Abstract
Title of Dissertation: THE RADIO-FREQUENCY SINGLE-ELECTRON
TRANSISTOR DISPLACEMENT DETECTOR
Matthew D. LaHaye, Doctor of Philosophy, 2005
Dissertation directed by: Adjunct Professor Keith C. Schwab
Department of Physics
For more than two decades, the standard quantum limit (SQL) has served
as a benchmark for researchers involved in ultra-sensitive force and displacement
detection. In this thesis, I discuss a novel displacement detection technique which
we have implemented that has allowed us to come within a factor of 4.3 from the
limit, closer than any previous e?ort. Additionally, I show that we were able to use
this nearly quantum-limited scheme to observe the thermal motion of a 19.7 MHz
in-plane mode of a nanomechanical resonator down to a temperature of 56 mK. At
this temperature, the corresponding thermal occupation number of the mode was
?n
th
??60. This is the lowest thermal occupation number ever demonstrated for
a nanomechanical (or larger) device. We believe that the combination of these two
results has important and promising implications for the future study of nanoelec-
tromechanical systems (NEMS) at the quantum limit.
The detection scheme that we used was based upon the single-electron tran-
sistor (SET). The SET has been demonstrated to be the world?s most sensitive
electrometer and is considered to be a near-ideal linear amplifier. We used stan-
dard lithographic techniques for the on-chip integration of the SET with both a
microwave-matching network and nanomechanical resonator. The SET served as a
transducer of the resonator?s motion: fluctuations in the resonator?s position mod-
ulated the SET impedance. The microwave-matching circuit allowed us to read-out
the modulation of the SET?s impedance with ? 75 MHz bandwidth. The com-
bination of microwave-matching circuit and SET is known as the radio-frequency
single-electron transistor (RFSET). Including the nanomechanical resonator, the
configuration is called the radio-frequency single-electron transistor displacement
detector.
In this thesis, I discuss the basics of quantum-limited measurement and some
of the subtleties of observing mechanical quantum phenomena. I then discuss the
basics of the RFSET displacement detector, its ultimate limits, its engineering and
operation, the first generation results, and finally what improvements could be made
to future generation devices.
THE RADIO-FREQUENCY SINGLE-ELECTRON
TRANSISTOR DISPLACEMENT DETECTOR
by
Matthew D. LaHaye
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements of the degree of
Doctor of Philosophy
2005
Advisory Committee:
Professor Fredrick C. Wellstood, Chair
Professor Keith C. Schwab, Co-Chair
Professor Christopher J. Lobb
Professor James R. Anderson
Professor Donald L. DeVoe
c? Copyright by
Matthew D. LaHaye
2005
Dedication
...tomylovelywife,Juhee,and herheart ofgold.
ii
Acknowledgements
This thesis is dedicated to my wife, Juhee Kothari LaHaye, for her enduring
support and patience throughout these six years. She bore the brunt of the long
hours, lost weekends, and endless uncertainty, yet still kept a smile on her face and
on mine. Juhee is the binding that holds this book together.
I am forever grateful to my family, Mom, Dad, Mark, and Chris. They gave
me the world, taught me the essentials, and then let me go on my way. With each
success and blunder, they have stood behind me the same. I have had a great life,
and they made that possible.
I am truly fortunate to have such wonderful in-laws. Mom, Dad, and Pooja
have o?ered to me nothing but love and enlightenment since the day we met.
I am especially indebted to my advisor Professor Keith Schwab. Keith has
been an inspiration, a mentor, and a friend. Along with the opportunity to travel
and meet other leading researchers in the field, he provided a lab full of equipment
and five years of ideas. From scratch, he built the LPS Nanomechanics group into
a first-class research team. I am fortunate and proud to have been a part of this
experience.
Also, thanks to Drs. Bruce Kane, Keith Miller, and Marc Manheimer for their
support over the years and the hard work they put into the quantum computing
group.
I owe many thanks to my labmates, fellow students, and sta? in both the
iii
physics department at the University of Maryland and at the Laboratory for Physical
Sciences.
Thanks to the the nanomechanics group, a very unique collection of talented
and friendly individuals: Emrah Altunkaya, Harish Bhaskaran, Dr. Olivier Buu,
Benedetta Camarota, Jared Hertzberg, Akshay Naik, and Patrick Truitt.
Particularly, I would like to thank Dr. Olivier Buu. Olivier provided me with
advice on everything from Matlab to cryogenics to life in general. He is a selfless
and tireless individual who contributed greatly to my education as an experimental
scientist. He played an integral role in these experiments, assembling the dilution
refrigerator and SET-Gain feedback circuit, and sharing the measurement responsi-
bilities with me.
Also, I would like to thank Benedetta Camarota for many, many insightful
conversations, much sound advice, fabrication support, and half-a-million laughs.
Benedetta also played an essential role in developing the device fabrication recipe.
Finally, thanks to Jared Hertzberg for reading drafts of this thesis and always
o?ering his opinion or a helping hand.
I must also take the opportunity to thank Dan Sullivan, Carlos Sanchez, and
Kenton Brown for their support, integrity, and friendship. Essentially, we all started
the program together and have been through many years of ups and downs. Dan?s
humor, wit and insight into everything from physics to politics to sports will not
be forgotten. Carlos was always willing to share his extensive knowledge, whether
it was music, politics, electronics or fabrication. I must particularly recognize the
advice he provided for SET fabrication. Kenton?s intelligence and generosity will
iv
also be greatly missed.
I would like to thank Josh Higgins, Ben Palmer, Jonghee Lee, and Nathan
Siwak for both friendly and technical conversations over the years.
I am grateful for the technical support and machine work that Les Lorenz,
J.B Dottellis, and Russell Frizzell provided. As well, I must thank them for many
pleasant and informative conversations.
Thanks to Toby Olver for keeping the cleanroom running and being so patient
with my requests.
Thanks to Butch Bilger for keeping the Helium flowing.
I would also like to thank Jane Hessing for her work over the years keeping
me up-to-date on paperwork, informed of deadlines, and answering my questions.
As well I must thank Margaret Lukomska for her work expediting purchase
orders, travel requests and various other questions.
I would like to thank Profs. Miles Blencowe and Aashish Clerk for their help
clarifying the subtleties of SET back action and the quantum limit.
Finally, I would like to sincerely thank the members of my thesis committee,
Profs. Keith Schwab, Fred Wellstood, Bob Anderson, Chris Lobb, and Don DeVoe
for taking the time to read through 200 pages of technical information and run-on
sentences, as well as to participate in my defense and provide very helpful feed-
back. I am especially grateful to Prof. Wellstood for accepting the responsibility of
?Chairman? and meeting with me on multiple occasions to discuss corrections.
v
TABLE OF CONTENTS
List of Figures ix
List of Tables xiv
List of Symbols xv
1Overview 1
1.1 ContextandMotivation ......................... 1
1.2 StructureoftheThesis.......................... 8
2 Introduction 11
2.1 TheQuantumLimitI:ThermalNoise.................. 12
2.2 TheQuantumLimitII:IdealDetection................. 20
2.3 Nanomechanical RFSET Displacement Detection . . . . . . . . . . . 38
3 Design and Fabrication 56
3.1 TheWafers ................................ 56
3.2 Silicon Nitride Membrane Fabrication . . . . . . . . . . . . . . . . . . 56
3.3 Fabrication of the Bond Pads and Tank Circuit . . . . . . . . . . . . 59
3.4 Fabrication of the Nanomechanical Resonator and SET . . . . . . . . 70
4 Apparatus 82
4.1 TheSamplePackage ........................... 82
4.2 TheDilutionRefrigeratorandShieldedRoom ............. 84
4.3 RefrigeratorWiring............................ 90
vi
5 The RFSET 102
5.1 RFSETReflectometry ..........................102
5.2 Gain Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 RFSET Displacement Detection 124
6.1 Methodology ...............................126
6.2 MechanicalNoiseThermometry.....................140
6.3 DiscussionofSETBackAction .....................157
7 Conclusions 169
7.1 ShotNoiseLimitedDetection ......................169
7.2 SampleThermalization..........................174
7.3 PartingMotivation ............................187
A Useful Mechanics Information 188
A.1 Euler-Bernoulli Theory and The Simple Harmonic Oscillator . . . . . 188
A.2 SpringConstantsforSETDetection...................193
A.3 CorrectionstoFrequencyDuetoTension................197
A.4 The Driven-Damped Harmonic Oscillator . . . . . . . . . . . . . . . . 201
A.5 TheMagnetomotiveTechnique .....................203
B Useful SET and RFSET Information 208
B.1 SETParameters..............................208
B.2 MeasurementCircuitParameters ....................227
B.3 CalibrationofChargeSensitivity ....................232
vii
Bibliography 238
viii
LIST OF FIGURES
1.1 Generic circuit schematic and SEM image of an RFSET displacement
detector................................... 6
2.1 Thermal occupation factor for typical nanoresonators . . . . . . . . . 14
2.2 Estimated decay time for a nanoresonator in a superposition state . . 16
2.3 Generic circuit schematic and SEM image of an RFSET displacement
detector................................... 40
2.4 Ultimate detection limit of a normal-state SET displacement detector 51
2.5 Ultimate detection limit of a normal-state SET displacement detector 53
3.1 Silicon nitride membrane fabrication . . . . . . . . . . . . . . . . . . 57
3.2 SEM image of RFSET displacement detector: tank circuit and bond
pads .................................... 67
3.3 SEM image of RFSET tank circuit inductor . . . . . . . . . . . . . . 68
3.4 Double-angleevaporation......................... 76
3.5 Fabricationofthenanoresonator..................... 79
3.6 SEM image of two RFSET displacement detectors . . . . . . . . . . . 80
3.7 SEM image of an RFSET displacement detector: Device 2 . . . . . . 81
4.1 Imageofasamplepackage......................... 83
4.2 Image of the dilution refrigerator . . . . . . . . . . . . . . . . . . . . 85
4.3 Images of the dilution refrigerator mixing chamber and sample stage . 86
ix
4.4 Picture of shielded room and optical table. The
4
He dewar is visible
behind the front pillar. A winch is used to raise the dewar over the
refrigerator. ................................ 87
4.5 Picture of the top of the optical table. DC connections at the top of
the fridge are made through ?-powder filters. Battery-powered pre-
amps and voltage sources as well as break-out boxes are shelved on
theelectronicsrack............................. 88
4.6 Image of ?-powderfilterandtransfercharacteristics.......... 89
4.7 Circuit schematic of wiring in the shielded room and refrigerator . . . 91
4.8 Transfercharacteristicsofpowderfilters ................ 92
4.9 Transfercharacteristicofgateline.................... 93
4.10 Image and transfer characteristics of microstrip heat sink . . . . . . . 96
4.11 Transfer characteristics of the microwave circuit: input. . . . . . . . . 98
4.12 Transfer characteristics of microwave circuit: output. . . . . . . . . . 99
5.1 CircuitschematicforRFSETreflectometry...............103
5.2 Plot of reflection coe?cient and I
SD
versus Q
g
.............106
5.3 RFSET reflectometry: plot of 1 MHz sideband modulation . . . . . . 110
5.4 I
SD
V
SD
V
g
map in the superconducting state. . . . . . . . . . . . . . . 113
5.5 ThederivativemapandRFSETgain..................114
5.6 Plot of sideband dependence on V
g
...................117
5.7 Modulation of sideband amplitude for RFSET gain stabilization . . . 118
5.8 Circuit schematic for RFSET gain stabilization . . . . . . . . . . . . 120
x
5.9 Circuitschematicforaudiofeedbackcircuit ..............121
5.10 Plot demonstrating RFSET gain stabilization . . . . . . . . . . . . . 123
6.1 Circuit schematic of RFSET displacement detector . . . . . . . . . . 125
6.2 Circuit schematic for RFSET detection of capacitivelyexcited nanores-
onators...................................130
6.3 Response of capacitively excited nanoresonator measured with RF-
SET,Device2...............................133
6.4 I
SD
V
SD
V
NR
mapofDevice2 ......................134
6.5 Measurement of time-domain response of a nanoresonator, Device 2 . 137
6.6 FFToftime-domainresponse,Device2.................138
6.7 Power spectrum measurements of a nanoresonators thermal motion . 145
6.8 Plot of a nanoresonator?s integrated thermal response versus refrig-
eratortemperature ............................147
6.9 Plot of a nanoresonator?s integrated thermal response versus refrig-
erator temperature and coupling voltage: Device 2 . . . . . . . . . . . 149
6.10 Plot of a nanoresonator?s integrated thermal response versus refrig-
erator temperature and coupling voltage: Device 1 . . . . . . . . . . . 150
6.11 E?ective resonator temperature, T
e
=56mK:Device2........151
6.12 Plot of the integrated resonator response scaled with coupling and
measurementcircuitgain:Devices1and2 ...............153
6.13 Plot of the lowest noise temperature achieved for RFSET displace-
mentdetection:Device2.........................155
xi
6.14 Plot of noise temperature versus coupling voltage: Device 1 . . . . . . 158
6.15 Plot of noise temperature versus coupling voltage: Device 2 . . . . . . 159
6.16 Plot of nanoresonator frequency shift versus coupling voltage . . . . . 161
6.17 Plots of nanoresonator e?ective damping versus temperature . . . . . 163
6.18 Plot of damping versus coupling: Devices 1 and 2 . . . . . . . . . . . 165
7.1 Noise temperature versus coupling: the shot noise limit. . . . . . . . . 170
7.2 Illustration of Nb-Nb semi-rigid coax for heat-flow calculation. . . . . 176
7.3 Thermal circuit for the SET and resonator on the SiN membrane . . 181
7.4 Numerical calculation of the temperature of phonons near the nanores-
onator as a function of bath temperature and power dissipated in the
SET ....................................183
A.1 Schematic of a prismatic, doubly-clamped nanoresonator . . . . . . . 189
A.2 Plot of the first four transverse modes of a doubly-clamped resonator 191
A.3 Plot of the e?ect of tension on the frequency of the first six modes of
adoubly-clampednanoresonator.....................199
A.4 Schematic of magnetomotive detection . . . . . . . . . . . . . . . . . 204
A.5 Plot of the response of a nanoresonator measured using magnetomo-
tivedetection................................206
B.1 Circuit diagram for I
SD
V
SD
V
g
measurement..............209
B.2 Normal-state I
SD
V
SD
V
g
map ......................210
B.3 Extraction of SET capacitance parameters: normal-state . . . . . . . 215
xii
B.4 ExtractionofSETresistanceparameters ................217
B.5 Failure of the normal-state SET parameter extraction . . . . . . . . . 220
B.6 Superconducting I
SD
V
SD
V
g
map ....................222
B.7 Extraction of SET junction capacitance: superconducting state . . . . 224
B.8 Circuit schematic for the measurement of the shot noise ring-up of
theRFtankcircuit............................228
B.9 Tankcircuitresponse...........................230
B.10Experimentaldeterminationofchargesensitivity............233
B.11 Charge sensitivity calibration: the Bessel function fit . . . . . . . . . 236
xiii
LIST OF TABLES
3.1 Tableoftankcircuitparameters:Devices1-4............. 69
3.2 Table of geometric parameters: Devices 1 - 4 . . . . . . . . . . . . . . 73
3.3 Table of e?ective spring constants: Devices 1 - 4 . . . . . . . . . . . . 73
3.4 TableofSETjunctionparameters:Devices1-4............. 77
3.5 Table of SET charging energy and gap energy: Devices 1-4 . . . . . . 77
4.1 Table of measurement circuit parameters: Devices 1 - 4 . . . . . . . . 100
A.1 Table of geometric constants for first six modes of a doubly-clamped
nanoresonator...............................192
A.2 Table of geometric parameters: Devices 1 - 4 . . . . . . . . . . . . . . 196
A.3 Table of e?ective spring constants: Devices 1 - 4 . . . . . . . . . . . . 196
B.1 TableofSETjunctionparameters:Devices1-4.............226
B.2 Table of SET charging energy and gap energy: Devices 1-4 . . . . . . 226
B.3 Table of measurement circuit characteristics: Devices 1 - 4 . . . . . . 231
B.4 Tableofchargemodulation:Devices1-4 ...............237
B.5 Tableofchargesensitivities:Devices1-4 ................237
xiv
LIST OF SYMBOLS
Symbol Definition
a
1
Relationship between resonator?s mid-point motion and
the average motion over the length of the SET island
?a
?,?
,?a
?
?,?
Lowering and raising operators of the input signal into
ideal linear amplifier
A Interaction strength between resonator and detector; also
charge-to-voltage ratio for SET gate in electrons; total
linear amplifier noise contribution in units of quanta; am-
plitude of resonator?s thermal response; cross-sectional
area of resonator
A
i
Linear amplifier noise contribution referred to input for
each quadrature in units of quanta; also cross-sectional
area of SET junctions
A
Q
Charge gain of RFSET electrometer based upon the 1
MHz calibration signal
A
r
Amplitude of reflected microwave signal
b Geometric constant for the first derivative of C
NR
with
respect to y
avg
b Geometric constant for the first derivative of C
NR
with
respect to y
avg
B Bandwidth; magnetic field magnitude
xv
Symbol Definition
?
b
?,?
,
?
b
?
?,?
Lowering and raising operators of the output signal from
ideal linear amplifier
?b Total rms noise in the amplitude at the output of ideal
linear amplifier
?b
i
Total rms noise for each quadrature at the output of ideal
linear amplifier
C
g
SET gate capacitance
C
gnr
Capacitance between SET gate and resonator
C
i
Individual SET junction capacitances
C
n
E?ective electromechanical capacitance for magnetomo-
tive detection, n
th
mode
C
NR
Capacitance of resonator to SET island
C
T
Tank circuit capacitance
C
?
Total SET island capacitance
? Quasiparticle gap energy of SET leads and islands (as-
sumed to be equal)
d
gnr
Spatial separation between SET gate and resonator
d
NR
Spatial separation between SET island and resonator
DJQP Double Josephson Quasiparticle Resonance
?epsilon1 Energy sensitivity of RFSET plus measurement circuit
?epsilon1
SET
Energy sensitivity of SET
xvi
Symbol Definition
e magnitude of electron charge
?
1
Projection of force on resonator?s fundamental mode
E Young?s Modulus of resonator
E
C
Charging energy of SET
E
J
Josephson energy of SET junctions
?f Half-width of tank circuit resonance, bandwidth of RF-
SET detection scheme; also used as noise equivalent
bandwidth of the resonator
F
i
Noise contribution to each quadrature of input signal
from ideal linear amplifier
?F
i
RMS noise contribution to each quadrature of input sig-
nal from ideal linear amplifier
?F
?
i
Change in free-energy of SET circuit when an electron
tunnels on or o? of the SET island
?
b
Damping of the resonator?s fundamental mode due to
dissipation in the thermal bath T
bath
?
d
Damping of the resonator?s fundamental mode due to
dissipation in the e?etive thermal bath T
det
?
e
Total e?ective damping including both damping from en-
vironment and SET
xvii
Symbol Definition
? Reflection coe?cient; also decoherence time of superpo-
sition of coherent states
?
max
Reflection coe?cient at maximum SET conductance
?
?
i
Tunneling rates onto and o? of SET island through SET
junctions
G Gain of measurement circuit including pre-amplifiers and
loss
?h Planck?s constant divided by 2?
I Moment of inertia for transverse, in-plane vibration of
rectangular bar
IR Integrated resonator response
I
SD
SET source-drain current
JQP Josephson Quasiparticle Resonance
K
m
Spring constant of resonator defined with respect to av-
erage motion over the length of the SET island for the
fundamental mode
k
B
Boltzman?s constant
k
n
Roots of the characteristic equation for the transverse
vibration of a doubly-clamped rectangular bar
K
1
Spring constant of resonator defined with respect to mid-
point motion for the fundamental mode
xviii
Symbol Definition
? Admittance or response of detector current to resonator
displacement; also wavelength of microwave applied to
RFSET; also roots for characteristic equation for the
transverse vibration of a doubly-clamped rectangular bar
under tension
L Length of resonator; also, gain of linear amplifier in num-
ber quanta
L
n
E?ective electromechanical inductance for magnetomo-
tive detection, n
th
mode
L
T
Tank circuit inductance
m Geometric mass of the resonator
M Depth of modulation of reflection coe?cient
M
m
E?ective mass of resonator with respect to average dis-
placement over the length of the SET island for the fun-
damental mode
m
eff
E?ective mass of resonator with respect to mid-point mo-
tion for the fundamental mode
m
i
Slopes of JQP resonance; and slope of the onset of tun-
neling in the normal state
n Number of electrons on SET island; also mode number
of transverse, in-plane resonator motion
xix
Symbol Definition
N
a
Number of quanta in the input signal to ideal linear am-
plifier
N
b
Number of quanta in the output signal from ideal linear
amplifier
?n
th
? Thermal occupation number of resonator?s fundamental
mode
? Volume of SET island
?
1
Resonator?s fundamental mode frequency multiplied by
2?
?
T
Resonant frequency of tank circuit multiplied by 2?
P Resonator?s thermal response
P
o
Background or noise-floor of detection scheme
P
1MHz
Power in 1 MHz sideband gain calibration
? Potential of SET island
Q Total e?ective quality factor of tank circuit
?
Q Heat transfer down Nb microwave coax and dissipation
in SET
?
Q
1
Power conducted to bath through SiN membrane
?
Q
2
Power conducted through SiN membrane from SET to
region near nanoresonator
xx
Symbol Definition
?
Q
3
Power conducted through SiN membrane from region
near nanoresonator to bath
?
Q
4
Power conducted through electron-phonon coupling to
electrons in Au film on top of the nanoresonator
Q
b
Quality factor of the resonator?s fundamental mode, ac-
counting only for ?
b
Q
d
Quality factor of the resonator?s fundamental mode, ac-
counting only for ?
d
Q
e
Total e?ective resonator quality factor
Q
g
Polarization charge on SET gate electrode
?Q
g
Charge modulation on SET gate
Q
NR
Polarization charge on resonator electrode
Q
S
Quality factor of tank circuit due to loading from SET
Q
T
Quality factor of tank circuit due to loading from trans-
mission line
RFSET Radio Frequency Single-Electron Transistor
? Mass density of resonator
R
i
Individual SET junction resistances
R
n
Real-component of electromechanical impedance for
magnetomotive detection, n
th
mode
R
Q
Resistance quantum
xxi
Symbol Definition
R
?
Total resistance of SET junctions at large V
SD
R
S
Total resistance of SET junctions
SET Single-Electron Transistor
?
S
F
Symmetrized back action noise spectral density of linear
detector
?
S
F
b
, S
f
Symmetrized thermal noise spectral density
S
y
F
Back action contribution to the spectral displacement
sensitivity
?
S
I
Symmetrized forward coupling noise spectral density of
linear detector
? Electron-phonon coupling
?
int
Intrinsic axial stress in SiN film
S
II
SET shot noise spectral density
S
I?
Correlations between SET shot noise and island-potential
fluctuations
S
y
I
Shot noise contribution to the spectral displacement sen-
sitivity
SNR Signal-to-noise ratio of 1 MHz sideband gain calibration
S
??
SET island-potential fluctuations spectral density
S
Q
, S
Q
NR
Spectral density of charge sensitivity in units of e
2
/Hz
S
y
Total spectral displacement sensitivity
xxii
Symbol Definition
? E?ective resonator damping time; also used as measure-
ment time
T Intrinsic tension on SiN resonator
t
Au
Thickness of resonator?s Au layer
T
b
Temperature of the thermal bath, the?environment? or
?external bath?
T
d
E?ective temperature of the SET noise environment
T
e
Total e?ective temperature of the resonator due to con-
tact to T
bath
and T
det
T
island
SET island electron temperature
T
N
Noise temperature of displacement detection scheme
T
det
n
Noise temperature contribution of measurement circuit
including pre-amplifier and circuit loss
T
o
Noise temperature of measurement circuit plus the SET
shot noise contribution
T
QL
Quantum limit of noise temperature for continuous linear
detection
T
S
Temperature of the dilution refrigerator?s sample stage
T
SET
Noise temperature contribution of the SET shot noise
t
SiN
Thickness of resonator?s SiN layer
v
c
Incident microwave signal, or carrier
xxiii
Symbol Definition
V
D
Capacitive excitation voltage
V
g
SET gate bias
?V
g
Voltage modulation on SET gate
v
n
Average transverse, in-plane velocity of doubly-clamped
rectangular bar over the bar?s length, n
th
mode
V
NR
Resonator bias, also referred to as coupling voltage
v
r
Reflected microwave signal
V
SD
SET source-drain bias
V
?,i
t
Threshold voltage for tunneling of electrons onto (o? of)
the SET island through junctions i, normal state SET
w Width of resonator
? frequency in rad/s
?
1
fundamental mode frequency in rad/s
X
i
Signal quadratures at the input of ideal linear amplifier
Y
n
Eigenfunctions of the wave-equation for in-plane dis-
placement of the resonator
?y
QL
Quantum limit for continuous linear position detection
?y
ZP
Zero-point fluctuations of mechanical resonator for in-
plane, fundamental mode
y
1
In-plane displacement of the resonator?s mid-point for the
fundamental mode
xxiv
Symbol Definition
y
1,m
, y
m
Average, in-plane displacement of resonator over the
length of the SET island for the fundamental mode
?y, ?y
m
Total displacement sensitivity in resonator?s noise-
equivalent bandwidth
Z
o
50 ? Transmission line impedance
Z
LC
Characteristic impedance of tank circuit
Z
LCR
Impedance of tank circuit and SET at the resonant fre-
quency of the tank circuit
Z
m
Electromechanical impedance for magnetomotive detec-
tion
xxv
Chapter 1
Overview
1.1 Context and Motivation
The Heisenberg uncertainty principle [1] places a limit on the precision with
which one can measure an object?s position [2]. For the case of two successive
measurements of a mass M undergoing simple harmonic motion, this limit, known
as the ?standard quantum limit?, is neatly expressed as [2]
?y
SQL
=
radicalBigg
?h
2M?
, (1.1)
where ?/2? is the frequency with which the mass oscillates, and ?h is Planck?s con-
stant.
Since the 1970?s, researchers have been engaged in both theoretical and experi-
mental e?orts to understand and implement mechanical detectors and measurement
strategies for displacement detection at (or even below) the standard quantum limit
[2-17].
Inititally, the impetus for quantum-limited displacement detection arose out
of the hunt for gravitational waves [2]. Through nearly three decades of e?ort,
the gravitational-wave community has moved quantum-limited detectors from mere
thought-experiments to nearly practicable measurement devices. For example, the
4 km L1 interferometer of the LIGO I project has demonstrated a sensitivity, at
200 Hz, of ?y ? 150?y
SQL
[11] for the displacement detection of its 10 kg test-
1
masses. Researchers at Laboratoire Kastler Brossel, employing a tabletop Fabry-
Perot interferometer, recently achieved a displacement sensitivity of ?y ? 25 ?y
SQL
for the read-out of the 2 MHz surface modes of a silica mirror [7]. The SQUID-based
amplifiers developed for the Auriga project have demonstrated noise temperatures
of ? 10?s ?K, corresponding to sensitivities of ?y ? 100 - 200 ?y
SQL
for the read-
out of the vibrational modes of ? 2000 kg acoustic bar resonators [9]. As well,
researchers in the Supeconductivity Center at the University of Maryland used a
scheme based on a Paik-style transducer [19] to achieve a noise temperature of ? 1
mK at 900 Hz, yielding ?y ? 200 ?y
SQL
[8].
In the last decade, the development of nanoelectromechanical systems (NEMS)
has generated a second wave of interest in the standard quantum limit. Driven
by potential applications to ultra-sensitive imaging [20] [21], mass detection [22],
and quantum computing [23] [24], as well as, ultimately, the possibility to study
mechanical quantum systems in the macroscopic limit [23-33], the NEMS community
has rapidly pushed mechanical transduction to the quantum frontier. In the last
year alone, several important results have been generated. For example, researchers
at IBM used magnetic resonance force microscopy (MRFM) [20] to detect the spin
of a single electron [21]. Using a magnetomotive technique [22], researchers at
the California Institute of Technology demonstrated mass sensivity on the order of
zeptograms, su?cient for the detection of a single molecule [34]. Finally, our group
in the Laboratory for Physical Sciences at the University of Maryland used the radio-
frequency single-electron transistor displacement detector [12] [13] to demonstrate
both displacement sensitivity approaching closer to the standard quantum limit
2
than any previous measurement scheme (?y ? 5.8 ?y
SQL
1
) [15] and an approach
to low thermal occupation numbers (?n
th
??60) for a 20 MHz nanomechanical
resonator[15].
It is important to note that the achievement of low thermal occupation num-
bers is a general and significant point of distinction between NEMS devices and
the resonators used in gravitational-wave detection. We can see why this is by first
looking at the definition of the thermal occupation number. For a resonant mode
with frequency ? in thermal equilibrium with a bath of temperature T, the mode?s
thermal occupation number is given by [35]
?n
th
? =
1
2
+(e
?h?/k
B
T
?1)
?1
, (1.2)
where k
B
is Boltzman?s constant and ?
1
2
? accounts for the mode?s zero-point fluc-
tuations. This quantity provides a simple ?rule-of-thumb? for gauging whether one
should be able to observe a mode?s quantum properties:
k
B
T
?h?
? 1, (1.3)
If Eq. 1.3 is satisfied, the mode is said to be ?frozen out?. That is, the mode is in it?s
ground state and the contribution of the thermal energy to the mode?s total energy
is comparable to or less than the zero-point contribution. As k
B
T/?h? grows, so too
does the contribution from thermal fluctuations, making it more di?cult to observe
the mode?s quantum attributes. There is no general prescription for how small the
ratio k
B
T/?h? must be before quantum behavior becomes observable (see Chapter
1
In Chapter 2, I make the distinction between ?y
SQL
and ?y
QL
. In terms of ?y
QL
,amore
appropriate gauge for continuous position detection, we achieved ?y ? 4.3?y
QL
3
2). For the purpose of observing the resonator in a pure quantum state such as
a Fock state or superposition state, or for detection of the resonator?s zero-point
fluctuations, the smaller the ratio the better (see Chapter 2).
For the above-mentioned gravitational-wave detectors, the operating temper-
atures were su?ciently high (> 1 K) and resonators? frequencies were su?ciently
low (< 5MHz), that, at a minimum, ?n
th
??3x10
6
(the Auriga project at 1.5
K and the Fabry-Perot scheme at 300 K). In contrast, because NEMS devices have
demonstrated resonant frequencies as high as ? 1 GHz [36] [23] and are routinely
installed on cryogenic probes for measurement at mK temperatures, it should be
possible for researchers to observe ?n
th
??1.
The demonstration of nearly quantum-limited position detection and low ther-
mal occupation numbers promises NEMS researchers the opportunity to push the
study of quantum mechanics to a significantly larger realm. For example, one recent
proposal to prepare and measure a nanomechanical resonator mode in a superposi-
tion of position states could be implemented if one could cool the mode to ?n
th
??50
[25] (please see Refs. [23-33] for other recent proposals). This is signficant because,
while NEMS devices are, by definition, nanoscopic, they are typically composed of
? 10
10
atoms. With a few exceptions, such as the measurement of the quantum
of thermal conductance [37], previous demonstrations of mechanical quantum phe-
nomena have been limited to the scale of molecules and atoms (for exmaple Refs.
[38] [39] [40]).
In this thesis, I discuss the details of the first generation of radio-frequency
single-electron transistor (RFSET) displacement detectors. The technique was first
4
proposed by Miles Blencowe and Martin Wybourne [12] and utilizes the RFSET?s
large bandwidth (demonstrated to be > 100 MHz) [41] and near-ideal noise char-
acteristics [42] to perform displacement detection near the quantum limit. Figure
2.3(a) shows an SEM image of an RFSET displacement detector, and Fig. 2.3(b)
shows a generic circuit schematic for the transduction process. Here, a metallized
SiN nanoresonator is positioned within 1 ?m of an SET island, resulting in a cou-
pling capacitance C
NR
on the order of 10?s aF. Displacement of the nanoresonator
from its equilibrium position linearly modulates the coupling capacitance through
?C
NR
?
C
NR
d
NR
?y, (1.4)
where d
NR
is the separation between the nanoresonator and the SET island, and ?y
lessmuch d
NR
is the displacement of the resonator from equilibrium. Establishing a voltage
V
NR
between the resonator and the SET converts the capacitance fluctuations into
charge fluctuations:
?Q
NR
?
C
NR
V
NR
d
NR
?y. (1.5)
The charge fluctuations modulate the SET impedance which is then monitored by
performing microwave reflectometry [41]. The use of an on-chip tank circuit (L
T
and
C
T
in Fig. 2.3(b)) allows for matching between the large SET impedance (typically
10?s k?) and 50 ? transmission line.
Ultimately, the sensitivity of the RFSET displacement detector is limited by
the intrinsic noise of the SET [12] [13]. This is composed of two sources [43] (1)
the SET shot noise and (2) the potential fluctuations of the SET island. The SET
shot noise is forward coupling. That is, it simply adds to the signal, resulting in
5
(a)
(b)
Figure 1.1: (a) SEM image of the RFSET displacement detector and (b) Circuit
schematic
6
a contribution to the total displacement noise that is inversely proportional to the
coupling C
NR
V
NR
/d
NR
. The island-potential noise is back-acting. That is, the
island-potential fluctuations couple to the resonator through C
NR
and drive it, re-
sulting in a contribution to the total displacement noise that is linearly proportional
to the coupling. A minimum in the total displacement noise is found at a coupling
strength where the two sources contribute equally. For such optimal coupling, and
typcial device parameters (see Chapter 2), the total displacement noise has been
predicted to be ?y ? 2?y
QL
[13].
In the measurement of the first generation of RFSET displacement detectors
(LPS), we were not limited by the intrinsic noise of the SET. Instead, we were lim-
ited by the 80 pV/
?
Hz noise (referred to the input) of our cryogenic pre-amplifier
(see Chapter 4 and Chapter 7), which set our charge sensitivity at approximately a
factor of 4 - 6 from the SET?s intrinsic shot noise limit. Consequently, the lowest dis-
placement sensitivity which we observed was on the on the order of a factor of 4 from
the quantum limit [15]. Nevertheless, this is the closest approach to the quantum
limit of displacement detection that anyone before or since has demonstrated, and
marks a factor of 30 improvement over the SET-mixer technique previously demon-
strated by Robert Knobel and Andrew Cleland at the University of California, Santa
Barbara [14].
An additional improvement of the LPS detectors over the Santa Barbara SET-
mixer technique was the ? 75 MHz bandwidth provided by the rf-matching network.
