposed to a stable
oxide like Nb2O5 at these temperatures.
Fig. 3.7 shows an Al and SiNx passivation layer was introduced into Process C. This
Al/SiNx prevented a NbOx layer from forming.
63
Figure 3.5: EDS data on damascene process B 0.25-micron MTL showing a) semi-quatitative
plot of O, Si, and Nb atomic% in a top-down line scan. b) A STEM image with two superimposed
EDS maps (Nb and O) showing spatial variation and NbOx layer. c) A STEM image with a Si
EDS map superimposed, showing areas with low Si concentrations.
Figure 3.6: Plots of a) EDS top-down line scan showing calculated respective O/Nb ratios vs
position for damascene Process A and B. b) Replotted Tc vs O/Nb ratio from [Hulm1972] [162]
64
Figure 3.7: EDS data on Process C 0.25-micron MTL showing a) semi-quatitative plot of
atomic% in a top-down line scan. b) EDS maps for each element.
3.3 Theory and Method
The method to deconvolve superconductor and dielectric losses will be described in this
section, and is termed the Dispersive Loss Deconvolution (DLD) method. The method described
here will be demonstrated in Chapters 4 and 5.
65
3.3.1 RLGC Telegrapher?s equations
The internal Q-factor of a superconducting transmission line resonator can be expressed as
[95]
1 1 1 R G
= + = + (3.1)
Qi Qc Qd ?L ?C
where Qc is the partial Q-factor associated with the conductor loss, Qd is the partial Q-factor
associated with the dielectric loss,R, L,G, andC are the transmission line resistance, inductance,
conductance, and capacitance per unit length, respectively, ? = 2?f is the angular resonant
frequency, and the radiation loss is ignored as justified in Appendix B [163]. The telegrapher?s
expressions for R, L, G, and C of a superconducting transmission line, assuming ?1  ?2,
embedded in an interlayer dielectric are [95, 138, 164],
2 ?
R = R 2
| |2 s
?|J| ds (3.2)
I
S
? ?0
L = (|H|2 + ?2|J|2)ds (3.3)
|I|2
S
?? ?0
G = ?r tan ?|E|2ds (3.4)|V |2
S
? ?0
C = ?r|E|2ds (3.5)|V |2
S
where I is the line current, V is the potential difference between the strip and ground plane,
J is the vector (longitudinal) current density, H is the (transverse) magnetic field vector, ?0 is
the permeability of free space, ?0 is the permittivity of free space, Rs, ?, tan ?, and ?r are the
66
spatially dependent superconductor intrinsic resistance, magnetic penetration depth, dielectric
loss tangent, and relative dielectric constant, respectively, in the lateral plane and this treatment
assumes everything is homogeneous in the longitudinal direction. The integrals are over the entire
cross-sectional area S of the line.
3.3.2 Geometric Factors in Inhomogeneous Transmission Line
Inserting Eqs. 3.2-3.5 into Eq. 3.1 yields
1 ? Rsm ? tan ?n
= + (3.6)
Qi m ?cm n ?dn
where Rsm is the effective intrinsic resistance of the m-th conductor, ?cm is the partial geometric
factor associated with losses in that conductor, tan ?n is the effective loss tangent of the n-th
dielectric layer or tube, and ?dn is the partial geometric factor associated with losses in that
dielectric layer or tube. These are defined as follows
? ?
2
R = ? S Rs?|J| dsmsm ? (3.7)
? ? 2S ?|J| dsm
S ?r tan ?|E|2dstan ? nn = ? ? (3.8)
? ? ? |E|2S dsn r
2 2 2
S(|?H?| + ? |J| )ds?cm = ?0? (3.9)
2 ? ??|J|2S dsm? ? S ?r|E|2ds?dn = (3.10)
S ?r|E|2dsn
where Sm and Sn are the cross-sectional areas of the mth wire and nth dielectric layer or tube,
respectively.
67
It shall be noted that ?cm defined by Eq. 3.9 differs from the cavity geometric factor? ?
definition for conductor loss (Q = R /?cavity) where ?cavityc s = ?0? V |H|2 dv/ S |H 2? | ds
defined as the ratio of the volume integral due to magnetic energy stored inside the cavity to the
surface integral due to energy loss into the cavity walls [63, 131] associated with the tangential
fields on the conductor. In Eq. 3.9, the first addend under the numerator integral includes resonant
energy stored in the magnetic field penetrating the superconductor, while the second addend is
due to resonant energy stored in the kinetic energy of the superconducting carriers [138]. The
?cavity does not account for them because in voluminous high-Q superconducting resonators
[70, 165, 166] those energies can be ignored due to the penetration depth (typically, nm to um)
being much smaller than the resonator characteristic size (mm to m). Also, unlike ?cavity that
depends on the mode index, Eq. 3.7 effectively defines ?cm per unit length of transmission line
because a transmission line resonator only supports a quasi-TEM mode, making it independent of
the mode index. Likewise, ?dn defined in Eq. 3.10 is the cross-sectional fractional electric field
energy stored in the n-th dielectric and is equivalent to the reciprocal of both (i) the electrical
energy volume filling factor described by Krupka et al. [132] and (ii) the participation ratio
definition used by the Quantum Computing community [145], [146].
Due to the assumption ?1  ?2 for isotropic superconductors, low transition temperature s-
wave superconductors (e.g. Nb) exhibit intrinsic resistanceRs with quadratic frequency dependence
[63], which per Eq. 3.6) and Eqs. 3.7-3.10 leads to a Q ? ??1c dependence [100]. At the
same time, at GHz frequencies and above, amorphous dielectrics commonly exhibit a universal
frequency dependence tan ? ? ?b?1, with b = 0.6?1 [97, 98]. Thus, insertingR = R (?/? )2s s0 0
and tan ? = tan ? (?/? )b?10 0 into Eq. 3.6 and assuming frequency-independent ?dn, allows one
68
to factor out the frequency as follows
( )
1 ? ?M R ? b?1 Ns0m ? tan ?0n
= + (3.11)
Qi ?0 m=1 ?c0m ?0 n=1 ?dn
where Rs0m and tan ?0n are the intrinsic resistance and the loss tangent at a reference frequency
?0, ?c0m is the partial geometric factor at that frequency associated with loss in themth conductor,
and ?dn is the partial geometric factor associated with loss in the nth dielectric.
3.3.3 Geometric Factors in Homogeneous Transmission Line
Consequently, Eq. 3.11 implies that the experimental dependence of Q?1i on the resonant
frequency ?n = 2?fn of a multi-mode transmission line resonator, with n = 1, 2, 3. . . being the
mode index, can be fit to
Q?1i = A? +B?
b?1 (3.12)
?
where the first term corresponds to the superconductor loss A = ??1 ?10 mRs0m?c0m, while the?
second term corresponds to the dielectric loss B = ?1?b ?10 n tan ?0n?dn .
Generally, thin film fabrication methods may cause the ground plane and conducting strip to
have dissimilar superconducting properties, especially for submicron width strips [72]. Similarly,
the stack may involve multiple dielectrics. In a basic case of a homogeneous superconducting
resonator formed by MTL embedded into non-dispersive dielectric (?dn = 1, ? = ?0, b = 1)
69
Eq. 3.11 reduces to
1 ? Rs0
= + tan ? (3.13)
Qi ?0 ?c0
where the net conductor geometric factor ?c0 is
?
? 20GP?0CS S(|H|? + ?2|J|2)ds?c0 = = ?0?0 (3.14)
? + ? 2 ?|J|20GP 0CS S ds
Here ?0GP and ?0CS are the partial geometric factors of the ground plane (GP) and conducting
strip (CS), respectively. We choose ?0/2? = 10 GHz because it is commonly used for scaling
and comparison of the surface resistance obtained by different superconducting resonators, and
is also a convenient frequency to perform HFSS simulations. Per Eq. 3.13 the experimental
dependence of Q?1i on the resonant frequency can be fit with
Q?1i = A? +B (3.15)
where the fit y-intercept immediately yields the dielectric loss tangent as tan ? = B [100], while
the fit slope coefficient A corresponds to the intrinsic resistance as follows
Rs0 = A?0?c0 (3.16)
where ?c0 is found either from Eq. 3.14 if the fields and currents are known, or from the HFSS
simulation, as described in Section 3.4. A linear fit of Eq. 3.15 enables easy analysis of large
data sets and is advantageous for engineering. Moreover, the fit y-intercept yields tan ? even for
70
the loaded Q-factor data since the capacitive coupling loss becomes zero at zero frequency. On
the other hand, a dispersive model per Eq. 3.12 accounts for fundamental physical properties of
dielectric material. Initially, the data will be fit to the non-dispersive loss tangent model using
3.15 and Eq. 3.16 in Chapter 4. Then, the linear model will be tested against the dispersive model
and results from reference [99].
To provide a simple analytical limiting case for Eq. 3.14 where fringing fields are not a
factor, consider a resonator formed by a parallel-plate waveguide with identical superconducting
plates of the width w and thickness d, separated by the dielectric layer of thickness s. This
geometry approximates MTL withw  swhere the fringing fields can be ignored. The geometric
factor is
s+ 2? coth(d/?)
?PP (?) = ??0 (3.17)
2[coth(d/?) + (d/?)csch2(d/?)]
utilizing the expression for the effective surface resistance of thin superconducting films, and by
Reff ? Rs[coth(d/?)+(d/?)csch2(d/?)] that ignores transmission loss through the superconducting
films [69]. In the thick film limit the geometric factor reduces to ? 1PP = ??0(s + 2?) and the2
geometric factor is no longer dependent on the superconductor plate thicknesses d. For the thin
film limit where ? d the full expression needs to be used in Eq. D.11 and a term including the
dielectric substrate impedance may be needed [70].
Eq. 3.14 implies the resonator optimized for characterization of both losses calls for a
geometric factor ??c0 = Rs/ tan ?. For Rs ? 17?? representative of high quality Nb at 4.2 K, 10
GHz [138], and tan ? ? 10?3 [99], the optimal geometric factor is ??c0 ? 17 m?. At the same
time, applying Eq. D.11 to our fabrication stack in Fig. 4.1(b) yields ?PP ? 12 m?, which is
71
close to ??c0. Incidentally, this affords us the opportunity to design an MTL resonator capable of
deconvolving the superconducting and dielectric losses at GHz frequencies.
3.3.4 RLGC and Q-Factors From Impedance Matrix [Z]
In the last section, it was established that the net conductor geometric factor ?c0 needs
to be known to extract accurate values from measured data using Eq. 3.16. The net conductor
geometric factor can be approximated analytically using Eq. D.11 if a parallel plate waveguide
geometry can be assumed. The right hand side of Eq. 3.14 can be solved numerically using
finite element solvers like HFSS [142]. This section will describe the method to extract RLGC
parameters and Q-factors from the output impedance matrix [Zij] from HFSS, where i and j refer
to the i-th row and j-th column in the impedance matrix. The geometric factors can then be found
from these extracted parameters and will be shown in Section 3.5.
The ABCD matrix for a generalized, linear transmission line is [95]
?? ??A B ?
?? ? ? ?? ? cosh(?l) Z0 sinh(?l)?? ?? 1 + (?l)
2/2 Z0?l
= ? ?? (3.18)
C D sinh(?l)/Z0 cosh(?l) ?l/Z
2
0 1 + (?l) /2
where ? is the propagation constant, l is the length of transmission line, andZ0 is the characteristic
impedance. The approximation on the right holds for electrically short line ?l  1 using
the quadratic Taylor series expansion for hyperbolic functions. The propogation constant and
characteristic impedance of a general transmission line are [57]
?
? = ? + i? = ZY (3.19)
?
Z0 = Z/Y (3.20)
72
where Z and Y are the series impedance and shunt admittance per unit length of the line.
Inserting Eq. 3.19 and Eq. 3.20 into approximation Eq. 3.18 and assuming the two-port network
is reciprocal yields
?? ??A B ?
? ?
? ?? 1 + ZY l
2/2 Zl
= ?? (3.21)
C D Y l 1 + ZY l2/2
[Z ?
? ? ?(Y l) 1 + (Zl)/2 (Y l)?1 ?
ij] = ? ? (3.22)
(Y l)?1 (Y l)?1 + (Zl)/2
Z = 2(Z11 ? Z12)/l (3.23)
Y = (Z l)?112 (3.24)
where [Zij] is the 2?2 impedance matrix. See Fig. 3.8 for the simplified diagram representations
of the 2-port network parameters.
Without assuming a reciprocal two-port network, the series impedance and shunt admittance
per unit length of the transmission line can also be found from HFSS solved [Zij] using a very
simple yet intuitive differential voltage method in the following way [57]. The propagation
constant can be solved in terms of series impedance Z and shunt admittance Y per unit length of
?
the line ? ? ZY by taking advantage of the current at each end of the ports being approximately
equal for an electrically short line. The current at one end (port 1) is equal and the opposite
polarity to the current at the other end (port 2) I = I1 = ?I2. In a two-port network the
relationship between current and voltage using the impedance matrix [Z] is
?? ?? ?? ?? ??V1? = ?Z11 Z12???? I ?? (3.25)
V2 Z21 Z22 ?I
73
Figure 3.8: Block diagrams of 2-port network parameters showing (a) ABCD block diagram
(Eq. 3.21), (b) ABCD can be transformed into the impedance matrix [Zij] (Eq. 3.22), and the
2-port network can be represented as a single series impedance Z (Eq. 3.23) or shunt admittance
Y (Eq. 3.24) [95].
The differential voltage between the two ports is equal to [57]
?V = V1 ? V2 = ZI (3.26)
From Eq. 3.25 and Eq. 3.26, Z can be written as
Z = R + i?L ? (Z11 ? Z12 ? Z21 + Z22)/l (3.27)
where Zij are the elements of the impedance matrix, and R and L are the series resistance and
series inductance per unit length, respectively. Equivalently, by utilizing I = I1 = ?I2, the
74
differential current being equal to the admittance and voltage product [57]
?I = I1 ? I2 = Y V (3.28)
and the relationship between current and voltage using the admittance matrix [Y ]
?? ? ? ?? ?? I1?? ??Y11 Y12= ???? V ?? (3.29)
I2 Y21 Y22 ?V
the admittance can be expressed as
Y = G+ i?C ? (Y11 + Y12 + Y21 + Y22)/l (3.30)
where Yij are the elements of the admittance matrix, G and C are the shunt conductance and
shunt capacitance of the network per unit length, respectively. Now with expressions for both Z
and Y ,RLGC can be separated and analyzed separately in two-port HFSS simulations. Eqs. 3.23
and 3.24 assume a reciprocal 2-port network, and Eqs. 3.27 and 3.30 do not.
For Q ? 1i ???1 [95], inserting Eq. 3.23 and Eq. 3.24 into Eq. 3.19 and Eq. 3.20, and2
? ?
utilizing the identity Im[ z]/Re[ z] = tan [(arg z)/2] for a reciprocal 2-port network, one can
simplify Qi in terms of the impedance matrix Eq. 3.22 by
[ ]
1 1
Qi = tan arg(2Z
?1
2 2 21
(Z11 ? Z21)) (3.31)
Eqs. 3.27, 3.30, and 3.31 will be used in Section 3.5 to extract numerically solved geometric
factors of microstrip transmission line resonators measured in Chapters 4 and 5. The above
75
method can be applied to any electrically short, 2-port transmission line geometry.
3.4 HFSS Modeling
In this section I describe how to model superconductors embedded in dielectrics by using
impedance boundary conditions or by a Solve-Inside method. These distinctly different model
types to be simulated are shown in Fig. 3.9, and are evaluated here to show (i) the limitations
of Fig. 3.9(a) due to assuming boundary conditions, and (ii) the capability of the Solve-Inside
method Fig. 3.9(b) to extract geometric factors analytically for any arbitrary geometry. Here
we start by modeling a parallel plate waveguide with superconducting impedance boundary
conditions. This is how superconductors have been defined in finite element modelers in the
past [139, 140]. A microstrip transmission line (MTL) will be modeled using the solve-inside
approach described here. From HFSS solved telegrapher?s expressions (RLGC) described above,
the superconducting and dielectric net geometric factors ?c0 and ?d0, respectively, are calculated.
Analytically solving these factors can be difficult as dimensions become comparable to
? and geometries increase in complexity. Here we utilize this method to extract the intrinsic
resistance Rs, and determine RLGC and ? sensitivities to fabrication variation.
Modeling dielectric loss tangent tan ? comes standard in HFSS and is straight forward. The
critical differences in modeling a superconductor in HFSS as opposed to a normal conductor are
(i) the definition of the superconducting material with a real conductivity ?1 and a negative real
permittivity ?r < 1 (? = ?1? i?2 or Zs = Rs+ i??0? cannot be defined directly for 3D material
[142]), (ii) the use of Perfect Electrical Conductor material to connect superconductor materials
to a wave port due to wave ports not being able to handle materials with ?r < 1 (see Fig. 3.12),
76
and (iii) selecting the solve-inside option for the defined 3D lossy superconducting material due
to a real conductivity much smaller than normal conductors. The first critical difference will be
discussed in detail, followed by the process of converting the HFSS impedance matrix into useful
parameters, and results validation.
This method can be used for any linear 2-port electrically short transmission line geometry,
as long as the ratio of largest to smallest dimensions defined in the simulation are less than a factor
of 104. Other possible geometries that can be simulated include coplanar waveguides (CPWs) and
striplines.
3.4.1 Parallel Plate Waveguide
A method to simulate a superconductor in HFSS is by defining the superconductor as
impedance boundary conditions [140, 141, 167, 168]. To demonstrate this, a parallel plate
waveguide (PPW) will be modeled and validated against analytical equations. A range of superconductor
thicknesses d and penetration depths ? are simulated to show HFSS yields proper solutions for
the bulk, thin film, and intermediate limits.
As described in Section 2.2, the complex surface impedance of a parallel plate waveguide
(or resonator) can be modeled using Eq. 2.24 and Eq. 2.25 [70]. The expressions in brackets
in Eq. 2.24 and the term coth (d/?) in Eq. 2.25 are modifications to the intrinsic impedance to
account for superconducting plate thicknesses on the order or less than the penetration depths
d ? ?. For a system composed of bulk superconductor with thicknesses d  ?, the intrinsic
impedance of each conductor simplifies to Zs0 = R ( ? )2s0 + i?0?0?, where R? s0 is the intrinsic0
resistance at reference frequency ?0. Here, a PPW system has been enforced, and we are
77
Figure 3.9: A diagram showing the different HFSS model geometries simulated in this section: a)
Parallel Plate Waveguide (PPW) with superconducting complex impedance boundary conditions
on the top and bottom faces labeled ?ImpBound SC? and PerfH boundary conditions on the
side faces. b) Microstrip Transmission Line (MTL) with Solve-Inside ground plane (GP) and
conducting strip (CS). The conducting strip has width w. These models have superconductor
thickness d and interlayer dielectric thickness s. The nominal HFSS simulation frequency is set
to 10 GHz and simulated lengths ranging from 10-100 ?m. For all Ls, Cs, and l simulated here
meet electrically short condition ?l  1.
interested in the resulting transverse electromagnetic (TEM) mode propagating along the longitudinal
length l in the x direction (see Fig. 3.10). The accuracy of HFSS solutions will be tested using the
following equations for RLGC by inserting the superconducting impedance definitions Eq. 2.24
and Eq. 2.25 into PPW analytical solutions [95]
2R
R = eff (3.32)
w
L = ? s+2? coth (d/?)0 (3.33)w
78
G = ??0?r tan ?
w (3.34)
s
C = ? w0?r (3.35)s
where ?0 is the vacuum permeability, ?0 is the vacuum permittivity, ? is the angular frequency,
w is the PPW width, s is the plate separation dielectric thickness, Reff is the effective resistance
defined in Eq. 2.24, and tan ? is the dielectric loss tangent. The RLGC are the series resistance,
series inductance, shunt conductance, and shunt capacitance per unit length of a generalized
transmission line defined in Eq. 3.2-Eq. 3.5 in Section 3.3.
Using Eq. 3.1, the partial conductor and dielectric Q-factors for a PPW, Qc = ?L/R and
Qd = ?C/G, respectively, are
Qc = ??
s+2? coth (d/?)
0 (3.36)2Reff
Qd = 1/ tan ? (3.37)
Notice, the PPW width w drops out in Qc, and ? drops out in Qd with no geometry dependence.
Then, the only parameters needed to model the superconducting PPW, analytically and with
HFSS impedance boundaries, are s, d, ?, ?, Rs, and tan ?.