In contrast, at best, the maximum bandwidth of the Santa Barbara technique would
have been on the order of kHz, either limited by the DCSET or the dc electronics
7
at room temperature. Either way, as the quality factor and resonant frequency of
the resonator were ? 1.5 x 10
3
and 116 MHz respectively [14], the half-width of the
resonator?s spectral response was ? 10
5
and thus much larger than the detection
bandwidth.
The large bandwidth of the RFSET technique allowed us to observe the res-
onator?s full spectral response, facilitating the detection of the resonator?s thermal
motion. In the end, we were able observe the thermal motion of the nanoresonator
down to a temperature as low as ? 56 mK, corresponding to a thermal occupation
number of ?n
th
??60, and demonstrating, that, indeed, NEMS is on the verge of
the quantum regime.
1.2 Structure of the Thesis
The structure of this thesis is as follows.
Chapter 2 provides the basic definitions and theoretical concepts upon which
the rest of the text is based. First, the quantum limits of a mechanical resonator are
defined, and the criteria for reaching these limits are presented. This is followed by
the introduction of the RFSET displacement detector and a discussion of its basic
operating principles. In the final section, the intrinsic noise properties of the SET
are reviewed and used to demonstrate that, in principle, the RFSET is capable of
performing as a nearly quantum-limited displacement detector.
Chapter 3 presents a detailed account of the fabrication steps we developed
and followed to produce our first generation of RFSET displacement detectors.
8
Chapter 4 discusses the details of the apparatus which we constructed for the
measurement of our samples.
Chapter 5 presents and explains the RFSET reflectometry technique, the back-
bone of the detection scheme.
Chapter 6 describes the implementation of the RFSET displacement detection
technique and presents our main research results. Relying heavily on the results of
Chapter 5, it begins with a treatment of the basic methodology. This is followed by
a discussion of the RFSET detection of capacitively driven nanoresonators. Next,
the topic of nanomechanical noise thermometry is introduced. It is in this section
that the central results of the thesis are put forth. Finally, the chapter finishes by
addressing the issue of SET back action.
Chapter 7 concludes the main body of the thesis with a discussion of the
technical improvements and future prospects.
The remaining chapters I label as Appendix A and Appendix B. They contain
information that I think is either essential for understanding the basic concepts
and limitations of the RFSET displacement detector or is useful for the actual
implementation. Included in these chapters are tables of the various parameters for
the devices around which this thesis is built.
Two of the devices, Device 3 and Device 4, are included even though they are
not discussed in the main body of the dissertation. Initially, my intent was to pro-
duce a work that fully addresses the noise characteristics of the RFSET displacement
detector, including the SET back action. Devices 1 and 2 were to be used for treat-
ing the forward-coupling limit. Devices 3 and 4 were to be used for discussing the
9
back action limit. However, the physics involved with SET back action, particularly
the superconducting SET, are more complicated and interesting than I originally
imagined, and their investigation would constitute an entire thesis. Furthermore,
we do not understand all the observations that we have made of Devices 3 and 4.
I have left Devices 3 and 4 in the thesis mainly for illustrative purposes and for
technical explanations of useful information (ie. RFSET gain calibration and RF
tank-circuit characterization). Additionally, I would like to have the parameters and
characteristics of all four devices and accompanying measurement circuits cataloged
in one place.
Finally, Device X and Device Y, devices which are not in any of tables, I have
also used for illustrative purposes in Chapter 5. The nanoresonator in Device X met
an early demise, however, the data taken for the gain-feedback circuit and sideband
amplitude versus V
g
is the best data I have to illustrate these techniques. Device
Y is actually from the latest generation of devices (courtesy of Akshay Naik). I
used this data to illustrate the equivalence of the reflection map and the numerical
derivative of the IV map. In the earlier devices, either this data is incomplete
(for Devices 1 and 2 I have no simultaneous measurements of reflection map and
numerical derivative) or the IV maps were less ?photogenic? (for Devices 3 and 4
the DJQP and JQP resonances are either smeared or faint). I also used data from
Device Y to help illustrate the principle of amplitude modulation.
10
Chapter 2
Introduction
At first glance, it might not be obvious why one can treat a nanomechanical
device, such as a doubly-clamped resonator, as a simple harmonic oscillator. After
all, a typical structure might have dimensions ranging from nanometers to microns,
and be comprised of tens of billions of atoms and three times as many normal
vibrational modes.
The situation is simplified, though, if one is only interested in the lowest-
frequency transverse modes. In this case, the ratio of the wavelength-to-lattice
spacing is su?ciently large, ? 10
4
, that deformation of the lattice occurs slowly over
the length of the device, allowing for the use of continuum elasticity theory to model
the mode?s behavior [44] [45]. For deformations smaller than a critical amplitude
[46], non-linear e?ects are negligible. Below the critical amplitude, the system can be
reduced to a simple harmonic oscillator with an e?ective mass and spring constant
determined by the mode shape and the portion of the oscillating structure that one
considers (see Appendix A). The critical amplitude for the resonators measured in
this research can be calculated to be ? nm?s [46]. The typical displacements we
measure are ? pm?s.
Peering at such a structure, for example, through an optical microscope, if
our eyes and brains had the temporal resolution, we would expect to see it jumping
11
about, its motion driven by thermal fluctuations and other classical interactions.
We might not expect to observe any deviations from the classical behavior we are so
familiar with from our daily experiences. However, the question arises: what would
it take to observe one of these structures exhibiting quantum behavior?
In this chapter, I present some basic criteria which, when met, could allow for
the observation of quantum phenomena in macroscopic mechanical resonators [4].
The first criteria, which I call Quantum Limit I, establishes an approximate level
to which classical interactions must be reduced in order to observe the resonator?s
quantum dynamics. It is implicit in my discussion that thermal fluctuations are the
biggest problem and that all other classical forces are negligible. The second criteria,
which I call Quantum Limit II, establishes the characteristics that a linear amplifier
must possess in order that it minimally disturb the resonator during the process
of measurement. It is seen that quantum mechanics requires such an amplifier to
add a minimum of one-half quanta of noise power in the bandwidth of the signal.
In the final section, I present and discuss the basics of the radio-frequency single-
electron transistor (RFSET) displacement detector, a detection scheme which we
have implemented and which has allowed us to come closer than any previous scheme
to satisfying both criteria.
2.1 The Quantum Limit I: Thermal Noise
The question of how cold a mechanical mode must be before thermal fluctua-
tions are reduced to a level that does not obscure the mode?s quantum dynamics is
12
rather subtle. To thoroughly treat the topic is beyond the scope of this section and
thesis. However, here, I present some basic, case-specific constraints on temperature
with which I can later guage our experimental results (Chapter 5 and Chapter 6).
I first discuss the freeze-out of a mechanical mode to its ground state. This is the
simplest case to treat and provides a back-of-the-envelope estimate of how ?quan-
tum? a particular mode at a given temperature is (ie. whether or not the mode?s
dynamics can be described by classical equations of motion). Second, I briefly ex-
amine the issue of decoherence. In particular, I discuss the decoherence of a pure
harmonic oscillator state due to linear coupling to a thermal bath, and present an
expression for the decoherence rate of a superposition of position states in terms of
the mode?s temperature. Finally, I consider the detection of a mechanical mode?s
zero-point motion in the presence of thermal noise. I show that, even if k
B
T
b
greatermuch ?h?
1
,
depending on the duration of the measurement and the coupling of the resonator to
the thermal bath T
b
, it is possible to reduce the change in amplitude due to thermal
fluctuations below that due to zero-point fluctuations.
Freeze-Out
The simplest constraint to consider is a resonator?s ?freeze-out? to the ground
state. This is equivalent to determining the temperature at which a mode?s thermal
occupation number is reduced signifcantly below 1. The average thermal occupation
of an oscillator mode with frequency ?
1
/2? is given by [35]
?n
th
? =(e
?h?
1
/k
B
T
b
?1)
?1
, (2.1)
13
Figure 2.1: Thermal occupation number ?n
th
? plotted as a function of temperature
for a range of nanomechanical resonant frequencies. The dashed lines represent the
large-T
b
limit given of Eq. 2.1. Note that the zero-point contribution of
1
2
has not
been included.
14
where ?h is Planck?s constant, k
B
is the Boltzman constant, T
b
is the temperature of
the mode, and the criteria for freeze-out is just
k
B
T
b
?h?
1
? 1. (2.2)
Note that I have neglected the zero-point contribution of
1
2
.
Figure 2.1 shows the thermal occupation number for mode frequencies ranging
from 100 KHz to 1 GHz. This range roughly represents the realm of demonstrated
doubly-clamped, nanomechanical mode frequencies. Examination of the plot re-
veals that achieving freeze-out with passive refrigeration techniques (eg. dilution
refrigeration) requires working with resonant frequencies in excess of 100 MHz. Of
course with adiabatic demagnetization, the limit could be pulled down toward 10
MHz. Note, though, that I have not taken into account the issue of thermalization
of the mechanical mode of interest. Whether a mechanical mode at 100 MHz can
be tightly coupled to, say, the mixing chamber of a dilution refrigerator is a compli-
cated problem that depends on both the experimental apparatus (eg. connections,
?heat leaks?, etc.) as well as the resonator?s geometry and material (essentially the
parameters that determine the resonator?s quality factor), and one that I address in
Section 7.2.
Decoherence of a Mechanical Superposition
The temperature constraint for the observation of a mechanical superposition
state depends on the quality factor of the resonator under measurement and the
desired duration of the superposition. A theoretical treatment of the harmonic
15
Figure 2.2: Decay time of a superposition of coherent states versus temperature for
a range of nanomechanical resonant frequencies and quality factors. It is assumed
that Gaussian peaks of the coherent states are separated by 2?y
zp
.
16
oscillator states suggests that a mechanical superposition of two coherent states
with spatial separation ?y between the Gaussian peaks will preferentially decay to
a single coherent state at a rate given by [47]
?=
4k
B
T
b
?hQ
b
parenleftBigg
?y
?y
zp
parenrightBigg
2
, (2.3)
where Q
b
= ?
1
? is the resonator?s quality factor, assumed to be determined strictly
from coupling to the thermal bath, ?y
zp
=
radicalBig
?h/2M
m
?
1
is the resonator?s zero-point
deviation, and M
m
is the resonator?s e?ective mass.
Figure 2.2 displays a plot of the inverse of the decay rate versus temperature
for resonators with quality factors in the range of 10
2
to 10
5
. Here I assume that
the superposition has been prepared so that ?y =2?y
zp
for each case. Thus the
resonant frequency does not factor into Eq. 2.3. However, the quality factor for each
of the resonators has been chosen to roughly reflect what has been demonstrated
experimentally with real nanomechanical resonators. Ideally, one would want to
engineer a nanoresonator with both large Q
b
andhighfrequency,sayQ
b
? 10
5
and
f
1
? 1 GHz, so that the decay would occur over many cycles at 50 - 100 mK. From
Fig. 2.2, for such a device at 50 mK, the decay time would be on the order of 10
3
cycles. In practice, achieving such a large quality factor and high resonant frequency
might prove di?cult. To date, the only published, doubly-clamped 1 GHz resonator
demonstrated a quality factor of approximately 10
2
[36], which would yield one cycle
over the decay time at 50 mK. On the other hand, a 10 MHz resonator with quality
factor in excess of 10
5
has recently been demonstrated [48], which would yield ? 10
coherent cycles at 50 mK.
17
Thermal Amplitude Flutuations
In this subsection I estimate the temperature below which thermal fluctuations
in a resonator?s amplitude become negligible with respect to the resonator?s zero-
point motion.
The amplitude of a resonator in contact with a thermal bath T
b
is seen to
undergo a ?random-walk? with a variance approximated by [49]
?y
2
m
??
k
B
T
b
M
m
?
2
1
parenleftBig
1?e
?t/?
parenrightBig
, (2.4)
where ?=Q
b
/?
1
is the resonator?s thermal-relaxation time and I have assumed that
at time t = 0 that the amplitude is known precisely, ie.that?y
2
m
? = 0. I note that
the subscript ?m? is used for consistency with later portions of the thesis. It denotes
the mean displacement of the neutral surface over the segment of the nanoresonator
that couples to the SET detector, essentially the length of the SET island.
For times t greatermuch ?, Eq. 2.4 reduces to the standard equipartition relationship.
In this case, I expect thermal fluctuations of the amplitude to become small with
respect to the resonator?s zero-point fluctuations when [4]
k
B
T
b
M
m
?
1
2
? (?y
zp
)
2
(2.5)
or
k
B
T
b
?
?h?
1
2
. (2.6)
This is a rather strict condition, and nearly identical to the criteria for freeze-out.
On the other hand, for t lessmuch ?, the fluctuations in the resonator?s amplitude
18
are seen to increase linearly with t:
?y
2
m
??
k
B
T
b
M
m
?
1
2
t
?
. (2.7)
The condition for thermal fluctuations to be small with respect to ground-state
uncertainty in position is then [4]
k
B
T
b
M
m
?
2
1
t
?
?
?h
2M
m
?
1
(2.8)
or
T
b
?
?hQ
b
2k
B
1
t
. (2.9)
Clearly this is a less stringent requirement; and it implies that, if one could
prepare an high-Q resonator in a well known position at time t =0andthenmakea
measurement in a time t lessmuch 1/?, the exchange of energy between the resonator and
the thermal bath would be a fraction t/? smaller than k
B
T
b
. Strictly speaking, then,
the temperature to which one would have to cool a particular resonator for thermal
fluctuations to become negligible would be inversely proportional to how quickly
one could make a measurement of the resonator?s position and linearly proportional
to the quality factor.
This is just an order-of-magnitude analysis, and it begs a couple of questions:
can one specify the position of the resonator with ?y
m
2
? = 0? And, what is the
e?ect of the detector on the resonator during the measurement process? These are
questions that I address in the following section.
19
2.2 The Quantum Limit II: Ideal Detection
In this section I consider a second aspect of the quantum limit dealing with
optimizing measurement precision. Ultimately, quantum mechanics places a limit
on the precision with which certain information (ie. conjugate coordinates) can be
extracted from the measurement of an object
1
ignore here, and throughout the
thesis. This fundamental measurement limit is a direct result of the Heisenberg
uncertainty relations for both the measured object?s coordinates of interest (eg.?y and
?p
y
of an oscillator) and the measurement device?s detection coordinates (eg.
?
I and
?
V of transistor). The purpose of this section, then, is to develop an understanding
of such constraints in the context of the measurement of the displacement of a
mechanical mode, and determine the conditions necessary to perform detection at
this fundamental limit.
Initially, I consider the simple case of ?quick?, repeated measurements of an
harmonic oscillator?s position coordinate ?y, and derive the so-called Standard Quan-
tum Limit for position detection. I then discuss the case of continuous linear de-
tection of a generic narrow band signal, and derive the corresponding quantum
constraints on amplifier noise temperature. Finally, I use linear response theory
to phrase the quantum constraints on position detection in terms of an amplifier?s
intrinsic noise characteristics. The result is thus a prescription which an amplifier
must fulfill in order to operate as a quantum-limited position detector; and fur-
1
Techniques (eg. squeezed states, QND measurement, and contractive states) have been pro-
posed for beating the quantum limit (for example, see Refs. [2] [4] [5] [6] [17]). However, these
advanced measurement strategies are beyond the scope of the research presented here.
20
thermore, a guage by which I can assess our experimental results (Chapter 6 and
Chapter 7).
The Standard Quantum Limit
Following reference [2], I start with a crude derivation of the Standard Quan-
tum Limit. Consider a mechanical mode undergoing simple harmonic motion with
frequency ?
1
. The hamiltonian for such a system is given by
H =
p
2
y
2M
m
+
1
2
M
m
?
2
1
y
2
m
, (2.10)
where p
y
and y
m
are the conjugate momentum and displacement of the resonator
and M
m
is the e?ective mass for the motion of interest (See Appendix A). I note
that, as in the previous section, y
m
is used for consistency with later sections in
which it denotes the average displacement of the neutral surface over the length of
the SET detector.
The task at hand is to determine how precisely one can measure y
m
by mak-
ing two measurements such that the measurement time ? lessmuch 1/?
1
. For a classical
resonator, in principle, there is no limit on how precisely one can measure y
m
or p
y
.
However, for a quantum resonator, the resonator?s position and momentum are de-
scribed by the operators ?y
m
and ?p
y
, which are constrained through the commutation
relation [50]
[?y
m
, ?p
y
]=i?h
21
to obey the Heisenberg uncertainty principle
?y
m
?p
y
?
?h
2
. (2.11)
Simply put, the more precisely one specifies ??y
m
?, the less precisely one can
know ??p
y
?. This is not really a concern for one quick measurement of ??y
m
?;in
principle, it can be done with arbitrary precision. However, if one intends to make
two or more measurements of ??y
m
? with the highest precision possible, the e?ect of
the measurement on ?p
y
, or the quantum mechanical back action, must be taken into
account.
From Eq. 2.10, in the Heisenberg representation, the equations of motion for
?y
m
and ?p
y
are given by [51]
?y
m
(t)=?y
m
(0)cos ?
1
t +
?p
y
(0)
M
m
?
1
sin?
1
t (2.12)
and
?p
y
(t)=?M
m
?
1
?y
m
(0)sin ?
1
t +?p
y
(0)cos ?
1
t. (2.13)
If the resonator is not in an energy eigenstate, then the expectation values
??y
m
(t)? and ??p
y
(t)? will be oscillatory functions of time with the respective variances
given by [2]
(?y
m
(t))
2
=(?y
m
(0))
2
cos
2
?
1
t +
parenleftBigg
?p
y
(0)
M
m
?
1
parenrightBigg
2
sin
2
?
1
t (2.14)
and
(?p
y
(t))
2
=(?y
m
(0)M
m
?
1
)
2
sin
2
?
1
t +?p
y
(0)
2
cos
2
?
1
t. (2.15)
I see that if, at time t = 0, I make an initial measurement??y
m
(0)? with precision
?y
m
(0), the uncertainty in the resonator?s position due to the initial measurement
22
at a time t later is
(?y
m
(t))
2
? (?y
m
(0))
2
cos
2
?
1
t +
parenleftBigg
?h
2M
m
?
1
parenrightBigg
2
sin
2
?
1
t, (2.16)
where I have assumed that there is no correlation between the uncertainties in ??y
m
?
and ??p
y
?, only that the rms amplitudes are related through the uncertainty principle,
Equation 2.11.
To minimize the uncertainty in position due to the initial measurement, it is
clear that I must have
?y
m
(0) =
?h
?y
m
(0)2M
m
?
1
(2.17)
or
?y
m
(0) =
radicalBigg
?h
2M
m
?
1
. (2.18)
This is known as the Standard Quantum Limit (SQL) for position detection
[4] [2]. From Equation 2.16, for such a measurement, ?y
m
(t) is constant in time,
implying a resonator state with phase-insensitive noise. One set of phase-insensitive
states, with the additional stipulation that the equality in Eq. 2.16 be satisfied, is
the set of coherent states [52]. Thus I can conclude this section by stating that,
to minimize the error in each of two consectutive quick measurements of ??y
m
(t)?,
it is necessary that the first measurement projects the resonator into a minimum
uncertainty state.
The Ideal Linear Amplifier
While the analysis of the preceding section provides us with an idea of the
role of the Heisenberg uncertainty principle in measurement, it is unsatisfactory
23
for at least two reasons. First, one is not always interested in simply making two
consecutive quick measurements of a system. For example, the results presented
in this thesis were obtained in the continuous measurement limit (ie. the limit in
which the time interval between measurements becomes small with respect to the
time scale of the dynamics of the measured system). Second, the analysis makes no
reference to a measuring device, relying only upon the uncertainty relation for the
measured oscillator, or, essentially, its wave nature. Which begs the question: what
istheroleofthedetector?
In this section, I paraphrase a work of Carleton Caves [3] and derive the quan-
tum measurement limit for the case of a quantum signal continuously measured by a
linear quantum amplifier. It is seen that such a detection scheme necessarily adds a
minimum of one-half of a quanta of noise to the measured signal. As this minimum
is imposed only by the assumptions of linearity and the appropriate commutation
relations invoked by unitarity, the limit is known as the ideal linear amplifier limit,
and such an amplifier is referred to as an ideal linear amplifier.
In Cave?s model [3], the input signal and the amplifier are represented by
Bosonic modes with noise power per unit bandwidth per mode given in terms of
the number quanta
?
N
a
=?a
?
?a
?
?
and N
b
=
?
b
?
?
b
?
?
respectively. Here ?a
?
,?a
?
?
and
?
b
?
,
?
b
?
?
are
annihilation and creation operators for the respective modes of the oscillator and
detector, and obey the commuation relations
bracketleftBig
?a
?
, ?a
?
?
bracketrightBig
= ?
?
?
,
bracketleftBig
?
b
?
,
?
b
?
?
bracketrightBig
= ?
?
?
(2.19)
24
and
[?a
?
, ?a
?
]=0,
bracketleftBig
?
b
?
,
?
b
?
bracketrightBig
=0. (2.20)
The analysis proceeds in the Heisenberg representation where it is assumed
that the evolution of the output (detector) operators can be expressed as a linear
superposition of the input (oscillator) modes [3]:
?
b
?
=
summationdisplay
?
parenleftBig
M
??
?a
?
+ L
??
?a
?
?
parenrightBig
+
?
F
?
(2.21)
and
?
b
?
?
=
summationdisplay
?
parenleftBig
?a
?
?
M
?
??
+?a
?
L
?
??
parenrightBig
+
?
F
?
?
, (2.22)
where M
??
and L
??
are matrices related to the amplifier?s gain and
?
F
?
is an operator
representing the amplifier?s noise contribution, which is assumed to be random in
time with a Gaussian distribution.
It is further assumed that
?
F depends only on the internal modes, or the internal
state, of the amplifier and thus commutes with the input mode operators. It turns
out that this assumption has rather important consequences, which I will discuss in
the end. For a less ideal amplifier M
??
and L
??
would be replaced by operators to
account for any time dependence in the gain (ie. gain fluctuations).
Next several assumptions are made. First, the analysis is restricted to the case
of single mode detection
2
so that Eqs. 2.21 and 2.22 simplify to
?
b =
parenleftBig
M?a + L?a
?
parenrightBig
+
?
F (2.23)
2
I note that Caves also treats the more general multi-mode case. The purpose of this section,
however, is to give a brief demonstration of how the quantum limit arises in the context of continous
measurement. For this purpose, presentation of the single-mode analysis is su?cient.
25
and
?
b
?
=
parenleftBig
?a
?
M
?
+?aL
?
parenrightBig
+
?
F
?
. (2.24)
Second, it is assumed that the amplifier is phase-conjugating, ie. that a phase-
shift in the input signal generates the opposite sign phase-shift in the output signal.
That is, if
?a
prime
=?ae
?i?
,
then
?
b
prime
?
?
F =
?
be
i?
?
?
F,
This requires that M =0.ThusIamleftwith
?
b =?a
?
L +
?
F (2.25)
and
?
b
?
= L
?
?a +
?
F
?
. (2.26)
This assumption is made arbitrarily. I could have just as easily assumed phase-
preseving. In the end, Caves demonstrates that for large gain amplifiers, the ultimate
limit is the same.
Finally, it is assumed that the amplifier noise is phase-sensitive. That is,
the amplifier?s rms noise contribution is split unequally between the input signal?s
quadratures. Thus one must break up the input and output signals into their re-
spective quadratures:
?a =
?
X
1
+ i
?
X
2
(2.27)
26
and
?
b =
?
b
1
+ i
?
b
2
, (2.28)
where
?
b
1
= L
?
X
1
+
?
F
1
?
b
2
= L
?
X
2
+
?
F
2
, (2.29)
where
?
F=
?
F
1
+i
?
F
2
. To recover the phase-insensitive amplifier, simply set F
1
= F
2
.
With these assumptions, one can now express the total output noise for each
quadrature as [3]
(?b
1
)
2
= |L|
2
(?X
1
)
2
+(?F
1
)
2
, (2.30)
(?b
2
)
2
= |L|
2
(?X
2
)
2
+(?F
2
)
2
, (2.31)
where ?X
1
(?X
2
)and?F
1
(?F
2
) are the signal and detector rms noise contribu-
tions to the quadratures respectively.
The amplifier contribution referred to the input for each quadrature is thus [3]
A
1
=
(?F
1
)
2
|L|
2
, (2.32)
A
2
=
(?F
2
)
2
|L|
2
, (2.33)
where |L|
2
plays the role of the amplifier?s power gain in number of quanta.
27
Using the Schwartz inequality [3],
?F
1
?F
2
?
1
2
|?
bracketleftBig
?
F
1
,
?
F
2
bracketrightBig
?|, (2.34)
and the relation
bracketleftBig
?
F,
?
F
?
bracketrightBig
= ?2i
bracketleftBig
?
F
1
,
?
F
2
bracketrightBig
, (2.35)
one arrives at the uncertainty relation for a phase-sensitive linear amplifier [3]
A
1
A
2
?
1
16
|?[
?
F,
?
F
?
]?|
2
, (2.36)
and the total amplifier noise contribution
A = A
1
+ A
2
?
1
2
|?[
?
F,
?
F
?
]?|. (2.37)
As they stand, Eqs. 2.36 and 2.37 are not very illuminating. However, from
the commutation relation for
?
b, Eq. 2.20, one finds [3]
[
?
F,
?
F
?
]=1+|L|
2
. (2.38)
Thus
radicalBig
A
1
A
2
?
1
4
(1 +|L|
?2
), (2.39)
and
A ?
1
2
(1 +|L|
?2
). (2.40)
For large gain, |L|
2
greatermuch 1, Eqs. 2.39 and 2.40 tell us two things: noise in
one quadrature can only be reduced at the expense of signal-to-noise degradation
in the other quadrature [3]; and the absolute minimum total noise power per unit
bandwidth that an amplifier can add to a narrow band signal is one-half quanta [3].
28
Using Eqs. 2.30, 2.31, 2.32, 2.33, and 2.37, one can re-express the total output
noise as
?b
2
= |L|
2
parenleftBig
?X
2
+ A
parenrightBig
. (2.41)
If the signal contributes one-half quanta of noise, ie.?X
2
=
1
2
,then
?b
2
= |L|
2
parenleftbigg
1
2
+ A
parenrightbigg
. (2.42)
From Eq. 2.40, this then yields
|?b|
2
?
1
2
|L|
2
+
1
2
(1 +|L|
2
)=|L|
2
+
1
2
, (2.43)
which simply states that, for large gain, the minimum total noise at the output of an
ideal amplifier is composed of two parts: one-half quanta contributed by the internal
amplifier modes, and one-half quanta contributed by the input mode; both of which
are amplified by |L|
2
[3]. The fact that the two noise sources add in quadrature
is a consequence of the assumption that the internal states of the amplifier and
the initial input signal state are independent. As a result, their fluctuations are
uncorrelated.
Finally, Caves defines the noise temperature, T
QL
, of the ideal linear amplifier
by assuming that the total input noise is given by the Planck distribution (plus the
zero-point energy),
|?a|
2
=
1
2
coth
parenleftBigg
?h?
1
2k
B
T
b
parenrightBigg
, (2.44)
and asking: by how much would one have to increase T
b
to observe |?b|
2
at the
output of the amplifier? In the limit of large gain, after working through the algebra,
29
Caves finds:
T
QL
=
?h?
1
k
B
ln(3)
, (2.45)
if T
b
=0; and
T
QL
=
?h?
1
2k
B
, (2.46)
if k
B
T
b
greatermuch ?h?
1
.
This minimum is imposed only by the assumptions of linearity and the appro-
priate commutation relations invoked by unitarity, and is known as the ideal linear
amplifier limit. An amplifier that meets this condition is referred to as an ideal
linear amplifier.
In the low-T
b
limit, then, the resulting minimum position sensitivity is
?y
QL
=
radicalBigg
T
QL
k
B
K
m
=
radicalBigg
?h
ln(3)M
m
?
1
, (2.47)
which is greater than the standard quantum limit, Equation 2.18:
?y
QL
=
radicalBigg
2
ln(3)
?y
SQL
. (2.48)
Quantum-Limited Position Detection
In the previous sections it was demonstrated that quantum mechanics places
a limit on the minimum rms uncertainty in the knowledge of a resonator?s position;
it was also shown that quantum mechanics requires that there be an additional
minimum noise contribution from the amplifier itself.
3
However, the discussion
up until this point has been rather abstract; it is not obvious how to extend the
formalism or the results to a solid-state position amplifier such as the SET.
3
I implicitly mean an amplifier with linear, time-indepedent coupling to the resonator?s position.
30
In this section, I present and discuss the results of Aashish Clerk?s linear
response approach to quantum-limited position detection [16]. This approach ar-
rives at the same conclusion as Cave?s Bosonic-mode model, with the advantage of
phrasing the quantum constraints on continuous linear amplification in terms of an
amplifier?s intrinsic noise properties.
Clerk considers a resonator with conjugate momentum, ?p
y
, and displacement,
?y
m
, and an amplifier with input and output characterized by the hermitian operators
?
F and
?
I respectively.
He further assumes that the resonator is coupled to both an equilibrium bath
with temperature T
b
and to the amplifier via the interaction
H
int
= ?A
?
F ? ?y
m
, (2.49)
where A sets the strength of the interaction and
?
F can be thought of as the inter-
action force or equivalently the back action of the amplifier on the resonator.
The analysis is restricted to the case of weak coupling (H
int
? 0) the relevant
parameter regime for our experiments (see Section 2.3). There are two consequences
of this. First, the response of the output of the amplifier ?
?
I? to a small change ??y
m
?
can be determined using linear response theory [53]. From Liouville?s theorem, to
first-order in H
int
, Clerk finds that the amplifier?s output response is given by
??
?
I(t)? = Tr(
?
I??(t)) = A
integraldisplay
?
??
dt
prime
?(t?t
prime
)??y
m
(t)?, (2.50)
where ??(t) is the first-order density matrix term in the iterative solution of the
Liouville equation.
31
The admittance or amplifier gain, ?, is given by [16]
?(t?t
prime
)=
1
i?h
?(t?t
prime
)?
bracketleftBig
?
I(t),
?
F(t
prime
)
bracketrightBig
?; (2.51)
and the expansion is done about the amplifier?s zero-coupling configuration.
The second consequence of the weak-coupling assumption is that the equation
of motion for ??y
m
(t)? reduces to a ?Langevin-like? expression. Specifically, Clerk
finds that
M
m
?
2
??y
m
(t)?
?t
2
= ?M
m
?
1
2
??y
m
(t)???
b
???y
m
(t)?
?t
? (2.52)
?A
2
integraldisplay
dt
prime
?(t?t
prime
)
???y
m
(t
prime
)?
?t
prime
+ F
b
(t)+A ?F(t), (2.53)
where ?
b
and F
b
describe the damping and fluctuating forces provided by the bath
respectively and are related through the fluctuation-dissipation theorem:
?
S
F
b
= ?
bath
?h? coth(
?h?
k
b
T
b
). (2.54)
The detector?s influence is manifest in the damping term ?(t ? t
prime
)andthe
back action force F(t). In the limit where the resonator?s frequency is small with
respect to the intrinsic time-scale of the amplifier, Clerk demonstrates that the
detector-induced damping and back action force are related in a manner simliar to
the fluctuation-dissipation theorem:
2k
B
T
d
=
?
S
F
?
. (2.55)
32
Here,
?
S
F
= lim
??0
S
F
(?)+S
F
(??)
2
(2.56)
and
? = lim
??0
S
F
(?)?S
F
(??)
2?h?
, (2.57)
where,
S
F
(??)=
integraldisplay
?
??
dt?
?
F(t)
?
F(0)?e
?i?t
(2.58)
are the positive and negative frequency components of the amplifier?s back action
noise spectral density, with (+) referring to energy transfer from the resonator to the
amplifier, and (-) referring to energy transfer from the amplifier to the resonator. See
reference [54] for a nice explanation of positive and negative frequencies in quantum
noise.
Classically, ?F(t)F(0)? =?F(0)F(t)?,sothatS
F
(?)= S
F
(??). However, this
is not generally true for a quantum mechanical system, ie.
bracketleftBig
?
F(t),
?
F(0)
bracketrightBig
negationslash= 0 [54] . It
is convenient then to use the relation
?
F(t)
?
F(0) =
1
2
parenleftBigbraceleftBig
?
F(t),
?
F(0)
bracerightBig
+
bracketleftBig
?
F(t),
?
F(0)
bracketrightBigparenrightBig
to break-up Eq. 2.58 into two components: a real component representing the total
force spectral density experienced by the resonator due to the amplifier?s back action
noise (Eq. 2.56); and an imaginary component representing the energy-loss rate of
the resonator due to the interaction with the amplifier, or detector-induced damping
(Eq. 2.57). With these definitions, the e?ective amplifier temperature T
d
is thus
interpreted as gauging the asymmetry between the amplifier?s positive and negative
frequency back action noise.
33
For the case of normal-state SET?s and tunnel-junctions, it has been found
that T
d
is positive and propotional to the average energy lost by an electron as
it traverses the device?s junctions [55] [56] [57]. On the other hand, due to the
myriad tunnelling processes, the case of the superconducting SET (SSET) is much
more complicated [58] [59] [60]. For example, both positive and negative e?ective
temperature and dissipation are possible when the SSET is biased near the single and
double Cooper-pair resonances (see Appendix B for brief discussion of the SSET);
the ?direction? of the exchange of energy depending on whether energy needs to be
removed or added for the resonant tunneling of Cooper-pairs to occur .
Regardless of whether T
d
is positive or negative, the total e?ective tempera-
ture, T
e
, of the resonator is given by the sum of T
d
and T
b
, weighted by the respective
coupling to each reservoir [16] [55]:
T
e
=
1
?
e
(?
b
T
b
+ ?
d
T
d
), (2.59)
where
?
e
= ?
b
+ ?
d
=
M
m
?
1
Q
e
, (2.60)
?
d
= A
2
?, (2.61)
and Q
e
is the e?ective quality factor of the resonator due to damping induced from
both the detector and the environment.
From the above definitions, and Eqs. 2.54 and 2.55, Clerk expresses the spec-
tral density of the resonator?s motion as
34
?
S
y
(?)=
?
S
F
b
(?)+A
2
?
S
F
(?)
|M
m
(?
2
??
1
2
+ i??
1
/Q
e
)|
2
=
= |g(?)|
2
parenleftBig
?
S
F
b
(?)+A
2
?
S
F
(?)
parenrightBig
. (2.62)
Using Eqs. 2.51 and 2.62, the total noise-power density at the output of the detector
is thus
?
S
I,tot
(?)=
?
S
I
(?)+A
2
|?(?)|
2
?
S
y
(?)?2A
2
Re
bracketleftBig
?(?)
?
g
?
(?)
?
S
IF
bracketrightBig
, (2.63)
where
?
S
I
(?) is the symmetrized spectral density of the amplifier?s forward coupling
noise,
?