The HFSS Driven Modal network analysis design will be used [142] to model the PPW.
The top and bottom faces of the rectangular PPW are defined as impedance boundary conditions
(see Fig. 3.10). To ensure no fringing fields, the two sides are defined with Perfect H boundary
conditions. When defining the HFSS wave ports, a new line of integration needs to be defined
to explicitly set the expected electric field E direction for the propagating mode 1. For the PPW
case, the direction is defined to be normal to the impedance boundary plates. In this section, we
79
are concerned with the first mode of propagation only, where the electric field is exactly normal
to the conducting plates (impedance planes). Take note, this is enforced with the integration line,
allowing HFSS to converge to the correct solution IF and ONLY IF mode 1 is the only mode
present in the model defined. If other modes are present for a given geometry and simulation
frequency ?/2?, then phenomena such as mode conversion, the presence of non-propogating
evanescent modes, and parasitic reflections of higher-order modes can cause HFSS to converge to
non-realistic (bad) results [169]. This is true when modeling any material as complex impedance
boundary conditions.
To facilitate proper convergence of transmission line parameters (e.g. RLGC), the default
settings are the following:
(a) maximum number of passes is set to 40
(b) maximum delta S is set to 0.001
(c) output variables RLGC are set to a 1% convergence criteria
(d) lambda refinement and initial mesh operations are not used
(e) maximum refinement per pass of mesh is set to 50%
(f) solution options are set to mixed order with relative residual of 1? 10?4
(g) maximum delta Z0 set to 2% in the port options.
To aid HFSS in converging to accurate solutions quickly, the simulation geometry used in
Fig. 3.10 and Fig. 3.11 will be set to l = 1 ?m, w = 0.5 ?m, and nominal s = 150 nm. For
impedance definitions, the nominal ? = 90 nm, R = 20 ??, tan ? = 1 ? 10?3s , and simulation
frequency ?/2? = 20GHz.
80
Figure 3.10: Model of parallel plate waveguide (PPW) in HFSS using a Driven Model solution
type. The model a) isometric view, b) cross section view along x-axis, and c) cross section view
along y-axis. The top and bottom faces (z-axis) of the simulation box are the superconducting
plates defined as impedance boundary conditions using Eq. 2.24 and Eq. 2.25. The side faces
(y-axis) are Perfect H boundaries. The PPW length, width, and plate separation thickness are
l ? 1 ?m, w ? 0.5 ?m, and s = 0.15 ?m, respectively. The nominal simulation frequency is
?/2? = 20GHz.
For a wide range of d and ?, the RLGC and partial Q-factor errors from HFSS solutions
relative to Eq. D.3-Eq. D.6 are plotted in Fig. 3.11. The d = 1 ?m (pink) at minimum ? = 1 nm
represents the bulk case (d ?), whereas the d = 10nm (red) at maximum ? = 1?m represents
the thin film limit (d  ?). Everything in between is considered the intermediate regime. A
positive (negative) error corresponds to HFSS converging to a larger (smaller) results relative to
Eq. D.3-Eq. D.6. It can be seen the errors in all parameters do not exceed 0.35% over the bulk
limit, thin film limit, and intermediate regime. This validates HFSS is converging to the expected
results.
Modeling superconductors as impedance boundary conditions has been used in the community
and has been further validated in HFSS for a parallel plate waveguide (PPW) here. Although, this
81
Figure 3.11: HFSS results of a PPW using geometry in Fig. 3.10. The (a) series inductance L,
(b) series resistance R , (c) partial conductor Q-factor Qc0 , (d) shunt capacitance C , (e) shunt
conductance G, and (f) dielectric partial Q-factor Qd error relative to parallel plate analytical
Eq. D.3-Eq. D.6 as a function of penetration depth ? and effective plate thickness d. In HFSS the
penetration depth ? (with name $lambda on the x-axis) and plate thickness d are parameterized
variables. The error was calculated using the equation Xerror = 100 ? (XHFSS ?XPP )/XPP ,
where XHFSS is solved value by HFSS simulation and XPP is the calculated value from parallel
plate model analytical Eq. D.3-Eq. D.6. The simulation frequency was ?/2? = 20GHz.
is only a good approximation under the assumptions of (i) fringing fields can be ignored and (ii)
current distributions inside the superconductor are known. If this cannot be assumed and a two-
fluid model can still be used [40, 43], then one must resort to numerically solving inside a 3D
superconductor.
3.4.2 Defining Solve-Inside Superconductor
A good, isotropic superconductor (?1  ?2) can be defined as a material with bulk
conductivity ?1 and negative relative permittivity [170]. Inserting the superconductor current-
field constitutive relationship J = (?1 ? i?2)E into Maxwell?s curlH equation in phasor form
82
yields
?? ? ?H E 2= ?1 + i??0(?r )E (3.38)
?0?
where ?sc = ?r ? ?2/?0? can be seen as the superconductor real relative permittivity, with ?r
being the ordinary dielectric constant associated with displacement current. The ?r value can be
estimated by approximating a superconductor as a collisionless neutral plasma with Langmuir
2
frequency ?p = c/? and dielectric function ?
?
?p = 1
p
2 = 1 ? (??/c)?2, where c is the speed?
of light [171]. Comparison of ?sc and ?p implies that ?r = 1 is the superconductor permittivity
in the limit ?  ?p (approached from below), and ?2 = (?? 2 ?10? ) is the London conductivity
applicable at frequencies well below the gap and temperatures not too close to Tc. Therefore,
a superconductor can be defined in HFSS as a material with real relative permittivity ?HFSSsc =
1??2(T, ?)??1??10 and real conductivity ?HFSS1 = ?1(T, ?), which is applicable over the entire
range of temperatures and frequencies considered in this work. At frequencies below the gap,
and temperatures below Tc, a superconductor relative permittivity is substantially negative [172].
According to Eq. 3.38, the superconductor complex permittivity is  = ? (1 ? ?2 ? i?1sc 0 ),?0? ?0?
where the three addends are associated with displacement, superconducting, and normal current,
respectively. Due to the fact that ?/?0?  1?the displacement current is negligibly small, and
the intrinsic impedance general?ly defined as??0/sc reduces to a value much smaller than the
vacuum wave impedance Zs = i??0/?  ?0/0 = 120? ?.
Generally, ?1 and ?2 can be calculated from the Mattis-Bardeen theory [49] and the respective
?HFSS HFSS1 (?) and ? (?) are tabulated and implemented in HFSS as datasets [49]. To meet
the assumption ?1  ?2 made in Sections 2.2.2 and 3.3, I will model ?1 as a Drude-like
83
conductivity, which is related to the experimental intrinsic resistance found in Eq. 3.16 via
?HFSS = 2R ??2??2??31 s0 0 0 [138]. Likewise, we will model ?2 as the London conductivity which
gives ?HFSS = ?(??/c)?2. Such an approach allows one to implement ?HFSS1 and ?HFSS
analytically. Forcing HFSS to solve inside the superconductor overcomes the limitations of an
impedance boundary approximation [140], [167].
3.4.3 Solve-Inside Model Setup
When simulating electromagnetic fields inside highly conductive materials with ?/?0? 
1, a proper modeler setup is vital for accurate results. The HFSS model and setup were optimized
for accuracy (meeting condition ?lHFSS  1) and simulation time, in order to complete parametric
sweeps on the order of 100 simulations in a few hours for sensitivity analysis and MTL design
optimization. The network analysis driven terminal solution type was used [142] to simulate 2-
port MTLs with nominal lengths lHFSS = 10?50?m, cross-section geometry shown in Fig. 3.12,
simulation frequencies fHFSS = 10 ? 100 GHz, and perfect electrical conductor port length
lPEC = 1 ?m (see Fig. 3.12) [36]. The MTL is surrounded by a radiation bounding box with
walls sufficiently spaced to avoid impinging magnetic and electric fields shown in Fig. 3.13. Ports
were de-embedded by a length lPEC to calculate only the network parameters pertaining to the
superconductor MTL. After defining a superconducting material with negative real permittivity
?HFSS and real conductivity ?HFSS1 described in last section, the solve inside option is selected,
to override HFSS default to solving inside when real conductivity ?1 < 105 S/m. All geometries
and material properties were parametrized and the Optimetrics option was used for sweeps and
sensitivity analysis. To allow for fast and accurate convergence, the analysis setup was set to
84
Maximum Delta S of 0.001 and 0.1-1 % convergence criteria for RLGC output variables.
3.4.4 Microstrip Transmission Line (MTL)
The (embedded) microstrip transmission line (MTL) geometry is the one measured in
Chapter 4 and Chapter 5. The MTL geometry simulated here consists of a ground plane (GP) and
a conducting strip (CS) embedded in a dielectric having thickness d, and separated by a thickness
s (see Fig. 3.12). The GP and CS are defined as solve-inside superconductors as explained above.
To stay within the assumption ?l  1, the simulated MTL simulation lengths and frequencies
are l < 100 ?m and ?/2? < 100 GHz, respectively. Unless otherwise specified, the nominal
conductor and separation thicknesses are d = 200 nm and s = 150 nm, respectively. The
GP width is nominally 20 ?m when simulating CS w ? 4?m, to be sufficiently far from the
electromagnetic fringing fields. Similarly, the simulation box height separation from the GP and
CS is roughly 5-10s to be sufficiently far from electromagnetic fields. The perfect electrical
conductor (PEC) strips (yellow) connecting the GP and CS to the waveports are lpec = 1 ?m.
The wave ports are surrounding the CS cross section and only touch the corner of the GP to
properly act as references. Extending the wave port into the cross section of the GP will induce
an error and HFSS will not be able to solve. Additionally, the GP cannot touch the edges of the
simulation, as this will effectively cause a short. Lastly, all faces of the simulation box are defined
as Radiation Boundary conditions.
Two-port driven terminal simulations for MTL widths w = 0.25 ? 4 ?m were done and
magnetic field profiles in cross section can be seen in Fig. 3.13. Up to this point, the lossy
telegraph?s R and G for an MTL at these dimensions and ? have not yet been reported. For
85
the 4 ?m MTL, the magnitude of the magnetic fields (H) at the edge (fringing fields) are only
a small fraction relative to the H fields underneath the CS. Therefore, a parallel plate model is
a good approximation for this width (assuming w >> s). For the 0.25 ?m MTL, the fringing
fields become a significant fraction. It can also be seen from Fig. 3.13 for there is a transition in
fringing field character between 0.5 and 1 ?m. At 0.5 ?m, the fringing fields on each side begin
to overlap. At 0.25 ?m, a significant fraction of the CS is penetrated by magnetic field, but the
current is not yet uniform, making it difficult to analytically model. When considering the GP,
the fraction of fields inside reduce significantly relative to the CS. This will be discussed further
in the next section. Chang et al. analytically solved telegraph?s L and C acounting for a fringing
factor for microstrip widths w ? 1 ?m [134].
86
Figure 3.12: MTL geometry modeled in HFSS. The MTL conducting strip (blue) has width w
and a dielectric separation thickness s from the ground plane (gray). Both the strip and ground
plane have thickness d and simulation length l. In HFSS, the MTL is surrounded by a simulation
box with all six faces defined with Radiation boundaries. To properly launch waves into a
superconductor with negative real permittivity, small lengths of Perfect Electrical Conductors
lpec ? 1 ?m (yellow) are used to connect the superconducting ground plane and conducting strip
to the wave ports. To cut down on simulation time and reduce the simulation size in half, one can
utilize a Symmetry boundary condition.
87
Figure 3.13: Cross section MTLs simulated using HFSS and geometry shown in Fig. 3.12. The
dielectric thickness is s = 150 nm. The ground plane and conducting strip thicknesses are both
d = 200 nm. The MTL width varied from 0.25? 4 ?m going from top to bottom. The inputted
intrinsic resistance Rs = 20 ??, TEOS loss tangent tan ? = 1? 10?3, and the SiNx loss tangent
tan ? = 1? 10?4. The magnetic penetration depth was set to ? = 90 nm. The relative dielectric
constant of TEOS and SiNx relative dielectric constant were set to ?TEOS = 4.5 and ?SiNxr r = 7.5,
respectively. The simulation frequency was set to ?/2? = 10GHz.
88
3.5 MTL Geometric Factor ? Extraction
For an electrically short transmission line ?l  1, the propagation constant of a general
?
transmission line ? = ? + i? = ZY [57], and the internal Q-factor of a transmission line
resonator is Q = 1???1i [95] This can be expressed in terms of the series impedance Z and2
? ?
shunt admittance Y per unit length yielding Q 1i = Im( ZY )/Re( ZY ). Inserting here Z ?2
2l?
? ?
1(Z11?Z21) and Y ? (Z ?112l) found in Section 3.3.4, and using an identity Im( z)/Re( z) =
tan[Arg(z)/2], yields
[ ( )]
1 1 2(Z11 ? Z21)
Qi = tan Arg (3.39)
2 2 Z21
where Arg(z) is the argument of a complex variable z, and Zij are the respective elements of the
model Z matrix solved by HFSS. For a model of arbitrary length, one shall resort to a general
solution for propagation factor e??l of a transmission line with mismatched ports [36].
Setting the dielectric loss to zero, tan ? = 0 throughout the entire HFSS model leads to
Qi = Qc. Then, the conductor geometric factor ?c0 can be computed as
? = RHFSSQHFSSc0 s c (3.40)
where RHFSSs0 is the superconductor intrinsic resistance defined in the HFSS model, and Q
HFSS
c
is the quality factor found from the HFSS solution using Eq. 3.39, and the simulations are
done at a reference frequency ?0/2?. Here ?c0 is independent of intrinsic resistance and the
product QHFSSRHFSSc s of each cross-sectional geometry can be found from a single simulation
89
at any intrinsic resistance value meeting RHFSSs  ?0?0?. This simplifies the experimental data
analysis by removing the need to simulate Qc as a function of Rs for every MTL width.
Similarly, by setting the superconductor intrinsic resistance to zero throughout the entire
HFSS model leads to Qi = Qd. Then the dielectric geometric factor ?d can be computed as
? = tan ?HFSSQHFSSd d (3.41)
where tan ?HFSS is the dielectric loss tangent defined in the HFSS model, and QHFSSd is the
quality factor associated with the dielectric loss.
The HFSS results can be validated by comparing the HFSS solved geometric factors ?c0
using Eq. 3.40 to the analytical parallel plate model using Eq. D.11 for MTL widths much larger
than the dielectric thickness w/s  1. For wHFSS = 4 ?m the partial Q-factor associated with
the conductor loss solved by HFSS QHFSSc0 (4 ?m) = 586 using Eq. 3.13 and an input value of
RHFSSs0 = 20 ??. Using the parallel plate model, the partial Q-factor is found to be Q
PP
c0 = 584,
yielding agreement within 0.34%. This can also be seen directly in Fig. 3.13, where ?totc0 (green
solid line) approaches ?PPc0 (dashed line) forwHFSS ? 1?m. Intuitively this can be understood by
looking at Fig. 3.14. The fringing fields at MTL left/right edges of cross-section begin to overlap
for wHFSS ? 1 ?m, making the parallel plate model inapplicable as it does not account for field
fringing, concentration at the edges, and penetration above the conducting strip. Furthermore, it
is important to note the geometric factors calculated from HFSS solutions were independent of
the input value RHFSSs = 0.1? 100 ??.
90
Figure 3.14: Geometric factors for the conducting strip ?CS , ground plane ?GP , and ?totc0 c0 c0 . This is
assumingRGPs0 = R
CS
s0 from Eq. 3.14 for all MTL widths. The parallel plate analytical calculation
?PPc0 is compared (see Eq. D.11).
91
3.6 MTL RLGC and ? Sensitivity to Fill
Spatially dense areas of metal polishes (etches) faster than lower density areas in the
damascene CMP process. Floating fill patterns (metal strips) are used to maintain a certain metal
density, achieving uniformity across the chip and wafer, mitigating dishing effects [54].
Fig. 3.15 shows the MTL strip below the ground plane and surrounded by floating fill
patterns length (left to right) and width (into the page) ? 1.5 ?m and ? 0.5 ?m, respectively, to
reach minimum density requirements of ? 65%.
Figure 3.15: STEM images of (a) full MTL showing the via walls and surrounding fill, (b)
measurements of fill proximity to conducting strip (CS), and (c) is a zoom in of the conducting
strip showing the resulting inverted trapezoid geometry.
Using the measurements in Fig. 3.15, three HFSS simulations were run and compared (see
Fig. 3.16 and Table 3.2). The top row in Fig. 3.16 is the nominal cross sectional geometry. The
middle row is the actual cross-sectional geometry of the MTL without surrounding fill. Note, the
92
sidewall angle was introduced in the HFSS simulation to model the damascene Process A. The
bottom row is the same as the middle row but with the additional Nb fill to reflect the geometry
shown in Fig. 3.15(b).
93
Figure 3.16: Cross sections of three different HFSS simulations, where the (top) is the nominal
designed geometry of the 250 nm wide conducting strip (CS) underneath the ground plane (GP).
The (middle) row is the actual cross section geometry as measured by TEM data for wp432-
TL6 with CS shaped more like a Reeses Cup. The (Bottom) row is the same MTL geometry as
(middle) but with added and repeated superconducting strip fill patterns.
Table 3.2 summarizes the results from simulations shown in Fig. 3.16. The RLCG values
were calculated as described in Section 3.3.4. The partial geometric factors associated to the
ground plane (GP), conducting strip (CS), TEOS dielectric layer, and SiNx dielectric passivation
layer are calculated based off of geometry shown in Fig. 3.16.
Immediately, it is important to point out that Fig. 3.15 shows the CS is in the layer below
the GP (also shown in Fig. 3.1 and summarized in Table 3.1). Therefore the electric fields are
masked from the SiNx. This is confirmed in Fig. 3.2, showing ?SiNx is a factor of 10 larger than
?TEOS , and is not participating. This is also true for Process A and B where the GP is below the
CS. Referencing Fig. 3.13, only a small fraction of the fields of significant magnitude reach the
SiNx layer (magnetic field is shown, but electric field profile is almost identical outside of the CS
94
and GP). Therefore, for MTL widths and process geometries, ?SiNx  ?TEOS, and therefore is
ignored in the remainder of this work.
Table 3.2 shows the effective geometric factor ?c0 decreased 5% for actual geometry and
further decreased by 2% when floating fill was introduced. This change is primarily due to the
large change in geometry of the GP (?GP ) having thickness half of nominal and the dielectric
thickness ? 8% thinner whereas the CS geometric factor ?CS was almost unchanged.
Due to? 8% change in dielectric separation thickness bothG andC changed by? 8?10%.
When fill was introduced, bothG andC increased by? 36?38%, due to the removal of dielectric
replaced by metal fill. BothG andC changed the same way and therefore the dielectric geometric
factor ?d0 was almost unchanged. Although, the notable increase in C by about 36% and almost
no change inLwill considerable decrease the data propagation velocity by about 15% where vp ?
(LC)?1/2. This result is relevant for synchronous clock timing designs where the interconnect
inductance and capacitance across long distances need to be meet required values, and fill patterns
around the interconnect can have a notable affect.
95
Table 3.2: Table of RLCG for three different HFSS simulations shown in Fig. 3.16 using
geometry measured from Fig. 3.15 fabricated using Process A (see Table 4.1). The material
properties used for all three simulations areRs0 = 20??, ? = 90nm, ?TEOS = 4.5, tan ?TEOS =
1? 10?3, ?SiNx = 7.5, and tan ?SiNx = 1? 10?4.
Overall, measuring RLGC sensitivities to fill showed that geometric factors are fairly
insensitive and can be approximately ignored. However, propagation velocity is sensitive to fill
placement and geometry and should be taken into consideration. The method described here can
be used to check the change in propogation velocity for a given fill geometry. Most importantly,
the largest changes in RLGC and ? were seen in the difference between nominal (expected)
cross sectional geometry and as fabricated (measured by STEM). In the next section, we will do
a sensitivity analysis on MTL geometries.
96
3.7 MTL ? Sensitivity to RQL Fabrication
This section will discuss howRLGC parameters (see Eq. 3.2-Eq. 3.5) and resonator geometric
factors (see Eq. 3.9-Eq. 3.10) in MTLs vary at parametric corners of the fabrication process (see
Fig. 4.1). A simple sensitivity analysis will be presented to get a sense of (i) the precision of
extracted Rs0 from measurements in Chapter 4 (ii) the application of this method to determine
sensitivity to fabrication variation. This procedure can be used for any arbitrary two port transmission
line cross-sectional geometry (e.g. coplanar wave guide, strip line, etc).