S
IF
(?) is the symmetrized spectral density of the cross-correlations between
forward and back-acting noise sources.
Finally, Clerk shows that Eq. 2.63 can be converted into an equivalent dis-
placement noise density, referred to the input of the amplifier:
S
y,tot
(?)=
?
S
I
(?)
|?(?)|
2
A
2
+ A
2
|g(?)|
2
?
S
F
(?)?
2Re
bracketleftBig
?
?
(?)g
?
(?)
?
S
IF
(?)
bracketrightBig
|?(?)|
2
+
+|g(?)|
2
?
S
F,b
(?). (2.64)
The first three terms represent the amplifier?s contribution to the total dis-
placement noise; whereas the last term represents resonator fluctuations due strictly
to the equilibrium bath.
Minimization of Eq. 2.64 is a rather involved process, requiring the optimiza-
tion of the noise sources
?
S
I
(?),
?
S
F
(?), and
?
S
IF
(?) and the coupling A. However,
35
Clerk imposes two important constraints that allow for the determination of a min-
inum on resonance, ?=?
1
.
First,
?
S
I
(?)and
?
S
F
(?) are constrained by the amplifier uncertainty principle:
?
S
I
(?)
?
S
F
(?) ?
?h
2
4
(Re [?(?)])
2
+
parenleftBig
Re
bracketleftBig
?
S
IF
(?)
bracketrightBigparenrightBig
2
(2.65)
with the equality fulfilled for the case of a quantum-limited amplifier.
Strictly speaking, Eq. 2.65 states that in the presence of gain, even an ideal or
quantum-limited amplifier must add a minimal amount of back-acting and forward-
coupling gain. An alternative interpretation of the equality in Equation 2.65 is that
no signal information is lost in the process of measurement [61]. One can see this
by recognizing that the rate at which information is attained from the output of the
detector, ?
meas
, is inversely proportional to
?
S
I
(?)(ie. the smaller
?
S
I
(?), the better
the signal-to-noise, and the less time for which one needs to integrate); whereas, the
rate at which information ?enters? the detector, ?
?
, is proportional to the interaction
S
F
. At the quantum-limit, Clerk et al. demonstrated that ?
meas
=?
?
, implying
a tight coupling between the amplifier input and output degrees of freedom. In a
sub-ideal amplifier, ?
?
> ?
meas
, implying that some information about the input
signal is lost to internal degrees of freedom which do not influence the amplifier?s
output.
Second, Clerk demands that the total power available at the output of the
amplifier be much greater than the total power delivered from the amplifier to the
resonator. He shows that, for a quantum-limited amplifier, this is equivalent to
36
requiring that
k
B
T
d
?h?
1
greatermuch 1. (2.66)
With these two constraints, the requisite conditions for minimization can be
stated. First, it is necessary that the amplifier noise terms satisfy the equality in
Equation 2.65. That is, the amplifier must be quantum-limited.
Second, the symmetrized cross-correlation term
?
S
IF
(?) must vanish, minimiz-
ing the product
?
S
I
(?)
?
S
F
(?).
Third, the back action and forward-coupling must contribute equally to the
total displacement noise. This requirement falls out of the optimization of the
coupling A,
A
opt
=
radicaltp
radicalvertex
radicalvertex
radicalbt
?
S
I
(?)
|?(?)g(?)|
2 ?
S
F
(?)
, (2.67)
and can be thought of as being analagous to noise impedance matching for opti-
mization of signal-to-noise.
A consequence of this third condition is that the detector-induced damping, ?
d
,
must be small with respect to ?
bath
to ensure that the resonator is more tightly cou-
pled to T
b
than T
d
(a consequence of the second constraint and Eq. 2.59). Explicitly,
Clerk shows that the third condition implies
A
opt
2
?
?
b
+ A
opt
2
?
=
?h?
1
4k
B
T
d
, (2.68)
which is necessarily much less than one due to the assumptions I have made.
If all three conditions are satisfied, Clerk shows, then, that an amplifier must
contribute at least the equivalent of the resonator?s zero-point contribution to the
measured signal. From Eqs. 2.68 and 2.59, it is evident that half of the contribution
37
is in the form of back action or heating of the resonator:
T
e
=
A
opt
2
?T
d
+ ?
b
T
b
A
opt
2
? + ?
b
=
?h?
1
4k
B
+ T
b
, (2.69)
as one would expect, having optimized with respect to the coupling A. The other
half of the amplifier contribution is necessarily forward-coupling noise. Thus, the
noise temperature of a such an optimized detector is
T
QL
=
?h?
1
2k
B
. (2.70)
Essentially, this is the same result derived in the previous section: an amplifier
must add at least one-half quanta of noise power per unit bandwidth to the mea-
sured signal. However, the advantage of the present approach is that it provides a
prescription (ie. the three conditions listed above) which an amplifier must fulfill in
order for quantum-limited displacement detection to be possible. Additionally, the
e?ect of the amplifier?s back action on the resonator is explicitly manifest as heating
of the resonator by one-half its zero-point energy.
2.3 Nanomechanical RFSET Displacement Detection
With the development of the RFSET [41] [62] [63], came the suggestion that,
it could be implemented as a nearly quantum-limited nanomechanical displacement
transducer [12] [13]. The realization was spurred by a combination of factors. For
one, theoretical treatments of the SET suggested that the electrometer could achieve
near-ideal noise characteristics required for quantum measurement schemes [43] [42]
[64]. Secondly, the RFSET had been demonstrated to be capable of operating with
38
over 100 MHz bandwidth [41], a pre-requisite for the read-out of the high frequency
nanoresonators thought to be necessary to demonstrate freeze-out. And third, the
similarities in size-scale and fabrication between SETs and nanoresonators suggested
that sub-micron positioning of the devices and, hence, tight coupling should be
possible.
In this section I first review the basic idea behind nanomechanical RFSET
displacement detection. I then review the theoretical work on the noise characteris-
tics of the SET and apply the results of the previous section to discuss the ultimate
limits of the detection scheme.
The RFSET Displacement Detector
In essence, the RFSET displacement detector is a capacitive microphone: me-
chanical fluctuations are converted into an electrical signal via the capacitive mod-
ulation of an SET?s di?erential resistance; the di?erential resistance of the SET is
then read-out using microwave reflectometry. A generic circuit schematic for the
transduction process and an SEM micrograph of an RFSET displacement detector
are displayed in Fig. 2.3.
By application of a large DC bias, V
NR
, between the nanoresonator and the
SET island, mechanical displacement of the resonator, y
m
, results in modulation of
the polarization charge on the SET island through the relation
?Q
NR
=
?C
NR
?y
m
V
NR
y
m
? b
C
NR
V
NR
d
NR
y
m
, (2.71)
where C
NR
and d
NR
are the capacitive-coupling and spatial separation between the
39
(a)
(b)
Figure 2.3: (a) Circuit schematic and (b) SEM image of the RFSET displacement
detector
40
resonator and the SET island respectively. In principle, C
NR
can be found to high
precision from measurements of the SET conductance versus V
NR
(see Appendix B).
However, the derivative of C
NR
with respect to y
m
must be calculated numerically.
Thus, in the last step in Eq. 2.71, I have used a capacitance extraction program [65]
and found, for typical device parameters, that ?C
NR
/?y
m
? bC
NR
/d
NR
where b is
of order unity.
In Fig. 2.3(a), y is the in-plane displacement of the mid-point of the neutral
surface (solid line) from the equilibrium position (dashed line). The quantity y
m
is
defined as the average displacement of the neutral surface over the region a to b,
the length of the SET island (the relationship between y
m
and y is calculated in
Appendix A). It is straight-forward to show that the resonator?s fundamental mode
couples most strongly to the SET island (ie.bothy
n,m
and ?C
NR
/?y
n,m
decrease
with increasing number, n, of resonator nodes). For the remainder of the thesis, I
will assume that y
m
represents the average displacement of resonator?s fundamental
in-plane mode over the length of the SET island.
It should also be noted that, in Fig. 2.3(b), the resonator?s displacement,
the lengths of the SET island and the resonator, and the separation between the
resonator and SET are not drawn to scale. For our samples the resonator?s displace-
ment is about 10
?6
? d
NR
, the length of the SET island is about 0.33 - 0.5 times
the length of the resonator, and d
NR
is typically about 0.02 - 0.05 times the length
of the resonator.
The modulation of the SET-island charge by ?Q
NR
results in the modula-
tion of the SET?s di?erential resistance, R
S
. For a normal-state SET, biased at
41
the edge of the Coulomb blockade, the relationship between Q
NR
and R
S
can be
approximated by (see Chapter 5 and Appendix B)
4
R
S
?
V
SD
I
SD
?
2R
?
sin(?Q
NR
/e)+1
, (2.72)
where V
SD
and I
SD
are the source-drain voltage bias and current respectively.
For small displacement and, hence, small charge modulation, ?Q
NR
lessmuch e,the
maximum modulation of R
S
is given by
?R
S
??2?R
?
?Q
NR
e
??b
2?R
?
C
NR
V
NR
ed
NR
y
m
, (2.73)
where R
?
is the SET?s di?erential resistance at large source-drain bias V
SD
and e is
the magnitude of the electron charge.
The modulated di?erential resistance ?R
S
is measured by applying a mi-
crowave signal v
c
(t) to the SET drain and measuring the modulation in the reflected-
signal (sideband microwave reflectometry is discussed in Chapter 5). Because R
S
greatermuch
Z
o
,whereZ
o
is the characteristic transmission line impedance of 50 ?, an LC circuit
is inserted in series with the SET for impedance matching. Ideally, the values of L
T
and C
T
are chosen so that, at the carrier frequency ?
T
=1/
?
L
T
C
T
, the impedance
of the L
T
C
T
R
S
circuit is
Z
LCR
=
L
T
R
S
C
T
= Z
o
. (2.74)
4
As is discussed in Chapter 5 for both normal-state and superconducting-state SET?s no analytic
expression for the di?erential resistance for an arbitray bias point is known. One must either solve
the SET master equation numerically or use a measured IV curve and take the numerical derivative
to find the relationship between ?Q
NR
and ?R
S
.
42
Equivalently, this can be seen as transforming the transmission line impedance
so that Q
2
T
Z
o
= R
S
where Q
T
= ?
T
L
T
/Z
o
is the external quality factor.
The resulting modulation of the reflected-signal ?v
r
(t) is well approximated
by linearizing it with respect to ?R
S
(t) [66], so that, on resonance ?
T
,
?v
r
(t) ? v
c
(t)??(t), (2.75)
where
??(t) ??b
Q
2
T
Z
o
C
NR
V
NR
R
?
?
?
2ed
NR
y
m
(t) (2.76)
at a bias-point of maximum Q
NR
-response (see Chapter 5 for details).
As stated, in Eq. 2.76, I have assumed that the reflected-signal frequency is at
the tank-circuit resonance, ?
T
, so that the Z
LRC
= L
T
/R
S
C
T
. Typically, the carrier
frequency is tuned to ?
T
. However, y
m
(t) might be modulated at, say, 5 MHz. The
reflected signal will then have sidebands at ?
T
?(2??5MHz). The magnitude of the
sidebands will depend on the half-width of the L
T
C
T
resonance. This is determined
by loading from both Z
o
and R
S
:
?f =
?
T
4?Q
(2.77)
where
1
Q
=
1
Q
T
+
1
Q
S
=
Z
o
?
T
L
T
+
?
T
L
T
R
S
. (2.78)
?f is essentially the bandwidth of the RFSET. For optimal matching, it reduces to
?f =
?
T
2?Q
T
. (2.79)
In practice, the desired bandwidth, along with the matching-condition, sets the
choice of the tank-circuit inductor and capacitor values.
43
The role of the tank circuit is clear from Eq. 2.76: near resonance, it e?ectively
serves to amplify the modulation of the reflected-signal by Q
2
T
. Without the tank-
circuit ?? ? Z
o
/R
S
? 10
?3
, and thus
?v
r
??10
?3
b?
?
2
C
NR
V
NR
e
y
m
d
NR
v
c
. (2.80)
On the otherhand, with the tank circuit, and for optimal matching, Q
2
T
Z
o
= R
S
,I
have
?v
r
??
b?
?
2
C
NR
V
NR
e
y
m
d
NR
v
c
. (2.81)
Finally, I close this subsection with some remarks about optimizing the reflected-
signal modulation response. First, it is obvious that maximizing the reflected-signal
response requires optimizing the impedance matching. However, it also requires op-
timizing v
c
and the coupling C
NR
V
NR
/d
NR
. The optimal carrier amplitude depends
on the bias-point, tank-circuit quality-factor Q [66] [67], as well as whether the SET
is superconducting or normal. In practice, it is simplest to determine the optimal
value by tuning the amplitude manually and looking for the maximum response.
The optimization of the coupling is more subtle. This is because I am not simply
interested in maximizing the SET reponse to fluctuations in position of a nearby
resonator. I am primarily interested in optimizing the SET displacement sensitivity,
which, because of the SET back action, is a separate issue. In the following sec-
tion, then, I consider the intrinsic SET noise, address the issue of optimal coupling
C
NR
V
NR
/d
NR
, and present the predictions for the ultimate limit to RFSET position
sensitivity.
44
The Ultimate Limit
For simplicity, I develop the ultimate limit of RFSET displacement detection
by considering the intrinsic noise of a normal-state DCSET. At the end of the section,
I briefly address the much more complex intrinsic noise limits of the superconducting
SET (SSET). Also, I assume that the di?erence in position sensitivity between the dc
and rf modes of operation can be accounted for by applying the predicted reduction
factor of 1.4 - 1.9 for the RFSET?s optimal intrinsic sensitivity [67]. The authors
in Reference [67] state that the degradation of ultimate sensitivity in the rf mode
compared to the dc mode is simply a result of the increased bandwidth of the rf
mode.
I assume that the orthodox theory (see Reference [68] and Appendix B) is
applicable, and thus neglect the e?ects of co-tunneling [69]. Additionally, I assume
that k
B
T
b
lessmuch E
c
, and neglect any thermal contributions to SET tunneling. Finally,
I assume that the frequency range of interest is above the 1/f noise tail, ? 10 kHz,
and below the intrinsic SET tunneling rate, (R
?
C
?
)
?1
? 1 - 100 GHz.
In the relevant limits, the intrinsic noise of the SET is due to two white sources
[43] [42]: the shot noise in the source-drain current I
SD
; and fluctuations in the SET-
island potential, ?. The origins of both sources arise from the stochastic nature of
electron tunneling events, of which I
SD
is composed (see Appendix B).
First, consider the shot noise. From the Orthodox Model, if one could in-
sert an ammeter at each of the SET junctions, the tunneling-events would appear
as delta-function peaks, separated in time according to a correlated-Poisson distri-
45
bution. The correlations arise from the fact that the probabilities for successive
tunneling events are not independent but related through the accompanying change
in the SET?s free-energy. To calculate the spectral density of the current fluctua-
tions, one must solve the SET master equation and calculate the auto-correlation
function, taking into account all the relevant tunneling processes. However, near
the Coulomb-blockade threshold (the sequential tunneling regime), an approximate
analytic expression for the shot noise spectral density exists
5
, and is given by [43]
S
II
(?)=?
I
2eI
SD
, (2.82)
where
?
I
=
?
2
1
+?
2
2
(?
1
+?
2
)
2
. (2.83)
Here ?
I
accounts for the correlations between tunneling events, and ?
1
and ?
2
are the
tunneling rates through junctions 1 and 2 respectively. It is S
II
(?)thatultimately
sets the limit of the SET charge sensitivity,
S
Q
NR
(?)=S
II
(?)
parenleftBigg
?I
SD
?Q
NR
parenrightBigg
?2
. (2.84)
In terms of the SET parameters, for symmetric junctions, this is expressed as
[42]
S
Q
NR
(?) similarequal
(1??
2
)(1 + ?
2
)
8?
2
eV
SD
R
?
C
2
?
, (2.85)
where
? =
(2C
NR
V
NR
?e)
C
?
V
SD
. (2.86)
5
From the initial assumptions, this is a classical analysis of the amplifier?s noise sources, and,
thus, it follows that there is no need to symmetrize the sources as in Section 2.2.
46
Here, the parameter ? specifies I
SD
and the set of (V
NR
, V
SD
)biaspointswhich
yield that particular value of current. The expresion above is valid provided that
one restrict the bias-points to non-degeneracy points, 0 <?<1-R
Q
/?R
?
,where
R
Q
is the quantum of resistance. The expression requires that V
SD
<e/C
?
so that
only two charge states are involved in the tunneling process.
Next, I consider the SET-island potential fluctuations. If one could connect a
voltmeter to the SET island, the potential, ?, would be seen to decrease (increase)
by e/C
?
for each electron tunneling-event onto (o? of) the island. Calculation of the
spectral density of the potential fluctuations requires solving the master equation
for all contributing tunneling events. But, in the sequential tunneling regime, the
potential noise spectral density takes the simple form [43]:
S
??
(?) similarequal ?
?
e
2
C
?
, (2.87)
where
?
?
=
4?
1
?
2
(?
1
+?
2
)
3
. (2.88)
Similiar to ?
I
, ?
?
accounts for the correlations between successive steps in the island
potential. Essentially, these steps cause the SET back action. In terms of the SET
parameters, the back action can be expressed as [42]
S
??
(?) similarequal
(1 ??
2
)
4
e
3
R
?
C
2
?
V
SD
, (2.89)
where ? has the same definition and restrictions as above.
From these results for the SET shot noise and back action, I can now gauge the
overall noise performance of the SET, and determine if it meets the three conditions
47
necessary for quantum-limited linear position detection. It is straightforward to
show that [42]
radicalBig
?
2
S
Q
NR
(?)S
??
(?) ? 2.2?h? (2.90)
for an ?optimized? SET, where ? is the frequency of the mesured signal. Thus,
while it is apparent that the ?optimized? SET fails to meet condition (1) (Eq. 2.65),
this result demonstrates that the SET, in principle, is a nearly quantum-limited
detector. Note also that Eq. 2.90 does not include correlations between the shot
noise and back action.
The correlations between the shot noise and back action are given by [42]
|S
I?
(?)|
radicalBig
S
??
(?)S
II
(?)
?
radicalBigg
2
1+?
2
?
2
. (2.91)
For the range of ? allowed in this approximation, the relative magnitude of S
I?
(?)
can range from ? 0.0 to ? 0.5. Depending on the bias-point and on the source
impedance, the correlations could add to or reduce the total noise power added by
the SET amplifier [16] [43] [42]. Regardless, it appears that, at least in some cases,
the SET also fails to meet condition (2).
Lastly, I address the condition (3) (optimal coupling). I first assume that
the SET is biased such that the correlation term is negligible. Next, I convert the
SET?s intrinsic noise sources into a total e?ective position sensitivity by assuming
that the SET is coupled to a nanomechanical resonator as described in the previous
subsection. The shot noise contribution is simply:
S
y
I
(?) ? S
Q
(?)
parenleftBigg
d
NR
bC
NR
V
NR
parenrightBigg
2
. (2.92)
48
Equation 2.92 simply states that as I increase the coupling, bC
NR
V
NR
/d
NR
,
between the resonator and the SET island, the mechanically-induced charge signal
increases linearly above the SET?s shot noise background.
The contribution due to the back action is approximated by considering the
force of the SET-island potential fluctuations on the resonator:
F(t) ? b
C
NR
V
NR
d
NR
??(t), (2.93)
where ??(t) represents the step-wise potential fluctuations.
On resonance, the response of the resonator is given by the standard relation:
y
m
(?)=
Q
e
f
K
m
, (2.94)
where f is the normalized force, accounting for the mode shape as discussed in
Appendix A. Here, the quality factor Q
e
(Eq. 2.60) is used to account for the detector
induced damping
6
[16] [55]. For the case of the normal state SET, the e?ective
quality factor is given by [16] [55]
1
Q
e
=
1
Q
b
+
1
Q
d
=
=
1
Q
b
+
parenleftBigg
bC
NR
V
NR
C
?
V
SD
parenrightBigg
2
e
2
R
?
2M
m
?
1
d
2
NR
, (2.95)
where I have assumed symmetric junctions, and Q
b
is the quality factor relating to
damping due the equilibrium bath.
6
The detector also induces a shift in the resonator?s spring constant. I neglect this a?ect as the
resulting frequency shift is small with respect to the shift induced by the electrostatic softening of
the mode due to the coupling voltage V
NR
.
49
From Eqs. 2.93 and 2.94, the resulting back action contribution to the dis-
placement noise is thus
S
y
F
(?) ?
parenleftbigg
Q
e
K
m
parenrightbigg
2
parenleftBigg
bC
NR
V
NR
d
NR
parenrightBigg
2
S
??
(?). (2.96)
The total e?ective mean square displacement noise for the SET displacement
detector is now written simply as
(?y
m
)
2
=(S
y
I
(?)+S
y
F
(?)) ?f, (2.97)
where ?f = ?
1
/4Q
e
is the noise equivalent bandwidth of the resonator.
It is clear from Eqs. 2.96 and 2.92 that Eq. 2.97 can be minimized with respect
to the coupling. Thus condition (3) is satisfied. For a given sample, the only
parameter in the coupling which can be tuned is V
NR
. In the simple case where
damping from the equilibrium bath dominates, the weak-coupling limit, I find the
optimal coupling voltage to be
V
2
NR,opt
=
parenleftBigg
d
NR
bC
NR
parenrightBigg
2
K
m
Q
b
radicaltp
radicalvertex
radicalvertex
radicalbt
S
Q
(?)
S
??
(?)
. (2.98)
Figure 2.4 displays the total rms displacement noise of the SET displacement
detector. The parameter values are chosen to roughly represent Device 2. The
back action and shot noise are seen to contribute equally at a voltage of V
NR,opt
? 13 Volts. This so-called ?sweet-spot? yields an ultimate intrinsic displacement
sensitivity ? 1.5?y
QL
,where,
?y
QL
=
radicalBigg
?h
ln(3)M
m
?
1
, (2.99)
as determined from the definition of T
QL
, Eq. 2.45.
7
7
Choosing the low-temperature limit of T
QL
rather than the high-temperature limit is simply
a matter of convention.
50
Figure 2.4: Log-log plot of displacement noise of the SET displacement detector in
the weak coupling limit. Parameter values: Q
b
=5.0? 10
4
, C
?
= 450 aF, C
NR
=
26 aF, f
1
= 19.7 MHz, d
NR
= 600 nm, and K
m
? 15 N/m. The line representing
the measurement circuit limit is for S
Q
=15?e/
?
Hz.
51
For comparison, also included in Fig. 2.4 is the additive measurement circuit
noise (labeled ?measurement circuit limit?). As discussed in Appendix B and in
Chapter 5, this noise is due to the input voltage noise of the pre-amplifer. It is
uncorrelated with the intrinsic sources, and simply adds to the forward coupling
noise S
Q
(?) in Eq. 2.92.
Figure 2.5 demonstrates the limit where the damping crosses over from bath-
limited to detector-limited. Optimal coupling is achieved at ? .5 Volts. The cross-
over from the bath-limited regime to the detector-limited regime occurs at ? 15
Volts. In this regime, the e?ective resonator temperature T
e
, Eq. 2.59, is determined
primarily by the e?ective detector bath temperature T
d
, Eq. 2.55. For the case of
the normal-state SET [55],
T
d
?
eV
SD
k
B
(2.100)
where the proportionality constant is of order .5, and depends on the specific bias
SET bias point. For typical parameters, this should be on the order of .5 - 1 K.
Before concluding, it is necessary to comment on how the preceding dicussion
would have di?ered if I considered the case of a superconducting single-electron tran-
sistor (SSET). As discussed in Appendix B, the combination of Coulomb blockade
and Josephson phenomena leads to a large variety of potential tunneling processes.
Consequently, a general statement about the the di?erences between the noise char-
acteristics and response of the SSET and the SET cannot be made; it depends on
the particular bias point and the dominant tunneling processes at that point.
For example, at the onset of single quasiparticle tunneling (at 4 ?), the re-
52
Figure 2.5: Log-log plot of displacement noise of the SET displacement detector,
demonstrating the cross-over from bath-limited damping to detector-limited damp-
ing. Parameter values: Q
b
=2.0? 10
5
, C
?
= 550 aF, C
NR
=65aF,f
1
=9.5MHz,
d
NR
= 300 nm, and K
m
? 2 N/m. The line representing the measurement circuit
limit is for S
Q
=90?e/
?
Hz.
53
sponse or I
SD
Q
NR
transfer function of the SSET has been found to be greater than
the SET response at the onset of tunneling by a factor of ? 1.6?/E
C
[70]. Predic-
tions for the optimized ultimate charge sensitivity of the SSET for this particular
bias regime also show improvement over the normal-state [71].
As well, around the JQP resonance, it is expected that, because of the sup-
pression of the shot noise from resonant tunneling, the charge sensitivity should
improve over what is achievable in the normal-state [72].
Finally, recent theoretical investigations of the measurement e?ects of the noise
characterisics near the DJQP and JQP resonances have shown that, just as in the
normal-state, the SSET back action should appear as an e?ective thermal bath,
driving, damping and shifting the frequency of the measured resonator [59] [60]. It
is still seen that
Q
?1
d
? V
2
NR
(2.101)
and
T
d/
? V
SD
. (2.102)
However, unlike in the normal state, assymmetry in the quantum noise can
yield both negative and positive T
d
and Q
d
[59] [60]. Additionally, it has been pre-
dicted that the SSET biased near the DJQP should be able to approach more closely
to the quantum measurement limit than a normal-state device [58]. Preliminary ex-
perimental results appear to verify the predictions of negative and positive T
d
and
Q
d
near both the JQP and DJQP resonances [73].
I conclude the section and the chapter by noting that, while the normal-state
54
SET is not an ideal linear detector, it comes within a factor of 2. It is expected
that the SSET should perform even better. Configured as a displacement detector,
the SET or SSET can be optimally coupled to a nanomechanical resonator, and
perform detection near the quantum limit of position detection (Quantum Limit
II). Furthermore, implementation of the RFSET, which will result in the loss of
at most a factor ? 2 in detection sensitivity, should be capable of providing the
bandwidth necessary for the read-out of the high frequency nanoresonators required
to observe freeze-out (Quantum Limit I).
55
Chapter 3
Design and Fabrication
The following sections describe the design considerations and the sequence of
steps involved in the fabrication of our RF-SET displacement detectors.
3.1 The Wafers
Each sample was fabricated from 500 ?m thick, (100)-oriented, silicon wafers
(doped p-type with resistivity quoted to be 1 - 10 ?-cm at room temperature). The
wafers were purchased from The MEMS Exchange [74], and provided to us with a
100 nm low-stress, amorphous silicon nitride (SiN) coating on each side (Fig. 3.1(a)).
The SiN was deposited using low pressure chemical vapor deposition (LPCVD). Each
side of each wafer was polished.
3.2 Silicon Nitride Membrane Fabrication
The first step in the process was to fabricate the SiN membrane from which
the nanomechanical resonators would eventually be cut. Using silicon nitride mem-
branes provided two advantages: (1) a relatively large Young?s modulus (200 - 300
GPa) [75] [76], yielding, for a given geometry, larger nanoresonator frequencies than
any alternative processing material other than diamond; and (2) convenience in the
processing of the nanoresonator (see the final section in the chapter on etching the
56
Figure 3.1: (a) SiN-coated Si wafer. (b) After RIE etch. (c) After KOH etch.
57
nanoresonator).
Initially, a protective layer of photoresist was spun on the top of each wafer:
1. Spin protective layer of photoresist on top side of wafer at 2000 rpm.
2. Let dry in air at room temperature.
Next, square openings in each wafer?s bottom SiN layer were defined by means
of contact optical lithography and reactive ion etching (RIE). The etch was timed so
that all the SiN was etched away, leaving the underlying silicon substrate exposed
(Fig. 3.1(b)):
1. Spin negative photoresist NR7-1500py [77] on bottom of wafer at 3000 rpm
for 1 minute; bake at 150
?
C for 1 minute on vacuum hot-plate.
2. Expose with ? 400 nm wavelength light, for 20 seconds with an intensity of
12 mW/cm
2
3. Post-bake at 120
?
C for 1 minute on vacuum hot-plate
4. Develop in RD6 [77] for 15 seconds, and a rinse in deionized (DI) water.
5. RIE etch for 10 minutes in SF
6
with SF
6
flow rate and chamber pressure 20
sccm and 20 mTorr respectively; and RF power of 170 Watts.
Finally, an anisotropic wet etch was used to etch the Si substrate. Aqueous
potassium hydroxide was used, providing an etch rate of ? 70 ?m/hr on the [100]
Si-planes:
58
1. Place wafer in aqueous potassium hydroxide (KOH) solution of concentration
425 grams of K [78] per liter of DI H
2
O, at 80
?
C, stirred at 120 rpm, for ? 8
hours
2. Remove from the KOH bath, rinse wafer with DI water, clean with acetone,
methanol, and isopropanol (IPA), and then blow dry with compressed N
2
.
After ? 8 hours in the KOH solution, the result was a v-groove through the
silicon, stopping at the SiN top layer, and yielding a square SiN membrane on the
top of the wafer (Fig. 3.1(c)). The size of the membrane depended on the size of
the opening on the bottom of the wafer through the relationship
L
m
= L
o
?
2t
tan ?
, (3.1)
where L
m
was the length of each side of the membrane, L
o
was the length of each side
of the opening on the wafer?s bottom, t = 500 ?m was the silicon wafer thickness,
and ? = 54.7
?
was the angle between the (100) and (111) silicon planes. Typically,
we chose L
o
similarequal 765 ?msothatL
m
similarequal 55 ?m. This size was large enough to allow for
two complete resonator/SET devices on each membrane.
After this step, the wafers were diced into units containing four membranes
each.
3.3 Fabrication of the Bond Pads and Tank Circuit
The main design considerations for the fabrication of the bond pads and tank
circuits were as follows: (1) safegaurd the SET junctions and nanoresonator from
59
electrostatic discharge by implementing on-chip room-temperature electrical shorts;
(2) reduce dissipation in the rf circuit by constructing the entire circuit out of
superconducting materials; and (3) choose the appropriate values of tank circuit
capacitance and inductance in order to achieve both large bandwidth (50 - 100
MHz) and matching between 50 ? and the SET dynamic resistance R
S
.
It was equally important to choose the tank-circuit parameters so that the res-
onant frequency would be large enough to employ an rf amplifier, but small enough
to ?look? like a lumped element to the incident radiation. Thus, for the first genera-
tion of devices, we chose an rf amplifier with a band-pass between approximately 1.2
and 1.7 GHz (see Chapter 4), and we engineered the tank circuits to be resonant at
approximately 1.5 GHz. For a silicon substrate, the wavelength of 1.5 GHz radiation
is on the order of centimeters and is much larger than any of the components in the
microwave circuit (see the subsections below).
Through the relation
f
T
=
1
2?
1
?
L
T
C
T
, (3.2)
the choice of 1.5 GHz operating frequency set the product of the tank circuit in-
ductance and capacitance to be L
T
C
T
? 1x10
?20
. The individual inductance and
capacitance values would then have been set by the bandwidth condition ?f = 50 -
100 MHz and the optimal matching condition (Eq. 2.79). The optimal inductance
would have been given by
L
T
=
Z
o
?f?
. (3.3)
However, mistakes were made in the design phase, and the inductor value
60
we chose (50 nH) was approximately a factor of 3.2 less than the optimal value.
Through Eq. 3.2, this increased the value of capacitance by approximately a factor
of 3.2 (300 fF). As a result, the SET/tank-circuit impedance on resonance (Z
LCR
,
see Eq. 2.74) was approximately a factor of 10.3 smaller than 50 ? for R
S
=50k?.
Additionally, for Devices 1 - 3, R
S
was closer to 100 k? (see Section 3.4), making
the impedance mismatch nearly a factor of 2 worse. Finally, the tank-circuit quality
factor (Eq. 2.78) was dominated by loading from the 50 ? transmission line, resulting
in measurement bandwidth of approximately 70 MHz.
Fabrication of the Bonds Pads and Tank Circuit Capacitor
The first step in the process was to define the bond pads and the tank circuit
capacitor. This was done using optical lithography and metal lift-o? on the top-side
of the wafer:
1. Spin negative photoresist NR7-1500 at 4000 rpm for 1 minute; baked at 150
?
C
for 1 minute on vacuum hot-plate.
2. Expose with ? 400 nm wavelength light, for 20 seconds with an intensity of
12 mW/cm
2
.
3. Bake at 120
?
Cfor1minute.
4. Develop in RD6 for 20 seconds.
5. Rinse in deionized (DI) water.
61
Before the deposition of the metal layers, the SiN within the regions defined
by the optical lithography step was removed by an RIE etch:
1.RIEetchwithCHF
3
and O
2
at respective flow rates of 18 sccm and 2 sccm
and a pressure and RF power of 40 mTorr and 175W.
2. O
2
plasma etch for 30 seconds at 1.75 Torr and 100 W.
This step served to provide electrical contact between the bond pads and
the silicon substrate, and was implemented in order to reduce electrostatic buildup
across the SET junctions and resonator before the device was installed on the dilu-
tion refrigerator. At room-temperature, the ?short? between leads was measured to
be on the order of k?,
Next, a tri-layer of metals [80] was deposited using electron-beam evaporation.
The evaporation was done in a vacuum chamber at a pressure of less than 1 ?Torr.
The following layers were evaporated sequentially, without breaking vacuum:
1. Evaporate 190-200 nm of Al.
2. Evaporate 20 nm of Ti.
3. Evaporate 200 nm of Au.
4. Lift-o? in RR2 [77] at 80
?
C for 10 minutes.
5. Rinse in IPA and blow dry with compressed N
2
.
The purpose of the tri-layer was to reduce dissipation in the circuit components
and at the same time ensure oxide-free overlaps between the components. We found
that the tri-layer superconducted below 700 mK.
62
Fabrication of the Tank Circuit Inductor
The planar-coil inductors for the LC resonator were defined by electron-beam
lithography and electron-beam evaporation. Resist exposure was performed with a
Jeol 6500 scanning electron-beam field emission microscope (SEM) [81]. The pattern
for the exposure was generated using Design Cad LT 2000 [82]. The exposure was
controlled using the Nanometer Pattern Generation System version 8.001.77 [83].
1. Spin electron-beam resist PMMA 495K [84] on top of wafer at 5000 rpm for
60 sec; baked at 180
?
C for 5 minutes on vacuum hot-plate.
2. Spin electron-beam resist PMMA 950K [84] on top of wafer at 5000 rpm for
60 sec; baked at 180
?
C for 5 minutes on vacuum hot-plate.
3. Expose resist at an acceleration voltage of 30 KV, with beam current of 2 nA,
and magnification of 150x.