Table 3.3(a) shows the nominal parameters for the fabrication of samples from Process
A, B, and C (see STEM summary table Fig. 3.1). Table 3.3(b) are the assumed minimum to
maximum variations of the fabrication/material parameters. These parameters were inserted into
HFSS as parametric variables to calculate variations in ? and RLGC using methods described in
Section 3.3.4 and Section 3.5.
Table 3.3: Tables showing a) the nominal material properties and simulation lengths, and b) the
minimum, nominal, and maximum values chosen as the parametric corners based off fabrication
variation worst case scenario.
Fig. 3.17 shows the percent variation of the geometric factor ?c0 is roughly ?20% for
97
all CS widths shown. The 0.25 ?m has an additional ?2% variation. These are conservative
estimates for Process A, B, and C based off worst case geometry variations shown in Table 3.3(b).
Aside from the ground plane thickness offset in Process A, the summary Table 3.1 from STEM
measurement show the variations in thicknesses do not exceed 10%. Although, these are critical
dimension (CD) measurements on single chips and are not representative of the whole wafer
or lot of wafers. Thickness uniformity and line width targeting at minimum feature sizes are
one of the many challenges in fabrication. Therefore, a worst case 20% variation in ?c0 will be
incorporated as error in our extraction of Rs from measurements in Chapter 4.
Figure 3.17: Parametric Corners to show worst case max/min error in ?c0 for MTL widths
0.25? 4 ?m. This is assuming nominal geometry from Table 3.3(a) and worst case cross section
geometric variation in Table 3.3(b), based off fabrication tolerances.
The variation in ?, or Qc assuming a constant Rs, can be further scrutinized by doing
a sensitivity analysis on each parameter contributing to changes in RLGC. Fig. 3.18 shows
RLGC % change as a function of the parameter % change for a 0.25 ?m MTL starting at
98
nominal geometry Table 3.3(a). Note,Rs and tan ? were held constant in Fig. 3.18 showsRLGC,
and therefore any variations in Qc or Qd are variations in the geometric factors exclusively.
Immediately, it can be seen from Fig. 3.18(f) that ?d is insensitive to all parameter variations.
This is expected for an MTL embedded in a homogeneous dielectric, as the material property
tan ? is geometry independent. If phase velocity sensitivity to fabrication variation is desired
(v ? (LC)?1/2p ), then the L and C sensitivities can be simply multiplied from Fig. 3.18(a)-(b).
Here, the discussion will be focused on ?c sensitivity to fabrication variation.
The conducting strip (CS) magnetic penetration depth ?CS varies L and R by as much as
?10%, but change in the same way and therefore ?c (Qc with constant Rs) is insensitive to ?CS .
Interestingly, ?c can change by 3% for -30% change in ?GP . This is most likely due to magnetic
field becoming more concentrated in and around the CS (see Fig. 3.13). Evidently, this suggests
?c is relatively insensitive to changes in ?.
The most surprising result is ?c is most sensitive to thickness variations in the conducting
strip dCS . The Lmaximum change is?6%, whereas theR can change by as much as?12%. The
lower L sensitivity to dCS is most likely due to the internal inductance term only being one of two
contributions to the total inductance (see 3.3). Whereas R is completely defined by ? (current
distribution inside wire), and is greatly affected when the CS (or film) thickness is on the order of
the ? ? dCS , i.e. ?eff = ? coth (d/?). This argument can also explain ?c secondary sensitivity
to MTL widths wbottom and wtop. Although, ?c has a slightly lower sensitivity to wbottom relative
towtop. The dielectric separation thickness s can vary ?c by?5%, as this can significantly change
the current distribution and fringing fields.
Using HFSS simulations and the methods described here, it has been determined ?c can
vary as much as ?20% for MTL widths 0.25 ? 4 ?m. A sensitivity analysis on a 0.25 ?m
99
Figure 3.18: RLGC Sensitivity Analysis on MTL minimum width 0.25 ?m as well as the %
change in Qc and Qd (see Eq. 3.1 and Eqs. 3.2-3.5). The Nominal geometry shown in Fig. 3.12,
where wbottom and wtop are the dimension closest and furthest from the ground plane (GP),
respectively. The absolute parameter ranges can be found in Table 3.3.
with trapezoidal geometry suggests ?c is most sensitive to the conducting strip thickness dCS , s,
wbottom, and wtop. This analysis affords us the ability to bound our error in extracted Rs from
measurement in Chapter 4 and Chapter 5.
100
3.8 Summary
In this chapter, I discussed the importance of numerically solving inside a lossy superconductor,
and reviewed the work that has been done thus far. I presented a method to extract the intrinsic
superconductor and dielectric RF losses coupled with HFSS modeling.
I discussed the morphologies (non-idealities) in Nb-TEOS 0.25 ?m and 1 ?m MTLs from
Process A, B, and C were study using STEM and EDS. For the damascene Process A and B,
line defects and voiding occur at the edge of the fill trenches, quintessential to the sputtered fill
step in damascene processes. These defects occur in a region of highest magnetic field (current
density) concentration for Process A. For Process A and B, up to 1at% Ar was detected inside of
the conducting strips. No Ar was detected in MTLs from Process C. Process A shows a total Nb
oxide layer approximately 10 nm with 5 nm layer of NbO ? NbO2. Process B shows the same
total Nb oxide thickness of approximately 10 nm, but no stable oxide was detected. Process C
shows no detectable Nb oxide layer due to the incorporated SiNx/Al passivation layer.
I presented a method to deconvolve the partial Q factorsQc ? ?/Rs from the superconductor
and Qd = 1/ tan ? from the dielectric by relying on their respective dispersion relationships. To
extract the intrinsic resistance Rs from Qc, I presented a procedure using HFSS to calculate the
geometric factors ? numerically. The HFSS procedure was validated by comparing to analytical
models and other work. The geometric factor is insensitive to the surrounding fill patterns used
in Process A, B, and C. The geometric factor is notably sensitive to primarily the conducting strip
thickness, interlayer dielectric thickness, and the conducting strip width, but is not sensitive to
the magnetic penetration depth. The conservative error in ?c will be bounded to ?20%.
101
Chapter 4: Characterization of Microwave Loss in RQL Interconnects at 4.2 K
In the previous chapter, non-idealities in the morphology were measured in TEM/EDS,
and could be causing added extrinsic resistive losses in Nb interconnects. The geometric factor
methodology and solving numerically in HFSS was established as a means to accurately extract
the intrinsic resistance Rs from measurements of the partial conductor Q-factor Qc. This enables
the ability to compare Rs across processes A, B, and C for all MTL widths.
In this chapter, a brief history on Q-factor measurements in different systems and assumptions
will be presented in the introduction. The MTL resonator design will then be described, followed
by the measurement and fitting method to measure the internal Q-factor Qi. The Qi can then
be deconvolved in the dielectric Qd and conducting Qc partial Q-factors by relying on their
respective dispersive relationships. The analysis of data will be presented using two different
models. The non-dispersive model and analysis will show a clear difference in Rs between the
three processes and a nearly constant tan ? for all samples measured. The dispersive model is
difficult to confirm or deny, but there is some agreement found with previously measured results
and fundamentals expectations. In the discussion section, the morphology characterization and
HFSS simulations will then be reference to help explain the difference between the three pro-
cesses.
102
4.1 Introduction
As said before, resonator remains the only practical way to accurately measure Rs [131],
[62] and tan ? [100, 132] at microwave frequencies. Because Q?1c and Q
?1
d are additive, the two
losses are inseparable. Hence, existing resonant measurement techniques exploit regimes where
the net Q-factor is dominated by either the superconductor or dielectric loss. The former case with
Qc  Qd, favors the measurements ofRs. The latter case withQc  Qd, favors the measurement
of tan ?. Tuckerman et al. take advantage of the condition Qc  Qd to measure tan ? of Nb-
polyimide flexible transmission line tapes at 2 K [100]. Dielectric measurements by Krupka et
al. optimize their resonant cavity design to satisfy the condition Qc  Qd and measure the
dielectric loss tangent [132],[173]. Oates et al. Q-factor measurements on Nb-SiO2 sub-micron
stripline resonators show they are limited by dielectric loss except for the narrowest strips, but
they do not report measurements of tan ? or Rs [174]. Superconducting cavities measure Rs
under circumstances where Qc  Qd [165], [175]. Taber overcomes the above limitations by
varying the geometric factor (dielectric thickness) of the parallel plate resonator [136], [70], and
extracts bothRs and tan ?, but his approach is impractical for characterization of superconducting
integrated circuits if one wants to deconvolve losses in a single structure and measurement. To
measure the frequency dependence of tan ? of thin film dielectrics commonly used as Josephson
Junction barriers, Kaiser et al. neglects the metal loss by making use of lumped element LC
resonators satisfying the condition Qc  Qd.
In contrast with most prior work, to measure Rs and tan ?, I designed and tested microstrip
transmission line (MTL) resonators having comparable superconductor and dielectric losses such
that Qc ? Qd ? 1000. To be sensitive to Qc and Qd, the MTL was designed into fabrication
103
Process A, B, and C with multiple resonances and sensitive to intrinsic Q-factor Qi. In this
chapter, I compare the extract Rs and tan ? from measurements of the frequency dependent
internal Q-factor Qi(?) for MTL widths 0.25-4 ?m. I was most interested in analyzing Rs MTL
width dependence and difference between Process A, B, and C, and seeing how Rs correlated
with morphology measured by STEM and EDS in Chapter 3.
4.2 Resonator Design
To implement the proposed concept, we designed a chip with five open-ended half-lambda
MTL resonators shown in Fig. 4.1. The conducting strip width varies from 0.25 to 4 ?m.
Each 15 mm long resonator was folded into meander shape to preserve space. Each resonator is
reactively coupled to a 50? feedline via a coupling capacitor Cc. Via walls were placed about ten
dielectric thicknesses (? 10s) away from the edge of the conducting strip, to reduce cross-talk
between two adjacent resonator meander sections (see STEM cross-section in 3.15).
The method to deconvolve tan ? and Rs0 can be found in Section 3.3. In this section,
deconvolution will be demonstrated by comparing extracted RF losses for two damascened Process
A and Process B) and a cloisonne? Process C. Results are summarized in Table 4.1).
4.2.1 Resonant Frequency
MTL resonators support a TM00n [59] mode with the eigen frequency
n
fn = ? ? ? nc (4.1)
2lres LC
?
2lres ?r 1 + 2? coth(d/?)/s
104
Table 4.1: Results summary of Extracted Rs and tan ? from Process A, B, and C. The MTL
resonator measurements were done at 4.2 K and are shown in Fig. 4.5. A homogeneous MTL,
with ? = 90 nm, and a non-dispersive tan ? were assumed.
Figure 4.1: a) Physical layout of five MTL resonators of varying widths w showing (bottom)
contact pad arrangement, feedline wiring to Cc, and resonator meander geometry taking up chip
area ? 0.5 ? 1mm2. The (Top) of a) is a zoom in of how the VNA makes connections to the
feedline, followed by the coupling capacitor Cc connection to the MTL. This also shows the via
walls that serve as isolation for the feedlines and MTL conducting strips. b) A cross-section of
fabrication process showing the nominal critical dimensions of a two metal layer process: the
ground plane (dark gray), conductiong strip (blue), surrounding TEOS dielectric (light gray),
and SiNx passivation layer (gold). c) Representative diagram of the open ended MTL resonator
connected to Cc, 50 ? feedline and VNA
105
where n = 1, 2, 3. . . is the mode index, lres is the geometrical length of the resonator, and L
and C are the conducting strip series inductance and shunt capacitance per unit length defined
by Eq. 3.3 and 3.5. The approximation on the right in Eq. 4.1 holds for a wide MTL where the
width is much greater than the dielectric thickness (w  s) [70], where c is the speed of light,
and ?r is the dielectric constant. The resonator frequency was designed based upon the nominal
material properties (?r, ?) and the fabrication process thicknesses (s, d). The selection of lres
was a trade-off between placing many resonators on a single 5 ? 5 mm2 chip, and a convenient
frequency range to measure multiple resonant modes using a vector network analyzer (VNA)
while avoiding the standing-wave ripple due to impedance mismatch. For a resonator length of
lres = 15 mm the fundamental mode resonant frequency f1 ? 3.27 GHz. The average mode 1
index resonant frequency for Process C w = 4?m resonators f1 = 3.2? 0.07 GHz and is within
0.8% of wide MTL estimate using Eq.. 4.1, and this agreement supports our nominal ? ? 90 nm
[54] and ?r ? 4.2 [73].
4.2.2 Critical Coupling
To overcome parasitic ripple seen in the transmission coefficient |S21| in Fig. 4.4(a), which
is caused by inevitable impedance mismatch within the dip probe, we tune the coupling between
the resonator and feedline to provide about 6dB insertion loss (IL) at the resonant frequency. For
a reactively coupled resonator this corresponds to critical coupling, Qi = Qe, where Qe is the
external (coupling) Q-factor.
The feedline is directly connected to the top plate of the coupling capacitor while the MTL
resonator is directly connected to the bottom plate. The Cc value is tuned to reach a critical
106
coupling g=1 using the following equation [176]
R 1? |S21| Qi
g = = = = 1 (4.2)
2Z0 |S21| Qe
where g is the coupling factor, R and Z0 are the series resistance and characteristic impedance
of the MTL resonator, |S | = 10?IL/2021 with IL being the insertion loss at resonance in dB,
Qi is the resonator internal (or intrinsic) Q-factor, and Qe is the resonator external (or coupling)
Q-factor. The input impedance Zin looking into a coupling capacitor towards a resonator is [95]
?Z0
Zin = ; bc = ?CcZ0 (4.3)
2Q 2ibc
where ? is the angular frequency. Converting the ABCD transmission matrix for a shunt
admittance Y = Z?1in into respective S21 gives [95]
2Zin
S21 = (4.4)
2Zin + Zfeed
where Zfeed is the feedline characteristic impedance. Assuming a parallel plate geome?try (w 
?s) and thick plates (d ?), the MTL characteristic impedance can be estimated Z0 = L/C ?
?0(s+ 2?eff )/ (?0?rs). Eq. 4.4 can be used to calculate the Cc needed to achieve the desired
coupling factor g to a resonator with a given Qi, characteristic impedance Z0, and resonant
frequency f0 [95]
( )?
1 2?g
Cc = (4.5)
2?f0 ZfeedQiZ0
107
Using Eq. 4.5 and the following parameters expected for a 4 ?m conducting strip resonator
f0 = 3 GHz, Qi = 685, Z0 = 8.6 ?, and Zfeed = 50 ? gives a Cc = 245fF to critically
couple the resonator and achieve an IL ? 6 dB. The measured IL ? 6 dB for this geometry
giving g ? 0.7, which means the resonator is under coupled and the measurement is sensitive to
material losses. Error is most likely due to variations in the feedline and resoantor characteristic
impedance due to fabrication. Fig. 4.2 shows four different coupling strengths for a 0.25 ?m
MTL resonator. Each color is data averaged over 2 wafers and 9 chips (see Table 4.1. Each
symbol is a different mode index of the MTL resonator (see Fig 4.4). Fig. 4.2(a) shows the
extracted Qi for each coupling design (colors) overlay fairly well with the 12 GHz modes having
the most spread. Fig. 4.2(b) shows Qi for calculated coupling factor g from measurement of IL
(see Eq. 4.2). Similar calculations are found when taking the ratio of Qi/Qe within 15%. This
data suggests the extracted Qi for all coupling designs are equal with error from wafer spread,
and confirms our extraction method is valid and is fairly insensitive to coupling strength up to
g = 3.
Hence, the coupling capacitor design targets the following value
?
1 2?
Ccrit = (4.6)
2?f1 ZfeedQiZ0
where f1 is the fundamental resonant frequency given by Eq. 4.1, Zfeed = 50? is the characteristic
impedance of the feedline, Qi is the internal quality factor of the resonator that can be estimated
from Eq. D.11, and Z0 is the characteristic impedance of the MTL resonator. The characteristic
impedance can be estimated using Eq. D.8 assuming a parallel plate transmission line geometry or
simulated in HFSS as described in Chapter 3. In this MTL design, the Cc varies from 50?250fF
108
Figure 4.2: Demonstration of the coupling design on a 0.25 ?m MTL resonator design showing
fitted Qi on four resonant modes for four different couplings as a function of a) resonant
frequency (or mode index), and b) the coupling factor g (see Eq. 4.2). The measurement method
is described in Section 4.3. Representative S21(?) data and Qi fitting can be seen in Fig. 4.4.
These measurements are on samples fabricated using Process C taken at 4.2 K
for MTL w = 0.25? 4 ?m, respectively.
A plaid capacitor design (see Fig. 4.1(a)) was used for the coupling capacitors. This
capacitor design enabled the fabrication of large capacitors in a damascene process, where wide
metal patches are disallowed to minimize under (over) etching of low (high) metal density during
CMP. The capacitor is composed of multiple parallel lines (with minimum allowed spacing) in
one metal layer and many more in adjacent metal layer, running perpendicular to the lines in
the first layer. The two metal layers are separated by TEOS of thickness s. Every other line in
each layer are connected with vias thus creating two interwoven (plaid) electrodes. While the
structure complies with design rules that limit wire width and metal density it exhibits electrical
performance of a parallel plate capacitor; it has low parasitic inductance and approximately the
same capacitance (within ? 30%) per unit area.
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4.3 Measurement
Chips were selected within the inner 80 mm diameter of 150 mm wafers (see Fig. 4.3). The
number of chips and wafers measured from each Process can be found in summary Table 4.1.
Measurements were taken at 4.2 K with a cryogenic, RF dip probe outfitted with a 32 pin
(contact pad) test fixture. The same test fixutre was used in previous works [35], [34], [36],
[10]. To achieve RF signal integrity and avoid wire bonding for fast sample chip exchanges,
a non-magnetic Cu/Au flip chip press contact technology is used. The fixture PCB connects
the flip-chip bump pads to the probe .047 inch semi-rigid coaxial cables. The chip and roughly
30 cm of the cables are submerged in L-He bath during measurements. The temperature was
monitored with a calibrated Lakeshore Cernox sensor. The resonators were measured by 2-port
S-parameters using a Keysight N5222A vector network analyzer (VNA). The network analyzer
was calibrated up to the top of the probe.
Typical measurements were taken over 2? 14GHz frequency span with 200 kHz spacing
with input power ? ?20 dBm to minimize possibility of non-linear effects. The probe usable
bandwidth accommodates resonant modes up to n = 4. Measurements above 14 GHz are
dismissed due to the standing-wave Q-factor becoming comparable to the resonance Q-factor.
Fig. 4.4(a) shows the first four resonant modes for each MTL width w. From the complex S-
matrix circle fits (see Fig. 4.4(b), theQi, Qe, and fn are found using the methodologies described
in [177, 178], where the Q-factors are related to the loaded Q-factor by Q?1 = Q?1 +Q?1L i e in the
complex plane. The first mode fitting for each width is shown in Fig. 4.4. Measurements shown
in Fig. 4.4 were taken across 4 wafers and a total of 22 chips (5 MTLs per chip). All resonances
from 2-14 GHz were detected, fitted, and analyzed using a custom Python script.
110
Figure 4.3: Wafer Map Diagram used for all three Processes showing (a) the 16 mask reticles
(blue) and the chip (pink) fabricated with the resonator chips. (b) A zoom in of the MTL
resonators chip (see Fig. 4.1) for more details on chip layout and design. All measured chips
from Process A,B, and C were picked from the inner 80 mm diameter of 150 mm wafers shown
here.
4.4 Method Demonstration
The statistical mean and standard deviation for all modes, conducting strip widths, and
fabrication processes are shown in Fig. 4.5. The increased spread in Q?1i for n = 3, 4 is due to
the fitting error due to standing wave ripples and reduced measurement repeatability. A separate
gauge study was performed by installing, testing, and re-installing the same chip 4 times and
yielded max percent variations in extracted Qi and fn to be < 10% and < 0.1%, respectively.