4. Develop for 25 seconds in MIBK [84], diluted in IPA to one-part-in-three by
volume.
5. Rinse in IPA for 25 seconds and blow dry with with compressed N
2
.
6. O
2
plasma etch for 15 seconds to remove any residual resist in defined regions.
The metal deposition was performed using an electron-beam evaporator at a
pressure of less than 1 ?Torr.
1. Deposit 100 nm Al at .5 nm/s.
63
2. Lift-o? in acetone at 80
?
C for 10 minutes.
3. Rinse with IPA and blow dry with compressed N
2
.
Contact between the inner coil of the inductor and the appropriate bond pad
was achieved using an Al cross-over bridge (Fig. 3.3). The cross-over and the inner
coil made contact via a tri-layer connector
1
(Fig. 3.3). The cross-over was insulated
from the coils of the inductor by a layer of SiO
2
. The bridges were defined in two
steps: (1) electron-beam exposure followed by deposition of SiO
2
; (2) electron-beam
exposure followed by deposition of Al.
Deposition of SiO
2
layer:
1. Spin electron-beam resist MMA EL 11 [84] diluted in anisole to one-part-in-one
by volume at 2500-3000 rpm for 60 seconds; bake at 200
?
Cfor5minutes.
2. Spin PMMA 950k at 4000 rpm for 60 seconds; and bake at 180
?
Cfor5minutes.
3. Expose at accelerating voltage of 30 kV with 1 nA current and magnification
of 150x.
4. Develop for 30 seconds in MIBK/IPA (1:3).
5. Rinse for 30 seconds in IPA.
6. O
2
plasma etch for 15 seconds.
1
The tri-layer connector was deposited in the same step as the bond pads, leads, and tank-circuit
capacitor.
64
7. Evaporate 220 nm of SiO
2
at a rate of 0.5 nm/s using electron-beam evapora-
tor.
8. Lift-o? in acetone at 80
?
C for 10 minutes.
9. Rinse in IPA and blow dry with compressed N
2
.
Deposition of Al layer:
1. Spin electron-beam resist MMA EL 11 [84] diluted in anisole to one-part-in-one
by volume at 2500-3000 rpm for 60 seconds; bake at 200
?
Cfor5minutes.
2. Spin PMMA 950k at 4000 rpm for 60 seconds; and bake at 180
?
Cfor5minutes.
3. Expose at accelerating voltage of 30 kV with 1 nA current and magnification
of 150x.
4. Develop for 30 seconds in MIBK/IPA (1:3).
5. Rinse for 30 seconds in IPA.
6. O
2
plasma etch for 15 seconds.
7. Evaporate 330 nm of SiO
2
at a rate of 0.5 nm/s using electron-beam evapora-
tor.
8. Lift-o? in acetone at 80
?
C for 10 minutes.
9. Rinse in IPA and blow dry with compressed N
2
.
65
Figures 3.2 and 3.3 are scanning electron micrographs of Device 2, displaying
the completed bond pads and tank circuits. For Devices 1 - 4 , the inductor coils
consisted of 13 - 14 turns and were 130 x 130 ?m
2
at their outer edge. Each turn of a
coil was 1 ?m wide, and the separation between turns was 1 ?m. The corresponding
inductance was found to be approximately 50 nH. The capacitor was made up of
7 inter-digitated fingers, each 300 ?m long and 25 wide. The resulting capacitance
was found to be approximately 300 fF.
Table 3.1 lists the values of inductance L
T
, capacitance C
T
, the resonant fre-
quency f
T
, bandwidth ?F, the tank circuit characteristic impedance Z
LC
,andthe
impedance of the SET and tank circuit on resonance Z
LCR
. In Appendix B, I discuss
how these parameters were determined.
For Devices 3 and 4, we found that the values of L
t
agree to within 5% of
values calculated using microwave circuit simulation software. For Device 2, the
quality of the tank resonance was very poor due to additional ?sub-resonances? in
the peak. Hence it was di?cult to fit with precision better than ? 20%, and thus
the Q fell somewhere between 8 and 10, putting bounds on L
T
of 46 nH and 58 nH
respectively.
The capacitance values C
T
for Devices 3 and 4 were greater than the simula-
tions by 25 - 40%. This was probably a result of not including in the simulation the
contribution to C
T
from the lead connecting C
T
to L
T
and the parasitic capacitance
of L
T
.
66
Figure 3.2: Scanning electron micrograph of the tank circuit and bond pads, Device
2. The SiN membrane is seen as the black square at the center of the sample.
67
Figure 3.3: Scanning electron micrograph of tank circuit inductor, Device 2.
68
Table 3.1: Tank circuit parameters Devices 1 - 4.
Device turns L
T
(nH) C
T
(fF) f
T
(GHz) ?F(MHz) Z
LC
(?) Z
LCR
(?)
1 13 - - 1.43 - - -
2 13 46 290 1.37 68 357 2.2
3 14 52 287 1.30 76 425 2.5
4 14 51 320 1.23 79 400 5.6
The tank capacitance for Device 2, depending on Q, could have ranged from
230 - 290 fF. As the capacitor design and fabrication were identical for all three
samples, it is more likely that C
t
? 290 fF. This would imply, then, that L
T
? 46 nH.
For the Devices 2 - 4, Z
LCR
? 0.04 - 0.12Z
o
. As a result, the reflection coe?-
cients at maximum conductance ?
max
ranged from ? 0.8 - 0.92, yielding maximum
depths of modulation, M = 20log(?
max
)of? 0.7 - 2.0 dB. Additionally, the SETs
were ?over-coupled? to the transmission line, and, hence, the Q?s were dominated
by Z
o
, setting the measurement bandwidth to ?F ? 70 - 80 MHz.
69
3.4 Fabrication of the Nanomechanical Resonator and SET
The main design considerations for the fabrication of the nanoresonators and
the SETs were as follows: (1) produce resonators with in-plane fundamental-mode
resonant frequencies large enough to approach low thermal occupation numbers
on the dilution refrigerator, yet small enough so that the response could be easily
detected using both magnetomotive (Appendix A) and SET techniques; (2) produce
SETs with as large of a charging energy as possible to minimize the intrinsic SET
charge sensitivity (Eq. 2.85); and (3) engineer as small a separation as possible
between the nanoresonator and the SET to maximize the coupling C
NR
and the
displacement-induced charge modulation (Eq. 2.71).
Fabrication of Nanoresonator and SET Gate
The nanomechanical resonators and the central coupling gate were defined on
the SiN membrane using electron-beam lithography and Au deposition. Defining
the nanoresonator from gold served two purposes: (1) the gold layer acted as a gate
electrode with which we could establish a voltage bias between the resonator and
SET island (ie. couple the resonator?s motion to the SET?s conductance); and (2)
the gold layer acted as an etch mask for the freeing of the nanomechanical resonator
from the SiN membrane in the final step (see Fig. 3.5).
The geometry of the nanoresonators was determined by the desired nanores-
onator frequency. We assumed that the in-plane fundamental-mode frequency of
70
each of the nanoresonators was described by
f
1
=
k
2
n
w
2?L
2
radicalBigg
E
Au
t
Au
+ E
SiN
t
SiN
12(?
Au
t
Au
+ ?
SiN
t
SiN
)
, (3.4)
where k
1
= 4.73 from the clamped-clamped boundary conditions, E
Au
and E
SiN
were
the Young?s moduli of Au and SiN respectively, t
Au
and t
SiN
were the thickness of
the Au and SiN layers respectively, ?
Au
and ?
SiN
were the densities of the respective
layers, w was the width of the nanoresonator, and L was the nanoresonator?s length.
In Appendix A, I list the values of the Young?s moduli and density for each layer;
as well, there, I discuss the calculation of the resonator?s e?ective mass and spring
constant k
m
for SET detection.
The sequence of steps used to define the nanoresonators and gates were as
follows:
1. Spin electron-beam resist PMMA 495K [84] on top of wafer at 5000 rpm for
60sec; baked at 180
?
C for 5 minutes on vacuum hot-plate.
2. Spin electron-beam resist PMMA 950K [84] on top of wafer at 5000 rpm for
60sec; baked at 180
?
C for 5 minutes on vacuum hot-plate.
3. Expose resist at accelerating voltage of 30 kV, with 50 pA beam current, and
a charge dose of 400 ?C/cm
2
at a magnification of 500x.
4. Develop in MIBK/IPA (1:3) for 40 seconds.
5. Rinse in IPA for 40 seconds.
6. O
2
plasma etch for 15 seconds.
71
7. Electron-beam evaporate 10 nm of Ti at a pressure of less than 1 ?Torr.
8. Electron-beam evaporate 700 nm of Au at a pressure of less than 1 ?Torr.
9. Lift-o? in acetone at 80
?
C for 10 minutes.
For Devices 1 - 4, Table 3.2 lists the geometrical parameters and raw mass,
m
r
. Table 3.3 lists the corresponding calculated and measured resonant frequencies
and e?ective spring constants.
Fabrication of the SET
Each SET was fabricated on the SiN membrane, with the SET island parallel
to a nearby nanoresonator (see Fig. 3.7). The separation between the SET and
the nanoresonator d
NR
, as well as the length of the SET island L
i
,werechosen
to maximize the capacitive coupling C
NR
. Due to limitations in the alignment of
successive layers of electron-beam lithography, the minimum separation d
NR
that
we were able to achieve was on the order of 300 nm.
2
Because we wanted to ensure
that the SET island self-capacitance did not become comparable to the SET junction
capacitances [85], we chose L
i
=5?m.
The widths of the SET island and leads varied between 50 - 100 nm. Ideally,
to minimize junction capacitances C
i
and, hence, the charge sensitivity of the SETs
(see Eq. 2.85), we would have liked to have made these narrower. However, we were
not able to develop a consistent recipe for such on the SiN membrane.
2
More recently other members of our group have been able to engineer the nanoresonator and
SET in the same lithography step and decrease d
NR
to less than 100 nm.
72
Table 3.2: Geometry and raw mass, m
r
, of Devices 1-4. The error in the length and
width of the resonator comes from 10% quoted error in the SEM calibration. The
error of 30% in the thickness of the Au layer on the resonator comes from the spread
in etch rates over time, and is a very rough estimate.
Device w(nm) L(?m) t
Au
(nm) t
SiN
(nm) m
r
(pg)
1 300 ?30 10 ?1 30 ?20 100 ?2 2.6 ?1.2
2 200 ?20 8 ?.8 30 ?20 100 ?2 1.4 ?.7
3 200 ?20 15 ?1.5 30 ?20 100 ?2 2.6 ?1.2
4 225 ?23 18 ?1.8 30 ?20 100 ?2 3.2 ?1.5
Table 3.3: E?ective masses, M
m
, and spring constants of Devices 1-4. ?a? corre-
sponds to frequency calculated using either Eqs. A.15 and A.16 or Eq. A.17. ?b?
corresponds to the frequency measured using SET detection at a temperature of
100mK and coupling voltage V
NR
.
Device M
m
(pg) ?
1
/2?(MHz)
a
K
a
m
(N/m) ?
1
/2?
b
K
b
m
1 1.5 ?.7 17 ?5 17 ?8 17.976648(3) 19 ?9
2 .96 ?.45 18 ?6 12 ?6 19.654505(7) 15 ?7
3 1.2 ?.54 6 ?1.8 1.5 ?.7 9.37163340(2) 4 ?2
4 1.4 ?.66 4 ?1 .9 ?.4 4.8976624(2) 1.4 ?.6
73
The oxidation time and pressure for the fabrication of oxide junctions was
determined by trial and error. We sought to make the total SET resistance R
?
=R
1
+R
2
= 50 k?. For this value of resistance, tank-circuit matching could be
performed with on-chip components; at the same time, co-tunneling processes would
not be a dominant e?ect (see Appendix B).
We fabricated the SETs using electron-beam lithography and the bi-layer/double-
angle evaporation technique [86] (Fig. 3.4):
1. Spin MMA EL (8.5) 11 at 5000 rpm for 60 seconds; bake at 140
?
Cinanoven.
2. Expose membrane to 60 ?C/cm
2
electron-beam dose (30 pA beam current,
30 kV accelerating voltage, and magnification of 500x)to insure quick devel-
opment of the MMA layer.
3. Spin PMMA 950K A4 at 5000 rpm for 60 seconds; bake for 30 minutes at
140
?
Cinanoven.
4. Evaporate 100 nm layer of Al in thermal evaporator to serve as both an anti-
charging layer and a focusing standard for the electron-beam exposure.
5. Expose SET leads and island each with an area charge dose and line charge
dose of 50 ?C/cm
2
and 1.0 nC/cm respectively; 30 kV accelerating voltage,
beam current of 30 pA, and magnification 1000x.
6. Remove Al layer with OPD4262 [87], 40 seconds.
7. Rinse in DI water and blow dry with compressed N
2
gas.
74
8. Develop in MIBK/IPA (1:3) for 1 minute.
9. Rinse in IPA for 1 minute.
10. O
2
plasma etch for 10 seconds to remove residuals.
Figure 3.4(a) displays a schematic of the result of the exposure and develop-
ment process.
The double-angle depositions were done in a thermal evaporator. The system
was connected to building ground through a copper-braided strap to prevent sam-
ple damage due to electrostatic discharge. Both evaporations were performed at
a chamber pressure less than 1 ?Torr. Three-nines Al [88] was evaporated from a
tungsten boat [89] at a rate of .2 - .4 nm/s:
1. Evaporate 30 nm Al at ?
1
=12-15
?
with respect to the sample surface normal
(Fig. 3.4(b)).
2. Oxidize Al by introducing oxygen into chamber for 2-3 minutes. For Device
1 and 2, the pressure in the chamber during oxidation was 10 mTorr. For
Devices 3 and 4, a di?erent chamber was used and the requisite pressure was
90-100 mTorr (Fig. 3.4(c)).
3. Evaporate 60 nm Al at ?
2
=12-15
?
(Fig. 3.4(d)).
4. Lift-o? in acetone for 30 minutes at 90
?
C.
Tables 3.4 and 3.5 lists the values of C
i
,R
i
,C
g
,C
NR
,d
NR
, and the charg-
ing energy for Devices 1 - 4. In Appendix B, I discuss how we determined these
parameters.
75
(a) (b)
(c) (d)
Figure 3.4: Cross-section schematic of the double-angle evaporation of an SET junc-
tion. (a) Developed bi-layer of resist. (b) First deposition of Al. (c) Oxidation of
Al. (d) Second deposition of Al.
76
Table 3.4: Junction capacitances and resistances of Devices 1-4.
Device C
1
(aF) R
1
(k?) C
2
(aF) R
2
(k?) C
g
(aF) C
NR
(aF) d
NR
(nm)
1 81 61 84 59 14 61 600
2 250 21 100 53 10 26 600
3 173 47 341 24 14 64 300
4 ? 600 ? 15 ? 600 ? 15 19 63 300
Table 3.5: Total capacitance, charging energy, gap energy and Josephson energies
Devices 1 - 4. (a) Total capacitance found by summing C
i
, C
g
, C
NR
.(b)Total
capacitance found from position of JQP peak. (c) Total capacitance found from
position of DJQP peak.
Device C
a
?
(aF) C
b
?
(aF) C
c
?
(aF)
E
c
e
(?V )
?
e
(?V )
E
j,i
e
(?V )
1 241 - 279 287 220 12, 12
2 386 410 435 184 220 37, 14
3 592 577 540 148 200 15, 21
4 - 1310 - 61 180 ? 40, 40
77
Etching the Nanoresonator
The last step in the fabrication of the samples was the etching of the nanome-
chanical resonator from the silicon nitride membrane (Fig. 3.5):
1. Spin PMMA 950K A4 [84] at 4000 rpm for 60 seconds; bake for 5 minutes at
180
?
C on a hot-plate, no vacuum.
2. Spin PMMA 950K A4 [84] at 4000 rpm for 60 seconds; bake for 5 minutes at
180
?
C on a hot-plate, no vacuum (Fig. 3.5(a)).
3. Expose resist with charge dose of 350 - 400 ?C/cm
2
(30 pA beam current,
accelerating voltage of 30 kV, and magnification of 1000x).
4. Develop resist in MIBK/IPA (1:3) for 1 minute and rinse in IPA for 1 minute.
This defined a 400 nm wide window along the length of the beam (Fig. 3.5(b)).
The remaining resist served to mask the SET and accompanying circuit while the
resonator was etched (Fig. 3.5(c)):
1. RIE etch in CHF
3
/O
2
(18 sccm/2 sccm, 40 mTorr, and 175 Watts) for 6
minutes to etch through SiN membrane (Fig. 3.5(c)).
2. O
2
plasma etch for 1 minute to remove remaining resist (Fig. 3.5(d)).
The selectivity of the etch between SiN and the PMMA was approximately
1:1 at about 20 nm/min. The etch rate on gold was about 10 nm/min.
The final result (Device 2) is displayed in Fig. 3.6 and Fig. 3.7.
78
(a) (b)
(c) (d)
Figure 3.5: Etching of the nanomechanical resonator. (a) Sample coated with resist
(PMMA). (b)Etch mask defined. (c) Resonator suspended after RIE etch. (d) Resist
removed with O
2
etch.
79
Figure 3.6: Two detectors were fabricated on each sample.
80
Figure 3.7: SEM image of an RFSET displacement detector: Device 2.
81
Chapter 4
Apparatus
In the following sections I discuss the measurement equipment which we used
for the implementation of our RF-SET displacement detectors. The chapter is di-
vided into three main sections. The first section is a brief description of the packages
in which the samples were mounted for measurement. The second section describes
the dilution refrigerator and shielded room. In the final section, I discuss the refrig-
erator wiring, including the dc and microwave circuitry and thermometry.
4.1 The Sample Package
Each sample was placed in a homemade, gold-plated silver sample package
and held in place atop a gold-plated circuit board with copper clips (Fig. 4.1). Con-
nections from the sample to the circuit boards were made using 0.025 mm diameter
Au wire and a commerical wedge bonder. The circuit boards [90] were designed to
provide four 50 ? microstrip transmission lines; each microstrip was soldered to an
SMA connector for external connections.
Gold-plating of the circuit boards and sample packages was done in a fume
hood with a standard electroplating technique: 20 grams of KAu(Cn)
2
[91] and 50
grams of (NH
4
)
2
HC
6
H
5
O
7
[92] per liter of DI H
2
O, at 65
?
C, constantly stirred, with
a current density of 2-10 mA per square-cm of the surface area to be coated.
82
Figure 4.1: Homemade sample package (without the cover). Scale in inches.
83
4.2 The Dilution Refrigerator and Shielded Room
The sample packages were thermally anchored to the mixing chamber stage of
an Oxford Kelvinox 400 dilution refrigerator (Figs. 4.2 and 4.3). The base temper-
ature of the refrigerator was measured to be 9 mK. For operation, the refrigerator
was inserted into an Oxford 175 liter helium storage dewar, equipped with a 9 T
superconducting magnet. The entire cryostat was suspended from a 750 Kg optical
table supported by air dampers on four pillars, each of which was loaded with 750
Kg of lead and sand (Fig. 4.4).
The set-up was enclosed in a shielded room with its own ground connection
(Fig. 4.4), providing 100 dB isolation between 10 KHz and 5GHz. All control and
measurement electronics was located outside the shielded room, except for home-
made battery-powered voltage sources. Low-frequency signals were applied from
outside the shielded room through opto-isolators [94]; microwave signals were trans-
mitted into the shielded room through DC blocks [95].
At the input to the cryostat, low-frequency signals and DC biases were di-
vided by a factor of 100 to 1000 and filtered (Fig. 4.5). Commercial RF filters
[96] with cut-o? frequency in the MHz range were used in conjunction with two
home-made powder filters [97] connected in series. Each filter consisted of a box
containing a ?-filter [98], which connected two RF-tight compartments (Fig. 4.6(a)).
The compartments were filled with a mix of epoxy resin [99] and 10 ?m grain copper
powder [78] in equal weight proportions. The ?-powder filters provided more than
90 dB attenuation above 1 GHz (Fig. 4.6(b)). Commercial high-pass filters [100]
84
Figure 4.2: Image of dilution refrigerator: 4 K flange to mixing chamber.
85
(a)
(b)
Figure 4.3: Dilution refrigerator. (a) Mixing chamber. (b) Sample stage.
86
Figure 4.4: Picture of shielded room and optical table. The
4
He dewar is visible
behind the front pillar. A winch is used to raise the dewar over the refrigerator.
87
Figure 4.5: Picture of the top of the optical table. DC connections at the top of the
fridge are made through ?-powder filters. Battery-powered pre-amps and voltage
sources as well as break-out boxes are shelved on the electronics rack.
88
(a)
(b)
Figure 4.6: (a) Homemade ?-powder filter (scale in inches). (b) The 3 dB point is
2.5 - 3.0 MHz for one filter (solid curve) and 1.0 - 1.5 MHz for two filters in series
(dashed curve).
89
and attenuators [101] were used on the input microwave lines.
4.3 Refrigerator Wiring
A generalized schematic of the wiring inside the refrigerator is shown in Fig.
4.7.
Low-Frequency Circuitry
Between room temperature and the 1K pot stage, low-frequency signals were
transmitted through 1 meter of lossy, flexible, stainless steel, coaxial cables [102].
These lines were heat sunk at 4K and connected to a bank of microwave filters at
the 1K pot flange (Figs. 4.2 and 4.3). The filters were built in banks of four and
were based on the same principle as the room temperature microwave pi-powder
filters, but they did not include any pi-filters. The room temperature transfer char-
acteristics of one microwave powder filter bank are displayed in Fig. 4.8(a). I do not
have the transfer characteristics for the filters at low temperature.
Below the 1K pot flange, the lines consisted of 1 meter of lossy CuNi coaxial
cables [103], and another bank of microwave filters at the mixing chamber flange.
The transfer characteristics of the lossy coax are displayed in Fig. 4.8(b). For Devices
1 and 2, from the filters to the sample package, the connections were made via semi-
rigid Cu coaxial cables. The total attenuation through each of these lines was ? 90
dB above 800 MHz (Fig. 4.9). At 1 MHz and 10 MHz, the loss through the gate
line, including the pi-powder filters at the top of the fridge, was ? 2dBand? 11
90
Figure 4.7: General schematic of circuitry inside the shielded room and the refrig-
erator.
91
(a)
(b)
Figure 4.8: (a) S21 measurement of powder filters. (b) S21 measurement of 1 meter
of lossy CuNi coax. Measurements made at room temperature.
92
Figure 4.9: S21 measurement of gate line, including CuNi coax and two banks of
powder filters. Measurement made at room temperature.
93
dB respectively.
Additionally, a dc heat sink was added at the mixing chamber after the powder
filters (Fig. 4.3). The heat sink consisted of a center conductor of alternating Au
and Al sections (see the microstrip heat sinks below) on top of a .4 mm thick quartz
substrate inside a Au-plated RF-tight Cu package. We found 5 - 6 dB attenuation
from 1 MHz to 1 GHz. For Devices 3 and 4, the total loss through the gate line at
1 MHz was found to be ? 7dB.
Microwave Circuitry
For clarity, the discussion of the microwave circuit below is broken up into
three regions defined by the ports of the directional coupler: ?rf in?, ?coupled?,
and ?rf out?. Please refer to the general schematic in Fig. 4.7. However, note
that the configuration of the refrigerator microwave circuit in Fig. 4.7 is for the
measurement of Devices 2, 3 and 4. The configuration of the microwave circuit for
the measurement of Device 1 is not shown but can be recovered by changing the
position of the bias tee and heat sink as explained below.
At the input to the microwave circuit, at room temperature, the signal was
attenuated through a 20 dB attenuator. From the room temperature attenuator to
4.2 K, the input microwave signal was transmitted through a semi-rigid stainless
steel coaxial cable. The attenuation through the stainless steel coax was measured
at room temperature to be ? 6 dB. The signal was attenuated by 20 dB in the
cryostat before being fed into a directional coupler [104] at the 1K pot flange (Fig.
94
4.2). Between the input port of the directional coupler and the coupled port, the
signal was attenuated by an additional 20 dB at 1.5 GHz at a temperature of 1 K.
The coupled region of the circuit was modified several times during the course
of the measurements of Devices 1 - 4. For Device 1, the coupled port of the direc-
tional coupler was directed to a bias tee [105] at 1.7 K where the input rf signal
was added to the dc bias V
SD
. The output of the bias tee was directed to the mix-
ing chamber plate via a niobium semi-rigid coaxial cable [106], clamped at various
stages of the refrigerator for thermalization purposes. At the mixing chamber, the
Nb coaxial cable was clapmed and connected to a Cu semi-rigid coaxial cable, which
ran to the sample holder at the sample stage.
For the measurement of Devices 2, 3 and 4, the coupled port of the directional
coupler was connected to a microstrip heat sink through a Nb semi-rigid coaxial
cable. The heat sink was clamped at the 1K pot on the dilution refrigerator. The
output of the heat sink was connected to the bias tee via a section of Nb semi-rigid
coaxial cable. The bias tee was clamped at the mixing chamber. The output of the
bias tee was connected to the sample package through a Cu semi-rigid coaxial cable.
The purpose of the microstrip heat sink was to thermalized the inner conductor
of the coax while providing 50 ? transmission. For the measurement of Device 2,
the heat sink consisted of a 1.5 inch long, Au, 50 ? microstip transmission line on
a 0.4 mm thick quartz wafer housed in an RF-tight, Au-plated Cu package (similiar
to the microstrip shown in Fig. 4.10(a)). At 1.5 GHz and a temperature of 77 k,
the attenuation through the microstrip was less than 1 dB. For Devices 3 and 4 the
center conductor of the microstrip was made of alternating sections of Au and Al
95
(a)
(b)
Figure 4.10: (a) Au/Al RF microstrip. Scale is in inches. (b) S21 measurement of
Au/Al RF microstrip. Measurement made at 77 K.
96
(Fig. 4.10(a)). At 1.5 GHz and a temperature of 77 k, the attenuation through the
Au/Al microstrip was also less than 1 dB (Fig. 4.10(b)).
The transfer characteristics from the input of the fridge, through the direc-
tional coupler, to the bias tee, for the wiring of Devices 3 and 4 are displayed in
Fig. 4.11. At 1.5 GHz, including the losses in the attenuator (20 dB), the direc-
tional coupler (20 dB) and the SS semi-rigid coax (6 dB), 4 dB of attenuation is
unaccounted for in this section of the circuit.
On the return path from the directional coupler (rf out) the signal was sent
through another section of niobium semi-rigid coaxial cable to an ultra-low noise
cryogenic HEMT-amplifier [107], sitting in the helium bath at 4.2 K. The HEMT-
amplifier had a gain of 39 dB between 1 and 1.8 GHz (Fig. 4.12(a)), and noise
temperature of 2 K. A semi-rigid stainless steel coaxial cable connected the HEMT-
amplifier output to a Mini-Circuits room-temperature amplifier [108] with a gain
of 36 dB. The attenuation in the stainless steel coax was found to be 6 dB at 1.5
GHz at room temperature. The transfer characteristics of the room-temperature
amplifier are shown in Fig. 4.12(b).
From Fig. 4.12, at 1.5 GHz, we see that the total gain of the amplifiers plus
the stainless steel coaxial cable was approximately 72 dB. From Fig. 4.11, we see
that approximately 4 dB was lost in the coupled portion of the microwave circuit
for the measurement of Devices 3 and 4. There was an additional loss of ? 4dBin
the coax leading out of the shielded room. We would thus expect a measurement
circuit gain of ? 64 dB. For Devices 3 and 4, this agrees to within 3 dB of the gain
determined using the shot noise calibration (see Appendix B). Unfortunately, we
97
Figure 4.11: S21 measurements of the microwave circuit from the input of the fridge
to the bias tee, for the measurement of devices 3 and 4. Measurements done at
room-temperature. Including the 20 dB lost through the attenuator at the input of
the directional coupler, the 6 dB lost through the stainless steel semi-rigid coaxial
cable, and the 20 dB lost between the input port and coupled port of the directional
coupler, ? 4 dB attenuation was unaccounted for at 1.5 GHz. This was probably
due to connections, the bias tee, and the RF microstrip.
98
(a)
(b)
Figure 4.12: Gain of microwave amplifiers. (a) S21 measurement of cryogenic HEMT
amplifier plus 4 ft of stainless steel semi-rigid coaxial cable. Measurement made at
4 K. (b) S21 measurement of Mini-Circuits room-temperature amplifier.
99
Table 4.1: Measurement circuit gain and noise temperature Devices 1 - 4.
Device T
det
n
(K) G (dB)
1 - -
2 20 64
3 12.6 67
4 13.4 66.5
do not have the data for the transfer characteristics of the coupled portion of the
microwave circuit for the measurement of Devices 1 and 2. Table 4.1 lists the noise
temperature and gain of the measurement circuit for Devices 1 - 4.
Thermometry
The temperature of the sample package was monitored using a RuO chip
resistor [109]. The chip resistor was glued to the sample package using GE varnish
[93]. The resistance of the chip was monitored using four-terminal measurements
and a battery-powered resistance bridge [110]. Twisted pairs connected the bridge
to the resistor. From room temperature to 1 K, the pairs were made of Constanan.
From 1 K to the mixing chamber, CuNi-clad NbTi pairs were used.
The calibration of the RuO resistance was checked below 20 mK using nuclear
orientation thermometry [111] [112]. At a mixing chamber temperature of 10 mk,
the uncertainty in the calibration was approximately ? 1.5 % for 300 seconds of
100
counting. Above 20 mK, the calibration of the RuO resistance was checked with dc
SQUID thermometry [113]. The dc SQUID [114] was installed on the 1 K stage,
with the input connected to a 10 m? Cu film resistor at the mixing chamber via
homemade NbTi twisted pair. The SQUID, twisted pairs, and film resistor were
all shielded using various diameter Nb tubing. At 100 mK, the uncertainty in the
sample-stage temperature was dominated by the uncertainty in the measurement of
the RuO resistance and was approximately ? 1.0 mK.
101
Chapter 5
The RFSET
In this chapter I discuss our RFSET reflectometry measurement scheme. The
chapter is broken into two sections. In the first section I discuss the basics of
the reflectometry measurement and circuitry. In the second section I discuss the
feedback technique with which we used to stabilize the RF-SET gain in the presence
of low-frequency charge noise.
5.1 RFSET Reflectometry
RF-SET reflectrometry was performed using the measurement circuit shown
in Fig. 5.1. The basic process was first described by Schoelkopf et al. [41] and can be
summarized in three steps: a microwave carrier resonant with the microfabricated
L
T
C
T
circuit is applied to the port ?rf in? and directed to the sample via the
directional coupler; a fraction of the incident signal is reflected from the L
T
C
T
R
SET
circuit and directed back through the directional coupler and the two amplifier stages
to the port ?rf out?; from rf out, the signal is fed into a microwave mixer [115] for
homodyne detection.
In essence, the measurement of the reflected signal provides a measurement of
the SET impedance R
S
via R
S
= ?I
SD
(V
SD
,Q
g
)/?V
SD
. This is to be distinguished
from the quantity R
?
defined in Appendix B which is R
S
in the limit of large SET
102
Figure 5.1: Circuit schematic for RFSET reflectometry. Components left unlabeled
here are labeled in Figure 4.7.
103
source-drain bias, V
SD
.AsI
SD
is a sensitive function of the gate charge Q
g
(see
Appendix B), both R
S
and the reflected signal are as well. As a result, modulation
of Q
g
is detected as modulation of the amplitude of the reflected microwave signal,
and can be recovered by mixing the reflected signal with the original carrier signal
(homodyne detection). It is the sensitive dependence of the reflected signal on Q
g
which we exploit to perform our displacement detection. The remainder of the
section is dedicated to developing a quantitative understanding of this dependence
and discussing how, experimentally, we maximize it.
The Relationship Between ? and Q
g
The dependence of the reflected signal on Q
g
can be made explicit by first
writing the reflected voltage signal as
v
r
=?v
c
cos ?
T
t (5.1)
and the reflection coe?cient as
?=
Z
LCR
?Z
o
Z
LCR
+ Z
o
, (5.2)
where v
c
is the amplitude of the incident microwave carrier, ?
T
=
radicalBig
1/L
T
C
T
is
the tank circuit resonant frequency, Z
LCR
= L
T
/R
S
C
T
is the transformed SET
impedance at the tank-circuit resonance, and Z
o
= 50 ? is the characteristic trans-
mission line impedance.
For the measurement of Devices 1 - 4, Z
LCR
lessmuch Z
o
(see Appendix B), so I can
expand Eq. 5.2 as
104
? ? 1 ?
2Q
2
T
Z
o
R
S
(5.3)
where Q
T
?
radicalBig
L
T
/C
T
1Z
o
. Using Eq. 5.1 I find then that
v
r
?
parenleftBigg
1 ?
2Q
2
T
Z
o
R
SET
parenrightBigg
v
c
cos ?
T
t. (5.4)
In general, there is no closed-form expression for R
S
in terms of Q
g
. However,
from the considerations of Appendix B, I know we can approximate I
SD
near the
onset of tunneling as a sinusoidal function of Q
g
with period-e:
I
SD
?
e
2R
?
C
?
(sin(?Q
g
/e)+1). (5.5)
IcanwriteR
S
as
R
S
=
?V
SD
?I
SD
similarequal
V
SD
I
SD
?
2R
?
sin(?Q
g
/e)+1
, (5.6)
where I have assumed that V
SD
? e/C
?
.
From Eq. 5.4, I expect v
r
also to be sinusoidal in Q
g
but with a phase shift of
?-radians from I
SD
:
v
r
=
parenleftBigg
1 ?
Q
2
T
Z
o
R
?
(sin(?Q
g
/e)+1)
parenrightBigg
v
c
cos ?
T
t. (5.7)
Figure 5.2 shows a plot of v
r
vs. Q
g
for Device X. While not exactly sinusoidal,
the reflection coe?cient is seen to be e-periodic in Q
g
, with maxima occuring at
minima in I
SD
. For this particular sample R
?
? 50 k? and Q
T
? 10. Plugging
these values into Eq. 5.7, I see that the depth of modulation, M = 20log(?
max
/?
min
),
is on the order of 2 dB, consistent with the ? 2.5 dB seen in the plot, and of the
same order of magnitude as the values calculated for the Devices 1 - 4.
105
(a)
(b)
Figure 5.2: (a) Plot of I
SD
vs. Q
g
at V
SD
?
e
C
?
and (b) the corresponding reflection
coe?cient ?. Due to an o?set in Q
g
, the maxima in I
SD
do not occur exactly at Q
g
=.5e.DeviceX.
106
Charge Sensitivity
The utility of the transfer function ?-Q
g
in Fig. 5.2 is that it specifies, for a
given V
SD
bias, the Q
g
bias points at which one can expect ? and hence v
r
to be
maximally responsive to small changes in the SET-gate charge. Because the noise
floor of our charge-detection scheme is dominated by measurement circuit noise (see
Appendix B), the bias points which yield maximum charge responsivity also yield
the maximum charge sensitivity [66].