From Fig. 4.5, it can immediately be seen that Process A has larger slopes for all conducting
strip widths relative to Process B and Process C, and most convinently seen in Fig. 4.5(f). Using
Eq. 3.16, a larger slopeA yields an approximately larger intrinsic resistance assuming the slope is
111
Figure 4.4: Data for a chip from center of the wafer fabricated using process Process A. (a)
Insertion loss versus frequency for modes n = 1 ? 4 on MTL widths 0.25 ? 4 ?m at 4.2 K. (b)
Complex circle fitting of the fundamental mode resonant peaks from (a) using methodology from
[177, 178].
exclusively and the equal geometric factors between each process. The linear dependence of Qi
with frequency ? in Fig. 4.5 supports the inclination that Eq. 3.15 is valid, but will be scrutinized
in section 4.4.2. At the the same time, for lines 1?m and narrower Process B has relatively higher
Rs relative to Process C, while for the 2 and 4 ?m lines Process B and Process C have nearly
the same superconductor loss. It is also evident from Fig. 4.5 that all MTL widths and processes
yield roughly the same linear y-intercept, which is related to the dielectric loss tan ? ? 10?3.
Although, this is all assuming the ground plane and conducting strip have the same intrinsic
resistance Rs, a non-dispersive tan ?, and the same geometric factors between each process.
The fitted linear slope using Eq. 3.15 from Fig. 4.5 as a function of process and MTL width
is shown in Fig 4.6. For reference, the analytical geometric factor assuming a parallel plate model
?PP (solid black line) is also shown calculated using Eq. 3.16 with nominal dimensions shown in
the parallel plate MTL diagram. Primarily for processes A and B, the slope A is approximately
112
Figure 4.5: The reciprocal internal Q-factor Q?1i at 4.2 K as a function of the resonant frequency
for each of the five MTL widths and for three fabrication processes. The bottom right graph is
five linewidths and processes plotted together.
113
constant for MTL widths 1 ? 4 ?m. A noticeable upturn in A can be seen for MTL widths 0.25
and 0.5 ?m for processes A and B. Additionally, the y-offset of ? 1.5 ? 10?14 Hz?1 between
process A and B is a strong indication the intrinsic resistance is larger for process A with respect
to process B. Furthermore, process C has a near MTL width independent slope A and ? 1 ?
10?14 Hz?1 lower A at MTL width 0.25?m with respect to process B. Empirically, it can be
concluded that process C has a lowest conductive loss and does not have a width dependence.
Although, accurate conclusions about the intrinsic resistance Rs cannot be drawn due to the
convoluted width dependence of MTL geometric factor ?c0 found from HFSS simulations in
Chapter 3. As an example of the worst case scenario, without knowing the cross-section of the
MTL fabricated it can be conceivable the differences between process B and C are completely
due to geometry differences as opposed to the intrinsic resistance Rs.
Utilizing the methods and procedures described in Chapter 3 as well as the measurements
of cross-section dimensions from TEM and FIB, the geometric factors were numerically solved
in HFSS and are plotted in Fig. 4.7. Similar to Fig. 3.14 for nominal cross-sectional geometries,
Fig. 4.7(a) shows ?c0 solved for the as measured cross-sections. The solid black line is referenced
for the nominal geometries. Errors are calculated between measured relative to nominal and can
be seen in Fig. 4.7(b). Process A and B have percent errors increasing from of approximately
3-6% as the MTL width is reduced from 4 to 0.25 ?m, respectively. Process C has an almost
consistent error for MTL widths 0.5-4 ?m of -2%, but increase up to 6% for 0.25 ?m (delta
of 8%). Referring to the STEM summary Table 3.1, maximum error (variation) of 8% in ?c0
correlates fairly well to the maximum dimension variation of approximately 7-9% of the dielectric
thickness. From an engineering and fabrication perspective where targeting consistent critical
dimensions is important, 8% variation is manageable and is a good result confirmed by HFSS.
114
Figure 4.6: Plots showing the fit slope A as a function of MTL width for all three processes
(assuming non-dispersive model). (solid black line) The predicted slope A using Eq. 3.16
assuming the minimum BCS intrinsic resistance RminBCS and geometric factor for a parallel plate
MTL (very wide conducting strip) using Eq. D.11 at 10 GHz and 4.2 K.
115
In other words, an 8% variation in dielectric constant only varies the ?c0 by roughly the same
amount. This affords a two-fold understanding: (1) the Q-factor of the MTL resonator (RQL
interconnect) related to power dissipation is fairly resilient to fabrication variation geometrically
and (2) any variations in Q-factor above the fabricated geometric variation can be attributed to
extrinsic loss variation (i.e. Rs and tan ?).
Deconvolving the geometric factor further, the partial geometric factors of the conducting
strip ?0CS and ?0GP are plotted in Fig. 4.7(c) and Fig. 4.7(d), respectively. For the conducting
strip, there is about a 5% variation between all three processes at 4?m MTL width. Surprisingly,
as the MTL width is reduced, ?0CS becomes comparable for all widths and almost equal at
0.25?m MTL width. The total MTL geometric factors for all processes are also equal at 0.25 ?m
MTL shown in Fig. 4.7(a). Note, the microstrip geometries are notably different and diagrams are
shown on the right of Fig. 4.7(b) and better seen in Fig. 3.1. This is an important conclusion as
any changes of the partial Q-factor associated to the conductor Qc, or Qi since tan ? is geometry
independent, can be attributed to changes in the intrinsic resistance Rs and dielectric loss tangent
tan ?.
Referring back to STEM summary Table 3.1 on the large discrepancy in ground plane
thickness of 50% in process A, the partial geometric factor of the ground plane ?0GP has a
marginal 10% maximum difference. This is most likely due to the ground plane thickness not
falling below the estimated penetration depth of ? ? 90nm. Referring to Fig. 3.13, the magnetic
fields are penetration the 4 ?m MTL an approximate distance of ?. If process A ground plane
thickness dGP to magnetic penetration depth ratio becomes much less than one dGP/? 1, then
significant change would be seen in ?GP (see Eq. 2.24). Conversely, for a 0.25 ?m MTL, the
magnetic fields (currents) are more concentrated in the conducting strip (Fig. 3.13), so dGP/?
116
1 will have less of an effect, but an effect to keep track of nonetheless.
The next two sections will extract Rs and tan ? more accurately by assuming one of three
models: (i) non-dispersive tan ? and homogeneous superconductor, (ii) non-dispersive tan ? and
in-homogeneous superconductor, and (iii) dispersive tan ? and homogeneous superconductor.
4.4.1 Non-Dispersive tan ? Model
In this subsection, the non-dispersive tan ? model will be assumed throughout unless otherwise
specified using Eq. 3.13-3.16 and assumptions therein.
Taking the MTL geomtric factors solved by HFSS in Fig. 4.7, the intrinsic resistance Rs
from process A is extracted from the fitted slopes A as a function of MTL width in Fig. 4.6.
The red data set extracts Rs assuming a parallel plate model with ?PP ? 12 m? (see dashed
black line in Fig. 4.7(a)). The blue data set extracts Rs assuming a homogeneous MTL where
Rs0GP = Rs0CS and solved ?c0 in Fig. 4.7(a). The black data set (overlapping the blue data set)
extracts Rs0 assuming a non-homogeneous MTL where Rs0GP =6 Rs0CS for MTL widths less
than 4 ?m.
Immediately it can be seen that the parallel plate model (red data set) has approximately
25% high Rs for 0.25 ?m MTL relative to the homogeneous model using HFSS solve geometric
factors. This means that if one was to use the parallel plate analytical model alone, the extracted
Rs would be an upper bound of the actual intrinsic resistance. As MTL width dimensions
shrink below 0.25?m the induced error in extracted intrinsic resistance will also increase. This
makes it difficult to distinguish whether changes in Q-factor or intrinsic resistance Rs are due to
extrinsic effects caused by processing as the conducting strip goes to minimum dimensions or
117
Figure 4.7: Plots showing the calculated geometric factors ? from HFSS using STEM/FIB
measurements. a) The total MTL geometric factor calculated by HFSS from measured MTL
geometries versus MTL width. These will be used to extractRs from the slopeA found in Fig. 4.5
b) The % difference (or error) of ?c0 from measured geometry versus assuming nominal cross
section dimensions. c) The partial conductor geometric factor ?0CS as a function of MTL width
calculated by HFSS. d) The partial ground plane geometric factor ?0GP as a function of MTL
width calculated by HFSS. HFSS simulations and method can be found in Chapter 3. STEM
measurements can be found in Section 3.2. FIB cross section measurements can be found in
Appendix A
118
if its completely due to geometrical effects. Going a step further, if one wants to monitor the
process of the conducting strip alone and the intrinsic resistance of the ground plane Rs0GP is
known with some certainty and can be assumed to not be changing, then one can employ the
homogeneous model with the following equation
Rs0CS = (A?0 ?Rs0GP/?0GP )?0CS (4.7)
where Rs0CS is the intrinsic resistance of the conducting strip, Rs0GP is the intrinsic resistance of
the conducting strip, A is the fit slope from Fig. 4.6 and Eq. 3.15, ?0GP is the partial geometric
factor of the ground plane, ?0CS is the partial geometric factor of the conducting strip, and
?0/2? = 10GHz is the reference frequency.
Here, for the in-homogeneous model, it is assumed at 4 ?m MTL width the ground plane
and conducting strip have equal intrinsic resistances Rs0CS = Rs0GP . The coinciding of the
homogeneous (blue data) and in-homgeneous (black data) for all MTL widths suggests that the
Rs0CS is not significantly larger than the Rs0GP . Although, as the MTL width decreases, the
geometric factor of the conducting strip ?0CS decreases and therefore dominates the conductive
loss. The geometric factor can be looked at as a participation ratio similar to methods shown in the
quantum community to determine which interface dielectric losses participate more [145, 146].
In this case, a large partial geometric factor for the m-th conductor ?0m, its respective intrinsic
resistance Rs will participate less in the overall Q-factor dissipating less power and vice versa.
For 0.25 ?m MTLs, ?0GP ? 10?0CS so the loss in the conducting strip is dominating the
microstrip conductor loss.
Referring now to Fig. 4.9, to compare the power loss in the MTL resonators (MTL interconnects)
119
Figure 4.8: (red) Extracted effective resistance Reff from Qc assuming Eq. D.11 for a parallel
plate. (black) Extracted intrinsic resistance Rs0 from Qc using ?c0 calculated from HFSS at
each MTL width. (blue) Extracted intrinsic resistance of the conducting strip Rs0CS assuming
Rs0CS(4 ?m) = Rs0CS(4 ?m) using calculated ?cGP and ?cGP from HFSS at each MTL width
(see Fig. 3.14). The reference frequency is ?0/2? = 10 GHz. The cross section geometries of
each MTL was measured in TEM (see Section 3.2).
120
for each fabrication process, the homogeneous model with a non-dispersive loss tangent is assumed
from Eq. 3.13-3.16 using the HFSS solved MTL geometric factors ?c0(w). Fig. 4.9(a) and
Fig. 4.9(b) show the extracted intrinsic resistance Rs0 of Nb at reference frequency ?/2? =
10 GHz and assumed frequency independent tan ? of TEOS as a function of MTL width at 4.2
K, respectively. The error bars are?20% as worst case error of 20% variation in geometric factor
across a 150 mm wafer (see Fig. 4.3). The black dashed line is the BCS minimum intrinsic
resistance RminBCS as measured by Benvenuti et al. [64] for thin film Nb samples at 4.2 K and
scaled to 10 GHz using quadratic frequency scaling [62]. Since a 4 ?m MTL resonator can be
approximated as a parallel plate resonator, the Rs0 can be compared to parallel plate resonator
measurements at 4.2 K and 10 GHz. The intrinsic resistances for a 4 ?m MTL width ranges
from Rs0 ? 15 ? 25, and is in good agreement with Rs0 ? 20 measured by Taber [136] using
his parallel plate resonator technique. To get accurate measurements of Rs0, Taber varied the
dielectric spacer thickness in the parallel plate resonator, which requires about 1-2 days worth
of measurements. Here, we extracted Rs0 in a single measurement by relying in the frequency
dependence of Qc and assuming a non-dispersive loss tangent.
The data will now be compared between processes in Fig. 4.9. Process A, Rs0 is relatively
high for all MTL widths relative to process B and C. This was also inferred in Fig. 4.6 and
discussion. There might be a noticeable upturn in Rs0 as the MTL width goes down to 0.25 ?m,
but cant be claimed with the large error bars. Recall, process A has an inverted MTL geometry
where the ground plane is above the microstrip and will be discussed further in the discussion
section (see Fig. 3.2 and summary Table 3.1). Process B, a damascene CMP process, has an
Rs0 ? 15 ?? for MTL widths 0.5 ? 4 ?m and an upturn in Rs0 ? 20 ?? for 0.25 ?m MTL.
Process B falls below the BCS minimum intrinsic resistance measured by Benvenuti et al. for
121
MTL widths 0.5 ? 4 ?m within some error. This is an outstanding result for a damascene CMP
processed RQL interconnect at submicron dimensions and demonstrates its possible to fabricate
Nb submicron lines with the theoretically lowest power dissipation. Based on process C results
for Rs0 and small error bars, a cloisonne? process with a metal etch and dielectric CMP, it can
be argued that Nb 0.25 ?m wires can have Rs0 below what has been measured before. For all
processes, especially process C, a dip (or minimum) in Rs0 at about 1 ?m can be seen. At this
time, a robust model of this minimum at 1 ?m has not been developed and still needs to be
looked into further. Although, it is worth noting that from Fig. 3.13, the magnetic fields begin to
overlap on-top of the conducting strip and a redistribution of magnetic field circulating at the top
of the strip rather than be concentrated at the bottom edges may induce lower loss. Due to the
uncertainty in Rs0 from the assumed worst case error bars, it can be inferred that process C Rs0
does not have a width dependence for MTL widths 0.25? 4 ?m.
Now focusing on the power dissipation in the TEOS dielectric, Fig. 4.9(b) shows all three
processes have approximately the same loss tangent tan ? ? 1.2 ? 0.1 ? 10?3 and virtually no
MTL width dependence. This is not surprising as the deposition parameters of the TEOS were
unchanged for all three processes and tan ? is ideally geometry independent. In spite of this, it
is plausible that the presence of a NbOx layer in process A and B (or lack thereof in process
C) could participate as an additional loss mechanism if the NbOx was a dielectric. Also, the
TEOS RIE etch process in processes A and B could affect tan ? differently relative to the TEOS
CMP process using chemical mechanical polishing in process C (reduce or increase loss). In
retrospect, this is an compelling result that TEOS is not sensitive to small changes in the process
and therefore has a desirably large processing window (see Section 2.4).
122
Figure 4.9: (a) Extracted intrinsic resistance Rs0 at ?0/2? = 10 GHz, and (right) tan ? at 4.2
K as a function of MTL conducting strip width w and fabrication process recipe. The Rs0 from
slope A (see Fig. 4.6) HFSS calculated ?c0 (see Fig. 4.7). The Rs0 error bars are the worst-case
scenario of +/-20% using the parametric corners results in Fig. 3.17. These are conservative error
bars relative to measurement error, fitting error, and linear fitting errors in Fig. 4.5. The black
dashed reference on the left is the known BCS limit of the intrinsic resistance at 4.2 K and 10
GHz. The BCS Rs can be calculated using [179]. The minimum BCS Rs has been measured at
4.2 K for Nb thin films [64]. This may be the first time it has been reported for sub-micron Nb
wires.
123
4.4.2 Homogeneous Superconductor, Dispersive tan ? Model
In the last section, we assumed the loss tangent tan ? was constant in the frequency range
? 1?15GHz. It was found that Process C produces MTLs having an intrinsic resistanceRs near
or below BCS limit of 17 ?? at temperature T=4.2 K and ?0/2? = 10 GHz reference frequency.
It was also found that the frequency dependence for Qi scaled approximately linear. This affords
the opportunity to assume a model for Q?1c and subtract it from Q
?1
i in order to analyze the
frequency dependence of Q?1d ? tan ?. Assuming Q?1c ? ? frequency dependence, Rs0 = 17??
at 10 GHz and 4.2 K, and a nominal geometric factor for 4 ?m MTL ?c0(4 ?m) = 11.5m?, the
superconductor loss slope can be calculated to be A = 2.35 ? 10?14 Hz?1 (see second term in
Eq. 4.8). In Fig. 4.10(a) below, the black solid line is the model for Q?1 = ? Rc0c just described?0 ?c0
and agrees nicely for 4 ?m MTL (purple) with a constant loss tangent tan ? = 9 ? 10?4. This
is more validation the method described here fits the data well. Although, it is interesting to note
the 4 ?m , 3 GHz (purple) data point does not coincide with the black line, suggesting some
non-linearity. To further analyze this, and compare with Kaiser et al. [99], utilizing Eq. 3.13, the
frequency dependent loss tangent tan ?(?) can be inferred from Fig. 4.10(a) by
? Rc0
tan ?(?) = Qi(?)? (4.8)
?0 ?c0
where the subtracted term is the partial Q-factor associated to the conductor Q ? Rc0c = . Using?0 ?c0
Eq. 4.8, Qc was subtracted from the Qi data and can be seen in Fig. 4.10(b). The data measured
here is compared with Kaiser et al. measurements of tan ?(?) for Nb2O5 and SiO2 amorphous
dielectrics (dashed lines), the dielectrics most closely related or involved in our system. For
124
Figure 4.10: a) Inverse internal Q-factor Qi versus frequency for Process C taken from Fig. 4.5.
The black solid line is assuming Rs0 = 17?? at 10 GHz and 4.2 K, and nominal ?c0(4 ?m)
calculated by HFSS b) Qd ? tan ? inferred using Eq. 4.8 and compared to Kaiser et al. data [99].
TEOS fabricated here is non-ideal as the deposition temperature is limited to 150 C, preventing
crystal like growth kinetics and the presence of voiding (see Fig. 3.3(d) feature 5). Therefore, it
is expected the loss tangent of TEOS is 1-2 orders of magnitude higher than Kaiser et al. SiO2.
The Process C inferred tan ?(?) for all MTL widths in Fig. 4.10(b) agree well and fall in
between Nb2O5 and SiO2 loss tangents measured by Kaiser et al.. From 3 to 10 GHz, the inferred
tan ?(?) decreases by a factor of 2 for 0.25?m MTL, and only decreases by about 10% for the 4
?m MTL. In comparison, the Qc increases by a factor of 2 from 3 to 10 GHz for all MTL widths.
Using Eq. 2.44, the loss tangent data is fit to a power law and plotted as a function of MTL
width in Fig. 4.11(a). The fit parameter bd agrees well with Kaiser for 0.25 ? 2 ?m MTLs in
the regime of 0.5-0.75. The 4 ?m MTL has a relatively higher bd = 0.86, giving a loss tangent
frequency dependence of tan ? ? ??0.14.
It can be argued that the 4 ?m MTL, at low frequencies ?, is the most sensitive to tan ?, as
?c0 is the largest at this width (see Fig. 4.7 and Eq. 3.13). This could explain the non-linearity in
Fig. 4.10(a) mentioned eariler, and can also be seen in Fig. 4.10(b) (purple data). Additionally,
125
Figure 4.11: Results plotted from Fig. 4.10 showing the following as a function of MTL width:
a) fitted exponent bd from Eq. 3.12 and 1 standard deviation error and b) fitted loss tangent tan ?
at 10 GHz. Kaiser et al. measured bd ? 0.6 and tan ? ? 3? 10?4 for SiO2 at 10 GHz.
the first mode for 4 ?m MTL falls on top of all other MTL widths, but then deviates at higher
frequencies.
From the dispersive fits in Fig. 4.10(b), the loss tangent tan ? at ?/2? = 10GHz reference
frequency as a function of MTL width is plotted in Fig. 4.11(b). In comparison to extracted non-
dispersive loss tangents in Fig. 4.9, the dispersive loss tangent is approximately 20-60% lower at
10 GHz.
The agreement with Kaiser et al. of tan ? magnitude and power dependence as a function
of frequency for similar dielectrics is a good first indication that there is dispersion the TEOS
dielectrics measured here at 4.2 K. Yet, we had to assume an Rs0 and geometric factor ?c0.
To scrutinize the dispersion relationship further and demonstrate the superconductor and
dielectric loss deconvolution method described here, I used Eq. 3.12 to fit the Qi(?) data to 3
free fitting parameters shown in Fig. 4.12. Except the process A 1 ?m MTL, all fits (dashed
lines) fall in between the error bars, which are one standard deviation of Qi variation across the
wafer.