For the small-signal analysis, I set Q
g
= Q
g,o
+Q
g
(t), where Q
g,o
is the bias
point set by the external source V
g
,andQ
g
(t) lessmuch Q
g,o
is the time-dependent charge
signal we wish to measure. From Fig. 5.2, I see that the maximum response occurs
for Q
g
? n+e/4. Plugging Q
g
into Eq. 5.7 and expanding about Q
g,o
=e/4, I find
v
r
? (?
o
+??)v
c
cos ?
T
t, (5.8)
where
?
o
=
parenleftBigg
1?
Q
2
T
Z
o
R
?
?
2
parenleftBig
1+
?
2
parenrightBig
parenrightBigg
(5.9)
and
?? = ?
Q
2
T
Z
o
R
?
?
?
2e
Q
g
(t). (5.10)
From Eq. 5.8, I see that this measurement technique is equivalent to amplitude-
modulation: the signal, a time-dependent charge on the SET gate, modulates the
amplitude of an externally applied carrier. In most of our measurements, Q
g
(t)was
a sinusoidal signal. So I set Q
g
(t)=Q
1
cos(?
1
t), where Q
1
and ?
1
are the charge
107
modulation amplitude and frequency respectively. Thus Eq. 5.8 becomes
v
r
? v
c
bracketleftbigg
?
o
cos ?
T
t +
?
Q
2
T
Z
o
?
2
?
2R
?
e
Q
1
parenleftbigg
cos(?
T
??
1
)t +cos(?
T
+ ?
1
)t
parenrightbiggbracketrightbigg
(5.11)
After homodyne mixing, the reflected signal takes the form:
v
r
??v
c
Q
2
T
Z
o
R
?
?
2
?
2e
Q
1
cos ?
1
t, (5.12)
where I have just retained the terms with frequency ?
1
, which, in practice, can be
achieved with simple filtering.
It is perhaps helpful to stop here and comment briefly on the importance
of the tank circuit in the RFSET measurement. Without the tank circuit and
the impedance matching between the transmission-line impedance Z
o
and SET
impedance R
S
, v
r
would be reduced by a factor of Q
2
T
? 100. If you consider
the RFSET electrometer to be a charge amplifier, then this can be thought of as
losing a factor of 100 or 40 dB in the amplifier gain.
Eq. 5.12 specifies what signal to expect for a given gate charge modulation.
I can estimate the signal-to-noise ratio of the charge-detection scheme using the
measurement circuit noise temperature T
det
n
, discussed in Appendix B. The rms
voltage noise per frequency band referred to the input of the detector is given by
[116]
?
S
V
=
radicalBigg
k
B
T
det
n
Z
o
2
. (5.13)
Using Eq. 5.13 and the rms amplitude of Eq. 5.12, the signal-to-noise ratio
108
can be defined roughly as
S
N
?
vextendsingle
vextendsingle
vextendsingle
vextendsingle
radicalBigg
Z
o
4k
B
T
det
n
?B
Q
2
T
R
?
?Q
1
v
carrier
?
2e
vextendsingle
vextendsingle
vextendsingle
vextendsingle
2
, (5.14)
where ?B is the measurement bandwidth.
The charge sensitivity, in units of e
rms
/
?
Hz, is then defined by calculating
the charge modulation that yields
S
N
=1:
radicalBig
S
Q
=
?Q
e
?
?B
?
radicalBigg
8k
B
T
det
n
Z
o
R
?
?Q
2
T
v
carrier
. (5.15)
This result can be compared with what we have found experimentally. For
example, Fig. 5.3 shows the sinusoidal modulation of v
r
of Device X. Here, a carrier
signal with frequency ?
T
/2? =1.17GHzandpower
P
carrier
= 10log(
v
2
carrier,rms
50? ?mW
) ??29 dBm
is applied to rf in. Additionally, a modulation signal of ?
1
=1MHzandQ
1
=0.13
e
rms
is applied through rf mod (Fig. 5.1) to the SET gate.
Fig. 5.3(a) displays P
r
= 10log(v
2
r,rms
/50? ?mW) before homodyne detection.
Evident are the sidebands at ? 1 MHz from the carrier signal. Fig. 5.3(b) shows
the reflected signal, P
r
after mixing, and demonstrates the recovery of the 1 MHz
modulation. From the ratio of the power in the 1 MHz band P
1MHz
to the power
level of the background P
o
, I can find the charge sensitivity at 1 MHz:
radicalBig
S
Q
=
Q
1
?
?B
10
?
SNR
10
, (5.16)
where SNR(dBm) = P
1MHz
?P
o
is the signal-to-noise ratio, and ?B is the resolution
bandwidth of the spectrum analyzer. For the values in Fig. 5.3, I find
radicalBig
S
Q
? 65
109
(a)
(b)
Figure 5.3: (a) Power spectrum of rf out, before mixing. (b) Power spectrum of rf
out after mixing, demonstrating the recovery of the 1 MHz gate modulation. The
di?erence in background level between (a) and (b) is due to mixing (-8 dB), an
extra amplification stage before mixing (+16 dB), insertion loss in power-splitters
(-8 dB), and a change in resolution bandwidth from 1 kHz to 100 Hz. Device Y.
110
?e
rms
/
?
Hz. In comparison, using Eq. 5.15 and the parameters for this device (v
c
?
3 ?V
peak
after ? 72 dB attenuation, R
?
=80 k ?, T
det
n
? 22 K, Q
T
? 8), I calculate
radicalBig
S
Q
? 10
?3
e
rms
?
Hz.
The Derivative Map
There are two main factors contributing to the discrepancy between the the-
oretical estimate of the charge sensitivity and what we determined experimentally.
The first is the sinusoidal approximation of I
SD
(Q
g
). This approximation simply
assumes that the SET is in the normal state and biased at the onset of tunneling.
In fact, for the data displayed in Fig. 5.3, the SET was in the superconducting state
and biased around a DJQP resonance (see Appendix B). Second, the estimate does
not take into account the self heating of the SET, which, while not a problem at
that DJQP resonance, could certainly be a factor for bias points near the onset of
tunneling. Therefore, to obtain an accurate estimate of the charge sensitivity as a
function of the bias-points V
SD
and Q
g
(V
g
), I would need to model both super-
conducting processes and self heating on top of the Coulomb-blockade e?ects. This
wouldbeaninvolvednumericaltask.
For an existing device, the same information can be obtained more e?ciently
by simply taking an I
SD
V
SD
V
g
map and numerically calculating the curvature. Go-
ing back to Eq. 5.3, it is simple to see that measuring the modulation of the re-
flection coe?cient, ?, is equivalent to measuring the second-order derivative of I
SD
111
with repsect to V
SD
and V
g
:
? ? 1 ?
2Q
2
T
Z
o
R
SET
? 1?2Q
2
T
Z
o
?I
SD
?V
SD
. (5.17)
The small V
g
or Q
g
response of is thus
??(t)=
??
?Q
g
?Q
g
(t)=?2Q
2
T
Z
o
?
2
I
SD
?V
SD
?Q
g
?Q
g
(t). (5.18)
The charge sensitivity is then
radicalBig
S
Q
=
radicalBigg
k
B
T
det
n
Z
o
parenleftBigg
?
2
I
SD
?V
SD
?Q
g
parenrightBigg
?1
1
Q
2
T
v
c
, (5.19)
where Q
g
is in units of e and v
c
should be in units of V
peak
. Additionally, the response
has been divided by a factor of 2 to account for mixing.
Figures 5.4 and 5.5 display plots of I
SD
V
SD
V
g
, the numerical second-derivative,
and the modulated reflected signal for Device X. The I
SD
V
SD
V
g
mapwastakeninthe
superconducting state (see Appendix B for a description of how this is done). The
numerical derivatives of the IV-map (Fig. 5.5(a)) were then taken using Savitzky-
Golay filtering [117]. Finally, the reflected-signal modulation was measured (Fig.
5.5(b)) by applying a charge modulation of amplitude 0.02 e
rms
and frequency 1
MHz to the SET gate and a carrier signal of P
c
= -26 dBm to rf in. A computer was
used to increment V
SD
and V
g
. For each bias point, the modulated reflected-signal
amplitude was measured with an RF lock-in and recorded using GPIB (see Fig. 5.1).
Comparing Fig. 5.5(a) and Fig. 5.5(b), I find agreement to within a factor
of two between the numerically calculated derivative and the measured reflection
modulation. Using Eq. 5.19, I calculate the sensitivity at the bias point V
SD
?
0.4 mV and V
g
? - 9.8 mV (the DJQP resonance) to be ? 70 ?e
rms
/
?
Hz,which
112
Figure 5.4: I
SD
V
SD
V
g
map in the superconducting state.
113
(a)
V
g
 (mV)
V
SD
 (mV)
?10 ?5 0 5 10
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
?0.015
?0.01
?0.005
0
0.005
0.01
DJQP Resonances 
JQP Resonance
Calculated 2
nd
   Derivative (A/V 
2
) 
(b)
V
g
 (mV)
V
SD
 (mV)
?10 ?5 0 5 10
?0.8
?0.7
?0.6
?0.5
?0.4
?0.3
?0.2
?0.1
0
0.1
0.2
?0.01
?0.008
?0.006
?0.004
?0.002
0
0.002
0.004
0.006
0.008
0.01
Reflected Signal (A/V 
2
)
DJQP Resonances 
JQP Resonance
Figure 5.5: (a) Plot of the numerical derivative
?
2
I
SD
?V
SD
?V
g
of the superconducting IV
map in Figure 5.4. (b) Plot of the measured reflected signal modulation ?? for the
same range of V
g
and V
SD
as in (a). Here ?? has been scaled so that
??
2Q
2
T
Z
o
?V
g
is
the quantity plotted. Device Y.
114
is in good agreement with the 65 ?e
rms
/
?
Hz calculated from the signal-to-noise
ratio measured in Fig. 5.3. Here, a carrier amplitude of 4 ?V
peak
(-26 dBm applied,
-72 dB attenuatation in the circuit) was used to construct the reflection map. To
perform the conversion from V
g
to Q
g
, it is necessary to know that C
g
=11aFfor
this device. I also used T
det
n
? 22 K and Q
T
? 8 for the calculation.
The value of the derivative map is that it allowed us to quickly and accurately
determine which points in the V
SD
-V
g
plane yielded maximum gain and hence maxi-
mum charge sensitivity (the dark red and blue regions of Fig. 5.4(a). We then biased
the SET at one of these points, turned on the 1 MHz rf mod and the carrier signal,
and adjusted v
c
until we maximized the 1 MHz sideband, which we measured with
the rf lock-in or spectrum analyzer.
Close comparison of Fig. 5.5(a) and Fig. 5.5(b) reveals a problem. While the
calculated curvature and measured reflection amplitude are in close ageement, it is
clear that the reflection amplitude map is shifted by ? 3mValongtheV
g
axis.
This is a result of the slow SET background charge drift and the fact that the
I
SD
V
SD
V
g
map and the reflection map were taken several minutes apart. Especially
in the presence of large electric fields, which were necessary for RFSET displacement
detection, the SET background charge was so unstable that retuning of the V
g
bias point was necessary every few seconds. Clearly this is not ideal if one needs
to average for times longer periods of time. In the next section, we discuss the
engineering that allowed us to overcome this problem.
115
5.2 Gain Stabilization
In our SET?s, charge-drift and telegraph noise tended to destabilize the SET
gain, and limit the detection time to several minutes or less before retuning of V
g
was required. To overcome this, we implemented a feedback scheme to keep the
amplitude of the 1 MHz reflection sideband maximized.
The basic idea of the feedback technique can be understood by refering to
Fig. 5.6. The plot shows the amplitude of the 1 MHz sideband as a function of V
g
(Q
g
). As one would expect from the discussion in the previous section, the sideband
amplitude is a maximum with respect to V
g
(Q
g
)when?I
SD
/?V
g
is a maximum.
This occurs twice for each e-period, with the di?erence in height of the two peaks
directly related to the di?erence in the magnitude of the positive and negative I
SD
V
g
slopes. Obviously, for maximum gain, I would like to set V
g
to bias position A (or
an integer multiple of e from position A).
To keep the SET biased at point ?A?, I consider the curvature of the side-
band response. For small displacement from point ?A?, the sideband response is
approximately quadratic. Therefore, if I apply a small Q
g
(t) modulation, with am-
plitude C
g
?V
g
lessmuch e and frequency ?
A
, on top of the initial V
g
=A bias, I expect to
amplitude-modulate the 1 MHz sideband at 2?
A
(Fig. 5.7(a)):
v
r
? Q
g
(t)
2
cos?
1
t ?
(C
g
?V
g
)
2
4
[2cos ?
1
t +cos(?
1
+2?
A
)t +cos(?
1
?2?
A
)t].
Alternatively, if I bias the SET at points such as those labeled ?B? and ?C? in
Fig. 5.6, the small Q
g
(t) modulation response is approximately linear, resulting in
116
Figure 5.6: Dependence of the 1 MHz sideband amplitude on V
g
(1 MHz sideband
response) in Device 1 at a mixing chamber temperature of 35 mK. As expected
there are two maxima in the sideband amplitude per period. The arrows about bias
points A, B, and C indicate audio modulation of the sideband.
117
(a)
(b)
Figure 5.7: Power spectrum of the audio modulation of the 1 MHz sideband re-
sponse. (a) Bias condition A. Sidebands at ? 20KHzfrom1MHzsidebandare
prominent. (b) Bias condition B or C. Sidebands at ? 10 KHz from 1 MHz sideband
are prominent. Device X.
118
the amplitude moudlation of the 1 MHz sideband at ?
A
(Fig. 5.7(b)):
v
r
? Q
g
(t)cos?
1
t ?
(C
g
?V
g
)
2
[cos?
1
t +cos(?
1
+ ?
A
)t +cos(?
1
??
A
)t] .
In addition, if the bias point is o? maximum, at positions such as ?B? or ?C?,
the phase of the resulting audio sidebands depends on the sign of the slope of the
transfer function; this tells me whether to increase or decrease Q
g
to return to the
maximum, ?A?.
Thus by modulating the 1 MHz sideband with an audio signal, and monitoring
the harmonic content and phase of the resulting product frequencies, I can determine
the bias point Q
g
of the SET and whether Q
g
should be increased, decreased or left
alone to maximize the SET gain. I then apply the appropriate correction to Q
g
.
Obviously, we did not make the corrections to Q
g
by hand. A schematic of
the circuit we used is presented in Fig. 5.8 and Fig. 5.9. A charge modulation
of amplitude 0.02 e
rms
and frequency ?
A
= 10 kHz was applied to the SET gate
through the port labeled ?audio mod? for the modulation of the 1 MHz sideband.
The reflected signal v
r
was monitored at port ?rf out?, and directed to the
audio feedback circuit. In the audio feedback circuit (Fig. 5.9), v
r
was amplified
and then mixed with the orignial carrier signal to recover the sideband modulation.
The signal was then filtered and mixed with the original 1 MHz modulation
to recover the audio sidebands of the 1 MHz modulation. The resulting audio signal
was amplified and mixed with the original 10 kHz modulation.
The product of the audio mixing, or the error signal, was sent into a PID
controller with the reference set to 0.0 Volts (ie. it was set to minimize the 10 kHz
119
Figure 5.8: Circuit schematic for RFSET gain stabilization.
120
Figure 5.9: Feedback scheme for gain stabilization.
121
sidebands). The output of the PID was split: one branch directed the signal to the
SET gate to adjust the bias position; the other branch fed the signal into a digital
voltmeter that was monitored by the PC using GPIB. Using LabVIEW software,
the PC monitored the feedback signal. By adjusting the DAC output V
g
,itkept
the output of the PID below a predefined threshold voltage.
Finally, for the rapid charge drift which occured when a large electric field was
used to couple the nanoresonator and SET, the DAC output V
g
saturated within
minutes. To compensate for this, the same LabVIEW program reset the DAC output
by 1 electron (ie. one period in the I
SD
vs. Q
g
curve) at the SET gate.
The e?ectiveness of the feedback scheme is demonstrated in Fig. 5.10. With
feedback on (Fig. 5.10(a)), the 1 MHz sideband amplitude was stabilized to within
1% over minutes to hours, depending on the coupling voltage between the resonator
and the SET. In the frequency domain (Fig. 5.10(b)) the e?ect of the feedback was
seen as a reduction by nearly a factor of 100 in the spectral noise density of the 1
MHz sideband amplitude. The bandwidth of the feedback circuit was set by the
time constant of the integrator in the PID controller, which was typically on the
order of 1 - 10 ms.
122
(a)
(b)
Figure 5.10: (a) Sideband amplitude stability in time domain and (b) sideband
amplitude spectral noise density. Device X.
123
Chapter 6
RFSET Displacement Detection
In this chapter I discuss the implementation of the RFSET as a displacement
detector and present the main results of my thesis. The chapter is divided into three
sections. In the first section I draw on the results of the previous chapter to discuss
the basic principles of the detection scheme and the measurement methodology.
Some of our results for the RFSET detection of capacitively driven nanoresonators
is presented and discussed. In the second section I discuss the RFSET displacement
detector as a mechanical noise thermometer and use the results to characterize
the performance of our devices. In particular, I show a minimum resonator mode
temperature of approximately 56 mK for Device 2. Further, I demonstrate our
RFSET displacement detection scheme to be nearly an ideal one, approaching within
a factor of 5 from the quantum limit. In the final section I address the issue of SET
back action.
RFSET Displacement Response and Sensitivity
Having developed the small charge-signal response for the RFSET in the pre-
vious chapter, I discuss how we implemented the RFSET electrometer as a displace-
ment transducer. In principle, the main idea [12] [13] is straightforward (Fig. 6.1):
a static SET gate is replaced with a metallized nanoresonator and a large dc bias
124
Figure 6.1: A circuit schematic of the RFSET displacement detector. A metallized
nanoresonator serves as an SET gate to modulate the induced SET-island charge
and, consequently, the SET?s impedance. The impedance fluctuations are measured
using microwave reflectometry at the frequency of the L
T
C
T
tank circuit resonance.
125
V
NR
is applied between the suspended gate and the SET island; displacement of the
resonator from its equilibrium position modulates the induced SET-island charge;
the resulting modulation of the SET impdedance is then monitored by microwave
reflectometry.
6.1 Methodology
Using Eq. 5.18, I can express the amplitude modulation of the reflected mi-
crowave signal in terms of the mechanically-induced SET-island charge ?Q
NR
:
v
r
(t)=??(t)v
c
(t) ??2Q
2
T
Z
o
?
2
I
SD
?V
SD
?Q
NR
?Q
NR
(t)v
c
(t), (6.1)
where Q
NR
=C
NR
V
NR
. For small displacment y
m
(t) of the resonator, I can approx-
imate the corresponding charge modulation as
?Q
NR
?
?C
NR
?y
m
V
NR
y
m
? b
C
NR
V
NR
d
NR
y
m
(t) (6.2)
where d
NR
is the separation between the nanoresonator and the SET island, and
b is a geometrical constant of order unity and calculated using the capacitance
extraction software FastCap [65]. I consider y
m
(t) to be the in-plane fundamental
mode displacement of the resonator averaged over the length of the SET island.
I can rewrite Eq. 6.1 as
v
r
(t) ??2Q
2
T
Z
o
?
2
I
SD
?V
SD
?y
m
y
m
(t)v
c
(t). (6.3)
Here I have simply replaced ?Q
NR
with y
m
as the di?erential quantity. It is
important, though, to clarify a simple, yet, potentially confusing point. As discussed
126
in Appendix B, the SET current I
SD
is dependent on the change in free energy ?F
of the SET circuit for the respective electron tunneling events. This in turn is a
function of the SET-island potential,
?
island
C
?
= ?ne + C
1
V
SD
+ C
NR
V
NR
+ C
g
V
g
, (6.4)
where I have assumed an asymmetrical source-drain bias as in Appendix B.
In the previous chapter, as V
g
was the modulated quantity, I discussed the
RFSET response in terms of V
g
or Q
g
; in other words, ??
island
C
?
=?Q
g
.Inthis
chapter, the modulated quantity is C
NR
,sothat??
island
C
?
? ?Q
NR
- assuming
that C
?
greatermuch ?C
NR
. The important point is that, for ?Q
g
=?Q
NR
, the modulation
of ??
island
and the RFSET response are the same:
?
2
I
SD
?V
SD
?Q
g
=
?
2
I
SD
?V
SD
?Q
NR
?
d
NR
bC
NR
V
NR
?
2
I
SD
?V
SD
?y
m
,
Thus, I use the curvature ?
2
I
SD
/?V
SD
?Q
g
, which I numerically calculate from the
I
SD
V
SD
Q
g
map, to determine the RFSET displacement response:
v
r
(t) ??2Q
2
T
Z
o
bC
NR
V
NR
d
NR
?
2
I
SD
?V
SD
?Q
g
y
m
(t)v
c
(t). (6.5)
Assuming that the noise floor of our detection scheme is limited by the mea-
surement circuit noise, I estimate the displacement sensitivity to be
?y ?
radicalBig
S
y
?
d
NR
bC
NR
V
NR
1
Q
2
T
v
c
radicalBigg
k
B
T
det
N
Z
o
parenleftBigg
?
2
I
SD
?V
SD
?Q
g
parenrightBigg
?1
. (6.6)
As in the previous chapter, I divide the response by a factor of two to account
for the homodyne detection; and v
c
is the carrier amplitude given in peak units to
calculate the sensitivity.
127
Equation 6.6 can be expressed in terms of the charge sensitivity using Eq. 5.19
?y ?
radicalBig
S
y
?
d
NR
bC
NR
V
NR
radicalBig
S
Q
. (6.7)
Equation 6.7 simply states that, for a given resonator displacement, as I in-
crease V
NR
, I increase the charge signal detected by the SET. Of course, there are
obvious limitations that will a?ect how far I can turn the knob for V
NR
before Eq.
6.7 breaks down.
For one, I have not taken into account the SET back action, the contribution
of which to the spectral displacement noise density increases linearly with V
NR
.In
Chapter 2, I discussed how the coupling voltage at which back action becomes a
factor depends on many parameters, and can range from milli-volts to 10?s of volts.
A second limitation arises from the electrostatic force between the resonator
and the gate electrode and the resonator and the SET island. Increasing V
NR
will
eventually result in the resonator ?snapping? to either the gate or SET island. A
crude approximation of the voltage at which this will occur can be made by finding
V
NR
,foragivend
NR
, which yields an inflection point in the resonators potential
energy, U(y) ?
1
2
ky
2
-1C
NR
V
2
NR
/2. The result, V
snap
=
radicalBig
4kd
2
NR
/9C
NR
, yields 100?s
Volts for the devices presented here, and is not relevant. However, it could easily
be reduced to the order of Volts and be a greater concern for much more tightly
coupled devices.
A final limitation occurs if the resonator?s motion is so large that the swing
in the induced SET-island charge approaches one electron. For 1 pm of motion, a
typical thermal amplitude at 100 mK for the resonators we study, the corresponding
128
voltage at which this ?shuttling? should occur is V
NR
? .5ed
NR
/y
m
C
NR
? 100?s
Volts. Again, this is much larger than any voltage which we used. However, this
estimate does suggest, and experiments seem to confirm, that shuttling is a concern
for the SET measurement of resonators driven into the non-linear regime, where
amplitudes are ? nm?s.
RFSET Detection of Capacitively Excited Nanoresonators
In practice, we maximized v
r
by first applying V
NR
to the resonator and then
measuring the I
SD
V
SD
V
g
characteristics. After numerically calculating the curva-
ture, we then adjusted V
g
and V
SD
to the positions yielding maximum curvature.
Finally, we adjusted v
c
, and then implemented the feedback stabilization as de-
scribed in the previous chapter. There was a good deal of tuning involved, but
these were the general steps.
With the SET gain stabilized at maximum, we then tried to detect the res-
onator?s motion. Typically, before we cooled down a new sample with the dilution
refrigerator, we used the magnetomotive technique (Appendix A) to identify the
resonator?s fundamental in-plane mode frequency in a test probe at 4 Kelvin. The
resonant frequency measured at 4 K, thus, served as the starting point at which to
?look? for the resonance once we have cooled the sample down to mK.
Figure 6.2 shows a schematic of the circuit we used to probe the resonator?s
response with the RFSET. Essentially, we used the same circuit as the stabilized-
RFSET reflectometry circuit described in the previous chapter. However, we applied
129
Figure 6.2: Circuit schematic for RFSET detection of capacitively excited nanores-
onators.
130
an additional swept-sine signal V
D
cos ?t, to the SET gate to capacitively excite the
nanoresonator?s motion. The nature of the excitation can be understood by writing
the potential energy of the capacitor C
gnr
formed by the metallized nanoresonator
and the gate electrode,
U = ?
1
2
C
gnr
(V
NR
+ V
D
cos?t)
2
, (6.8)
and the force from the potential energy gradient,
F = ?1
1
2
?C
gnr
?y
(V
NR
+ V
D
cos?t)
2
?
b
2
C
gnr
d
gnr
(V
NR
+ V
D
cos ?t)
2
, (6.9)
where d
gnr
is the separation between the nanoresonator and the gate electrode.
There is a dc component of the force as well as terms at both ? and 2?.
Keeping just the ? term, I find
F ? b
C
gnr
V
NR
V
D
d
gnr
cos?t. (6.10)
From Eqs. A.19, A.25, and A.26, the fundamental mode response of the
resonator is described by
y(x,t)=b
C
gnr
V
NR
V
D
d
gnr
m
eff
?
1
Y
1
(x)
((?
1
2
??
2
)+j (?
1
?/Q
e
))
cos ?t, (6.11)
where Y
1
(x) is the fundamental eigenmode for a clamped-clamped rectangular bar.
For the calculation of ?
1
, I have assumed that the force is uniform over the section
of the resonator defined by the ends of the gate electrode, and that it is zero outside
of this section. The function Y
1
(x) has been normalized so that it is dimensionless
and the mid-point displacement, Y(L/2), is equal to 1.
131
From Eqs. A.11 and 6.11, the average displacement over the length of the SET
island is given by
y
m
(t) ?
b
a
1
C
gnr
V
NR
V
D
d
gnr
M
m
a
1
?
1
((?
1
2
??
2
)+j (?
1
?/Q
e
))
cos?t, (6.12)
where a
1
is the geometrical factor accounting for the averaging of Y
1
(x)overthe
length of the SET island.
Using Eqs. 6.5 and 6.12, the modulation of the reflected signal due to the
motion of the capacitively driven resonator is given by
v
r
(t) ? 2Q
2
T
Z
o
parenleftBigg
?
2
I
SD
?V
SD
?Q
NR
parenrightBigg
b
2
C
gnr
C
NR
V
2
NR
V
D
a
1
d
NR
d
gnr
M
m
?
?
a
1
?
1
((?
1
??
2
)+j(?
1
?/Q
e
)
v
c
(t)cos?t. (6.13)
On resonance, and using the definition of the K
m
from Appendix A, I find the
amplitude of the modulated reflected signal:
A
r
? 2Q
2
T
Z
o
parenleftBigg
?
2
I
SD
?V
SD
?Q
NR
parenrightBigg
?
1
b
2
Q
e
C
gnr
C
NR
V
2
NR
V
D
a
1
d
NR
d
gnr
K
m
v
c
. (6.14)
In practice, the real and imaginary components of the response were mea-
sured by directing the output of the homodyne detection to an RF lock-in (Fig.
6.2) and sweeping the drive V
D
cos?t through resonance ?
1
. A computer program
incremented the frequency of the sinusoidal drive step-wise and recorded the two
quadratures of the output of the amplifier at each step.
Figure 6.3 shows the response of the nanoresonator in Device 2 to continuous
swept-sine capacitive excitation as measured with an RFSET. The data was taken
at a sample-stage temperature of 30 mK, and the RFSET was biased near the JQP
resonance as illustrated in Fig. 6.4. A coupling voltage of V
NR
= 4 V was applied.
132
Figure 6.3: Response of the capacitively excited resonator as measured with an
RFSET, Device 2.
133
Figure 6.4: I
SD
V
SD
V
NR
map of Device 2 with color surface depicting numerically
calculated curvature. Bias point is near the JQP resonance ridge.
134
Fitting the quadratures to a harmonic oscillator response, the resonant fre-
quency, quality factor and peak amplitude were determined to be f
1
= 19.654486(3)
MHz, Q
e
= 44.85(6) x 10
3
,andA
r
= 75.95(6) ?V
rms
respectively.
I can compare the measured amplitude A
r
with what is expected from Eq.
6.14. For Device 2, d
NR
= 600 nm, d
gnr
=1.2?m, v
c
=7.9?V
peak
, ?
1
= .838, C
gnr
? 15 aF, V
D
? 350 ?V
peak
,andb? .47 - the other parameters are listed in Tables
A.3, B.1, and 3.1. From Fig. 6.4, I estimate that ?
2
I
SD
/?V
SD
?Q
NR
? 0.01/C
NR
=
3.8 x 10
14
. The gain of the detection circuit at the output of the shielded room is
+64 dB. The gain of the audio feedback circuit is -1 dB (attenuator -13 dB, amplifier
+36 dB, 2 power splitters -8 dB, and mixer -8 dB). Thus, using the rms value of Eq.
6.14, I estimate the amplitude at the input of the lock-in to be 110 ? 60 ?V
rms
.The
uncertainty in the estimate is due mainly to the ? 50% error in K
m
.Itdoesnot
include the uncertainty the attenuation and gain in the measurement circuit. These
can be estimated from the discrepancy between measured and calculated charge
sensitivity at the measurement bias point V
SD
= -.9 mV. The values are 60 and 30
?e/
?
Hz respectively.
I can also determine the amplitude of the resonator?s displacement from the
measured reflected signal and Eq. 6.5. I calculate that the amplitude A
r
=76?V
rms
roughly corresponds to 50 pm
rms
of motion. From Eq. 6.12, I expect approximately
20 pm
rms
of motion. Again, the uncertainty is at least 50 % due to uncertainties in
the spring constant and measurement circuit gain.
Given the uncertainty in the estimate of the amplitude of the reflected signal
and in the conversion of the reflected signal to resonator displacement, order-of-
135
magnitude agreement between the estimates and measurements is about the best I
can expect. The total uncertainty could be reduced by determining more precisely
the resonator?s spring constant and the measurement circuit?s transfer characteris-
tics. However, as I will discuss in the next section, it is not necessary to go to such
e?orts in order to assess the performance of the detection scheme in terms of the
quantum limit.
Before concluding this section, I want to discuss the measurement of the
nanoresonator?s response in the time domain. In addition to applying a swept-sine
to the gate and measuring the nanoresonator?s frequency response with a lock-in, we
also applied a pulse to the gate and measured the nanoresonator?s amplitude decay
with a digital oscilloscope. We used the same measurement circuit as in Fig. 6.2 ex-
cept we replaced the lock-in with a digital oscilloscope - the same function generator
was used to apply the drive signal except we switched it from ?swept-sine? mode to
?burst? mode. Also, an additional amplifier with gain of 36 dB and an high-pass
filter with a 10 MHz cut-o? were inserted between the mixer and the oscilloscope.
Typically, the pulse we applied was a 60 kilo-cycle sinusoid at the nanores-
onator?s fundamental mode frequency ?
1
. At the falling edge of the pulse, the
digital oscilloscope was triggered, and the resonator?s decay was recorded for 20 ms
at a sampling rate of 50 - 100 MS/s. The process was repeated 50 - 100 times and
the waveforms were averaged.
Figure 6.5 shows the ?ring-down? of the nanoresonator in Device 2. The signal
has been digitally filtered in three steps. First, in the time domain, I multiplied the
waveform by an exponential decay with time constant ? =Q
e
/?
1
[118]. Second,
136
(a)
(b)
Figure 6.5: (a) Filtered time-domain response of a nanoresonator to capacitive-pulse
at the fundamental mode frequency, captured using a digital oscilloscope. (b) Blow-
up of (a) to demonstrate time-resolved oscillations at 19.65 MHz. Device 2, T
S
=
30 mK, V
NR
=6 V.
137
19.642 19.644 19.646 19.648 19.65 19.652 19.654 19.656 19.658 19.66
0
1
2
3
4
5
6
7
8
x 10
?4
Frequency (MHz)
Amplitude (V
2
)
Figure 6.6: FFT of the raw time-domain response of a nanoresonator to a capacitive-
pulse, Device 2.
138
I numerically calculated the FFT of the waveform and multiplied the result by a
Gaussian with a width of ? = 10 MHz. Third, I transformed the data back into the
time domain by numerically calculating the inverse FFT. The decay time for the
exponential in the first step was determined by taking the FFT of the raw time-
domain data and fitting it to an harmonic oscillator response (Fig. 6.6). From the
fit, the frequency and quality factor were determined to be f
1
= 19.6515382(4) MHz
and Q
e
= 64300 ? 250 respectively.
Finally, I note that the di?erence in resonant frequency of ? 2.9 kHz between
Figs. 6.3 and 6.6 was due to the electrostatic softening of the resonator?s mode.
Expanding the electrostatic energy of the resonator to second order about the equi-
librium position of the resonator,
?
2
U
e
??
C
NR
d
2
NR
V
2
NR
y
2
, (6.15)
I find the correction to the fundamental mode frequency to be
??
?
1
??
C
NR
V
2
NR
K
avg
d
2
NR
. (6.16)
Thus, if I increase V
NR
from 4 V to 6 V, I expect a shift in resonant frequency
of ? - 2 kHz , which is comparable to what we measured. For a more precise
determination it would be necessary to include the capacitive coupling between the
resonator and the gate, and, as well, determine the geometrical coe?cient for the
second derivative of C
NR
.
139
6.2 Mechanical Noise Thermometry
The Equipartition of Energy
I now turn to discuss the measurement of the resonators? random thermal
motion. I assume that the resonator and RFSET detector are su?ciently weakly
coupled so that the back action e?ects of the detector are negligible with respect
to the thermal fluctuations. From Eq. 2.59 and Eq. 2.60, then, T
e
= T
b
and Q
e
=
Q
b
. In this case, then, the classical equation of motion for a resonator in thermal
contact with a heat bath is given by the Langevin equation [49],
M
m
?
2
y
m
(t)
?t
2
+ ?
?y
m
?t
+ K
m
y
m
= f
N
(t), (6.17)
where
f
N
(t)=
F
N
(t)
a
1
integraldisplay
L
0
dxY
1
(x) (6.18)
and
? =
?
1
a
1
(6.19)
Here, the equation of motion is written in terms of the average displacement
of the resonator over the length of the SET island, which is the measured quantity.
For each of the devices I discuss, M
m
, K
m
and a
1
are defined in Appendix A.
Simply put, the Langevin equation states that a resonator in thermal contact
with a heat bath is subject to two external forces: a frictional force proportional to
? and a random driving force represented by f
N
(t). Generally, it is assumed that
f
N
(t) is both Gaussian and Markovian, meaning it has a white power spectrum,
140
S
f
(?), and its auto-correlation function is given by
?f
N
(t
1
)f
N
(t
2
)? = S
f
(?)?(t
1
?t
2
). (6.20)
Here S
f
(?)=S
f
b
(see Chapter 2) in the limit that ?h?/k
B
T
b
? 0.
From these assumptions, and the fact that the resonator is considered to be in
thermal contact with the heat bath, it can be shown that the dissipative force and
the random driving force are related through [53]
? =
1
k
B
T
eff
integraldisplay
?
0
dt?f
N
(t
1
)f
N
(t
2
)? =
S
f
(?)
2k
B
T
e
, (6.21)
or
S
f
(?)=2?k
B
T
e
=
2M
m
?
1
k
B
T
e
Q
e
, (6.22)
where I have used the relationship ? = M
m
?
1
/Q
e
; T
e
=T
b
is the bath temperature;
and Q
e
= Q
b
is the quality factor due to coupling to the thermal bath.
Equation 6.21 is generally known as the fluctuation-dissipation theorem [53],
and is a fundmantal classical statement equating the random forces which define
a state of equilibrium to the dissipative forces which tend to bring a driven or
nonequlibrium state back toward equilibrium. Practically speaking, then, knowing
the resonator?s response, one can determine the thermal forces driving the resonator,
and, thus, determine the resonator?s temperature.
The resonator?s response is found by solving Eq. 6.17 for the time-averaged,
mean-square displacement, [31]:
?y
m
(t)
2
? =
1
2?
integraldisplay
?
??
d?S
y
(?), (6.23)
141
where
S
y
(?)=
S
f
(?)
M
m
2
parenleftBig
(?
2
1
??
2
)
2
+(?
1
?/Q
e
)
2
parenrightBig
(6.24)
is the mechanical displacement noise spectral density.
Separating into positive and negative frequencies and integrating Eq. 6.23, I
find
?y
m
(t)
2
? =
S
f
(?)Q
2
e
K
2
m
?f +
S
f
(?)Q
2
e
K
2
m
?f, (6.25)
where ?f = ?
1
/4Q
e
is defined as the resonator?s noise-equivalent bandwidth and I
have assumed that Q
e
greatermuch1.
Using Eqs. 6.22 and 6.25, I recover the equipartition theorem:
?y
m
(t)
2
? =
1
2
k
B
T
e
K
m
+
1
2
k
B
T
e
K
m
=
k
B
T
e
K
m
. (6.26)
Of course, Eq. 6.26 is a classical expression and ceases to be valid when
k
B
T
e
/?h?
1
similarequal 1. However, for the temperature and frequency ranges in which we
conducted our experiments, the resonators were still more than an order of magni-
tude above this limit. Consequently, we were able to use Eq. 6.26 and measurements
of ?y
m
(t)
2
? to determine T
e
.
Power Spectra
Equation 6.26 is the basis of mechanical noise thermometry. Of course, as I
discussed earlier in the chapter, the RFSET is sensitive to y
m
(t)ratherthan?y
m
(t)
2
?.
To recover?y
m
(t)
2
?, we used a spectrum analyzer with FFT capability to record and
average the power density spectrum of the output of the mixer (?signal out? in Fig.
6.2).
142
From Eqs. 6.5, 6.22, and 6.24, the power spectral density at the input to the
spectrum analyzer takes the form:
P
s
? 4G
Q
4
T
Z
o
v
2
c
(?
2
I
SD
/?V
SD
?Q
NR
)
2
(bC
NR
V
NR
/d
NR
)
2
?
1
parenleftbigg
(?
2
??
2
1
)
2
+(?
1
?/Q
e
)
2
parenrightbigg
M
m
Q
e
k
B
T
e
, (6.27)
where G represents the total gain of the measurement circuit up to the spectrum
analyzer.
In practice to compare each averaged power spectrum, we simultaneously
recorded the charge gain of the circuit by measuring the magnitude of the 1 MHz
reflection modulation with a lock-in (?rf lock-in? in Fig. 6.2). We then divided the
power spectrum by the average gain A
Q
(in units of W/e
2
rms
) recorded during the
measurement of the power spectrum. This converted the measured thermal response
of the resonator into units of e
2
rms
/Hz. I thus rewrite Eq. 6.27 as
P =
A
parenleftbigg
(?
2
??
2
1
)
2
+(?
1
?/Q
e
)
2
parenrightbigg
?
4
1
Q
2
e
, (6.28)
where ?A? is in units of
e
2
rms
Hz
and is given by
A = GQ
4
T
Z
o
v
2
c
parenleftBigg
?
2
I
SD
?V
SD
?Q
NR
parenrightBigg
2
parenleftBigg
bC
NR
V
NR
d
NR
parenrightBigg
2
k
B
T
e
K
m
1
A
Q
?f
. (6.29)
Figure 6.7 displays the power spectra of a nanoresonator?s thermal response
at sample-stage temperatures T
S
= 75 mK, 150 mK, 300 mK, and 500 mK and
a coupling voltage of V
NR
=10 V. The data was taken using Device 2. For each
temperature, 500 to 1500 traces were taken, averaged, and fit (lines) to Eq. 6.28
plus a background. From the fit we extracted the resonant frequency ?
1
, quality
factor Q
e
, peak amplitude A, and background P
o
, for each temperature. The error
143
bars for each of these quantities was determined using standard error propagation
with a uniform variance assigned to each point in the power spectrum plot. The
variance was taken from the variance in the background level far from resonance.
The background is Gaussian and due the 80 pV /
?
Hz input voltage noise of the
cryogenic microwave pre-amplifier.
E?ective Resonator Temperature
Knowing the resonator?s amplitude A, quality factor Q
e
and fundamental mode
frequency ?
1
/2?, I can calculate the e?ective temperature of the resonator?s funda-
mental mode. There are two ways that I can do this: (1) use Eq. 6.29 and the known
device parameters (eg.K
m
, G, etc.); or (2) use a a primary thermometer (eg.nuclear
orientation thermometer) or calibrated secondary thermometer (eg. RuO resistor)
to calibrate the temperature dependence of one of the resonator?s extracted param-
eters (eg.A,Q
e
, etc.). The total uncertainty in the device parameters is greater
than 50 % (due mainly to uncertainties in K
m
and G), whereas the uncertainty in
the extracted parameters is typcially 1 - 10 % (see Fig. 6.7). Provided I choose a
suitable parameter, it is clear that the second method is much more precise.
While not necessary, it is preferable that I choose a parameter that varies
linearly with T
e
. Neither the resonator?s amplitude A, quality factor Q
e
or funda-
mental mode frequency ?
1
/2? demonstrate this behavior. However, if I integrate
Eq. 6.28, for Q
e
greatermuch 1, I find
IR =
integraldisplay
?
0
Pd? =
?
1
4Q
e
A = GQ
4
T
Z
o
v
2
c
parenleftBigg
?
2
I
SD
?V
SD
?Q
NR
parenrightBigg
2
parenleftBigg
bC
NR
V
NR
d
NR
parenrightBigg
2
k
B
T
e
A
Q
K
m
, (6.30)
144
(a)
19.668 19.67 19.672 19.674 19.676 19.678 19.68
50
100
150
200
250
300
350
400
T
S
 =  75mK 
P ( 
?
e
rms2
 /Hz) 
Frequency (MHz) 
Q
e
  =  4.00(5) x 10
4
f
1
=19.674154(2) MHz 
P
o
  = 38.5(1) ?e
2
/Hz 
A  = 99.9(9) ?e
2
/Hz 
(b)
19.668 19.67 19.672 19.674 19.676 19.678 19.68
50
100
150
200
250
300
350
400
T
s
 = 150 mK
P ( 
?
e
rms2
 /Hz) 
Frequency (MHz) 
Q
e
  = 3.86(2) x 10
4
f
1
=19.674600(1) MHz 
A = 144.6(5) ?e
2
/Hz
P
o
 =31.1(1) ?e
2
/Hz
(c)
19.668 19.67 19.672 19.674 19.676 19.678 19.68
50
100
150
200
250
300
350
400
T
s
  =  300 mK 
Frequency (MHz) 
P  ( 
?
e
rms2
 /Hz) 
Q
e
 = 3.43(2) x 10
4
f
1
 = 19.674559(1) MHz 
A = 227(1) ?e
2
/Hz
P
o
 = 35.6(1) ?e
2
/Hz
(d)
19.668 19.67 19.672 19.674 19.676 19.678 19.68
50
100
150
200
250
300
350
400
T
s
 = 500 mK 
P  ( 
?
e
rms2
 /Hz) 
Frequency (MHz) 
Q
e
 = 2.92(1) x 10
4
f
1
=19.675225(1) MHz 
A  = 343(1) ?e
2
/Hz
P
o
 = 51.1(1) ?e
2
/Hz
Figure 6.7: Power spectrum measurements of a nanoresonator?s thermal response,
P, (squares) fit to harmonic oscillator response (lines). Data taken with coupling
voltage V
NR
=10V.Device2.
145
where IR stands for integrated response. Below 500 mK, the temperature depen-
dence of the parameters in Eq. 6.30 is negligible and, hence, IR is linear in T
e
.Above
500 mK, the temperature dependence of ?
2
I
SD
/?V
SD
?Q
NR
becomes significant, and
IR is no longer linearly proportional to T
e
.
In practice, to calibrate IR, we first measured the resonator?s frequency re-
sponse P as a function of sample-stage temperature T
S
(see the Thermometry sec-
tion in Chatper 4 to see how we measured T
S
)and extracted the parameters A, Q
e
,
and ?
1
/2? from an harmonic oscillator fit to the data at each temperature. We then
calculated IR via
IR =
?
1
4Q
e
A. (6.31)
From the scatter in the power spectra data, the error in the calculation of this
quantity was typically 1 - 15%. Finally, we plotted IR versus T
S
(Fig. 6.8).
For T
S
> 100 mK, we found IR exhibited a linear dependence on T
S
.The
y-intercept was within measurement error of the origin (Fig. 6.8). Furthermore, the
data scaled with V
2
NR
, Figs. 6.9 and 6.10. That is, when divided by the square of
the coupling voltage, for a given T
e
, IR exhibited no dependence on V
2
NR
.These
observations were su?cient evidence to conclude that IR was an accurate measure-
ment of the temperature of the fundamental mode of the resonator, and that the
mode was in thermal equilibrium with the sample holder and RuO
2
thermometer
(ie. T
e
= T
S
). Accordingly, in this temperature regime, the slope of IR versus T
S
could be used as a calibration for performing noise thermometry. It is evident in
both Fig. 6.9 and Fig. 6.10 that there was scatter of 10 - 20% in some of the data
146
Figure 6.8: Plot demonstrating the integrated resonator response, IR, versus T
S
at
a coupling voltage of V
NR
= 4 V. Data is for device 2.
147
points. I address this issue at the end of the section when I discuss evidence for
back action.
From Fig. 6.9 and Fig. 6.10, it is clear that, for T
S
< 100 mk, the data did not
exhibit a linear dependence on sample-stage temperature. From Eq. 6.12, several
microvolts at the SET gate could have driven the resonators to an rms amplitude
of ? 200 fm - approximately the thermal amplitude of device 2 at 50 mK. However,
this can be ruled out based on several facts. First, the data for 100 mK and above fit
to a straight line through the origin. Second, the integrated power data, both above
and below 100 mK, exhibited no obvious dependence on V
4
NR
, as one would expect if
the resonator was driven by a capacitively coupled signal. And third, we knew from
transmission measurements that the attenuation down the gate lead was around -
20 dB at 20 MHz, and, thus, the noise at the input to the fridge would have to have
been ? 10?s ?V
rms
, which was much greater than the expected Johnson noise from
the resistors in the voltage dividers (10?s nV/
?
Hz at most) or the output noise of
the optical isolators (also 10?s nV/
?
Hz at 20 MHz). It was more likely the result of
either power from the RFSET line or dissipation in the SET heating the resonator
- in Chapter 7, I address these possible reasons and solutions for the hang-up.
Regardless of the source of the heating, we could use the calibration of IR
at and above 100 mK to determine the e?ective temperature T
e
below 100 mK.
For example, T
e
, at a sample-stage temperature of T
S
= 35 mK was determined by
dividing the integrated response at 35 mK by the integrated response at 100 mK,
Fig. 6.11. For the left peak, IR
35mK
= 423 ? 43 ?e
2
/V
2
, and for the right peak,
148
Figure 6.9: A log-log plot demonstrating the integrated resonator response, IR,
versus T
S
temperature scaled by V
2
NR
for Device 2. Using the data from 100 mK
and above as a calibration, the minimum temperature of the resonator?s fundamental
mode is found to be 56 ? 7mK.
149
Figure 6.10: A log-log plot demonstrating the integrated resonator response, IR,
versus T
S
scaled by V
2
NR
for Device 1. Using the data from 100 mk and above as a
calibration, the minimum temperature of the resonator?s fundamental mode is found
to be 99 ? 4mK.
150
Figure 6.11: Using the data from the 100 mK peak as a calibration, the integrated
response at 35 mK is found to correspond to T
e
= 56 mK. The data is for Device 2.
Please note that the 100 mK peak has been shifted by 1.0 kHz for clarity.
151
IR
100mK
= 747 ? 47 ?e
2
/V
2
. Thus we obtained
T
e
=
IR
35mK
IR
100mK
=56?7mK., (6.32)
Incidentally, T
e
= 56 mK was the lowest mode temperature that we measured.
Using the the Planck distribution function,
?n
th
? =(e
?h?
1
/k
B
T
eff
?1)
?1
(6.33)
we calculate that this corresponds to a thermal occupation number of 59 ? 7. This
is the lowest thermal occupation number ever measured for a collective mechanical
mode [15].
Finally, I note that the slope of the IR/V
2
NR
versus T
S
for Devices 1 and 2 di?er
by approximately a factor of 7 (see Figs. 6.9 and 6.10). This is a result of several
factors: (1) the increased coupling capacitance of Device 1 compared to Device 2
(61 aF compared to 27 aF); (2) the increased bandwidth of the spectrum analyzer
for the measurement of Device 1 (a factor of 1.7); and (3) di?erent spring constants
for the two devices (19 N/m for Device 1 compared with 15 N/m for Device 2).
Figure 6.12 shows a plot of the integrated resonator response versus sample-
stage temperature for both Devices 1 and 2. The data has been scaled with respect
to the derivative of the capacitive coupling, ?C
NR
/?y
m
and the e?ective spring con-
stant K
m
for each device. That is, I have plotted IR/(K
m
?C
NR
V
2
NR
/?y
m
). Note that
the data for the two devices fall on the same line, confirming that we understand the
basic principles of the detection scheme, and have taken into account the dominant
parameters - eg. C
NR
, K
m
. However, the fact that the slope of the scaled response
deviates from k
B
by a factor of ? 3 tells us that there is systematic uncertainty in
152
Figure 6.12: A plot demonstrating that the integrated resonator response, IR, for
Devices 1 and 2 collapse onto the same line when the data for each device is scaled
by the corresponding A
Q
, V
NR
, C
NR
,andK
m
.
153
our estimate of the measurement circuit gain. This discrepancy is consistent with
the deviation in the calculation of the capacitively driven resonator response from
the measured response, Section 6.1, and the deviation between the measured and
calculated charge sensitivity. Also note that the errors bars on the data points in
Fig. 6.12 are considerably larger then the error bars in the previous integrated power
plots. The main source of the error is from uncertainty in the spring constant, ?
50%. I have not included uncertainty in the gain of the measurement circuit.
Noise Temperature
The noise performance of the displacement detection scheme can be evaluated
by defining the noise temperature T
N
. This quantity correponds to the e?ective
resonator mode temperature, T
e
, at which the resulting thermal displacement can
be transduced and detected with a signal-to-noise ratio of 1. In other words, it is
the temperature at which the rms amplitude of the resonator response A is equal
to the rms background level P
o
.
Figure 6.13 displays a plot of the resonator response (Device 2) for sample-
stage temperature T
S
= 35 mK and coupling voltage V
NR
= 15 V. The data was
fit to an harmonic oscillator response, and the frequency, quality factor, amplitude,
background, and integrated response (IR), were extracted. Using the integrated
resonator response versus T
S
(7.3 ? .1 ?e
2
/V
2
) as the calibration (Fig. 6.9), the
integrated response of the peak (535 ? 24 ?e
2
/V
2
) was found to correspond to a
resonator mode temperature of T
e
=73? 2 mK. From the ratio of the amplitude
154
Figure 6.13: A plot demonstrating the lowest noise temperature, T
N
achieved by
RFSET displacement detection. From the slope of the integrated response versus
sample-stage temperature, T
e
= 73 mK . The ratio of the amplitude to the back-
ground yields a noise temperature of 15.5 ? .4 mK. Data is taken at V
NR
=15V,
and is for device 2.
155
to the background (4.71 ? .01) the noise temperature for the measurement of the
peak was found to be
T
N
=
T
e
4.71
=15.5 ?.4mK. (6.34)
Similiarly, we found, for Device 1, that the minimum noise temperature achieved
was 43 ? 2mK.
From T
N
and the equipartition relation, a rough estimate of the corresponding
displacement noise spectral density within the resonator?s noise equivalent band-
width can be made:
S
y
=
k
B
T
N
K
m
4Q
e
?
1
. (6.35)
Thus, a noise temperature of T
N
=15.5?.4 mK corresponded to a displacement
sensitivity of 3.8 ? .9 fm/
?
Hz,forQ
e
? 3.5 x 10
4
. For Device 1, the minimum T
N
corresponded to a displacment sensitivity of 7.5 ? 2fm/
?
Hz. Notice, though, that
because the calculation of the displacement sensitivity requires knowledge of the
spring constant, K
m
, the error in the estimate was roughly 25 % for both devices.
To find out how close our detection scheme was to the ideal, we expressed the
displacement sensitivity in terms of the quantum limit for each device, Chapter 2:
parenleftBigg
?y
m
?y
QL
parenrightBigg
2
=
T
N
T
QL
=
ln3k
B
?h?
1
T
N
. (6.36)
Notice that the dependence on K
m
has dropped out. For Device 1 then,
?y
m
?y
QL
=7.4?.2, (6.37)
156
For Device 2,
?y
m
?y
QL
=4.3?.3. (6.38)
These numbers represented the closest approach to the quantum limit, to date,
achieved in the read-out of the displacement of a mechanical system [15].
Finally, Figs. 6.14 and 6.15 show the noise temperature, T
N
(left axis) and
mean-square displacement noise (right axis) of Devices 1 and 2 as a function of
V
2
NR
. The displacement noise was normalized with respect to the quantum limit for
each device, ?y
m
/?y
QL
. Also plotted in the figure are lines (dashed) representing
the expected displacment sensitivity for a measurement circuit charge sensitivity
of 10 and 20 ?e
rms
/
?
Hz. Thus as we increased the coupling voltage, the noise
temperature improved linearly with V
2
NR
, as expected from Eq. 6.7.
6.3 Discussion of SET Back Action
From the discussion in Chapter 2, SET back action produces three e?ects in
the measurement of a nanomechanical resonator?s displacement: a frequency shift,
damping, and displacement fluctuations. In this section, I argue that there is no
clear evidence of any of these e?ects in the measurement of Device 1 or Device 2.
First, I note that the SET-induced frequency shift and damping arise as a
result of the dependence of the SET-island potential ? on resonator position. A
change in the resonator?s position alters the island potential, which changes the
electrostatic force between the SET island and the resonator.
The in-phase component of the response shifts the resonator?s frequency ac-
157
Figure 6.14: A plot demonstrating the noise temperature, T
N
, of the RFSET dis-
placement detection scheme as a function of V
2
NR
for device 1. The right axis is
the corresponding square of the position sensitivity normalized with respect to the
quantum limit. A minimum noise temperature of 43 ? 2 mK was achieved. This
corresponds to a displacement sensivity of a factor of 7.4 ? .2 from the quantum
limit, or 7.5 ? 2fm/
?
Hz.
158
Figure 6.15: A plot demonstrating the noise temperature of the RFSET displace-
ment detection scheme as a function of V
2
NR
for device 2. The right axis is the corre-
sponding square of the position sensitivity normalized with respect to the quantum
limit. A minimum noise temperature of 15.5 ? .4 mK was achieved. This corre-
sponds to a displacement sensivity a factor of 4.3 ? .3 from the quantum limit, or
3.8 ? .9 fm/
?
Hz.
159
cording to
??
1
?
1
??
C
NR
V
2
NR
K
m
d
2
NR
C
NR
2C
?
. (6.39)
Comparison with Eq. 6.16 shows that this shift in the resonator?s frequency
should be ?
C
NR
C
?
smaller than the frequency shift due strictly to the electrostatic
softening from V
NR
. For Devices 1 and 2, the ratio is ? 0.1 - 0.2, so the e?ect
should provide a small correction.
Figure 6.16 shows the frequency shift of the nanoresonators from Device 1 and
Device 2 versus V
2
NR
.
1
From a linear fit to the data, a slope of 72 and 124 Hz/V
2
were obtained respectively. These values are to be compared with the estimates of
140 and 100 Hz/V
2
provided by Eq. 6.16.
From Eq. 6.39, I expect that the e?ect of the back action should have been
about 10 - 20% of these values or ? 10 Hz/V
2
, which was approximately the magni-
tude of the scatter in the data points. Little more can be said as we lacked the data
to make a more precise determination of the slope. Furthermore, for this small of
an e?ect, I would need to develop a more detailed model of both the frequency shift
due strictly to V
2
NR
(ie. calculate numerically ?
2
C
NR
/?y
2
m
) and the frequency shift
due to the back action (ie. calculate correlations between tunneling and position
fluctuations). Thus, the frequency shift of the resonator cannot be used as a gauge
of the level of SET back action in the measurement.
Second, I note that the out-of-phase component of the SET response produces
1
The data for Device 1 excludes the 6 V coupling data as it was taken on a separate cool-down,
and exhibited a shift of 15 kHz.
160
(a)
(b)
Figure 6.16: Plot of the frequency shift versus V
2
NR
for (a) Device 1 at 35 mK and
(b) Device 2 at 100 mK. The shift is measured with respect to the lowest voltage
data point.
161
damping:
?
det
?
?
1
Q
d
=
parenleftBigg
bC
NR
V
NR
C
?
V
SD
parenrightBigg
2
e
2
R
?
2M
m
d
2
NR
= A
NR
V
2
NR
, (6.40)
where A
NR
/2? ? 0.02 and 0.003 Hz/V
2
for Devices 1 and 2 respectively. The total
e?ective resonator damping is thus
?
eff
=
?
1
Q
eff
= ?
bath
+ AV
2
NR
. (6.41)
It is assumed that A
NR
is independent of the bath temperature, T
b
. I expect,
then, that the temperature dependence of ?
e
should follow the temperature depen-
dence of ?
b
. While the sources responsible for ?
b
in nanoresonators are not well
understood, ?
b
has generally been observed to obey a power-law dependence of T
a
b
in several di?erent materials, with a ? 0.2 [121]. Assuming that the power-law holds
down to mK temperatures, if ?
d
is comparable to or greater than ?
b
, then its e?ect
should be evident in the deviation of ?
e
from the T
1/5
b
dependence; the deviation
becoming more pronounced at lower T
b
as ?
b
decreases and ?
d
remains constant.
Figure 6.17 shows plots of ?
e
/2? versus T
S
for V
NR
= 6 V (Device 1) and 10
V (Device 2). The data sets each represent the largest bias voltage for each device
for which complete data sets (35 mK to 500 mK) were taken. If back action is
a factor, it should be most pronounced here. The solid line in each plot denotes
T
1/5
S
dependence, and was generated by forcing a fit of the data to
?
eff
= C + DT
.2
S
, (6.42)
where A and B are o?-set and slope parameters. It is clear from the plots that the
e?ective damping does not saturate as T
S
decreases to 35 mK.
2
2
The other complete data sets for both Device 1 and Device 2 exhibit similiar behavior.
162
(a)
(b)
Figure 6.17: Plots of ?
e
/2? Vs. T
S
for (a) Device 1 at V
NR
=6Vand(b)Device
2atV
NR
=10V.
163
Furthermore, at the lowest T
S
, it is seen that there was no obvious dependence
of damping on V
NR
(Fig. 6.18). For both devices, ?
e
/2? was scattered about 500
Hz, ranging from 400 to 900 Hz. Using the estimate of ?
d
from Eq. 6.40 and the
parameters for Devices 1 and 2, I find that ?
d
/2? should have become comparable
to this range of values when V
NR
> 100 V, which is well above the parameter range
explored.
Finally I turn to the third back action e?ect: position-fluctuations. In Chapter
2, I showed that, in the absence of coupling to any other environment, the SET will
drive the measured resonator, resulting in fluctuations in the resonator?s position
with a variance given by
?y
2
m
? =
k
B
T
d
K
m
, (6.43)
where T
d
is considered to be a measure of the asymmetry in the SET?s quantum
noise.
If, in addition, the resonator is coupled to a thermal bath, then the resonator?s
variance will be given by
?y
2
m
? =
k
B
K
m
T
e
, (6.44)
where
T
e
=
(?
b
T
b
+ ?
d
T
d
)
?
e
. (6.45)
For small V
NR
, one expects T
e
= T
b
. However, as one increases V
NR
, it is expected
that the dependence of T
e
on T
b
will become weaker until ?
d
T
d
greatermuch ?
b
T
b
,atwhich
point the resonator will hang-up at T
e
= T
d
.
Turning back to Figs. 6.10 and 6.9, there is no discernible evidence for this
164
(a)
(b)
Figure 6.18: A plot of damping versus coupling for (a) Device 1 and (b) Device 2.
165
e?ect in either Device 1 or Device 2. As V
NR
was increased, the dependence of T
e
on T
S
(or T
b
above 100 mK) remained constant.
From Eq. 2.96 and the parameters for Device 2, it is stright-forward to estimate
the expected contribution of the SET back action to T
e
at 15 V coupling. An order of
magnitude estimate of the SET potential fluctuations yields S
1/2
??
(?) ? 1nV/
?
Hz.
The SET-island potential fluctuations should have then driven the resonator to an
rms amplitude of 30 fm
rms
. This would have corresponded to an e?ective heating
of ? 100 ?K, and, at T
b
= T
S
= 100 mK, would have been a 0.1 % e?ect.
I close this chapter with a few comments. First, it is clear that, for each
of the three quantities (?f
1
, ?
e
,andT
e
), there is scatter beyond the statistical
uncertainty determined from the least-squares fit of the power spectrum data and
error propagation. For example, in Fig. 6.9, at 175 mK, the three data points for 10
V coupling each have error bars representing ? 2% relative uncertainty. However,
the scatter about the mean of the three points is ? 10 - 15 %.
While it is not shown, there may have been a correlation between the scatter
in T
e
and the scatter in ?
e
;thatis,forasetofdatapointsataparticularcoupling
voltage and temperature, the data points which exhibited larger T
e
-ascompared
to the other data points in the set - also exhibited larger ?
e
,andviceversa.
Additionally, for Device 2, the scatter in the data at 10 V and 15 V coupling
was accompanied by fluctuations in the RFSET gain. These fluctuations were sub-
stantial, and resulted in the RFSET gain-feedback unlocking. The frequency and
magnitude of the fluctuations appeared to increase with V
NR
.
Two-level charge fluctuators and back action are both possible explanations for
166
the observed scatter in the ? f
1
, ?
e
,andT
e
. The possibility of charge fluctuations is
bolstered by the observation that the magnitude of scatter appeared to increase with
increasing coupling, and that it was accompanied by RFSET gain fluctuations. The
possibility of back action being the culprit is weakened by the estimates above which
demonstrate that all three back action manifestations should be small with respect
to thermal noise and other factors. Of course, until the back action is measured,
limited confidence can be placed in these estimates.
Finally, in all of the back action estimates, we used normal-state SET ap-
proximations. Recent theoretical [59] [60] and experimental [73] investigations of
the SSET back action near the JQP resonances demonstrate significantly di?erent
behavior than what is predicted for a simple normal-state SET. For instance, both
the magnitude and the sign of T
d
and ?
d
are very sensitive functions of the SET?s
detuning from the JQP (DJQP) resonance ridges. Furthermore, it has been found
that the SET must be biased o? of the center of the JQP ridge and the RFSET
carrier amplitude (v
c
) must be reduced to a fraction of the JQP resonance-width
in order to avoid sampling both the stable (negative ?
d
, negative detuning) and
unstable (positive ?
d
, positive detuning) regimes.
At the time the measurements of Device 1 and Device 2 were made, we were
not aware of these details. We were also not particularly careful with maintaining a
consistent SET bias point. Typically, the intent was to choose whichever bias point
maximized the RFSET gain. The records we have for the bias points (Fig. 6.3 for
example) demonstrate that at least some of the time we were biased near the top of
the JQP ridge. As well, the peak-peak amplitude of v
c
at the SET corresponded to
167
? 2Q
T
8 ?V ? 140 ?V, which is on the order of the half-width of the JQP resonance.
There are significant di?erences though between the present work (Devices
3,4,Y) and the work I report in this Chapter (Devices 1 and 2). The level of cou-
pling is much greater. For the present generations devices, d
NR
, the resonator/SET
spacing, has been decreased to ? 100 nm. This is to be compared with the 600 nm
spacing of Devices 1 and Devices 2. Additionally, the spring constants of the present
generations devices have been reduced by as much as a factor of 2 - 3, making the
resonator more ?susceptible? to the SET?s back action forces.
Finally, the evidence for back action in the more recent samples, while not yet
fully understood, is much greater. For example, in Device 3, the e?ective quality
factor has been observed to decrease with V
2
NR
dependence from above 1 x 10
5
at
1Vcouplingto2x10
4
at 12 V coupling. For the same span in coupling voltage,
the relationship between T
e
and T
S
is shown to go from directly proportional at 1
V to independent at 12 V. Finally, in an even more recent sample (Device Y), both
positive and negative damping have been observed in the vicinity of the both the
JQP and DJQP resonances.
In the light of these facts, while we cannot completely rule out the influence
of SET back action in Devices 1 and 2, it is clearly not a significant e?ect.
168
Chapter 7
Conclusions
So far, I have demonstrated that we are capable of approaching the quantum
limit on two fronts: we have implementeda near-ideal displacement detection scheme
with sensitivity ?y
m
=4.3?y
QL
; and we have cooled a mode of a nanomechanical
oscillator to ?n
th
??60. These observations are the closest approach to the quantum
limit for a nanomechanical or macroscopic object to date. In this final chapter,
I discuss possible reasons why our observations were limited to these values and
suggest several technical improvements to push even closer to the quantum limit in
future work.
7.1 Shot Noise Limited Detection
Figure 7.1 demonstrates the deviations from ideality of the displacement de-
tection scheme for the measurement of Device 2. We found that as we increased
V
2
NR
, the noise temperature T
N
decreased linearly with a slope (dashed line) deter-
mined by S
Q
? 10?e
?
Hz. This is at odds with the charge sensitivity of 30?e/
?
Hz
we measured from the 1 MHz gain calibration (Chapter 5 and Appendix II). The
discrepancy could be a result of the improper calibration of charge sensitivity during
the measurement of Device 2. At that time we were not aware of the Bessel func-
tion calibration method. It could also be the case that the charge sensitivity was
169
Figure 7.1: A plot of T
N
Vs. V
2
NR
for Device 2. The solid line is the total noise
temperature including both SET shot noise and back action.
170
better at 20 MHz than at 1 MHz. Nevertheless, the solid lines denote the estimated
SET shot noise and back action contribution to the noise temperature, assuming a
normal-state SET.
It is clear that there is room for a factor of 5 reduction in the forward-coupling
measurement circuit contribution before the detection scheme becomes limited by
the SET shot noise. To achieve this, there are at least three improvements that we
could make.
First, we could account for and reduce the 4 - 5 dB attenuation in the portion
of the microwave circuit between the sample and the HEMT pre-amplifier.
1
From a
simple consideration of the measurement circuit noise performance, one can see that
a loss of 4 - 5 dB between the RFSET and the HEMT pre-amplifier is a significant
contribution to the mesasurement circuit noise temperature
T
det
n
? T
L
+
T
HEMT
10
?L/10
? 11 ?15 K, (7.1)
where T
L
=4(10
L/10
?1) is the equivalent noise temperature for the section of the
circuit where the 4 - 5 dB is lost, L = 4 - 5 dB is the loss, and T
HEMT
? 2Kisthe
equivalent noise temperature of the pre-amplifier. Thus, while it is still a significant
factor, the ultra-low noise HEMT accounts for less than 20% of the measurement
circuit noise temperature.
Second, we could replace the HEMT with a better cryogenic amplifier and use
the HEMT as a follower. One possibile replacement would be the nearly-quantum
limited microstrip SQUID amplifier [119], which has been demonstrated with an
1
For Devices 1 and 2, we do not have a record of the loss in this portion of the circuit. Consid-
ering that T
det
n
? 20 K for Device 2, the attenuation was probably closer to 6 dB.
171
equivalent noise temperature a factor of 2 from the quantum limit at 500 MHz
[119]. It has been used routinely with a noise temperature of 100 mK and gain of 20
dB up to 500 MHz. These amplifiers can operate below 100 mK, which would allow
for placement very close to the sample, and thus reduce the possibility of signal-loss
in the coupled-portion of the microwave circuit (See Chapter 5 and Appendix II).
I can calculate the overall improvement in the noise temperature of the mea-
surment circuit for the case that the RFSET is read-out with a microstrip SQUID.
Assuming a gain of 20 dB and noise temperature of 100 mK for the SQUID, and
using the 2K HEMT
2
as a follower, I calculate
T
det
n
? 120 mK, (7.2)
without loss in the circuit.
With 5 dB of loss following the SQUID, I calculate
T
det
n
? 250 mK. (7.3)
This is a factor of at least 40 improvement in the noise temperature of the
measurement circuit. Assuming, an SET with the identical parameters as Device 1
or Device 2, this should result in the reduction of the charge sensitivity to the SET
shot noise limit.
There are several concerns with using the microstrip SQUID amplifier. One
is that it might require the use of a lower carrier frequency for the RFSET, and
hence lower bandwidth. This problem could probably be circumvented by detuning
2
There is a HEMT amplifier available from Reference[107] with a quoted noise temperature of
.9 K at 650 MHz which could be used as a follower for the SQUID.
172
the carrier frequency from the tank circuit resonance by the expected mechanical
resonator frequency. The second concern deals with dynamic range. It isn?t clear
whether the reflected signal from the RFSET would swamp the SQUID amplifier.