126
An interesting visual result can be seen in Fig. 4.12(e) for the 4?mMTL, which is the most
sensitive to the loss tangent, shows process B is concave down and process A and C are concave
up. In other light, process B and process C 4 ?m MTLs have almost the same mode 1 and mode
4 Q?1i , but modes 2 and 3 in process B are higher. This discrepancy and false concave down
character in process B may just be due to the error bar for mode 4 in process B. Other than the
fits for 1 ?m MTLs and process B 4 ?m MTL, all of the fits show a minimum in Qi somewhere
below 2 GHz. If tan ? is expected to decrease as a function of frequency and the inverse partial
conductor Q-factor Q?1c is proportional to frequency, then a minimum in Q
?1
i is expected. This
minimum in the internal Q-factor Qmini (?min) is a useful parameter for application, as it is the
optimum frequency to operate RQL interconnects in terms of minimum power dissipation. Using
tan ?(?) for SiO2 measured by Kaiser et al. (Fig. 4.10(b)), and assuming minimum Qc(?) with
RminBCS = 17?? and ?c0 = 11.5m? (Fig. 4.10(a)), the frequency for mimum power dissipation
can be estimated to be ?min/2? ? 1GHz. In a future MTL design, the fundamental mode should
be decreased to ideally below 1 GHz to see if there is a minimum in Q?1i .
The dispersive fitting parameters Ad, Bd, and bd as a function of process and MTL width
are shown in Fig. 4.13(d-f), where A = Ad is the slope associated to the conductor loss, B = Bd
is ?non-dispersive? tan ? if b = 1, and b = bd is the dispersive power exponent (see Eq. 3.12).
The non-dispersive fitting parameters A = And, B = Bnd, and b = bnd are shown in the first row
for reference. For a 3 parameter fit on only data points, this system is nearly over-determined and
could explain a lot of the scatter in Fig. 4.13(d-f) and Fig. 4.13. Similarly said before, other than
1 ?m MTLs and process B 4 ?m MTL, all dispersive exponents are less than 1 b = bd < 1. This
is a notable result, as a 0.5 < b < 1 fitted from 3 fitting parameters without assuming loss in the
superconductor is in agreement in Kaiser et al.. Thus, to first approximations, this demonstration
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Figure 4.12: Internal Q-factor Qi(?) for Processes A (red), B (blue), and C (green) for MTL
widths a) 0.25 ?m (crosses), b) 0.5 ?m (diamonds), c) 1 ?m (triangles), d) 2 ?m (squares), e)
4 ?m (circles) and f) all plotted together. The 3-parameter dispersive tan ? fits (dahsed lines) use
Eq. 3.12. Diagrams of the MTL geometry for each process are shown in a) for convenience.
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Figure 4.13: The top row are fitting parameters for non-dispersive tan ? model assuming b = 1
(linear fits) shown in Fig. 4.5 and using Eq. 3.12. Width dependence of linear a) slope parameter
A due to conductive loss, b) assumed non-dispersive loss tangent B = tan ? where c) b = 1.
The second row are fitting parameters for dispersive tan ? model showing width dependence
of free fitting parameters: d) slope A proportional to conductive loss e) pre-exponent term B
f) and exponent term b. Diagrams of the MTL geometry for each process are shown in d) for
convenience.
serves as a validation of the deconvolution method described here because 1) the Rs results are
comparable to measured values found in literature, and 2) the tan ? dispersive relationship and
values agree with previous findings.
Aside from the 1?mMTLs and process B 4?mMTL, extractedRs0 and tan ? assuming the
dispersive model in Fig. 4.14 general agrees with the non-dispersive extraction method in Fig. 4.9.
By general, i mean the Rs0 magnitude is still in between 15-35 ?? using the dispersive model
and tan ? is within 50% of values relative to non-dispersive fits. This is a very important result,
regarding the Rs0 values falling below BCS limit in Fig. 4.9(a) being suspect RBCS = 17 ?? <
Rs0. The minimal change in Rs0 values using dispersive fits here only marginally increase and
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Figure 4.14: Extracted a) Intrinsic resistanceRs and b) tan ? versus MTL width using dispersive
tan ? 3-parameter fit model
support results in Fig. 4.9(a). MTL width dependence is not discernible in Fig. 4.14 due to the
noise, which is most likely due to the dispersive fitting model being close to over-determined
with 3 fitting parameters and only 4 data points.
Ultimately, exploring the dispersive tan ? model and fitting the data as a function of MTL
width and different processes, varying in loss significantly as a function of frequency, offered the
proposed gleaned information:
(1) TEOS tan ?(?) extracted from process C Qi data using the simple Q?1c subtractive method
agrees with Kaiser et al. in both magnitude and frequency dependence for SiO2
(2) TEOS tan ?(?) extracted from a 3-parameter fit for the majority of data (process and MTL
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width) also agrees with Kaiser et al. in both magnitude and frequency dependence for SiO2
(3) Nb Rs0 extracted from a dispersive 3-parameter fit give support of values measured below
the results measured by Benvenuti et al. [64]
4.5 Discussion
The previous two sections have demonstrated the method to deconvolve the losses by
assuming different models. In this discussion, we will focus on the results from Fig. 4.9, where
a non-dispersive tan ?, homogeneous (Rs0GP = Rs0CS) MTL model was assumed. Models
to explain the intrinsic resistance Rs difference as a function of (i) width dependence) and (ii)
fabrication process (A,B,C) will be proposed relying on the morphology and chemical analysis
in Section 3.2.
Up-turn of Rs at 0.25 ?m
In reference to Fig. 3.3(a) and (c), the damascene process induces two grain orientations,
sidewall horizontal grains and vertical trench grains, meeting along a diagonal line with approximate
angle of 30 degrees. Here, we will call this the sidewall grain intersection. At these intersection,
voiding (or lower metal concentrations) occur. The percentage of sidewall horizontal grain on
both sides of the microstrip remain fairly constant in size with maximum length of 100 nm at
the top of the microstrip. As the microstrip width decreases, the crossectional area fraction of
the sidewall grains increases. Here, we will call this the sidewall grain fraction. For a 1 ?m
microstrip, sidewall grain fraction is only about 20%, whereas for a 0.25 ?m microstrip the
sidewall grain fraction is approximately 50%. This is a model that could contribute to the upturn
in Rs of about 20-30% for 0.25 ?m MTL relative to 4 ?m for process A and B.
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Another model that could attribute to the upturn in Rs is the presence of voids, or areas of
lower Nb concentration (see Fig. 3.3). Voids become a more significant fraction as microstrip
width decreases. The effect of voiding on microwave surface resistance has not been studied
carefully. Although, there has been extensive studies on how Nb vacancies can facilitate Nb
hydride precipitate formation [180]. Consequently, the Nb voids could be acting as hydride
formation sites. In addition, Nb voids could have an effect on the electron mean free path, and
in turn increasing both the magnetic penetration depth and microwave surface resistance (see
Eqs. 2.20 and 2.8).
It is worth mentioning a model of the NbOx layer as the mechanism causing the upturn in
Rs at 0.25 ?m. In reference to Fig. 3.13, the fringing magnet fields do not start start to sample
(overlap) a significant percentage of the top of the microstrip until below 1 ?m. In process B,
the exact geometry shown in Fig. 3.13, the upturn in Rs does not happen until 0.25 ?m. It is
conceivable the Rs upturn here is due to NbOx, but would have expected a more gradual increase
in Rs. This argument is more unlikely when considering Process A, as the fields are sampling the
NbOx layer for all MTL widths of equal fraction, and is more likely to be a different mechanism.
An upturn in Rs in Process C is not visible.
Rs versus MTL width offset
In process A, due to the orientation of the GP relative to the fill morphology (sidewall grain
orientation), the magnetic field (current density) is more concentrated at the oxide layer. The
current is also concentrated where voiding occurs. Whereas in process B, the damscened surface
is not at a location of highest current density. In ?morphology section? we also found from
chemical mapping that the oxide layer is closer to NbO or Nb2O3, which is a bulk superconductor
with Tc approximately 6 K, but at 5 nm thickness is most likely a normal metal. Therefore, the
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oxide location relative to the current concentration is a model that can explain the Rs offset by
approximately a factor of 2 between process A and B. Process C does not have a detectable oxide
at the interface of highest current concentration, suggesting the offset of approximately 40%
between process B and C 0.25 ?m microstrip. Surprisingly, due to the etch undercut process, the
rough and irregular surface of the strip did induce Rs larger than damascene processes A and B.
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4.6 Summary
In summary, microstrip transmission line (MTL) resonators were designed into an RQL
fabrication process where the Q-factors associated to conductor and dielectric loss are comparable
Qc ? Qd ? 1000. The comparable Q-factors are not compatible with established techniques to
analyze the dissipative loss coming from the superconductor and dielectric separately. Using
the methods and HFSS numerical modeling procedures described in Chapter 3, the intrinsic
resistance Rs of Nb and loss tangent tan ? of TEOS were extracted from measured intrinsic
Q-factors as a function of frequency Qi(?) utilizing their unique dispersive relationships. The
uncertainty in extracted Rs0 was used from the HFSS sensitivity analysis in Section 3.7 with a
worst case error of ?20%. The overall uncertainty is dictated by the fabrication geometry where
the microstrip and dielectric separation thicknesses have the most significant impact on geometric
factor variation. The standard deviation from measured values of the internal Q-factor Qi across
a wafer was much less than ?20%, suggesting a well controlled fabrication process where both
the resonator geometry and the intrinsic surface resistances are not varying significantly across
the wafer. Different dispersive models were proposed and analyzed to demonstrate and give
validation to the Rs and tan ? extraction method from a single measurement at 4.2 K. Three
fabrication processes were analyzed using this method, and found that the cloisonne? process C
yields Rs0 of Nb near theorical minimums down to 0.25 ?m MTL widths. Near equivalent Rs
values can be achieved in the damscene process B down to 0.25 ?m MTL widths. A NbOx
layers in MTLs can induce an extrinsic increase in Rs0, and its location relative to current density
concentration inside the conducting strip can exacerbate this issue. An Al capping layer has been
shown to prevent NbOx formation at the conducting strip interface, and suggests this aided in the
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minimum and near width independent Rs0 measured in cloisonne? process C. For all fabrication
processes and MTL widths measured here, TEOS loss tangent tan ? is width independent, and
the measured frequency dependence agrees with previous work on similar amorphous dielectrics.
The materials characterization done here informs the implications on power dissipation
performance of RQL interconnect operating at 4.2 K and 10 GHz. The TEOS dissipative loss
is geometry independent, and only dependent upon fabrication. If the TEOS could be deposited
at temperatures higher than 150 C, then it is possible to reduce its dissipative loss 1-2 orders
of magnitude (Qd ? 104 ? 105). The Nb dissipative loss has arguably reached its theoretical
minimum (Qc ? 3 ? 4 ? 103). The current minimum energy efficiency is ? ? 0.3 of RQL
resonator clocks relative to the power dissipated by the RQL logic ? = Plogic/(Plogic + Pclock).
The efficiency is proportional to the resonator internal Q-factor by ? ? 1/(1 + Q?1i ), where
the power dissipated in the clock is proportional to the Q-factor by Pdiss ? Q?1i having nominal
valueQi = 250 at 4.2 K and 10 GHz. Taking a resonator composed of MTL interconnect down to
0.25?mwidths fabricated by process C, the efficiency would increase to ? ? 0.6. If the dielectric
loss could be reduced by a factor of 10 and 100, the effieciency would increase to ? ? 0.83 and
? 0.86, respectively. This is an exceptional decrease in power dissipation relative to the logic
at maximum operation (i.e. all logic units are active in the circuit), but it is important to note
that the efficiency is at its limit and cannot be pushed further unless materials other than Nb and
TEOS are used or changes in the transmission lines geometries are allowable.
At the current RQL clock efficiencies of ? ? 0.3, it can be estimated that the total power
consumption of RQL at 4.2 K would be a factor of 90 less than equivalent CMOS circuits, which
includes refrigeration costs [4]. For ? ? 0.86, RQL would consume approximately 260 times
less than equivalent CMOS circuits. In the next chapter, temperature will be used as a knob to
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explore further reduction in power dissipation in RQL interconnects. The determination of power
dissipation limits and an optimum temperature of operation will also be investigated.
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Chapter 5: Temperature Dependent Characterization of Microwave Loss in RQL
Interconnects
5.1 Overview
In the last chapter, I measured the internal Q-factor of microstrip transmission lines (MTLs)
and delconvolved microwave losses of the superconductor and dielectric at 4.2 K. It was found
that the theoretical minimum of Nb BCS intrinsic resistanceRs ? 17?? was reached for Process
C down to 0.25 ?m widths at 4.2 K. The TEOS dielectric loss tangent was found to be tan ? ?
1 ? 10?3 at 4.2 K and is MTL geometry independent. It is known both Rs and tan ? decrease
with decreasing temperature. Hence, to maximize energy efficiency in RQL interconnects, the
temperature could be lowered to reduce the RF losses. Although, refrigeration costs increase as
the operation temperature is reduced. It takes approximately 3 times more power to reduce the
operating temperature from 4.2 to 2 K [181].
As mentioned in the Chapter 2, the microwave loss of superconductors and dielectrics have
different scalings with temperature. As temperature is decreased below 4.2 K, The bulk and
thin film Nb Rs temperature and frequency scaling is well known as temperature is decreased
below 4.2 K. For amorphous dielectrics like TEOS, the temperature (below 4 K) and frequency
dependence strongly depend on the input power (electric field strength) and dominant mechanisms
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of loss (e.g. classical dielectric relaxation, TLS tunneling relaxation, and TLS tunneling absorption).
Furthermore, the dominant mechanism of dielectric loss in the dielectric can change rapidly
as temperature is decreased. Up until the 1990s, there had been limited microwave dielectric
measurements in the 1.5-4.2 K temperature regime, as most SCE communities operate at 4.2 K
(SFQ) or at mK temperatures (Quantum Computing).
In this chapter, the dispersive loss deconvolution (DLD) method will be applied to temperature
dependent measurements. In the 1.5-4 K temperature range, the extracted temperature dependence
ofR (T ) ? e?1/Ts and tan ?(T ) ? T 3 agree with expected behavior. The measurement sensitivity
of this method will be discussed. Specifically at temperatures below 3 K, the frequency dependence
of the Qi is anomalous with non-monotic behavior increasing in magnitude as temperature is
decreased. With only 4 modes in MTLs, it is difficult to draw conclusions for the Qi frequency
dependence. This warranted the design and measurement of multiplexed resonators in the subsequent
section using process C to do further microwave spectroscopy from 1.5 - 4 K. In the temperature
range 1.5-3 K, it is suggested the frequency and temperature dependence of Q (T, ?)?1i ?
tan ?(T, ?) is due to two-level system loss, and is supported by changes in the dielectric constant
(MTL phase velocity).
At the end of this chapter, a discussion of the temperature dependent superconductor and
dielectric microwave loss, and concluding statements will be made on how this affects performance
and power efficiency of resonator clocks.
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5.2 Temperature Dependent Resonator Measurements
Temperature dependent measurements in this section were performed with the same test
fixture described in Chapter 4, but retrofitted into a system similar to the Dry ICE 1.5 K Cryostat
[182]. The cryostat base temperature is approximately 1.4 K and has temperature cryostat temperature
stability of ?10mK from 1.4 to 10 K. It is important to note, this is a dry system and cooling is
accomplished by means of thermal conduction. See Appendix C for more information.
All temperature dependent measurements in this section were performed on MTL widths
0.25 ? 4 ?m from process B. An example of the temperature dependent data for 4 ?m MTL
can be found below in Fig. 5.1 Four resonant modes from 3-12 GHz were analyzed extracting
Qi(T ), Qe(T ), fn(T ) using methods from Khalil et al. [177] (see sections 4.2, 4.3).
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Figure 5.1: Temperature dependent measurements of 4?m MTL from process B using Dry ICE
1.5 K Cryostat equivalent [182] with ?10 mK temperature control. S-Parameter measurements
were with an input power of Pin = ?20 dBm. It will be shown this input power is too high
and is shown only as an example that no non-linearities or bifurcation of resonant peak was
found. Although, this relative high input power impacted the extracted internal Q-factors Qi. a)
S21 showing the temperature dependence of 4 modes were captured. b) Zoom in of mode one
showing the resonant frequency shift and widening of 3 dB width as temperature is increased. c)
The complex S?121 , which were fitted using [177]. d) The phase where with resonant frequency
?/2? at zero phase.
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The nominal input power from the vector network analyzer (VNA) to the feedline is Pin =
?20 dBm = 10 ?W at 4.2 K. It will be shown that -20 dBm is a relatively high power at
1.76 K. The effective power delivered to the resonator is dependent upon (i) the attenuation
from the VNA to the feed-line on chip and (ii) the resonator coupling factor g. The coupling
factor is dependent upon the resonator internal Q-factor Qi and external (coupling) Q-factor Qe
by g = Qi/Qe (see Eq. 4.2). Since Q-factors are temperature dependent, the coupling factor
is also temperature dependent g(T ), and therefore the effective input power to the resonator is
temperature dependent.
If it is desired to do temperature dependent measurements, the input power needs to be
taken into consideration to prevent non-linear effects. Assuming the VNA is calibrated up to
the shunt connection node, the incident power can be expressed in the following way by energy
conservation [176]
Pin = Pr + Pt + Pd (5.1)
where Pin is the incident power in Watts at the input of the resonator after cable attenuation (at
the reference node in Fig. 5.2), Pr is the reflected power to the VNA, Pt is the transmitted power,
and Pd is the power dissipated by the resonator.
The transmitted power Pt for a reactively coupled MTL can be calculated by [176]
Pt = Pin|S 221| (5.2)
where Pin is the incident power in Watts at the input of the resonator after cable attenuation, |S21|
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Figure 5.2: a) Representation diagram of the open ended MTL resonator. The reference point for
the power conservation Eq. 5.1 is the node between the two 50 ? lines and the coupling capacitor
Cc. b) Block diagram showing the capacitively coupled ?/2 microstrip transmission line shown
in a) is effectively an impedance inverter where Y is the shunt admittance. The node pointed out
in a) is the same point in b) in reference to the power conservation Eq. 5.1
is the transmission coefficient magnitude related to the insertion loss IL in dB by |S | = 10IL/2021
at the resonant frequency ?/2?. In this case of a reactively coupled microstrip resonator, the IL
is -6 dB for critical coupling g ? 1. Note, the typical insertion loss off resonance ranges from
-15 to -5 dB due to cable loss and impedance mismatch. This measurement is self calibrated in
the sense that |S21| is the size of the resonant dip relative to the baseline loss. For example, in
Fig. 5.3(b) the 3.3 GHz resonant mode has an insertion loss of?8.5dB at 1.76 K. The transmitted
power to the resonator is Pt ? 14 ?W for a -20 dBm input power. Assuming constant material
properties, the transmitted power can be tuned by varying the coupling capacitor Cc, temperature,
and geometry of the resonator (refer to chapter 4).
From the energy conservation law for microwave circuits [10, 176], the dissipated power
at resonance can be found regardless of the g value and calibration. The dissipated power can be
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calculated by [10, 176]
Pd = 2Pin(|S21| ? |S21|2) (5.3)
where Pd is the dissipated power in Watts. In Fig. 5.1(b) at a nominal Pin ? 10 ?W , IL ?
?8.5 dB at 1.76 K yields a dissipated power of Pd ? 50 ?W .
The circulating (stored) power in the resonator can now be calculated by
Pst = QiPd = 2QiPin(|S21| ? |S 221| ) (5.4)
where Qi is the internal Q-factor at resonant frequency ?/2?. Fitting the 3.3 GHz resonance at
1.76 K from Fig. 5.3(b) yields a Q 3i ? 1.3? 10 . The stored power is then Pst ? 18mW .
A superconductor becomes non-linear when RF current density comes close to the temperature
dependent critical current density Jc(T ). In fabricated superconducting electronic circuits, the
minimum critical current density at T ? 4.2 K of Nb 0.35 ? 0.5 ?m metal lines with 200 nm
thickness are typically Jc ? 20MA/cm2 [32]. Considering a worst case for a 4?m and 0.25?m,
at nominal 200 nm thicknesses, the MTL minimum critical currents are approximately 160 mA
and 10mA, respectively.
The peak RF current in the resonator Ipeak can be compared to the expected critical current
Ic to estimate onset of nonlinearity. It is also important to keep track of the peak voltage Vpeak
to monitor electromagnetic strengths in both the superconductor and dielectric. The temperature
dependent magnitude of the peak current Ipeak(T ) and voltage Vpeak(T ) can be calculated by
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[183]
?
I (T ) = 16 Pinpeak Pd(T )Qi(T ) (5.5)n? Z0
Vpeak(T ) = Z0Ipeak(T ) (5.6)
where Pin is the input power, n is the mode index of the resonance, Z0 is the characteristic
impedance of the MTL assumed to be constant at T ? 4.2K, and Qi(T ) and Pd(T ) are the
temperature dependent internal Q-factor and dissipated power, respectively. The rms peak current
?
is Irms = 2Ipeak. Referring back to Fig. 5.3(b), assuming a characteristic impedance Z0 ? 8 ?
for 4 ?m MTL, the peak current is Ipeak ? 190 mA. This peak current of 190 mA exceeds the
estimated Ic ? 160mA.
For completeness, a 4?m MTL from Process B was measured at two input powers Pin of
-20 dBm and -35 dBm and the temperature dependent results can be found below in Fig. 5.3. At
1.7 - 1.8 K, a clear difference in Qi as a function of frequency can be seen. The 6 and 9 GHz
modes at T < 1.8K show almost equivalent Q?1i for -20 and -35 dBm, whereas the 3 GHz has a
much higher loss with Pin = ?20dBm. Surprisingly, the 12 GHz mode for the lower -35 dBm
power shows a factor of two increase in Q?1i relative to the 12 GHz mode at -20 dBm. This will
be revisited.
The non-monotonic behavior in Fig. 5.3 gives motivation to analyze the presence of non-
linear effects. In Fig. 5.4, the peak current Ipeak was calculated from data in Fig. 5.3. The peak
current Ipeak in the 4 ?m was calculated from Eqs. 5.5, 5.3 using measured |S21| and Qi at each
resonance and temperature T . It can be immediately seen that mode 1 (yellow) Ipeak 0.5Ic,min
for Pin = ?20dBm at the lowest temperatures. Mode 4 has peak current Ipeak 0.15Ic,min. For
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Figure 5.3: Extracted Qi versus resonant frequency (mode index) as a function of temperature
for a) Pin = ?20 dBm and b) Pin = ?35 dBm. No visual bifurcation of the resonant peaks
were witnessed. Although, particularly at 1.7-1.8 K, there is a clear different in extracted Qi as a
function of mode number
an Pin = ?35dBm, the peak current drops to a factor of 10 and 20 lower for mode 1 and
mode 4, respectively, relative to Ic,min. Note, looking vertically at each temperature, the peak
currents in the resonator has an approximate monotonic dependence with mode index n, where
n=1 is the 3.27 GHz mode. This is as expected and should go as I ?1/2peak ? n . Although, for
Pin = ?35 dBm, the mode 3 peak current relative to mode 2 decreases by Irf,3 ? 1.7Irf,2 at
1.7 K, where a factor of 1.2 would be expected from mode dependence alone. Whereas the mode
4 decrease relative to mode 2 at 4.2 K is as expected with Irf,3 ? 1.7Irf,2. Referring back to
Fig. 5.3, the larger decrease in Ipeak,3 for mode 3 seems to be coupled with a factor of 2 increase
in Q?1i,3 relative to mode 2 at 1.7 K. With mode 2 and 3 peak currents more than an order of
magnitude less than the critical current Ipeak,3 < Ipeak,2  Ic for Pin = ?35 dBm, the non-
monotonic mode dependence of Q?1i in Fig. 5.3(b) suggests it is related to the voltage amplitude
or some other mechanism This will be revisited towards the end of the section.
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Figure 5.4: Calculated Peak currents in 4?m MTL as a function of temperature for 4 resonances
at input powers (a) -20 dBm and (b) -35 dBm. The peak currents were calculated using Eqs. 5.5,
5.3, and measured transmission coefficient magnitude |S21| and Qi for resonance at temperature
T . The characteristic impedance was calculated to be Z0 = 8.2 ? from HFSS measurements and
known cross-sectional geometry. The red horizontal lines are a referenced expected minimum
Ic ? 160mA estimated from Jc ? 20MA/cm2 [32].
It has now been established that the peak currents for all modes in 4 ?m MTL resonator
at Pin = ?35dBm are much less than the critical current Ipeak,n  Ic for temperatures T =
1.5 ? 4.2 K. Fig. 5.5 below is a plot of the inverse internal Q-factor Qi at 1.7 K for each mode
index and MTLs 0.25 - 4 ?m from process B. All mode 4 Q-factors Q?1i,4 have approximately the
same magnitude for all MTL widths. Surprisingly, all mode 2 Q?1i,2 have the same magnitude.
Conversely, modes 1 and 3 vary significantly by as much as a factor of 5. The same peak current
and voltage analysis will be performed to determine if this is driven by an abnormal stimulus
dependence.
Fig. 5.6 shows the Qi as a function of calculated peak currents Ipeak for process B 0.25 -
4 ?m MTLs at 1.7 (a) and 4.2 K (b). At 4.2 K, all MTLs except 0.25 ?m yield peak currents
at least a factor of two less than Ic. This suggests that 0.25 ?m MTLs should be measured with
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Figure 5.5: Qi as a function of mode index at 1.7 K and Pin = ?35 dBm for MTL widths
0.25? 4?m. Using fitting methods described reference [177], complex S21 fitting errors for Qi,
Qe, and f0 were less than 5% for all data shown here. Careful analysis of the raw data was
performed, and no non-linear effects (e.g. resonant peak bifurcation) were found.
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input powers Pin ? 35 dBm. At 1.7 K, the 4 and 2 ?m MTLs have peak currents much less than
the estimated critical current Ipeak,1, Ipeak,2  Ic. The 0.5 ?m MTL peak currents come close
and even cross over the estimated critical current, and could explain the decrease of the Q-factor
for the fourth mode Q?1i,0.5?m ? 3 ? 10?3 and peak current Ipeak ? 17 mA Similarly, for the 1
?m MTL a similar increase in Q?1i,1?m can be seen at the same Ipeak ? 17 mA. This could also
be explained by 1 ?m Ipeak within < 2Ic of its critical current, inducing an additional loss due
to some weak link at the peak current nodes in the resonator. Conversely, the 0.25 ?m MTL
resonant modes have peak currents residing up to 2Ic, but the Qi increases at increasing peak
currents.
In summary, the high peak RF currents for all MTL resonator widths cannot explain the
non-monotonic mode (frequency) dependence ofQi alone. As briefly suggested, the peak currents
occur at specific locations along the resonator, and could be exacerbating weak-links or material
defects in the Nb wires or ground plane. This also goes for material defects in the TEOS
dielectric. In light of this, the peak current/voltage technique and analysis could be used to
accentuate nonlinear properties of the superconductor or dielectric losses due to spatially varying
extrinsic material defects. Furthermore, the peak current analysis proved useful to determine an
input power of -20 and -35 dBm was was too high to analyze the intrinsic superconductor and
dielectric RF losses as a function of temperature and frequency with the current MTL resonator
design. Regretfully so, these large input powers relative to the high coupling of the current
MTL design was known, and measuring at lower powers posed difficulty in signal-to-noise in the
current test fixture. This facilitated the design of a multi-plexed MTL design to further analyze
RF losses in RQL interconnects at temperature T = 1.7 ? 4.2 K, and is the topic of the next
section.
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Figure 5.6: Calculated peak currents Ipeak at (a) 1.7 K and (b) 4.2 K for MTL widths 0.25?4?m
and mode indices 1-4 with respectiveQi(?n). The vertical lines are the assumed minimum critical
currents Ic based on a critical current density Jc = 20MA/cm2 of superconducting Nb wires at
4.2 K [32]. The input power was Pin = ?35 dBm for all measurements and data shown here.
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5.3 Multiplexed Microstrip Transmission Line (MUX-MTL) Resonators
As seen in the last section, there was an anomalous mode dependence of the internal Q-
factor Qi. Especially at the lowest measured temperatures T < 2 K. In this section, I will
introduce a different resonator design with more resonances from 4-10 GHz to investigate the
frequency and temperature dependence with better certainty.
Rather than measuring a single ?/2 resonator, the new design has many microstrip transmission
line resonators that are capacitively coupled to a single feedline. This multi-plexed MTL resonator
design will be termed MUX-MTL. Multiplexing resonators designs are typically used to measure
many sensors at once, e.g. SQUID multiplexing [184, 185]. Here, the MUX-MTL is designed
to measure the intrinsic RF losses in the Nb and TEOS. Conveniently, the frequency dependence
of the material properties can be measured since each resonator needs to have a unique resonant
frequency.
A layout of the MUX-MTL resonators can be found in Fig. 5.7. Here, the MUX-MTL
highlighted in Fig. 5.7c will measured. These MTLs are coupled with coupling capacitors having
the lowest value of Cc ? 25 fF . This yields a resonator coupling low enough to ensure (i) a high
enough signal-to-noise (resonant peak size) at an input power of ?35 dBm and (ii) transmitted
powers to the resonator low enough to avoid peak current/voltage effects discussed in the last
section. Fig. 5.7d shows a zoom in of a single MTL capacitively coupled (orange) to the MUX
feedline (blue horizontal line running across the bottom). The feedline and MTL have a line
width of w = 0.75 ?m. As opposed to MTLs in Chapter 4, each MTL here coupled to the
MUX feedline are ?/4 resonators grounded at the end. A 7 mm ?/4 MTL has a fundamental
mode of approximately 4 GHz. The grounding areas are enlarged to allow for current spreading.
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Additionally, the feedline and ?/4 MTL have TEOS dielectric thicknesses of 500 nm with
nominal 200 nm Nb ground plane and conductor strip thicknesses. A single MUX-MTL will
be analyzed here and was fabricated using process C. The pertinent resonator information and
results are summarized in Table 5.2.
Figure 5.7: Physical design of a MUX-MTL resonator showing a) a diagram and the b) physical
layout of a single MUX-MTL resonator. There are 21 ?/4 MTL resonators on a single MUX
feedline with resonator lengths ranging from 3-7 mm with a 0.2 mm step. There are c) 8 MUX-
Resonators designed on a single 5x5 mm chip. d) A zoom in of the first ?/4 MTL resonator
connected to the MUX-Feedline coupled through a coupling capacitor with nominal capacitance
of Cc = 25fF . The MTL and feedline width is 0.75 ?m. The end of the MTL is shorted to GND
with a wire fan out and many 0.25 ?m vias to prevent the wire going normal at the current node.
Via walls surround the feedline and MTL for isolation.
The MUX-MTL resonators were measured in the same way MTLs were measured and
analyzed in Section 5.2. The resulting temperature dependent S-parameter measurements can be
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Figure 5.8: a) The measured S21 versus frequency as a function of temperature b) Cross-section
diagram of the MUX-MTL geometry. The 21 resonances pertain are the fundamental modes of
?/4 MTLs with lengths ranging from 4-7 mm increasing in 0.2 mm increments.
found in fig. 5.8(a), and (b) is a cross section diagram of the MUX-MTL resonator.
All resonances in Fig. 5.8 were analyzed using an automated python script. The internal
Q-factors Qi, external (coupling) Q-factors Qe, and resonant frequencies f0 were extracted from
a complex circle fitting script following the procedure in reference [177]
5.4 Using the Dispersive Deconvolution Method to Simultaneously Measure
tan ?(T ) and Rs(T )
In this section, I will apply the dispersive loss deconvolution (DLD) method presented in
Chapter 3 and applied in Chapter 4 to extract the temperature dependent loss tangent and intrinsic
resistance simultaneously.
Fig. 5.9 shows the resulting fitted inverse internal Q-factorsQ?1 of MUX-MTL ?/4 resonators
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Figure 5.9: Frequency dependence of the inverse internal Q-factor Qi at successive temperatures
from 1.6-4.5 K at a constant input power of Pin = ?35 dBm. Complex S21 fitting procedure
was performed on data from Fig. 5.8. The 0.75 ?m MUX-MTLs were fabricated using process
C described in Chapters 3 and 4.
from S-Parameter measurements in Fig. 5.8. At 1.6 K, a non-monotonic frequency dependence
of Q?1i can be seen, and will be discussed in the next section.
It is important to first discuss the measurement sensitivity of the DLD method before fitting
the data in Fig. 5.9. For parallel plate resonator measurements, it was found by Taber et al. that
the measurement sensitivity of the intrinsic resistance was Rs0 ? 5 ?? [136], and he needed
to vary the resonator geometric factor to get accurate measurements of Rs0. To estimate the
DLD measurement sensitivity of Rs0 and tan ? using MTL resonators, Eqs. 3.12 and 3.13 from
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Section 3.3 can be re-written as follows
Rs0(T )
Q?1i,nd(T ) = (?/?0) + tan ?0(T ) (5.7)?c0
?1 Rs0(T )Qi,d (T ) = (?/?0) + tan ?0(T )(?/?0)
b?1 (5.8)
?c0
Rlims0 (T ) ? ?c0 tan ?0(T ) (5.9)
where ? is the measurement frequency, ?c0 is partial geometric factor of the MTL, b is the free
fitting power exponent assuming a dispersive loss tangent, and Rs0(T ) and tan ?0(T ) are the
intrinsic resistance and loss tangent at reference frequency ?0 at temperature T , respectively.
Eq. 5.7 can be used to fit data when a non-dispersive loss tangent can be assumed. Eq. 5.8 is
used in this work to fit data assuming a dispersive loss tangent where 0 < b < 1. Eq. 5.9 is an
estimate of R = Rlims0 s0 measurement sensitivity at reference frequency ?0 from reference [136].
Similarly, the approximate measurement sensitivity for the loss tangent is tan ?lim0 ? Rs0/?c0
In other words, when the Rs0 and tan ? are comparable for a given resonator geometry, it is
important to utilize the frequency dependence, temperature dependence, or some other means to
deconvolve the two loss mechanisms. The latter case was demonstrated in the previous chapter.
In this section, S-parameter measurements are performed down to 1.6 K, and it is expected
the Nb Rs0 will exponentially decrease Rs0 ? (1/T )exp(?1/T ) (see Eq. 2.27). Assuming TLS
relaxation proccesses dominate the dielectric loss below 4.2 K, the TEOS tan ?0 is expected to
decreases with ? T 3 dependence (see Section 2.3). At 4.2 K, the TEOS loss tangent is tan ? ?
1 ? 10?3. The geometric factor for the 0.75?m MUX-MTL is approximately 17.3 m?. The
approximate sensitivity ofRlims0 at ?0 and 4.2 K is then 17 ??. Taking advantage of the conductor
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Q-factor frequency dependence Q?1c ? ?, the intrinsic resistance can still be deconvolved at 4.2
K using the DLD method, as the losses are still comparable. Although, at temperatures far below
4.2 K, the case Rlims0  ?c0 tan ?0 will yield Qi ? tan ?, and vice versa when temperature is
much greater than 4.2 K Qi ? ?c0/Rs0.
The DLD method is applied to data in Fig. 5.9 and Rs0(T ) and tan ?(T ) are deconvolved
and shown in Fig. 5.10(a) and Fig. 5.10(b), respectively. The x-axis in Fig. 5.9(a) is Tc/T where
at superconducting critical temperature of Tc = 9.2 K was assumed. Both a disperive (black)
and non-disperive (blue) tan ? were assumed using Eq. 5.7 and 5.8, respectively. Note, the DLD
method was only applied to the resonant peaks between 4.5-5.5 GHz and 8-10 GHz, ignoring the
?loss peaks? at 6.5 and 9.5 GHz. These loss peaks will be discussed in the next sections.
The dashed lines in Fig. 5.9(a) are calculated using Eq. 5.9 and serve as a reference for
the approximate measurement sensitivity of Rs0 as a function of temperature. At Tc/T = 3
for the non-disperive data (blue), the Rs0 is approximately a factor of two below the sensitivity
limit R lim,nd lim,ds0 ? 2Rs0 . Whereas the dispersive Rs0 (black) is only about 10% below Rs0 for
non-dispersive (blue) dashed line and approximately a factor of 3 above the dispersive (black)
dashed line. This suggests that tan ? is frequency dependent and has a power exponent of b ? 0
yielding tan ? ? ??1. Furthermore, the dispersive model is suggested to be more sensitive to
Rs0 in the approximate temperature range Tc/T = 2? 3. The Rs0 plateau at approximately 3??
is an artifact of the measurement sensitivity, and is not a residual resistance. Typical residual
resistance for Nb thin films are Rres ? 1? 20 n? and do not exceed 100 n?.
The expected temperature dependence of Rs0 was fit to the dispersive data using Eq. 2.27
with fitted Nb superconducting gap of ? = 19.5 ? 2.2 K. Benvenuti et al. measured a
superconducting gap of ? = 19.6K for Nb thin films [64]. Assuming a superconducting critical
155
temperature of T = 9.2 K, the gap ratio is estimated to be ?/kBTc = 2.1 ? 0.2, and is within
10% of the expected gap ratio 1.97 for Nb [46]. Valente-Feliciano measures a slightly lower gap
ratio of 1.89 for Nb thin films [71].
Fig. 5.9(b), the assumed dispersive (black) and non-dispersive (blue) tan ? temperature
dependence was fit to a power law expected when dielectric loss is dominated by TLS relaxation
(see Eq. 2.37). Using the dispersive model fit to deconvolved TEOS tan ?, the power exponent
is n = 2.5. Microwave dielectric measurements of amorphous silica have found T 2.4 from 2-8 K
in the TLS relaxation regime at 10 GHz [84]. TEOS deposited at low temperatures of 150 C is
considered amorphous [74]. This is the first indication that TLS relaxation processes are present
in TEOS at temperatures below 3 K and will be discussed further in the next section.
In this section, I have demonstrated the DLD method is capable of deconvolve Rs and
tan ? as a function of temperature. From this, I estimated the the superconducting gap of the Nb
in the MUX-MTL in good agreement with typical Nb thin films. Additionally, from the fitted
temperature dependence i found that the TEOS tan ? shows signs of TLS relaxation.
156
Figure 5.10: The deconvolved a) Rs0 and b) tan ? as a function of Tc/T using the DLD method
at reference frequency ?0/2? = 10GHz. The critical temperature is assumed to be Tc = 9.2 K.
Data from Fig. 5.9 were fit to Eq. 5.7 (2 parameter fit) and 5.8 (3 parameter fit) assuming a non-
dispersive and dispersive (frequency dependent) loss tangent, respectively. The DLD method was
only applied to the resonances between 4.5-5.5 GHz and 8-10 GHz, ignoring the ?loss peaks? at
6.5 and 9.5 GHz. The dashed lines in a) are calculated using Eq. 5.9 showing the approximate
sensitivity of the intrinsic resistance for the dispersive Rlim,ds0 and non-disperive fits R
lim,nd
s0 . The
solid lines in a) are fits to the Rs0 temperature dependence using Eq. 2.27 in the range Tc/T =
2.2 ? 2.5. The solid lines in b) are fits to the tan ? temperature dependence using Eq. 2.37 for
the entire temperature range. In a) the Rs0 plateau in at approximately 3 ?? is an artifact of the
measurement sensitivity, and is not a residual resistance.
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5.5 TLS Absorption and Relaxation Microwave Loss in TEOS Above 1K
In this section, I will claim signatures of TLS loss in TEOS at temperatures below 3 K
supported by measurements of the the relative dielectric constant ?r and dielectric loss tangent
tan ?. Furthermore, as temperature is decreased, there is a clear transition from TLS relaxation
to TLS absorption dominating the RF loss at approximately 2.25 K. This temperature transition
as a non-monotonic frequency dependence. Below is a summary table for the resonant frequency
dependent measurements and fitting to TLS models.
The electric perturbation of a two-level systems (TLSs) interacting with the phonon lattice
is analogous to the acoustic case for [79, 83]. The Hamiltonian matrixH = H +H ?0 describes the
system where H0 is TLS without perturbation and H ? is due to the electric field. The magnitude
of the diagonal and off-diagonal elements in H ? are the permanent dipole moment ? and induced
dipole moment ??, respectively. The permanent dipole moment describes the TLS relaxation
process expressing the shift in energy splitting caused by the electric field and is characteristic of
a TLS relaxation process. The induced dipole moment describes the resonant absorption of TLSs
by the coupling between the TLSs and electric field. The latter TLS resonant absorption process
can be saturated by strong electric fields and dominates at temperatures approximately below
1K. Conversely, the former relaxation process cannot be saturated by an electric field and is the
dominant process at temperature approximately above 1K [79]. Consequently, depending on the
temperature, applied electric field, and the angular frequency ?, a non-monotonic dependence of
the relative dielectric constant ?r and loss tangent tan ? can occur (see Fig. 2.4 in Chapter 2).