For example, in Reference[120], the author calculates that the maximum output
power a typical SQUID in open-loop configuration could supply to a 50 ?load is
approximately 3 nW. Based on this calculation, if the SQUID amplifier has a gain
of 20 dB, then the maximum input power is limited to approximately 30 pW. For
RFSET operation, depending on the bias point, the reflected power could be as
large as several 100 pW. This requires further invsetigation.
Finally, the third improvement that we could make would be to the matching
characteristics of the RFSET LC circuit. For Devices 1 - 4, the transformed SET
impedance on resonance, Z
LRC
, was approximately 0.04 - 0.12 Z
o
.Asaresult,the
reflection coe?cients at maximum conductance, ?
max
, ranged from ? 0.8 - 0.92,
and yielded maximum depths of modulation M = 20log(?
max
)of? 0.7 - 2.0 dB.
The obvious solution to this is to begin using larger inductance coils. Keeping
C
T
fixed at ? 250 fF would require increasing the inductance up to 100?s nH for
optimal matching. This would also have the desired a?ect of reducing the tank-
circuit frequency down to an acceptable range for the microstrip SQUID amplifier.
Again, though, we would pay the price in bandwidth. For C
T
? 250 fF, optimal
matching at 400 MHz, would require L
T
? 600 nH. This fixes the tank-circuit
quality factor and bandwidth to be ? 15 and 27 MHz respectively. Obviously, the
optimization is tricky. One might not need to implement perfect matching. For
example, increasing L
T
so that Z
LRC
? 0.5 Z
o
would increase M by 6 - 8 dB, and
173
hence improve the charge sensitivity by approximately a factor of 2.
7.2 Sample Thermalization
Ultimately, the base temperature of the dilution refrigerator is ? 9mK.This
has been confirmed using nuclear orientation thermometry. In the previous chapter,
I showed that the minimum resonator mode temperature that we measured was
around 60 mK. Thus, it is clear that considerable improvement could be made to
bring the mixing chamber and the nanoresonator?s fundamental mode into thermal
equilibrium. There are at least three components to this problem: (1) minimizing
the thermal impedance between the mixing chamber and the sample stage; (2) min-
imizing the thermal impedance between the nanoresonator?s mode and the sample
stage; and (3) reducing heat to the device. I assume that the first component, while
not trivial, can be made negligible using the proper materials and connections. The
second component is also not trivial and should depend on both the material out
of which the samples are made and the geometry and mode of the nanoresonator.
While, for a given nanoresonator mode, the thermal coupling between the mode and
the substrate ?bath? can be inferred from measurements of the resonator?s quality
factory, the nature of the coupling is not well understood [121] and is deserving of
an entire thesis. Thus, in the section, I focus on the third component. In particular,
I discuss two possible sources of heating: the Nb-Nb microwave coax and the SET.
174
Heat Transfer Through Nb-Nb Coax
It is possible that heat was conducted via the microwave coaxial cable to
the sample stage. From 1 K down to the mixing-chamber, we used Nb-Nb UT-85
coax (see Chapter 4) [106]. Gold-plated copper clamps were used to thermalize the
outer-shield of the coax at the still-stage and cold-plate. At the mixing-chamber,
the coax connected to a bias tee. The connection from the bias tee to the sample
holder was made via a Cu UT-85 semi-rigid coax. From the clamps and the low
thermal conductivity of the superconducting Nb [111], it seems unlikely, then, that
heat flow through the outer-shield of the coax was responsible for heating of the
sample. However, because of the Teflon insulation between the outer shield and
inner conductor, it is possible that heat transfer through the coax?s center conductor
could have resulted in the center conductor being out of equilibrium with the shield
and the mixing chamber.
To estimate the temperature di?erence between the inner and outer conductor
of the coax at the mixing chamber, I consider the coax to be a cylinder of length
L and composed of three concentric regions (Fig. 7.2): (1) an inner Nb conductor
with thermal conductivity ?
1
and radius r
1
; (2) a Teflon insulator with thermal
conductivity ?
o
; and (3) an outer Nb shield with thermal conductivity ?
2
= ?
1
,
inner diameter 2?r
2
,andthicknesst. I assume that the inner and outer conductors
are in thermal equilibrium at the 1 K pot (z = 0). On our dilution fridge, a typical
1 K pot temperature was 1.7 K. Thus T
1
(0) = T
2
(0) = 1.7 K. I further assume that,
at the mixing chamber (z = L), the shield and the mixing chamber are in thermal
175
Figure 7.2: An illustration of the Nb-Nb coax semi-rigid coax for heat-flow calcula-
tion. It is assumed that the center conductor and shield are in thermal equilibrium
at 1 K (T
1K
) and that the mixing-chamber end of the shield is in thermal equilbrium
with the mixing chamber.
176
equilbrium. The problem is to find the temperature T
1
(L) of the inner conductor
at the mixing chamber.
As a first approximation, I assume that the heat transfer in the Teflon is
purely radial (ie. the heat transfer along the length of the coax between 1.7 K and
the mixing chamber is dominated by the Nb conductors, which have a much larger
thermal conductance due to ?
1,2
/?
o
greatermuch 1 below 1 K [111]). Thus, I write the heat
transfer per unit length between the inner conductor and the shield at a position z
along the length of the coax as [122]
?
Q =
2??
o
ln(r
2
/r
1
)
(T
1
(z)?T
2
(z)). (7.4)
I also assume that the heat transfer is purely axial (along z) within the in-
ner conductor and within the shield. Fourier?s law [122] for a segment dz of each
conductor thus yields
?
1
?r
2
1
?T
1
?z
vextendsingle
vextendsingle
vextendsingle
vextendsingle
z
?
?
Qdz = ?
1
?r
2
1
?T
1
?z
vextendsingle
vextendsingle
vextendsingle
vextendsingle
z+dz
(7.5)
and
2?r
2
t?
2
?T
2
?z
vextendsingle
vextendsingle
vextendsingle
vextendsingle
z
+
?
Qdz =2?r
2
t?
2
?T
2
?z
vextendsingle
vextendsingle
vextendsingle
vextendsingle
z+dz
, (7.6)
where I have assumed that r
2
greatermuch t.
Expanding about z, the steady-state temperature profile of the inner conductor
is found from
?r
2
1
?
1
?
2
T
1
(z)
?z
2
+
2??
o
ln(r
2
/r
1
)
(T
2
(z)?T
1
(z)) = 0. (7.7)
177
As well, the steady-state temperature profile of the shield is found from
2?r
2
t?
2
?
2
T
2
(z)
?z
2
+
2??
o
ln(r
2
/r
1
)
(T
1
(z)?T
2
(z)) = 0. (7.8)
If I assume that the thermal conductivities ?
1
, ?
2
,and?
o
are temperature
independent and that ?
1
= ?
2
= ?, then the temperature di?erence between the
inner conductor and shield is given by
T
1
(z)?T
2
(z)=(T
1K
?T
m/c
)
r
2
1
+2r
2
t
r
2
1
sinh(L/?)+(2r
2
tL/?)cosh(L/?)
sinh(z/?), (7.9)
where
? =
radicaltp
radicalvertex
radicalvertex
radicalbt
parenleftBigg
?ln(r
2
/r
1
)
?
o
parenrightBigg
r
2
1
r
2
t
r
2
1
+2r
2
t
. (7.10)
Using the ratio ?/?
o
? 1x10
3
at 100 mK [111], and measuring r
1
? 0.25 mm,
r
2
? 3.5r
1
and t ? 0.75r
1
, I calculate that, for T
m/c
=.01KandT
1K
=1.7K,
T
1
(L) ? 30 mK, (7.11)
where I have assumed that L = 0.5 m.
This analysis suggests that the inner-conductor of the microwave coax is heated
by approximately 20 mK above the mixing chamber. However, the assumption that
the thermal conductivities of Teflon and Nb are independent of temperature between
1 K and 10 mK (and that their ratio is given by their values at 100 mK) is not
accurate. In fact, both materials exhibit a strong temperature dependence below
1 K. Specifically, ? ? T
3
and ?
o
? T
2
[111]. Thus, the ratio ?/?
o
is a function
of position along the length of the cable, decreasing from ? 10
4
at 1.7 K to ? 10
2
at the mixing-chamber. It is likely, then, that T
1
(L) could be heated less than
178
indicated by the above considerations. To estimate a lower bound, I assume that
the ratio of the thermal conductivities is given by their values at 10 mK (? 100). I
calculate, then, that T
1
(L) ? 17 mK. To conclude, I note that, at a minimum, the
center conductor should have been heated by approximately 10 mK above the mixing
chamber. However, a more detailed analysis taking into account the temperature
dependence of the thermal conductivity of each component of the coax must be
done.
Dissipation in the SET
It is also possible that the heating of the resonator mode was due to the
dissipation of power in the SET. I can estimate the phonon temperature in the
vicinity of the resonator using a steady-state thermal circuit model (Fig. 7.3). I
make several assumptions. First, I assume that the power dissipated in the SET
was determined by the dc current I
SD
and the SET resistance R
?
:
?
Q ? I
2
SD
R
?
= 400 fW (7.12)
for I
SD
=2nAandR
?
= 100 k?.
Second, I assume that the power dissipated in the SET must have been con-
ducted from the electrons in the SET island to the phonons in the SiN membrane
beneath the SET via the electron-phonon coupling for normal metals [123] [124]
?
Q =?
1
V
1
parenleftBig
T
5
1
?T
5
2
parenrightBig
, (7.13)
where ?
1
similarequal 2x10
9
nW/m
3
K, V
1
similarequal 5x10
?21
m
3
is the volume of the SET island,
T
1
is the temperature of the electrons in the SET island, and T
2
is the temperature
179
of the phonons in the membrane in the vicinity of the SET. It is not known how
accurate this assumption is for superconducting metals.
Next, I assume that the phonon transport in the SiN membrane was di?usive
[125] and that the power delivered through the SiN membrane (beneath the SET)
to the Si substrate can be written as [125]
?
Q
1
= .0145
A
3L
parenleftBig
T
3
2
?T
3
0
parenrightBig
, (7.14)
where A similarequal 5x10
?12
m
2
is the cross-sectional area of the SiN membrane between
the SET and the edge of the mebrane, L =25?m is the distance between the SET
and the edge of the membrane, and T
0
= 30 mK is the bath temperature, assumed
to be the temperature of the Si substrate and sample stage.
Similarly, I write the power delivered through the SiN membrane, from the
SET island to the phonons in the vicinity of the resonator, as [125]
?
Q
2
= .0145
A
prime
3L
prime
parenleftBig
T
3
2
?T
3
3
parenrightBig
, (7.15)
where L
prime
= 600 nm is the distance between the SET island and the resonator, A
prime
similarequal
1x10
?12
m
2
is the cross-sectional area of the SiN membrane between the SET and
the resonator, and T
3
is the phonon temperature near the resonator.
Finally, I assume that there were two ?paths? between the region around
the resonator and the sample-stage bath: (1) di?usive transport through the SiN
membrane [125]
?
Q
3
= .0145
A
3L
parenleftBig
T
3
3
?T
3
0
parenrightBig
; (7.16)
and (2) electron-phonon coupling [123] [124] between the phonons in the SiN mem-
brane near the resonator and the electrons in the Au film, which are thermally
180
Figure 7.3: Thermal circuit for the SET and resonator on the SiN membrane. Dissi-
pation in the SET (
?
Q) results in the heating of the phonon temperature around the
resonator to T
3
. The thermal resistances in the circuit are: (R
EP
) electron-phonon
resistance for the Al SET island; (R
2D
) 2D thermal resistance of the SiN membrane
from the SET to the bath; (R?
2D
) 2D thermal resistance of SiN membrane between
the SET and resonator; (R?
EP
) electron-phonon coupling for the Au layer of the
resonator; and (R
WF
) Weidemann-Franz resistance of the Au layer of the resonator
to the bath.
181
connected to the bath through electron scattering (Weidemann-Franz) [111],
?
Q
4
=?
2
V
2
parenleftBig
T
5
3
?T
5
4
parenrightBig
=
A
primeprime
L
primeprime
?L
o
parenleftBig
T
2
4
?T
2
0
parenrightBig
, (7.17)
where, for simplicity, I set ?
2
=?
1
Here, V
2
similarequal 1x10
?18
m
3
is the volume of the Au
on the membrane (this does not include the Au film on top of the resonator, just
the Au film leading to the resonator), A
primeprime
=2x10
?14
m
2
is the area of the interface
between the Au layer and the SiN membrane, L
primeprime
=2?25 ?m is the length of the
Au film (essentially from the ends of the resonator to the edges of the membrane),
T
4
is the temperature of the electrons in the Au film layer, ? similarequal 1x10
8
1/?m is the
conductivity of Au, and L
o
=2.4x10
?8
W?/K
2
is the Lorenz number [111].
With the additional assumption that Kircho??s law applies (ie.
?
Q =
?
Q
1
+
?
Q
2
and
?
Q
2
=
?
Q
3
+
?
Q
4
), I use Matlab to solve Eqs. 7.13, 7.14, 7.15, 7.16, and 7.17
for T
3
. I find that, for T
0
=30mKand
?
Q = 400 fW , approximately 200 fW is
delivered to the SiN membrane near the resonator. This results in the heating of
the resonator region to
T
3
? 60 mK. (7.18)
Figure 7.4 shows a numerical calculation of the local resonator temperature
T
3
as a function of both the bath temperature T
0
and the power dissipated in the
SET
?
Q. For the calculation of the plot in Fig. 7.4(a), I assumed that the total power
dissipated in the SET is
?
Q = 400 fW. It is seen that T
3
saturates at approximately 60
mK. Above 100 mK, T
3
is linear with T
0
. It appears that this behavior results from
the increase in electron-phonon conductance above 100 mK. For example, at 10 mK,
?
Q
4
(the power delivered from the phonons in the vicinity of the resonator through
182
(a)
(b)
Figure 7.4: Numerical calculation of temperature T
3
of phonons near the nanores-
onator as a function of (a) bath temperature T
0
for
?
Q = 400 fW and (b) power
dissipated in the SET
?
Q for T
0
=30mK.
183
the electron-phonon coupling to the electrons in the Au film) is approximately 1.4
fW. On the other, at T
0
= 500 mK,
?
Q
4
? 80 fW. The electron-phonon coupling
throughout this temperature range (T
0
= 10 - 500 mK) is still weak enough, though,
that the electrons in the Au layer stay thermalized with the bath T
0
. I note that
the SET electron temperature T
1
saturates at 380 mK below T
0
= 200 mK, rising
to approximately 520 mK at T
0
= 500 mK.
For the calculation of the plot in Fig. 7.4(b), I assumed that the bath temper-
ature T
0
= 30 mK. It is seen that for
?
Q<10 fW, the local phonon temperature T
3
is heated by less than 2 mK. For
?
Q =1pW,T
3
? 80 mK.
The actual power dissipated in the SETs during measurement is not known
with high precision. Typically, the SET was biased near IV features like the JQP
and DJQP peak (see Appendix B). However, for Devices 1 and 2, we did not keep
a record of both the IV characteristics and the bias point for each measurement.
For most of the measurements, we simply recorded the value of V
SD
and adjusted
V
g
to maximize the gain. I can estimate the order-of-magnitude of the dissipated
rf and dc power from the existing I
SD
V
SD
V
g
maps (see Figs. 5.4 and 5.5). The
half-width (in V
SD
) and height of the JQP peaks were ? 50 - 100 ?VandI
SD
? 1-
2 nA respectively. Thus, the dissipated dc power should have been
?
Q
dc
? 100s fW.
Typically, the incident rf signal was on the order of 10s ?V. Thus, the dissipated rf
power should have been comparable to the dissipated dc power.
To conclude this section I make several remarks. First, the temperature T
3
calculated in the above analysis is not necessarily the e?ective temperature of the
resonator?s fundamental flexural mode T
e
.TocalculateT
e
, it would be necessary
184
to determine the thermal conductance between the flexural mode and the phonons
in the SiN membrane near the resonator. In general, the thermal conductance of a
suspended nano-bar is a complicated problem (see references [37] [126] [127]) and is
beyond the scope of this thesis. Future work could involve incorporating the existing
nanoresonator thermal models into the above circuit analysis and fitting the data
for Devices 1 and 2 to the theroetical predictions. This work could be important for
understanding the nature of dissipation in nanoresonators. For instance, from such
an analysis, I could extract the thermal conductance between the nanoresonator and
the bath and compare this with the measured quality factor Q
e
of the fundamental
flexural mode. If I assume that the heat capacity of the flexural mode is given by
[35]
C
v
=
?E
?T
0
= k
B
, (7.19)
and that the thermal time constant for the mode is given by
? =
Q
e
?
1
= R
1
C
v
, (7.20)
where R
1
is the thermal resistance between the fundamental flexural mode and
some dissipative environment, then I expect that the thermal conductance should
be related to Q
e
through the relationship
g
1
=
1
R
1
=
?
1
k
B
Q
e
. (7.21)
Of course, other sources of dissipation, such as the SET detector (see Chapters 2
and 6) or charge noise in the substrate (see Chapter 6), might contribute to Q
e
.
From the considerations of Chapter 6, I expect that the back action was a negligible
185
factor in the resonator?s dynamics. However, a comparison of the agreement between
Q
e
predicted from thermal conductance models and Q
e
measured over a range of
coupling voltages V
NR
might allow for a more precise determination of how small
the e?ect was.
Second, I note that it is also possible that the heating of the resonator mode
could have been a result of dissipation in the Au film on top of the resonator.
However, from an analysis similiar to the analysis for heating due to SET dissipation,
I have found that this would require the electron temperature in the Au film to be
approximately 200 mK, far out of equilibrium with the sample stage. Based upon the
thermalization of the wiring for the resonator connection (see Chapter 4), it seems
unlikely that the resonator lead on-chip would have been at such a temperature.
Future work is necessary, though, to rule out black body radiation from the copper
grains in the powder filters at the mixing chamber.
Finally, I note that several experimental implementations could be made to
determine the nature of the sample?s heating. First, heat sinks (see Chapter 4) could
be added to the microwave circuit below the 1 K pot to see if better thermalization
of the Nb coax?s center conductor reduces the sample temperature. Second, the SiN
membrane geometry could be eliminated or the SETs could be fabricated o? of the
membrane to allow for the dissipation from the SET to radiate ballistically to the
bath. Third, insight could be gained by operating an SET far from the back action
limit and monitoring the e?ective temperature of the resonator?s mode as a function
of V
SD
,V
g
, and carrier amplitude v
c
, at a given bath temperature T
0
.
186
7.3 Parting Motivation
Besides serving as a manual for myself and others, it was my hope that this
text would motivate the pursuit of quantum mechanics in nanomechanical systems
by demonstrating how close we are technologically to this possibility.
First, we have demonstrated that the RFSET displacement detector is a near-
ideal detector, with sensitivity a factor of 4.3 from the quantum limit. It is thus a
promising candidate to be used for advanced measurement techniques such as the
quantum squeezing of a mechanical oscillator [29] [30]
Second, we have demonstrated that it is possible to cool and measure me-
chanical resonators to low thermal occupation numbers, ?n
th
??60. With technical
improvements and the implementation of feedback cooling [28], or by moving to
higher frequency resonators, this number could be reduced toward unity where we
could implement techniques to see evidence for quantized harmonic oscillator en-
ergy levels and zero-point fluctuations. Even with slight improvement,?n
th
??50, a
proposal to use a Cooper-pair box to prepare a nanoresonator in a superposition of
coherent position states could be implemented [25]. As well, these results open up
the possibility of implementing various other proposals that could extend the study
of quantum mechanics to much larger size scales [23-33].
187
Appendix A
Useful Mechanics Information
In this appendix, classical elasticity theory is used to model the transverse
displacement of a nanomechanical beam. In the first section, I make the connection
between the vibration of an elastic body and simple harmonic motion. In the second
section, I calculate the spring constants for SET detection. In the third secion, I
solve for the eigenfrequenices and eigenmodes of an elastic bar under tension. In
section four, I discuss the case of an elastic body undergoing damped-driven motion.
Finally, in the fifth section, I briefly discuss the magnetomotive detection technique.
A.1 Euler-Bernoulli Theory and The Simple Harmonic Oscillator
I start by modeling our nanomechanical resonators as prismatic bars, clamped
at both ends, and composed of isotropic, linear elastic materials (Fig. A.1). If I
consider small displacements from equilibrium
1
and assume the cross-sectional area
of the bars remain deformationless and perpendicular to the neutral surface, the
Euler-Bernoulli assumptions, I can express the equation of motion for vibration in
1
I consider displacements su?ciently small so that the radius of curvature is large with respect
to the transverse dimensions of the resonator. The resonators measured in our research easily
satisfy this criteria as a typical ratio of displacement-to-length is ? 10
?6
.
188
Figure A.1: A schematic of a prismatic, doubly-clamped nanoresonator.
the y-direction as [128]
?A
?
2
y
?t
2
+ EI
?
4
y
?x
4
=0. (A.1)
Equation A.1 equates the inertial force-per-unit-length on a segment of the
bar with the elastic restoring force the segment experiences when deformed. Here
E is the Young?s modulus of the material, and I = w
3
t/12 is the moment of inertia.
Parameters ? and A are the material density and the rectangular cross-sectional
area respectively. For composite resonators, such as the metallized resonators in
our research, EI should be replace by E
1
I
1
+ E
2
I
2
,whereE
1
,I
1
and E
2
, I
2
are the
Young?s moduli and moments of inertia for the two layers, respectively. Additionally,
for such a resonator, ?A should be replaced by ?
1
A
1
+?
2
A
2
,where?
1
and ?
2
are the
layer densities, and A
1
and A
2
are the respective layer cross-sectional areas [129].
189
Equation A.1 can be solved using separation of variables,
y(x,t)=Y (x)[C
1
cos?t + C
2
sin?t](A.2)
where C
1
and C
2
are determined from the resonator?s shape and velocity at some
initial time, t =0.
Substituting Eq. A.2 into Eq. A.1, I eliminate the time depedence, and solve
for the normal modes. Assuming clamped-clamped boundary conditions, Y
n
(0) =
Y
n
(L)=0and
dY
n
(0)
dx
=
dY
n
(L)
dx
= 0 , the normal modes are found to be [128]
Y
n
(x)=C
n
bracketleftbigg
(sink
n
x?sinh k
n
x)?
?
parenleftBigg
sink
n
L ?sinh k
n
L
cosk
n
L ?cosh k
n
L
parenrightBigg
(cos k
n
x?cosh k
n
x)
bracketrightbigg
, (A.3)
where the constants C
n
, Table A.1, are determined by normalizing the neutral sur-
face displacement of the mode of interest to unity at maximum displacement. The
choice of normalization is arbitrary. I have chosen this particular normalization con-
vention so that, for the fundamental mode, the equations of motion represent the
motion of the resonator?s mid-point (mid-point with respect to resonator length).
Also note that for this normalization convention the mode functions Y
n
(x)aredi-
mensionless. The first four modes are plotted in Fig. A.2.
The normal mode frequencies, ?
n
, are found from
?
n
2
=
EIk
4
n
?A
, (A.4)
with k
n
determined by the roots of the eigenvalue equation
cosk
n
Lcosh k
n
L =1. (A.5)
190
Figure A.2: The functions Y
n
(x) for the first four modes of a doubly-clamped res-
onator. The x-axis has been normalized by the resonator length in each plot.
191
Table A.1: Geometric Constants: Modes 1-5.
n k
n
C
n
?
n
?
n
1 4.730 .619 .397 .295
2 7.853 .663 .439 .145
3 10.996 .661 .437 .081
4 14.137 .661 .437 .052
5 17.279 .661 .437 .036
6 20.420 .661 .437 .026
The eigenvalue equation can be solved graphically or numerically. Table A.1
lists k
n
for the first six modes.
In our research, we were typically concerned with resonator motion of purely
one mode. Using Eqs. A.2 and A.3, the solution for a given mode, n is simply
y
n
(x,t)=Y
n
(x)[C
n,1
cos ?t + C
n,2
sin?t] . (A.6)
I now make the connection between the dynamics of a particular mode and
the simple harmonic oscillator. To calculate the bending energy E
n
of the mode, I
consider the average work done in deforming the bar into the mode shape Y
n
(x):
E
n
=
1
2
integraldisplay
?d?
n
M
n
?, (A.7)
where, M
n
= EI?
2
y
n
(x,t)/?x
2
is the mode?s bending moment, and ?
n
= ?y
n
(x,t)/?x
is the slope of the bar?s deformation.
192
Plugging the expressions for M
n
and ?
n
into Eq. A.7, I find
?E
n
??
EI
2
integraldisplay
L
0
?(
?
2
y
n
(x,t)
?x
2
)
2
?dx =
?
n
2
A?
2
integraldisplay
L
0
?y
2
n
(x,t)?dx
E
n
=
?
n
2
?AL?
n
2
?y
2
n
(t)? =
1
2
K
n
?y
2
n
(t)?. (A.8)
It is apparent that this is just the potential energy of an object undergoing
simple harmonic motion, with e?ective spring constant, K
n
= m
eff,n
?
2
n
, and e?ective
mass, m
eff,n
= ?
n
?AL.Here?
n
are dependent on mode shape, and are listed
in Table (A.1) for the first 6 modes. For the fundamental mode, the paramater
?y
2
1
(t)? is the mean square amplitude of the resonator?s mid-point (length-wise),
with magnitude determined by initial conditions.
Finally, using the definitions of K
n
and m
eff,n
, I multiply Eq. A.1 by Y
n
(x)
and integrate over x to recover the simple relation
m
eff,n
?
2
y
n
(t)
?t
2
= ?K
n
y
n
(t). (A.9)
For the fundamental mode, this is simply the expression for the harmonic oscillation
of the mid-point of the resonator.
A.2 Spring Constants for SET Detection
While Eq. A.8 expresses the potential energy of a particular resonator in terms
of the motion of the resonator?s midpoint, in practice, the SET detector is sensitive
to the average displacement of the resonator over the length of the SET island.
It would be nice, then, to recast Eq. A.8 in terms of this quantity so that I am
193
able to easily calculate the potential energy of the mode from the observed motion.
For example, for a resonator undergoing brownian motion, knowing the relation
between the measured displacement and the potential energy of the mode, and using
the equipartition theorem, I could calculate the mechcanical mode temperature.
Alternatively, knowing the temperature of the mechanical mode, I could calculate
the displacement signal I should expect to measure with the detector. Essentially,
then, what I want to know is the spring constant K
n,m
for the mean motion of the
resonator over the length of the SET island. I can calculate K
n,m
from Eq. A.8. To
do this, I need to determine the relationship between the mean displacement over
the length of the SET and the displacement of the mid-point of the resonator:
y
n,m
(t)=a
n
y
n
(t), (A.10)
where
a
n
=
1
L
2
?L
1
integraldisplay
L
2
L
1
dxY
n
(x), (A.11)
and L
1
and L
2
define the section of the resonator which corresponds to the length
of the SET island. I assume that the SET island is centered about the mid-point of
the resonator.
I solve Eq. A.11 numerically for the fundamental mode of Devices 1 - 4 (Table
A.3). For each sample, the length of the SET island was approximately 5 ?m. The
length of the respective resonators is listed in Table A.2. For the first mode, Eq.
A.8 becomes
E
1
=
?
1
2a
2
1
?AL?
1
2
?y
2
1,m
(t)? =
1
2
K
1,m
?y
2
1,m
(t)?, (A.12)
where K
1,m
=M
1,m
?
2
1
,andM
1,m
=?
1
/a
2
1
?AL are the e?ective spring constant and
194
e?ective mass for the motion of the fundamental mode of the resonator averaged
over the 5 ?m length of the SET. As I am only concerned with the fundamental mode
frequency, the superscript ?1? is dropped from K
1,m
and M
1,m
for the remainder of
the section and througout the text.
To calculate K
m
for each device, I need to know both M
m
and ?
1
.Toestimate
M
m
, I use values of A = wt and L obtained from scanning electron micrograph
(SEM) images and knowledge of the etch rates in the processing of the resonator;
I assume typical densities of Au and SiN to be 19.3 x 10
3
kg/m
3
and 3000 kg/m
3
respectively [130]. The raw mass m
r
= ?ALt is calculated by including both the Au
and SiN layers (Table A.2). To estimate ?
1
(Table A.3), I use either Eqs. A.15 and
A.16 or Eq. A.17. The additional parameters required for this calculation are the
Young moduli E
Au
and E
SiN
, which I assume to be approximately 50 GPa [131] and
250 GPa [75] respectively. I have assumed ? 50 GPa uncertainty in the value of
the Young?s modulus for SiN to reflect the spread in values found in the literature
(see [76]) and neglected the uncertainty in Young?s modulus for the Au layer (50
- 90 GPa reported in Reference [131] depending on grain size and thickness) as its
contribution to the total error should be a factor of 5 - 10 times smaller than the
contribution from the uncertainty in E
SiN
(a consequence of t
Au
? 0.2t
SiN
for our
samples after etching).
195
Table A.2: Geometry and raw mass, m
r
, of Devices 1-4. The error in the length
and width of the resonator comes from 10% quoted error in the SEM calibration.
The error of 30% in the thickness of the Au layer on the resonator comes from the
spread in etch rates over time, and is a very rough estimate.
Device w(nm) L(?m) t
Au
(nm) t
SiN
(nm) m
r
(pg)
1 300 ?30 10 ?1 30 ?20 100 ?2 2.6 ?1.2
2 200 ?20 8 ?.8 30 ?20 100 ?2 1.4 ?.7
3 200 ?20 15 ?1.5 30 ?20 100 ?2 2.6 ?1.2
4 225 ?23 18 ?1.8 30 ?20 100 ?2 3.2 ?1.5
Table A.3: E?ective masses, M
m
, and spring constants of Devices 1-4. ?a? corre-
sponds to frequency calculated using either Eqs. A.15 and A.16 or Eq. A.17. ?b?
corresponds to the frequency measured using SET detection at a temperature of 100
mK.
Device a
1
M
m
(pg) ?
1
/2?(MHz)
a
K
a
m
(N/m) ?
1
/2?
b
K
b
m
1 .838 1.5 ?.7 17 ?5 17 ?8 17.976648(3) 19 ?9
2 .760 .96 ?.45 18 ?6 12 ?6 19.654505(7) 15 ?7
3 .941 1.2 ?.54 6 ?1.8 1.5 ?.7 9.37163340(2) 4 ?2
4 .947 1.4 ?.66 4 ?1 .9 ?.4 4.8976624(2) 1.4 ?.6
196
A.3 Corrections to Frequency Due to Tension
The nanomechanical resonators used in our research are made from amorphous
silicon nitride, which has been deposited using low pressure chemical vapor deposi-
tion (LPCVD). The resulting intrinsic stress ?
int
in the silicon nitride films is on the
order of MPa?s [132], and is largely tensile. We can model the e?ect of the stress by
including an e?ective tension T = ?
int
A in the equation of motion (Eq. A.1): [133]
?A
?
2
y
?t
2
+ EI
?
4
y
?x
4
?T
?
2
y
?x
2
=0. (A.13)
Using dimensional-analysis, I can estimate the order of magnitude of the con-
tribution of the tension to the restoring force. From Eq. (A.13), the ratio of tensile-
to-bending force is TL
2
/EI.ForTL
2
/EI lessmuch 1, we can expect the bending moment
to dominate, and the dynamics to be governed by the results of the previous section.
For TL
2
/EI greatermuch 1, the tenisle force will dominate, and the dynamics will be similiar
to the case of a tensioned string. For silicon nitride nanoresonators, however, the
dimensionless ratio can range from TL
2
/EI lessmuch 1toTL
2
/EI ? 1. For this range,
it would be helpful then to calculate the corrections to normal mode shape and
frequency .
Equation (A.13) can be solved exactly using separation of variables, yielding:
Y
n
(x)=C
n
bracketleftbigg
parenleftBigg
sin?
n
x?
?
n
?
n
sinh?
n
x
parenrightBigg
?
?
?
?
sin?
n
L?
?
n
?
n
sinh?
n
L
cos?
n
L ?cosh ?
n
L
?
?
(cos?
n
x?cosh?
n
x)
bracketrightbigg
, (A.14)
197
where
?
n
L = k
n
L
bracketleftBig
(a
2
+1)
1
2
?a
bracketrightBig1
2
,?
n
L = k
n
L
bracketleftBig
(a
2
+1)
1
2
+ a
bracketrightBig1
2
,
and
a =
T
2EIk
2
,
with
k
4
n
=
?A?
2
n
EI
. (A.15)
The values of k are found numerically from the characteristic equation:
cos(?
n
L)cosh(?
n
L)?
1
2
(
?
n
?
n
?
?
n
?
n
) sinh(?
n
L)sin(?
n
L)=1. (A.16)
It is straight-forward to verify that, as T ? 0, these expressions reduce to the
corresponding expressions in Section A.1.
Figure A.3 demonstrates the e?ect of tension on the frequency of the normal
modes of a doubly-clamped silicon-nitride resonator. In the Fig. A.3(a), the ratio of
frequency calculated with tension, ?
n
(T), to frequency calculated without tension,
?
n
(0), is plotted versus the dimensionless correction factor, TL
2
/12EI, for the first
six normal modes of a resonator with a fundamental frequency of 13.037 MHz and
length of 10 ?m. A bi-morph resonator with cross-sectional area A = 0.0375 ?m
2
,
width w = 250 nm, and a 50 nm thick layer of gold as the conducting layer are
assumed. Young?s moduli and densities of 300 GPa and 3000 Kg/m
3
and 50 GPa
and 19.3 x 10
3
Kg/m
3
are assumed for the silicon nitride and gold respectively.
It is apparent that for TL
2
/12EI < 0.01, tension increases the resonant fre-
quency of the first six modes by less than 0.1%. For the fundamental mode, this
corresponds to a frequency shift of ? 10 kHz.
198
(a)
(b)
Figure A.3: (a) The normalized mode frequency is plotted versus TL
2
/12EI,for
the first six modes of a doubly-clamped resonator with fundamental frequency of
13.037 MHz and a length of 10 ?m. (b) The normalized fundmental mode frequency
is plotted versus resonator length for intrinsic stress values of 1, 10, and 100 MPa?s.
199
As the tension is increased to TL
2
/12EI ? 1, the shift in resonant frequency
for the fundamental mode grows to ? 15% . The shift in the higher order modes is
smaller as they are e?ectively sti?er than the fundamental mode, and ranges from
? 1% to ? 8%.