Note, for a weak electric field as temperature is increased from 0 K, the cross over temperature
will shift to increasing (decreasing) temperatures as ? is increased (decreased), reminiscent of
158
an increased (decreased) sampling of higher energy resonant absorption TLSs prior to relaxation
dominating. Both the change in dielectric constant and loss tangent data will be fit assuming
absorption and relaxation TLS processes.
The temperature dependent change in relative dielectric constant ? assuming the superposition
of TLS absorption and relaxation contributions is typically written as [186]
??(T ) ?(T )? ?(T0) ??abs + ??rel
= = (5.10)
?(T0) ?(T0) ?(T0)
where ?(T ) is the temperature dependent relative dielectric constant of the material, T0 is typically
the lowest measured temperature and is arbitrary, and ??abs and ??rel are the change in dielectric
constant due to TLS resonant absorption and relaxation, respectively. The relative dielectric
constant of TEOS is approximately ? ? 4.2 [73] and agree with estimations from measured data
in Section 4.2.
At low temperatures, the resonant absorption contribution to the change in dielectric constant
can be fit to
[ ( ) ( ) ]
1 h?? 1 h?? T
??abs(T ) = 4? n?
? 2 Re? + ?Re? + ? ln (5.11)
2 2?ikBT 2 2?ikBT0 T0
where n?? 2 is the induced dipole coupling of TLSs having a density of states n, T and T0 are
the measurement and lowest arbitrary reference temperature, respectively, kB is the Boltzmann
constant, h? is the reduced Planck constant, and Re? represents the real part of the digamma
function [187]. Note, the first and second terms in brackets dominant at temperature  1 K
and a frequency dependence can be measured. For measurements here where T > 1.5 K,
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the natural natural logorithmic temperature dependence dominates and can be assumed to be
frequency independent.
Here, we assume the relaxation process is mediated by the so-called single-phonon assisted
tunneling as the TLSs relax back to equilibrium. This is also commonly referred to as the one-
phonon process and is in opposition to coherent unassisted tunneling motion[79]. Assuming a
constant density of states n, the TLS relaxation process responsible for the change in dielectric
constant can be modeled by
?
4? ? df ( )?1
??rel = n?
2 1 + ?2? 2 dU (5.12)
3 dU
0
??1 = a(UkBT )
3 coth (U/2) (5.13)
( )
U ?1f = e + 1 (5.14)
where n?2 is the permanent dipole coupling, f is the Fermi function, ? is the measurement
frequency under an applied electric field, and ? is the TLS relaxation time commonly referred
to as the radiative lifetime of TLSs T1 [79, 79]. Found from acoustic measurements the material
constant a is
M?2 1
a = (5.15)
?v5a 2?h?
4
M?2 = (M2/c5 + 2M2/c5)/(c?5l l t t l + 2c
?5
t ) (5.16)
where M? is the average coupling energy [188], va is the average acoustic velocity of two transverse
sound velocities and one longitudinal sound velocity, ? is the mass density of the dielectric, and
160
subscripts l and t in Eq. 5.16 are the longitudinal and transverse components, respectively. For
vitreous silica, M? ranges from approximately 0.28 ? 0.34 eV [188], ? ? 2.2 ? 103kg/m3, and
va ? 4.7 ? 103 m/s averaging the transverse and longitudinal components. Pohl et al. gives a
comprehensive review of these material constants for various glassy metals and dielectrics [93].
For glassy silicon oxides, ? and vt,l can decrease by approximately an order of magnitude for
high porosity.
The measured resonant frequency for reactively coupled microstrip transmission lines is
proportional to the relative dielectric constant by f ? ??1/20 . Using this relationship and Eq. 5.10,
the temperature dependent resonant frequency data in Fig 5.9 can be converted by
??(T ) f 20 (T )? f 2= 0 (T0) (5.17)
?(T0) f 20 (T0)
where f 20 (T ) is the temperature dependent resonant frequency of each resonator in the MUX-
MTL, and f0(T0) resonant frequency at the lowest temperature measured T0.
Fig. 5.11 shows the resulting fits to Eq. 5.10 to data from Fig 5.9. The data was fit with
a Python script using the ?Powell? minimization method in SciPy package [189]. The data was
fit at each data power where the measured resonant frequency and temperature were inputs. The
dielectric constant at T0 was assumed to be ? = 4.2 [73] and the material constant in Eq. 5.15 was
assumed to be a = 2? 1075 J?3s?1. The two fitting parameters were the permanent and induced
dipole coupling constants n?2 and n?? 2 in equations Eq. 5.11 and Eq. 5.12, respectively. The dip
in the data is reminiscent of a transition from TLS resonant absorption to relaxation processes
dominating the dielectric constant change as temperature is increased from T0. The magnitude
offset for the 7GHz (4.4 mm) resonator in the MUX-MTL is due to a slightly higher T0. Because
161
of this, the fitted data for the 7GHz resonator will be ignored due to minimization fitting artifact.
In this context, the maximum change in ??(T )/?(T0) is negative showing a slight decrease of
the dielectric constant of approximately 0.06%. Surprisingly, this value is in agreement with
Schickus et al. of vitreous silica with 1000 ppm OH- concentration [186]. Although, his
maximum change in dielectric constant occurs at a much higher temperature of approximately
7 K, whereas in Fig. 5.11 occurs at approximately 2.2 K. Starting from 3 K and decreasing in
temperature and following the color scale, the relaxation regime shows almost a monomonic
frequency dependence, where for increasing frequencies the dielectric constant is effectively
decreasing. At and below the dip around 2.2 K, a non-monotonic frequency dependence of
dielectric constant change is visible. This will be probed further in the fits of n?2 and n?? 2.
Fig. 5.12 is a plot of the data fitting (solid lines) from Fig. 5.11. Again, the 7 GHz data
here will be ignored due to a fitting artifact. Immediately it can be seen that the permanent
dipole coupling is 3 orders of magnitude larger than the induce dipole coupling (notice scale bars
on left and right y-axis). This is not understood and reported values for all permanent dipole
coupling constants for glassy dielectrics do not exceed 1 ? 10?3 [89, 93, 188]. Table 5.1 is a
summary of measured permanent and induced dipole coupling for various materials. From this
table, it can be seen that the permanent and induced diople coupling increases in vitreous silica
by approximately an order of magnitude when the concentration of hydroxide concentration is
increased. This is supported by the theory where hydroxide ions are the dominant TLS system
coupling to phonons and the electromagnetic waves [186]. The TLS states in borosilicate glass
have a higher permanent dipole coupling to phonons relative to vitreous silica [89]. Although
predominantly a crystalline solid electrolyte, it is assumed that the existence of low-energy
excitations in Na? ? ? Al2O3 is a consequence of a high degree of disorder in the Na+-sublattice
162
Figure 5.11: Temperature dependence of the dielectric constant ? for 0.75?m MUX-MTLs
fabricated from process C at a constant VNA input power of Pin = ?35 dBm. The data from
Fig 5.9 was converted into ?? = ?(T )??(T0)/? using Eq. 5.17. The data was fit using Eq.s 5.10,
where the total change in dielectric constant is the superposition of the TLS absorption and TLS
relaxation contributions using Eq.s 5.11 and 5.12, respectively. assuming material constant a =
2? 1075J?3s?1 [186]. Refer to Summary Table 5.2
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and is considered amorphous [94, 190]. Furthermore, Na? ? ? Al2O3 has the highest induced
dipole coupling with the electromagnetic waves and most likely due to its ionic nature. Li3N has
a much lower induced dipole coupling and the TLS low energy excitations are suggested to be
coming from hydrogen ions [190]. The permanent dipole coupling for Na? ? ? Al2O3 was not
calculated, but based on the high induced dipole coupling, it is possible the permanent dipole
coupling can be of order 10?2 but it is unlikely the coupling is as high as 10?1, as seen by the
extract values measured for TEOS in Fig. 5.12. Nevertheless, this can be further examined in
future experiments at lower temperatures and power dependent measurements as performed by
Schickfus et al. [84].
Table 5.1: A summary table of measured permanent dipole coupling n?2 and induced dipole
coupling n??2 in various materials at the respective measurement frequency. The values were
taken from references [Schickfus1976] [186], [SchickfusThesis1977] [89], [Strom1978] [94],
[Baumann1980] [190].
Care was taken in the evaluation of the integral in Eq. 5.12, and the material constant a
is not expected to change orders of magnitude. Furthermore, if this were real, the 3 orders of
164
magnitude increase would have to be comming from the permanent dipole moment, as the TLS
density of states n cannot increase beyond approximately an order of magnitude.
Aside from the magnitude difference, the results in Fig. 5.12, the data can still be discussed
qualitatively. The permanent dipole coupling increases with increasing frequency, which is
in agreement with the notion that higher frequencies accesses higher energy insatiable TLS
relaxation processes. The induced dipole moment decreases with increasing frequency, and is
also in agreement with intuition of TLS absorption in that higher energy absorption processes
contribute to the change in phase velocity (dielectric constant) in the resonator. Another way to
describe the decrease in induced dipole coupling is that at higher frequency, it is expected that
TLS absorption can dominate at higher temperatures [89].
The magnitude discrepancy for fitted values for n?2 measured via temperature dependence
resonant frequency will be investigated further and compared with temperature dependent data
of the dielectric loss tangent.
5.6 Dielectric Loss Spectroscopy
In this section, the dielectric loss tangent will be fit to TLS resonant absorption and relaxation
models to determine the frequency dependence of the permanent and induced dipole coupling
constants n?2 and n?? 2, respectively. The data fitting models will be introduced first, followed
by a suggested explanation of the anomalous loss peaks in Fig. 5.9 at 6.5 and 9.5 GHz.
As a consequence of the TLS tunneling and relaxation process scattering phonons, a non-
monotonic temperature dependence of the dielectric loss tangent tan ? can be measured and
depends on the microwave electric field strength [84].
165
Figure 5.12: Frequency dependence of the permanent and induced dipole coupling from the data
fits in Fig. 5.11. Refer to Summary Table 5.2 for assumed parameters and results comparison to
tan ? fits. MUX-MTL, Process B, Q?1i (T )
166
The superposition of TLS processes tan ? can be written as
tan ? = tan ?abs + tan ?rel (5.18)
where tan ?abs and tan ?rel are the resonant absorption and relaxation contributions, respectively,
and here I am assuming these are the only processes involved in the microwave loss in the
temperature range T = 1.5? 3K and frequency range 4? 10GHz.
The measured loss tangent from resonant absorption can be fit to the following
tanh h??4?
tan ? = n?? 2 2kBTabs (5.19)
3 (1 + J/J 1/2c)
where n?? 2 is the induced dipole coupling constant, J is the microwave energy intensity, and
Jc is the critical energy intensity. For J > Jc, resonant TLSs undergoing resonant absorption
are saturated and tan ?abs continually decreases as temperature is decreased. For J < Jc TLS
resonant absorption processes induce loss and an upturn in tan ? is observed as temperature is
decreased to absolute zero [84].
For microwave dielectric loss dominated by TLS relaxation, data can be fit to the following
equation [89]
[ ]
T 3 M?2 ?(4, 0.5)k3
tan ?rel = n?
2 B (5.20)
? ?v5 4a 8?h?
where n?2 is the permanent dipole coupling constant, ?(4, 0.5) = 97.8 is the Riemann zeta
function [89], ? is the mass density of the dielectric, M?2, va are the average coupling and acoustic
velocity introduced in Eqs. 5.15-5.16, respectively.
167
Before fitting to the above models and extracting n?2 and n?? 2, the power law fits to
tan ? data in Fig. 5.13 will be discussed. The resulting power law fits in Fig. 5.13(b) show the
power exponent is close to 3, strongly suggesting the TEOS dielectric loss is dominated by TLS
relaxation processes. The power exponent measured for vitrious silica is approximately 2.4 [84].
Power exponents below 3 are typical, as the curvature of the temperature dependence can change
rapidly as a consquence of either TLS absorption at lower temperatures or classical relaxation
processes at higher temperatures inducing the so called ?plataeu? effect [93].
Using Eqs. 5.18-5.20 above and data in Fig. 5.13, the TEOS dielectric loss tangent is fit for
MUX-MTL resonators and shown in Fig. 5.14. The fitting results of temperature dependence in
Q?1i (T ) and resonant frequency f0(T ) can be found in Summary Table 5.2.
Notice, the data (circles) and fits (solid lines) show non-monotonic frequency dependence
looking vertically at a particular temperature below 2.5 K. Assuming an onset of TLS resonant
absorption below 2.5 K, it can be inferred that different frequencies are coupling stronger to
resonant TLSs. The converse argument is that each resonator is coupling non-monotonically to
the feedline, inducing different power densities circulating in the resonator. The latter will be
explored in the next graph followed by frequency dependence of dipole coupling constant fits
from Fig. 5.14.
Fig. 5.15 shows the calculated stored power and peak currents in the MUX-MTL resonators.
It can be seen quantitatively that the circulating power in the resonators scale monotonically with
frequency at all temperatures. This is supporting the claim that the non-monotonic dependence
in Fig. 5.14 is not an artifact of the variation in resonator coupling. Although, the peak currents
in the resonator for the lowest frequency resonators are less than a factor of two away from the
assumed worst case critical current for these lines. It is conceivable that this could induce non-
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Figure 5.13: a) Temperature dependent inverse internal Q-factor of MUX-MTL resonators. At
these temperatures, It is assumed that the Q-factor is dominated by the TEOS dielectric loss
tangent by Q?1i = tan ?. Each color is a different resonator coupled to the MUX feedline with
resonant frequency listed in the legend. The dashed lines are power law fits in the temperature
range 2.5 to 3.5 K. b) Resulting power law fits for each resonator in the MUX-MTL including
one sigma error bars for each fit. These resonators were fabricated using process C and constant
VNA input power Pin = ?35 dBm. This is the same data from Fig 5.9.
169
Figure 5.14: Temperature dependence of the inversive internal Q-factor Q?1i for each MUX-
MTL resonator with unique fundamental resonant frequencies at a constant input power of Pin =
?35dBm. At these temperatures, It is assumed that the Q-factor is dominated by the TEOS
dielectric loss tangent by Q?1i = tan ?. The solid lines are fits to Eqs. 5.18-5.20 with n?? 2, n?2,
and J/Jc as free fitting parameters. A summary of results and assumed constants are listed in
Summary Table 5.2.
170
Table 5.2: A summary table of temperature dependent fitting results to models of TLS
absorption and relaxation. At these temperatures, It is assumed that the Q-factor is dominated
by the TEOS dielectric loss tangent by Q?1i = tan ?. The Q
?1
i (T ) and F0(T ) data were fit
with 3 and 2 free fitting parameters to Eq.s 5.18 and 5.10, respectively. The constants were
assumed from references [Kim2001] [72], [Pohl2002] [93], [Jackle1976] [188], [Rocha2004]
[73], [Schickfus1976] [186]. The constant a assumed in Q?1i (T ) fits was calculated from the
other assumed constants in the column using Eq. 5.15. Note, the extracted values of n?2
from resonant frequency fits are 3 orders of magnitude larger than typical values of amorphous
dielectrics [Phillips1981] [89]
linear enhancement of quasiparticle loss in the Nb. Although, the peak current qualitatively scales
mono-tonically with frequency, where the peak current is decreasing for increasing frequencies.
171
This suggests that a similar monotonic dependence would be seen in the internal Q-factor. This
is by no means a rigorous argument and should be explored further by doing power dependent
measurements. Although, it is also worth noting that since the peak voltage is linearly related
to the peak current by Vpeak = Z0Ipeak where Z0 ? 50 ? is the characteristic impedance of
the MTL resonators. The reduction in peak voltage as frequency increased could be giving
rise to the non-monotonic dependence of the loss tangent in the TEOS. This is postulated to
be due to the expected increase in tan ? when voltage is increased, characteristic of TLS resonant
absorption processes. Admittedly, during the time of these measurements, it was not expected
that TLS mechanisms would be present nor was the design peak current incorporated as a design
parameter. The subject of the original design was to investigate the coupling capacitor values for
the smallest capacitors, inadvertently producing weak enough coupling, and circulating power to
see unsaturated TLS resonant absorption loss.
The resulting 3-parameter fits in Fig. 5.14 are plotted as a function of frequency in Fig. 5.16b
and Fig. 5.16c. Fig. 5.16a is re-plotted data from Fig. 5.9 for convenience. The permanent dipole
coupling constant n?2 (black) increases for increasing frequencies by a factor of 3 from 4-10
GHz. This agrees qualitatively with values fitted from temperature dependent changes in the
dielectric constant in Fig. 5.12. Although, the magnitude n?2 ? 3 ? 10 ? 10?3 in Fig. 5.16b
is more agreeable with published data on various amorphous dielectrics. The values of n?2 for
TEOS measured here are of order values measured for borosilicate glass 6 ? 10?3 [89], and
approximately an order of magnitude higher than vitreous silica for the lowest concentrations of
OH? [186]. Similarly, the range of extracted induced dipole coupling n?? 2 = 4 ? 8 ? 10?4
in Fig. 5.16 also are in fairly good agreement with borosilicate glass having 3 ? 10?4 found in
Schickfus thesis reported in [89].
172
Figure 5.15: Temperature dependence of a) the stored circulating power Pst b) peak current Ipeak
in the MUX-MTL resonators for VNA input power of Pin = ?35dBm. The stored power in each
resonator and each temperature was calculated from Eq. 5.4. The peak currents were calculated
using Eq. 5.5. The solid red line in b) is the minimum critical current Ic of a 0.75 ?m wide Nb
wire with a 200 nm thickness assuming critical current density Jc = 20MA/cm2 [32].
173
It is an important distinction to note that measured dipole moment coupling in microwave
dielectric experiments are much more sensitive to impurities and the chemical composition of
insulating glasses relative to acoustic measurements. Therefore, it is claimed that absolute values
of n?? 2 and n?2 plotted in Fig. 5.16 may be off by factors of 2-3 due to assumed constants in
Table 5.2, but the the qualitative frequency variation n?? 2 is claimed here to be due to coupling
of TLSs with the electric field at distinct frequencies. In other words, the peaks of n?? 2 at 6.5
and 9.5 GHz in Fig. 5.16a align with the loss peaks in Fig. 5.16b. Furthermore, these peaks
align with relatively low J/Jc ratios in Fig. 5.16c, suggesting that the loss peaks are affiliated to
enhanced TLS resonant absorption loss. The maximums of J/Jc at 8.5 GHz shows a saturation
of circulating power to TLS resonant absorption, reducing Q?1i . An interesting data point is the
7 GHz point, where there is an increase in both the induced coupling to TLSs and the circulating
power exhibiting an overall constant Q?1i relative to the neighboring frequencies.
This demonstration shows a new microwave loss spectroscopy method to probe TLS loss by
relying on the frequency dependence. In future work, better care can be taken with the coupling
design as well as systematic input power dependence to further explore this technique and the
exhibited TLS character relative to the dielectric composition and impurities.
5.7 Temperature Dependence of the Power Dissipation in Nb and TEOS
The primary goal of this dissertation is to glean information on how to reduce intrinsic
microwave losses in Nb and TEOS to reduce the power dissipation in RQL resonator clock
networks. This chapter has focused on demonstrating techniques to better understand the dominate
loss mechanisms coming from the superconductor and dielectric materials as temperature is
174
Figure 5.16: a) Frequency dependence of Q?1i for MUX MTL resonators for constant VNA
input power Pin = ?35 dBm in the temperature range T = 1.6 ? 4.5 K (Re-plotted Fig. 5.9).
b) Frequency dependence of the fitted permanent (black) and induced dipole (blue) coupling
constants. c) Frequency dependence of the power density ratio J/Jc where J and Jc are the
circulating input power density in the resonator and the critical power density for TLS resonant
absorption saturation, respectively. The fitted parameters in b) and c) were fit to data in Fig. 5.14
using Eqs. 5.18. A summary of results and assumed constants are listed in Summary Table 5.2.
175
decreased below 4.2 K. This chapter will be concluded with direct measurements of power
dissipation in MUX-MTL resonators that can be directly related to RQL interconnects having
the same geometry and input power. Furthermore, estimates can be made from this data for
arbitrary geometries and input power.
Strong et al. measured the dissipated power of meta-material zeroth-order resonators
(ZORs) used in RQL clock networks and found a dissipated powers on the order of 20 ?W
[10]. Using Eq. 5.3, the temperature dependent dissipated power Pd of MUX-MTL resonators at
Pin = ?35 dBm is plotted in Fig. 5.17.
In Fig. 5.17 at approximately 4-4.5 K, the dissipated power for all resonators (frequencies)
is approximately 100?200nW for VNA input power Pin = ?35dBm. For a nominal input power
Pin = ?9 dBm used in ZOR resonators, the dissipated power in MUX-MTLs is approximated
to be 60 ?W , and is within a factor of 3 of ZOR power dissipation. As temperature is decreased,
the 4.5GHz resonator reduces in dissipated power down to Pd = 70 nW at 1.6 K. At 1.7 K, the
dissipated power decreases as frequency is increased up to 10 GHz with a minimum dissipation
of Pd = 26 nW . The frequency dependence of this dissipated power is due to the saturation
of TLS resonant absorption loss at higher frequencies and discussed in the previous section.
This is an expected and known result, and further supports the notion that the TEOS dielectric
microwave loss dominates the ZOR resonator clock power dissipation for the temperature range
1.5-3.5 K. There is a subtle maximum power dissipation at low frequencies around 3.5 K where
the Nb and TEOS microwave losses are comparable for this resonator geometry, and can shift
depending on the material used and the RQL clock geometry. Overall, this has established the
limits to power efficiency in resonator clocks due to material losses and demonstrates a method
of characterization.
176
Figure 5.17: Temperature dependence of the dissipated power Pd in MUX-MTL resonators at a
constant VNA input power Pin = ?35 dBm. The dissipated power was calculated from data in
Figs. 5.15 and 5.8 using Eq. 5.3.
177
5.8 Summary
At the beginning of this chapter, the temperature dependence of the resonant frequency
fn and internal Q-factor Qi in the range 1.5-4.5 K was measured for MTL widths 0.25 ? 4?m
from process B. The anomalous non-monotonic frequency dependence of Q?1i was ambiguous.
Measurements of the peak currents in the resonator suggested enhancements of Q?1i when the
currents in the resonator were close to the expected critical currents of the Nb strips Although,
the model did not apply when currents were more than a magnitude below the critical current.
This facilitated the design of multi-plexed microstrip resonators on a single feedline to have a
higher density of frequency points to analyze.
In the next section, a design of multiplexed 0.75 ?m MTL resonators was presented. In an
approximate range of 3-4.2 K, I was able to deconvolve the Rs(T) and tan ?(T ) temperature
dependence using the dispersive loss deconvolution (DLD) method. This was validated by
fitting to the expected Rs and tan ?(T ) temperature dependence e??/kBT and approximately T 3,
respectively. From the extracted tan ?(T) going to temperatures below 3 K, I proposed that the
change from T 3 dependence at approximately 2 K is due to a transition from TLS relaxation to
TLS absorption mechanisms. This was supported by a decrease in ?r of approximately 0.03-
0.04% at 2 K from fn(T ) measurements. Fits to the temperature change in resonant frequency
were fit to TLS resonant absorption and relaxation models, but the fitted values for permanent
dipole coupling were 3 orders of magnitude off from expected values and is not understood. Fits
to the temperature dependence of Q?1i to TLS models had much better agreement to expected.
From these fits, microwave loss spectroscopy analysis showed a non-monotonic dependence on
the TLS induced dipole coupling. This non-monotonic dependence agrees with loss peaks inQ?1i
178
and is claimed to be a combination of saturation of TLS absorption processes and enhancements
of specific TLSs coupled to impurities in the material resonant with the probed frequency. Although,
this is speculative and not a rigorous analysis, and systematic power dependent measurement
should be performed.
Aside from the anomalous behavior, the conclusions on temperature dependence power
efficiency of RQL clock resonators can still be summarized and found in the last section. Similar
to the conclusion drawn from the last chapter, tan ? is the limiting loss dominating power dissipation
in RQL resonators as temperature is reduced below 3 K. This is analogous to the dominant loss
mechanism in superconducting Qubit systems. However, care needs to be taken on resonator
clock input power requirements as temperature falls below 3 K. At high input powers, TLS
resonant absorption processes are saturated, allowing for reduced power dissipation in the resonator
clock. At increased powers, there are two additional considerations: (i) Input current (power) to
the resonator needs to remain far below the critical current of the MTL to mitigate nonlinear
effects and increased microwave loss from the superconducting wires and groundplanes. (ii)
At temperatures below 3 K, especially mK temperatures, the heat lift in cryostat decreases, and
heat dissipation designs may need to be implemented. At 2 K, the internal Q-factor Qi of RQL
interconnect increased by an order of magnitude, yielding an RQL interconnect efficiency of
? ? 0.9. Although, an order of magnitude increase of Qi at 2 K comes at an added refrigeration
cost by a factor of 3 relative to 4.2 K [181]. From this, it can be estimated that the total power
consumption of RQL operating at 10 GHz is 100x less than equivalent CMOS circuits operating
at 1-4 GHz (logic and interconnects) [4].
179
Chapter 6: Summary and Conclusions
6.1 Summary
The work performed in this dissertation was aimed at performing materials characterization
of microwave losses in Nb and TEOS to provide information on the limitations and mitigation
strategies to reduce power dissipation in RQL clock interconnects. The superconductor and
dielectric losses at temperature below 4.2 K and GHz frequencies have independent frequency
and temperature scalings. On one hand this makes it difficult to design a system to optimize power
efficiency performance. On the other hand, the unique frequency and temperature dependence
presents the opportunity to deconvolve the loss mechanisms in a single measurement. In this
dissertation I discussed the power efficiency limitations dictated by the Nb and TEOS supported
by direct measurements of the microwave losses using the Dispersive Loss Deconvolution method
(DLD). The DLD method confirmed the initially fabricated Nb material was of low quality, as
opposed to the TEOS dielectric, and helped the fabrication team rapidly fine tune the process to
yield the ultra-low loss Nb interconnect material.
At 4.2 K, the theoretical minimum of Nb microwave losses can be reached and heavily
depends on the microstrip geometry and the fabrication process used. At temperatures below
4.2 K, the microwave losses in the superconductor become negligible and the dielectric losses
dominate and is independent of the resonator geometry. It was found that TLS relaxation pro-
180
cesses dominate the dielectric loss in the temperature range 2.5 to 4.2 K. At temperatures below
2.5 K, TLS resonant absorption processes become comparable to the relaxation loss and has a
significant input power dependence.
At 2 K, it was found that the RQL interconnect efficiency was estimated to be ? ? 0.9
relative to the logic. Accounting for the refrigeration costs, it can be estimated that the total
power consumption of RQL operating at 10 GHz is 100x less than equivalent CMOS circuits
operating at 1-4 GHz (logic and interconnects) [4]. Changing the operating temperature of RQL
from 4.2 to 2 K would not create significant design changes since Nb inductance decreases by
less than 20%, Nb/AlOx/Nb Josephson junction critical currents increase by less than 20%, and
TEOS relative dielectric constant remains virtually constant.
RQL is a classical computing technology with notable energy efficiency advantages over
CMOS with circuits that can operate over 100 GHz. Although, circuit density still remains as
a significant limiting factor preventing the technology from competing with CMOS. Both the
size of the Jospehson junctions and the interconnects need to be pushed to dimensions below
0.25 ?m. Specifically for Nb interconnects, pushing the critical dimensions to be comparable
with the magnetic penetration depth ? ? 90 nm imposes strict requirements on the fabrication
process to produce low loss interconnects while increasing the processing window.
6.2 Future Work
The work performed in this dissertation was aimed at improving the energy efficiency of
RQL superconducting interconnects and help provide a path for RQL to meet computing power
demands. The power dissipation in RQL interconnects is inherently dependent on RF losses in
181
the superconductor and dielectric. Since resonator-based clock networks deliver power on the
same order of the total dynamic power of the Josephson junction logic, the RF losses in the
superconductor and dielectric can directly influence the efficiency and scaling of the technology.
Considering the superconductor and dielectric RF losses are comparable in RQL interconnect due
to the imposed transmission line geometry from the fabrication process, both RF loss channels
need to be minimized.
In this dissertation, through the lens of material science, I discussed some of the essential
work needed to improve the processing, properties, and performance of RQL interconnect. Furthermore,
the novel Dispersive Loss Deconvolution (DLD) method and resonator designs described here
serve as the groundwork for future groups interested in measuring RF losses in inseparable
superconductor-dielectric systems. The outlook and possible research directions include:
Processing
(a) Further improvement of the metal damascene fabrication process down to and below 0.25-
micron dimensions such as protective buffer and capping layers (thermal oxide diffusion
barriers), and deposition parameters that yield uniform trench filling (eliminate voiding and
facilitate conformal coverage).
(b) Tighter control of critical dimensions and material property variation across the wafer.
The sensitivity analysis in Section 3.7 showed that tan ? is geometry independent with a
variation of approximately 20% across the wafer. Whereas the conductor Q-factor is both
sensitive to the geometric factor ? with worst case 20% variation, and the intrinsic surface
resistance Rs, which seemed have a variation less than that (see Fig. 4.9).
(c) Increasing the thermal budget - (i) implement junction-on-top or more thermally resilient
182
Josephson junction barrier technologies, (ii) on wiring layers to promote ideal growth
kinetics if a damascene process is needed. A damascene process is used in CMOS industry
because it is difficult to etch Cu (as opposed to silicon dioxide). Nb is fairly easy to etch,
so it is conceivable to rely on Nb etch processes to define the wires and CMP the dielectric,
which yielded the lowest loss Nb reported in Chapter 4.
(d) Surveying different superconductor and dielectric materials with lower RF losses and/or
more resilient to process variations. Candidate superconducting materials are NbN and
NbTiN for their low RF loss properties and sputtering compatibility. On one hand, NbN
and NbTiN have higher inductance and can reduce circuit sizes, but on the other hand this
will decrease propagation speed relative to Nb and inductance uniformity needs to also
be considered. Silicon nitride is a candidate dielectric material as it can have an order of
magnitude lower loss than TEOS, but it is important to consider the change in propagation
speed (relative dielectric constant difference). The relative dielectric constant for TEOS
and silicon nitride are approximately 4.2 and 7.5, respectively. Therefore the propagation
velocity would decrease by approximately 25% switching from TEOS to silicon nitride.
Properties
(a) Reduce the fundamental mode frequency of the microstrip transmission line used in Chapters 4
and 5 down to below 1 GHz to increase the frequency range to validate the superconducting
frequency dependencies.
(b) Continue multi-plexed resonator technique to understand spatial distribution of properties
(and critical dimensions) across a chip/wafer
183
(c) Incorporate the impact of nonlinear dynamics in FEM of superconducting interconnects,
as I assumed the system displays linear response throughout this thesis, but is most likely
not the case since electromagnetic field concentration at the edges and will only become
exacerbated as dimensions are pushed below 0.25-microns and the input power will be
pushed to its limits.
(d) Incorporate common nonlinear measurements (vs power, vs applied magnetic field (in-
plane, out-of-plane).
(e) Physical characterization techniques with higher sensitivity to trace element contamination
both in interconnect cross-section as well as spatially (e.g. atom probe tomagraphy, etc).
(f) Determine if there is a correlation between dielectric loss tangent tan ? between 4.2 K and
mK. Conceivably, rapid measurements at 4.2 K may be more advantageous to evaluate a
superconducting interconnect.
(g) Use as a test vessel to measure vortex dynamic properties on wide microstrip MTLs with
temerpature dependent measurements in perpendicular magnetic field [137].
Performance
(a) In the nominal configuration, RQL power dissipation is less of an issue at mK temperatures
due to million level Q-factors (superconductor loss is negligible and TLSs saturate at RQL
powers).
(c) Continue work on exploring cross-sectional RQL interconnect geometries to mitigate RF
losses by engineering the geometric factor ?.
184
(d) Further explore mK-K temperature range of superconducting-dielectric RF losses in conjuction
with cooling costs to optimize energy efficient computing.
(e) Higher throughput test systems to get wafer statistics, a wafer prober can alleviate this
[191].
185
Appendix A: FIB Cross Sections of Process A and B
To accurately calculate the geometric factor ? of microstrip transmission line (MTL) resonators
(see Chapter 3), exact measurements of the MTL geometries were measured using a similar
instrument as the FEI Helios 660 Dual Beam FIB/SEM. The cross section of the MTLs were
exposed using a focused ion beam (FIB) and then imaged using a scanning electron microscope
(SEM).
Table A.1: FIB Results Summary table for processes A and B measured in this work. All of
these microstrip tranmission lines were measured using a similar instrument as the FEI Helios 660
Dual Beam FIB/SEM. Cross sections were taken at about the half way point along the meandor
resonator (? 7 mm). The second column are the nominal (designed) wire widths. Example FIB
cross sections can be found in Figs A.1 and A.2. The third column are the measured mean widths
averaged between the bottom and top of the wire. The ? values in the last three columns are the
calculated one standard deviations taken from measurements of all 5 wire widths.
Table A.1 is a summary of the measurements for MTLs made from process A and B. The
186
second column are the nominal (designed) wire widths. The third column are the measured mean
widths averaged between the bottom and top of the wire. The ? values in the last three columns
are the calculated one standard deviations taken from measurements of all 5 wire widths.
Figure A.1: Example cross sections and measurements of the microstrip tranmission line
resonator geometry for the 0.25?m and 4?m wire widths fabricated using process A. In this
process the conducting strip (CS) is below the ground plane (GP)
187
Figure A.2: Example cross sections and measurements of the microstrip tranmission line
resonator geometry for the 0.25?m and 4?m wire widths fabricated using process B. In this
process the conducting strip (CS) is above the ground plane (GP)
Example cross sections for 0.25?m and 4?m MTLs can be found in Fig. A.1 from process
A. In this process the conducting strip (wire) is below the ground plane.
188
Example cross sections for 0.25?m and 4?m MTLs can be found in Fig. A.2 from process
B. In this process the conducting strip (wire) is above the ground plane.
Appendix B: Estimation of Radiation Q-Factor
The radiation Q-factor in general is the ratio of the energy stored in the resonator over the
power radiated from the transmission line resonator, and can be estimated using the following
equation from [163]
Q 4?Z0rad = 2 (B.1)480(s?) F (?eff ) ( )
1/2
? +1 2? (? ? +1F (? ) = eff eff?1) ln effeff (B.2)? 3/2 1/2eff 2? ? ?1
eff eff
where Z0 is the characteristic impedance, s is the dielectric thickness, ? is the phase constant, and
?eff . For a Z0 = 55 ? and ? = 670m?1 found using HFSS simulation with inputs Rs = 20 ??,
? = 90nm, ?eff = 4.5, and geometry shown in Fig. 4.1, Qrad ? 230?106. For internal Q-factor
Qi and external (or coupling) Q-factor Qe on the order of 1? 103, it is safe to ignore Qrad using
189
the following relation of the loaded Q-factor Q?1 = Q?1 +Q?1 +Q?1L i e rad.
Appendix C: Temperature Dependent Resonator Measurements
C.1 Measurements System
For high quality RF measurements (up to 12 GHz) with precise temperature control (1.5
? 9 K), I utilize a system similar to the Dry ICE 1.5 K Cryostat [182]. This system has a base
temperature of 1.4 K and temperature can be set between 1.4-10 K with a temperature stability
of ?10mK. The temperature is controlled using a Lakeshore 336 PID control [192] with a 25 ?
heater.
190
Figure C.1: I utilized a system similar to the Dry ICE 1.5 K Cryostat [182] for temperature
control for the testing and results described in Chapter 5. The test fixture is similar to the RF
dip probe outfitted with a 32 pin (contact pad) described in Chapter 4. The same test fixutre
was used in previous works [35], [34], [36], [10]. To achieve RF signal integrity and avoid wire
bonding for fast sample chip exchanges, a non-magnetic Cu/Au flip chip press contact technology
is used. The fixture PCB connects the flip-chip bump pads to the probe .047 inch semi-rigid
coaxial cables. The chip and roughly 30 cm of the cables are cooled by the Dry ICE 1.5 K
Cryostat during measurements. The temperature was monitored with a calibrated Lakeshore
Cernox sensor. The resonators were measured by 2-port S-parameters using a Keysight N5222A
vector network analyzer (VNA). The network analyzer was calibrated up to the top of the probe.
191
Appendix D: Relevant Equations for Parallel Plate Transmission Line Resonator
In the following equations in this appendix section, a parallel plate transmission line geometry
is assumed. In this assumption with an infinitely wide ground plane, the microstrip width is much
greater than the dielectric thickness w  s.
The complex propagation constant ? = ?+ i? is assumed here to be that of a Swihart wave
and is a good approximation for a wide microstrip transmission line with the form [36, 59]
[ ]
? 2Reff
? = + tan ? (D.1)
2 ??0s+ 2(X?eff ?Reff tan ?)
? 2(Xeff ?Reff tan ?)
? = ? ?0?r?0 1 + (D.2)
??0s
whereReff is the effective surface resistance (see Eq. 2.24),Xeff is the effective surface reactance
(see Eq. 2.25), s is the dielectric thickness between the microstrip and groundplane, ?r is the
relative dielectric constant of the dielectric, tan ? is the loss tangent of the dielectric material, ?0
is the permittivity of free space, and ?0 is the permeability of free space.
Here, a PPW system has been enforced, and we are interested in the resulting quasi-
transverse electromagnetic (quasi-TEM) mode propagating along the longitudinal length l in the
x direction (see Fig. 3.10). The accuracy of HFSS solutions will be tested using the following
equations forRLGC by inserting the superconducting impedance definitions Eq. 2.24 and Eq. 2.25
192
into PPW analytical solutions [95]
2R
R = eff (D.3)
w
L = ? s+2? coth (d/?)0 (D.4)w
G = ?? w0?r tan ? (D.5)s
C = ? ? w0 r (D.6)s
where ?0 is the vacuum permeability, ?0 is the vacuum permittivity, ? is the angular frequency,
w is the PPW width, s is the plate separation dielectric thickness, Reff is the effective resistance
defined in Eq. 2.24, and tan ? is the dielectric loss tangent. The RLGC are the series resistance,
series inductance, shunt conductance, and shunt capacitance per unit length of a generalized
transmission line defined in Eq. 3.2-Eq. 3.5 in Section 3.3.
MTL resonators support a TM00n [59] mode with the eigen frequency
n nc
fn = ? = ? (D.7)
2lres LC
?
2lres ?r 1 + 2? coth(d/?)/s
where n = 1, 2, 3. . . is the mode index, lres is the geometrical length of the resonator, and L and
C are the conducting strip series inductance and shunt capacitance per unit length defined by Eq.
3.3 and 3.5. The right hand side in Eq. D.7 holds for a wide MTL where the width is much greater
than the dielectric thickness (w  s) [70], where c is the speed of light, and ?r is the dielectric
constant.
193
Using Eqs. D.4 and D.6, the characteristic impedance can be calculated by
? ?s ?0
Z0 = L/C = (1 + 2?eff/s) (D.8)
w ?0?r
where ?eff = ? coth(d/?) is the effective magnetic penetration depth.
Using Eq. 3.1, the partial conductor and dielectric Q-factors for a parallel plate resonator,
Qc = ?L/R and Qd = ?C/G, respectively, are
Q = ?? s+2? coth (d/?)c 0 (D.9)2Reff
Qd = 1/ tan ? (D.10)
Notice, the PPW width w drops out in Qc, and ? drops out in Qd with no geometry dependence.
Then, the only parameters needed to model the superconducting PPW, analytically and with
HFSS impedance boundaries, are s, d, ?, ?, Rs, and tan ?.
To provide a simple analytical limiting case for Eq. 3.14 consider a resonator formed by
a parallel-plate waveguide with identical superconducting plates of the width w and thickness d,
separated by the dielectric layer of thickness s. This geometry approximates MTL with w  s.
The geometric factor is
s+ 2? coth(d/?)
?PP (?) = ??0 (D.11)
2[coth(d/?) + (d/?)csch2(d/?)]
utilizing the expression for the effective surface resistance of thin superconducting films, and by
R ? R [coth(d/?)+(d/?)csch2eff s (d/?)] that ignores transmission loss through the superconducting
194
films [69]. In the thick film limit the geometric factor reduces to ? 1PP = ??0(s + 2?) and the2
geometric factor is no longer dependent on the superconductor plate thicknesses d. For the thin
film limit where ? d the full expression needs to be used in Eq. D.11 and a term including the
dielectric substrate impedance may be needed [70].
195
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