Figure A.3(b) demonstrates the shift in frequency of the fundamental mode of
a doubly-clamped resonator as a function of the resonator?s length for ?
int
= 1, 10,
and 100 MPa. For each curve, the tension is held constant, and the length of the
resonator is increased. Here, I used the same values of EI, ?, and cross-section as
were used in Fig. A.3(a). It is evident, that for silicon nitride resonators with lengths
less than 10 ?m, the shift in frequency due to tension can be expected to be less
than 10% for the fundamental mode - while not shown, the shift in the higher-order
mode frequencies is even smaller.
An alternative to solving Eq. A.13 exactly is to use a pertubative technique in
which it is assumed that the e?ect of tension on the mode shape is negligible, the
so-called Rayleigh Method [46]. Starting with Eq. A.13, I separate variables and
solve for ?
n
:
?
n
(T)=?
n
(0)
parenleftBigg
1+?
n
TL
2
12EI
parenrightBigg1
2
, (A.17)
with
?
n
(0) =
?
n
L
2
parenleftBigg
EI
?A
parenrightBigg1
2
,
and
?
n
=
12
L
2
integraltext
L
0
dx(?Y
n
(x)/?x)
2
integraltext
L
0
dx (?
2
Y
n
(x)/?x
2
)
2
,
200
where
?
n
= L
2
parenleftBigg
integraltext
L
0
dx (?
2
Y
n
(x)/?x
2
)
2
integraltext
L
0
dxY
2
n
(x)
parenrightBigg
.
I then make the approximation that Y
n
(x) are unaltered by the tension and
given by Equation A.3. For this case, ?
n
are found to be equal to k
n
determined
from Eq. A.5. The first ten ?
n
are listed in Table A.1.
For TL
2
/12EI ? 1, I have found that the agreement between Eq. A.17 and
the exact resonant frequency calculated by solving for the roots of Eq. A.16 is better
than .01%. This implies that the approximation that the Y
n
(x) are left una?ected
by tension T is a good one, and, thus, throughout the report, I simply use the
mode-shapes given by Eq. A.3.
A.4 The Driven-Damped Harmonic Oscillator
I can append Eq. A.1 to account for external non-dissipative forces by simply
adding in a term F(x,t) (in dimensions of Force/Length). I can account for damping
by also inserting a phenomenological term proportional to the resonator?s transverse
velocity. Dissipation in nanomechanical resonators is not well understood [121], and
several mechanisms including thermoelastic loss [134], attachment loss [135] [136],
and loss due to the measurement process itself [55] [137] have been proposed. Some
authors account for damping e?ects by defining a complex Young?s modulus where
dissipation is incorporated in the imaginary component (see [134] [138]). For the
purpose of this appendix, however, it is su?cient to account for the damping by
201
inserting a velocity-dependent term, as it captures the general physics. Thus,
?A
?
2
y(x,t)
?t
2
+ EI
?
4
y(x,t)
?x
4
+ ?
?y(x,t)
?t
= F(x,t), (A.18)
Following the approach of Reference [139], I assume that the damping and
driving force have a negligible e?ect on the mode shapes Y
n
(x). I then substitute
solutions of the form
y
n
(x,t)=Y
n
(x)y
n
(t), (A.19)
into Eq. A.18, multiply by Y
n
(x), and integrate over the length of the resonator,
obtaining the equation of motion for a mode, n,:
m
eff,n
?
2
y
n
(t)
?t
2
+ K
n
y
n
(t)+?
n
?y
n
(t)
?t
= f
n
(t), (A.20)
where
?
n
=
integraldisplay
L
0
dxY
2
n
(x)? (A.21)
and
f
n
(t)=
integraldisplay
L
0
dxF(x,t)Y
n
(x). (A.22)
Equation A.20 is the familiar damped-driven harmonic oscillator equation of
motion. A simple case to treat, and one which will be important for Section A.5, is
when F(x,t) is spatially invariant and has an harmonic time-dependence:
F(x,t)=
F
o
L
e
j?t
, (A.23)
where F
o
is the magnitude of the force.
202
In this case, f
n
(t)=?
n
F
o
e
j?t
,where?
n
is the average of Y
n
(x)overthelength
of the resonator,
?
n
=
1
L
integraldisplay
L
0
dxY
n
(x), (A.24)
and determines the projection of the force on a given mode (Table A.1). The steady-
state solutions, found for t/? greatermuch 1, where ? = m
eff,n
/?
n
, are then given by:
y
n
(t)=A
n
e
j?t
, (A.25)
where
A
n
=
?
n
F
o
m
eff,n
((?
n
2
??
2
)+j (?
n
?/Q
e
))
. (A.26)
For a general force F(x,t), Eq. A.26 is replaced by
A
n
=
integraltext
L
0
dxY
n
(x)F(x)
m
eff,n
((?
n
2
??
2
)+j (?
n
?/Q
e
))
. (A.27)
The resonant frequencies, ?
n
, are given by Eq. A.4. The e?ective quality
factor, Q
e
is defined as Q
e
= ?
n
m
eff,n
/?
n
, and sets the width of the resonator?s
frequency response. I assume that it could be a result of dissipation from coupling
to both the measurement environment and a thermal reservoir. Finally, I define the
phase di?erence, ?
n
, between drive signal and resonator response
?
n
= arctan
parenleftBigg
?
n
?/Q
e
?
n
2
??
2
parenrightBigg
. (A.28)
A.5 The Magnetomotive Technique
For the past decade, researchers have used various forms of magnetomotive
detection to study the properties of nanomechanical resonators [140] [141] [142]
203
Figure A.4: a). Schematic of Magnetomotive technique. b). Circuit diagram dis-
playing electromechanical impedance, Z
m
, current drive, I, and voltage amplifier -
assumed to have infinite input impedance.
204
[36]. In its simplest realization (Fig. A.4(a)), ac current I is applied, length-wise,
through a metallized resonator; in the presence of a transverse magnetic field B the
resonator is driven by the Lorentz force F = IBL andanEMFepsilon1
n
= BLv
n
, develops
across it?s length L. Here,
v
n
=
?y
n
(t)
?t
1
L
integraldisplay
L
o
dxY
n
(x)=?
n
?y
n
(t)
?t
(A.29)
is resonator?s mode-dependent velocity. From Eqs. A.25 and A.26, the electromotive
response takes the form:
epsilon1
n
=
j?
2
n
B
2
L
2
?
m
eff,n
(?
n
2
??
2
+ j?
n
?/Q
eff
)
I. (A.30)
The response of the resonator is measured by sweeping the frequency of the
applied current through the mechanical resonance, and simultaneously measuring
the induced EMF. The magnetomotive measurement is thus an impedance measure-
ment. In fact, it is apparent that the response function is equivalent to a parallel
RLC circuit (Fig. A.4(b)) with an electromechanical impedance defined as [137]
1
Z
m
=
Q
e
j?
n
?R
n
parenleftBig
?
2
n
??
2
+ j?
n
?/Q
e
parenrightBig
, (A.31)
where
R
n
= ?
2
n
Q
e
B
2
L
2
m
eff,n
?
n
, (A.32)
and ?
2
n
=(L
n
C
n
)
?1
,withL
n
= ?L
2
B
2
/m
eff,n
?
2
n
,andC
n
= m
eff,n
/?
n
L
2
B
2
.
Figure A.5 demonstrates the magnetomotive measurement of the fundamental
mode resonance of device 1. A 200 ?V
rms
voltage signal was applied through a 10
k? resistance to provide the current I. The data was taken at a mixing chamber
205
Figure A.5: Plot of the response of the fundamental mode resonance of Device
1, measured using magnetomotive detection. A lock-in was used to measure the
quadrature components of the resonator response. The response (solid line) was fit
to a driven-damped harmonic oscillator response. The data (circles) were taken at a
mixing chamber temperature of 15 mK and magnetic field B = 6 T. A drive current
of I ? 10 nA was used. The quality factor, resonant frequency, and amplitude were
determined to be 10881(3), 17.9756642(3)MHz, and 601.1(1) nV respectively.
206
temperature of 15 mK and a magnetic field of 6 T. A lock-in was used to measure the
quadrature components of the resonator?s response. These were then fit to a driven-
damped harmonic oscillator response. In the plot, the circles are the amplitude of
the response from the measured quadratures, and the line is the amplitude of the
least-squares fits to the individual quadratures. The resonant frequency and quality
factor were extracted from the fit and determined to be f
1
= 19.976 MHz and Q
e
=
10.9 x 10
3
. While the resonant frequency agrees very well with results of the SET
displacement detection technique, the quality factor is substantially lower. This is a
result of the loading from the capacitance of the co-axial cable and the 50 ? amplifier
impedance, which is much greater than the loading from SET detection. For more
details on the loading e?ect and the magnetomotive measurement in general, please
see reference [137].
207
Appendix B
Useful SET and RFSET Information
Knowledge of the SET and measurement circuit parameters is essential for
both the determination of the operating points of the RF-SET displacement detec-
tion scheme and for the characterization of its performance. In this appendix, using
the normal state and superconducting state current-voltage (IV) characteristics, I
first demonstrate how to extract the coupling-capacitance C
NR
, the gate capaci-
tance C
g
, the junction capacitances C
j
, the SET junction resistances R
j
,andthe
superconducting gap energy ?. I then discuss a technique to evaluate the frequency
response of the measurement circuit. Finally, I summarize a method that allows for
the calibration of the charge sensitivity.
B.1 SET Parameters
IV Map Measurement
To determine the SET parameters, four DC measurements of the SET source-
drain current, I
SD
, were made (Fig. B.1): in the normal state, I
SD
as a function of
the source-drain bias V
SD
and the resonator bias V
NR
; in the normal state, I
SD
as a
function of V
SD
and the gate bias V
g
; in the superconducting state (SSET), I
SD
as
a function of V
SD
and V
NR
1
; in the superconducting state, I
SD
as a function of V
SD
1
Both of the leads and the island are superconducting so the SET is in fact an (SSS) SET.
208
Figure B.1: Circuit diagram for IV map measurement .
and V
g
. The voltages are set and swept by a computer-controlled digital-to-analog
card. For each increment of the voltages, the current is sensed by a transimpedance
amplifier, and the output is measured by a digital voltmeter and recorded by the
computer through GPIB. A 1 Tesla magnetic field is applied to operate the SET in
the normal state.
The Normal-State Characteristics: The Orthodox Theory and Capac-
itance Calculations
Figure B.2 demonstrates a typical result of a normal-state IV map measure-
ment. Coulomb-blockade suppresses I
SD
for V
SD
below a threshold voltage V
t
,which
209
Figure B.2: Normal state IV map Device 2.
210
is periodic in V
NR
(V
g
) with period e.AboveV
t
, I
SD
asymptotes to a linear depen-
dence on V
SD
. The value and periodicity of V
t
are sensitive functions of C
j
and
C
NR
(C
g
), and the asymptotic behavior of I
SD
at large V
SD
is a function of the
series combination of the R
j
?s. Both limits, the onset of current and large V
SD
,
are described by the orthodox theory of single-electron tunneling [68], and can, in
principle, be used to extract the corresponding parameters [143] [144] [145].
In the orthodox theory, it is assumed that the charge state of the SET is-
land evolves stochastically through single-electron tunneling events, yielding, at any
instant of time, a value of n electrons with steady-state probability P(n)[68].
The tunneling events occur through either of the SET junctions, i,andin
either direction, on (+) or o? of (-) the SET island. They are characterized by the
tunneling rates ?
?
i
[68].
The net charge transfer or current through each junction is determined by
performing a weighted sum over all charge states, n, of the di?erence between the
(+) and (-) tunneling rates [68]:
I
1
= ?e
?
summationdisplay
n=0
P(n)
parenleftBig
?
+
1
??
?
1
parenrightBig
I
2
= ?e
?
summationdisplay
n=0
P(n)
parenleftBig
?
?
2
??
+
2
parenrightBig
.
As P(n) is assumed to be stationary in time, charge accumulation on the SET
island does not occur, and the current through each junction must be equal, yielding
[68]
I
SD
= I
1
= I
2
. (B.1)
211
The assumption that P(n) be stationary, also leads to the condition of detailed
balance [68],
P(n +1)?
?
(n +1)=P(n)?
+
(n), (B.2)
where
?
?
(n +1)=?
?
1
(n +1)+?
?
2
(n +1)
and
?
+
(n)=?
+
1
(n)+?
+
2
(n).
The tunneling rates ?
?
i
are calculated using Fermi?s golden rule [50]:
?
?
i
(n)=
?F
?
i
(n)
e
2
R
i
1
e
??F
?
i
(n)
?1
, (B.3)
where ? =k
B
T and ?F
?
i
(n) is the change in system free-energy accompanying a
particular tunneling event, and given, for the case of an asymmetrically biased SET,
by
?F
?
1
(n)=?E
c
bracketleftbigg
2
parenleftbigg
n?
C
NR
V
NR
e
?
C
g
V
g
e
parenrightbigg
?1+
+
2(C
2
+ C
NR
+ C
g
)V
SD
e
bracketrightbigg
(B.4)
and
?F
?
2
(n)=?E
c
bracketleftbigg
2
parenleftbigg
n?
C
NR
V
NR
e
?
C
g
V
g
e
parenrightbigg
?1 ?
2C
1
V
SD
e
bracketrightbigg
. (B.5)
where E
c
= e
2
/2C
?
is the charging energy or electrostatic cost for the tunneling of
a single electron, and C
?
= C
1
+ C
2
+ C
NR
+ C
g
.
Knowing ?
?
i
, P(n)andI
SD
can be calculated using Eqs. B.2 and B.1.
212
I can obtain a quantitative understanding of the Coulomb-blockade regime
without explicitly solving for P(n). For simplicity?s sake, I first set n =0,V
NR
=0
,andV
g
= 0 in Eqs. B.4 and B.5). For small |V
SD
|,?F
?
i
(n) > 0, and the work done
by the bias V
SD
is not enough to overcome the charging energy E
c
.Consequently
all four transisitions (?, junctions 1 and 2) are exponentially suppressed through
Eq. B.3, and no current is observed.
As V
SD
is increased from zero bias, eventually one of the transitions becomes
energetically favorable. That is, either ?F
+
2
(0)=0or?F
?
1
(0) = 0, depending on
which threshold is smaller, V
+,2
T
(0) = e/2C
1
or V
?,1
T
(0) = e/(2(C
2
+ C
NR
+ C
g
)). If
C
1
> (C
2
+C
NR
+C
g
), then an electron tunnels onto the island. As a result, n =1,
and the corresponding discharging step, ?F
?
1
(1), becomes energetically favorable,
and an electron tunnels o? through junction 1. After the discharge, n =0,andthe
charging step through junction 2 again becomes favorable, and so on. On the other
hand, if C
1
< (C
2
+ C
NR
+ C
g
), then V
?,1
t
(0) <V
+,2
t
(0), and the onset of current
begins with an electron tunneling o? through junction 1,
2
at which point ?F
+
2
(?1)
< 0, and an electron tunnels on through junction 2, and so on. Similiar processes
are observed if V
SD
is decreased from zero bias, as one can verify from Eqs. B.4 and
B.5.
Returning to Eqs. B.4 and B.5, it is apparent that the threshold voltage can
be tuned by adjusting V
NR
or V
g
.LeavingV
g
=0,increasing|V
NR
| lowers the
2
In practice there is an unknown background charge on the SET island which should be included
into Eqs. B.4 and (B.5). The designation of the SET island charge state as n thus refers to n
electrons induced above or below the background.
213
electrostatic tunneling cost and reduces |V
t
|.AtC
NR
|V
NR
| = e/2, the barrier is
completely removed, and current flows for infinitesimal |V
t
|.Atthisbiaspoint,the
charge configurations n =0andn = ? 1 are equally probable (n =1ifV
NR
is
positive and n =-1ifV
NR
is negative). As |V
NR
| is increased further, the n =
? 1 state becomes more favorable than the n =0state. |V
t
| thus increases, and
eventually returns to the maximum, V
t
= e/2C
1
or V
t
= e/2(C
2
+ C
NR
+ C
g
)at
C
NR
V
NR
= ? e. This process is repeated for higher n states as V
NR
is further
increased. I have thus found that the threshold for tunneling is periodic in C
NR
V
NR
with a period of one electron. The same is found to be true for tuning of V
g
with
V
NR
=0.
Turning back to Fig. B.2, the Coulomb-blockade regions are now understood
to be a result of ?F
?
i
(n) > 0 for all four transisitions (?, junctions 1 and 2) and all
n. From Eqs. B.4 and B.5, the width of the blockade in V
SD
is seen to be periodic
in C
NR
V
NR
with a period of e: minimized when C
NR
V
NR
= en/2; and maximized
when C
NR
V
NR
= 0 or a multiple of ne. Knowing the blockade to have period e,I
can calculate C
NR
from the relation
C
NR
?V
NR
= e, (B.6)
where ?V
NR
is the corresponding periodicity in mV (Fig. B.3). Similarly, I
can calculate C
g
, from the normal-state IV map with V
NR
=0(notshown),
C
g
?V
g
= e. (B.7)
214
V
NR
 (mV)
V
SD
 (mV)
?5 0 5
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
0.8
C
NR
? V
NR
 = e 
V
+,2
 
V
?,1
V
?,2
       
V
+,1
 
Figure B.3: Extracting capacitances from normal-state IV Device 2.
215
An expression for the edge of the blockade V
t
can be derived by noting that
the onset of tunneling occurs when at least one of ?F
?
i
satisfies ?F
?
i
=0:
V
?,1
t
=
C
NR
V
NR
+ C
g
V
g
(C
NR
+ C
g
+ C
2
)
?
e(1?2n)
2(C
NR
+ C
g
+ C
2
)
(B.8)
and
V
?,2
t
= ?
C
NR
V
NR
+ C
g
V
g
C
1
+
e(2n?1)
2C
1
. (B.9)
With the knowledge of C
NR
and C
g
, the junction capacitances C
1
and C
2
can
be extracted by equating the slopes of the experimental tunneling onset (Fig. B.3)
with the pre-factor of V
NR
in Eqs. B.8 and B.9.
Normal-State Characteristics: Junction Resistances
I cannot simply extract R
?
from electrostatic considerations, and must solve
for I
SD
using the detailed balance condition (Eq. B.2) and the definitions of ?
?
i
.For
V
SD
greatermuch 2E
c
/e, it is necessary to compute P(n) for several thousand n.IsolveEq.
B.2 numerically, and find that the slope of I
SD
vs. V
SD
asymptotes to R
?1
?
.The
serial resistance, R
?
is thus extracted from the normal-state IV map by a linear fit
of I
SD
at large V
SD
(Fig. B.4).
With the knowledge of the junction capacitances C
1
and C
2
and the serial
resistance R
?
, a rough estimate of the individual junction resistances R
1
and R
2
can be made [145] (Table B.1). Assuming that the thicknesses of the two junctions
are equal (both junctions are grown at the same time under similiar conditions of
pressure and temperature, see Chapter 3) and that C
i
? A
i
and R
i
? 1/A
i
,where
216
0 20 40 60 80 100
0
500
1000
1500
V
SD
 (mV)
I
SD
 (nA)
 
y = b*x ? a
b=13.6 ? .2
R
?
  = 74 ? 1 k?
Figure B.4: Extraction of junction resistance Device 2
217
A
i
is the cross-sectional area of junction-i, the individual junction resistances can
be expressed as
R
1
=
C
2
C
1
+ C
2
R
?
(B.10)
R
2
=
C
1
C
1
+ C
2
R
?
. (B.11)
Failure of Normal-State Extraction Method
Two e?ects combine to make the extraction of the junction capacitances from
the normal-state IV map unreliable: self heating of the SET island and quantum
charge fluctuations (co-tunneling).
Self-heating of the SET island results from the combination of dissipation in
the SET island and poor thermal coupling between the electron gas and phonon
bath [152]. Using the standard model for electron-phonon coupling [123] [124], and
assuming that ? 50% of the total power dissipated in the SET is dissipated in the
SET island, with the SET leads thermalized at the temperature of the phonon bath,
the SET-island electron-bath temperature T
island
at the onset of tunneling can be
estimated [152]:
T
island
=
parenleftbigg
P
island
??
parenrightbigg
1
5
, (B.12)
where ? = 0.2 nW/?m
3
K
5
[152] is the electron-phonon coupling for aluminum and
? = 0.05 ?m
2
is the total volume of the SET island. I have assumed that the
phonon-bath temperature, T
b
lessmuch T
island
.IfP
island
? I
SD
V
SD
/2, with V
SD
= e/C
?
and I
SD
? e/4R
j
C
?
from Eqs. B.3, B.4, and B.5, then
T
island
?
parenleftBigg
e
2
8R
j
C
2
?
??
parenrightBigg1
5
? 350 ?400mK (B.13)
218
for parameters C
?
=1fFandR
j
= 50 k?. This estimate does not take into
account cooling of the island due to the tunneling of electrons to and from the
lower-temperature SET leads and should thus be considered as an upper-bound
[152].
Nonetheless, the tunneling threshold is broadened, leaving the onset of current
V
t
indistinct [151]. This is modeled with the orthodox theory. Figures B.4(a) and
B.4(b) shows two simulations of the onset of tunneling of a normal-state SET with
charging energy E
C
/k
B
=1.5KandC
1
= C
2
greatermuch C
g
=10aF,andC
?
= 590 aF.
In Fig. B.4(a), T
island
= 10 mK, and the onset of tunneling is very clear and readily
fit, yielding onset contour slopes of 0.033 and thus E
C
= 1.5 K through Eqs. B.8
and B.9. In Fig. B.4(b), T
island
= 300 mK, the onset of tunneling is unclear, and
di?erent slopes are obtained depending on the contour chosen.
In addition to self-heating, quantum charge fluctuations, or co-tunneling, can
round the onset of current [69]. In general, co-tunneling is the process of charge
transfer through the SET via an energetically unfavorable intermediate virtual state,
and is the dominant charge transfer mechanism within the coulomb-blockade regime
[69]. From the energy-time uncertainty relation, for example, an electron may tunnel
through one junction to a forbidden island state, and dwell there for time ?t =
?h/?E,where?E ? E
C
is the energy required to make the transition. During
?t, it is energetically favorable for an electron to tunnel o? the island through the
second junction, resulting in a finite probability for the net transfer of one electron
across the SET.
The transition probability rate for the net process, and thus the contribution
219
(a)
0 2 4 6 8
x 10
?4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2 0
1
2
3
4
5
6
x 10
?9
|slope| ? .033
n
g
(e) 
V
SD
  (V) 
T=10mK 
I
SD
 (A)
E
C
  ?  1.5K 
(b)
0 2 4 6 8
x 10
?4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2 0
1
2
3
4
5
6
x 10
?9
T=300mK 
V
SD
 (V) 
n
g
(e) 
|slope| ? .123 
E
C
  ?  . 94 K
I
SD
 (A)
Figure B.5: Normal-state IV simulation for (a) T = 10 mK and (b) T = 300 mK.
220
of co-tunneling to I
SD
, can be estimated by multiplying the transition probability
rate for the forbidden transisition, ?
1
? 1/R
?
C
Sigma
, by the probability for the
energetically favorable tunneling event ?
1
?t:
?
co?tunneling
? ?
1
R
Q
R
?
, (B.14)
where R
Q
= h/e
2
is the quantum of resistance. The contribution of co-tunneling
is thus found to be a fraction R
Q
/R
?
of the sequential-tunneling current. For our
devices,
R
Q
R
?
? .2 ? .8. This is expected to result in the renormalization of the
charging energy by the same order of magnitude [150]:
E
prime
C
? E
C
(1 ?4
R
Q
?
2
R
?
). (B.15)
By not accounting for the e?ect of co-tunneling in the normal-state IV charac-
teristics, I thus expect to err by as much as 30% in the determination of the junction
capacitances. While modeling of the SET self-heating is straight-forward, account-
ing for the co-tunneling processes would require calculating 2nd-order (and higher,
depending on the precision desired) transition matrices for the tunneling rates and
could become both tedious and di?cult. Rather than going to all that trouble, a
simpler method is to use the features of the superconducting IV curve.
Superconducting IV Characteristics
Figure B.6 demonstrates a typical result of the superconducting IV map mea-
surement I
SD
V
SD
Vg. The complexity of the map reflects the large variety of tun-
neling processes that can occur in an SSET due to the combination of Coulomb
221
Figure B.6: Supreconducting IV map Device 2.
222
blockade, Josephson tunneling e?ects, and the superconducting gap energy [148]
[149]. Most prominent here are three distinct processes: the double Josephson-
quasiparticle resonance (DJQP) [146], the Josephson-quasiparticle resonance (JQP)
[147], and single quasiparticle tunneling. All three processes have been studied ex-
tensively, and consideration of the respective dynamics can be used to infer the
junction capacitances and the superconducting gap energy, ?.
Figure B.7 is a color contour plot of the superconducting IV map in Fig. (B.6.
The threshold for single quasiparticle tunneling defines the width of the IV plateau.
This threshold corresponds to the energy required for a quasiparticle to overcome
both the superconducting gap energy and the Coulomb charging barrier, and is seen
from simple electrostatic considerations to vary between 4? and 4? + e/C
?
.The
minimum width of the plateau is thus 8?.
Knowing ?, I can determine the Josephson coupling E
j
of each junction -
I assume ? is equal for the leads and the island, which should be the case con-
sidering that they have similiar cross-sections and composition - using the Ambe-
gaokar/Barato? relation [144]:
E
j,i
=
?h??
4e
2
R
i
. (B.16)
Within the plateau, current ridges are evident. Along these ridges, the bias
voltage is su?cient for the resonant tunneling of Cooper-pairs, followed by the tun-
neling of single quasiparticles [146] [147]. As the processes are initiated with the
tunneling of a Cooper-pair, the thresholds for the processes can be determined us-
ing electrostatic arguments similiar to the case of normal-state tunneling processes
223
V
g
 (mV)
V
SD
 (mV)
?20 ?10 0 10 20
?1.5
?1
?0.5
0
0.5
1
?8
?6
?4
?2
0
2
4
6
8
I
SD
 (nA)
          
4E
C
/e 8E
C
/e
m
1
m
2
 
8?/e
2E
C
/e
Figure B.7: Extracting junction capacitances from superconducting IV map Device
2.
224
(previous section). From these thresholds, the slope of ridges can be determined:
m
1
=
C
g
C
2
+ C
NR
+ C
g
(B.17)
m
2
= ?
C
g
C
1
. (B.18)
Having already determined C
NR
and C
g
,Icanusem
1
and m
2
to determine
the individual junction capacitances, Table B.1.
The intersection of the ridges at V
SD
= e/C
?
and V
SD
=2e/C
?
are known as
the DJQP and JQP resonance peaks respectively. In the case of the JQP resonance,
it is energetically favorable for a Cooper-pair tunneling event to occur through
either of the two junctions, followed by two sequential quasiparticle tunneling events
through the opposite junction (for example, the charge state of the island, n,goes
from 0 ? 2 ? 1 ? 0 and so on). In the case of the DJQP resonance, the Cooper-pair
tunneling event is followed, first, by a single quasiparticle, then the tunneling of a
Cooper-pair, and finally the tunneling of a single quasiparticle (0 ? 2 ? 1 ? -1 ? 0
and so on). For all four transitions in the DJQP cycle to be energetically favorable,
it is necessary for E
c
> 2?/3. The locations of the JQP and DJQP intersections in
terms of V
SD
give independent determinations of C
?
(Table B.2).
For Devices 1, 2, 3, I find that C
?
?s calculated from the slope of the current
ridges and the C
?
?s calculated from the JQP and the DJQP process agree to within
15%. Comparison of C
?
calculated using the JQP and DJQP process shows agree-
ment within 7%. For Device 4, the DJQP peak was absent and the ridges of the JQP
were very indistinct. The only available calculation of C
?
was from the location of
the JQP peaks. From the symmetry in the normal-state IV map with respect to V
g
225
Table B.1: Junction capacitances and resistances of Devices 1-4.
Device C
1
(aF) R
1
(k?) C
2
(aF) R
2
(k?) C
g
(aF) C
NR
(aF)
1 81 61 84 59 14 61
2 250 21 100 53 10 26
3 173 47 341 24 14 64
4 ? 600 ? 15 ? 600 ? 15 19 63
Table B.2: Total capacitance, charging energy, gap energy and Josephson energies
Devices 1 - 4. (a) Total capacitance found by summing C
i
, C
g
, C
NR
.(b)Total
capacitance found from position of JQP peak. (c) Total capacitance found from
position of DJQP peak.
Device C
a
?
(aF) C
b
?
(aF) C
c
?
(aF)
E
c
e
(?V )
?
e
(?V )
E
j,i
e
(?V )
1 241 - 279 287 220 12, 12
2 386 410 435 184 220 37, 14
3 592 577 540 148 200 15, 21
4 - 1310 - 61 180 ? 40, 40
226
for Device 4, we assume that C
1
? C
2
and R
1
? R
2
.
For Devices 1, 2, 3, I find that E
C
/e > 2?/3e, consistent with the observation
of the presence of the DJQP resonance in the superconducting IV maps for these
devices. For Device 4, E
C
/e ? ?/3, consistent with the absence of the DJQP
resonance in the superconducting IV map.
For Devices 1, 2, 3, I see that E
C
/e > E
j
/e. This is consistent with the absence
of a supercurrent in the superconducting IV map for these devices. For Device 4,
E
C
/e ? E
j
/e. This is consistent with the presence of a supercurrent modulated
with V
g
in the superconducting IV map of Device 4.
The reason for the chronologically increasing junction capacitances (decreasing
charging energy) is not known, but is consistent with the observed line-widths of
the SET leads and islands becoming progressively larger for each successive device
(see Chapter 3). As well, it is consistent with the general trend of decreasing R
?
and charge sensitivity.
B.2 Measurement Circuit Parameters
The measurement circuit parameters, including the RF tank circuit, were de-
termined by applying a large dc bias (V
SD
greatermuch V
t
) across the SET source-drain, and
recording shot noise ring-up of the tank circuit with a spectrum analyzer [63] ( Fig.
B.8).
For V
SD
>V
t
and time-scales slow with respect to the SET tunneling time, ?
R
?
C
?
? 0.1 GHz
?1
, the spectral density of the SET shot noise is white and given
227
Figure B.8: Circuit schematic for the measurement of the shot noise ring-up of the
tank circuit.
228
by S
II
= eI
SD
[43]. As the center frequency of the tank circuit was designed to be
? 1GHz< 1/R
?
C
?
, the shot noise served as a calibrated white noise source with
which we could probe the measurement circuit?s frequency response.
The resulting noise power density measured at the input of the spectrum
analyzer (Fig. B.9) takes the form:
P
in
similarequal GeI
SD
Z
o
f
4
o
(f
2
T
?f
2
)
2
+(ff
T
/Q)
2
. (B.19)
Here,
Q =
parenleftbigg
Z
LC
R
?
+
Z
o
Z
LC
parenrightbigg
?1
= Z
LC
parenleftbigg
Z
LCR
+ Z
o
parenrightbigg
?1
(B.20)
is the loaded quality factor of the tank resonance, Z
o
is the 50 ? transmission
line impedance, Z
LC
=
?
L
T
/C
T
is the tank circuit characteristic impedance, Z
LCR
=L
T
/(R
?
C
T
) is the transformed-SET impedance on resonance, and G is the power
gain of the measurement circuit.
Because the LC circuit was superconducting at the measurement temperature,
T ? 50 mK, and because the length of the circuit was at most 0.1?
1GHz
,Ihave
assumed that the tank circuit was a dissipation-less, lumped-element LC component.
Additionally, I have neglected the e?ect of loading on the resonant frequency, f
o
=
1/(2?
?
L
T
C
T
), as it was of the order Z
o
/R
?
? 0.001.
Including the noise of the measurement circuit, on resonance, the noise power
at the input of the spectrum analyzer thus takes the form:
P similarequal GB
parenleftBig
eI
SD
Q
2
Z
o
+ k
B
T
det
n
parenrightBig
, (B.21)
where T
det
n
, and B are measurement circuit noise temperature, and bandwidth re-
spectively. I have assumed that T
det
n
is independent of the SET bias point, at large
229
(a)
(b)
Figure B.9: Tank circuit response Device 3. (a) A lorentzian fit (red line) of the
noise power versus frequency for I
SD
= 120 nA (black circles). (b) A linear fit (red
line) of the integrated noise power versus I
SD
(black circles).
230
Table B.3: Measurement circuit matching, gain, and noise temperature Devices 1-4.
T
SET
measured at I
SD
? 120nA.
Device Z
LCR
(?) ?
max
M(dB) T
det
n
(K) G(dB) T
SET
(K)
1 - - - - - -
2 2.2-3.4 .87-.92 .72-1.2 20.2-31.7 62-64 4.3-6.6
3 2.5 .90 .82 12.6 67 5.8
4 5.6 .8 2.0 13.4 66.5 3.8
V
ds
[116]. The equivalent noise temperature of the detection scheme T
o
is then
defined by dividing Eq. B.21 by ?GBk
B
?:
T
o
=
eI
SD
Q
2
Z
o
k
B
+ T
det
n
. (B.22)
Figure B.9 displays a typical result of the shot noise measurement. Figure
B.9(a) is a plot of the noise spectrum as was measured using the spectrum analyzer.
From a fit to Eq. B.19, I can extract the width ?f and center frequency f
T
of the
resonance. From the width, I estimate the quality factor Q = f
T
/?f,andthe
detection bandwidth ?F = ?f/2. From the definitions of f
T
and Q,IcalculateL
T
,
C
T
, Z
LC
, and the impedance of the LRC on resonance Z
LCR
(Tables 3.1 and B.3).
Figure B.9(b) displays the integrated noise power and total noise temperature
of the detection scheme T
o
as a function of I
SD
. The integration was done over a
1 MHz band about the center frequency. As expected, the noise power increased
linearly with I
SD
. Fitting Eq. B.21 to the data, I estimate G and T
det
n
from the
231
slope and y-intercept respectively (Table B.3).
For Devices 2, 3, and 4, T
det
n
= 20.2 K, 12.6 K, 13.4 K respectively. The
decrease of ? 40% between Device 2 and Devices 3 and 4 is believed to have been
due mainly to the factor of ? 2 improvement in gain of the measurement scheme
for the measurement of Devices 3 and 4. For Devices 3 and 4, the calculated gain,
G,was? 2 dB less than the gain measured from room temperature transmission
measurements (see Chapter (4)). For Device 2, G is ? 5 dB less than expected.
Finally, the contribution of the SET shot noise T
SET
to the overall detection
noise can be estimated by subtracting T
det
n
from T
o
at a particular V
SD
.At? 120
nA, the smallest I
SD
at which we measured the output shot noise power, T
SET
ranged from ? 3.8 K - 5.8 K, or ? 20% - 50% of the total noise.
B.3 Calibration of Charge Sensitivity
The charge sensitivity of the detection scheme was measured using amplitude-
modulated (AM) reflectometry (Chapter 5). Microwaves resonant with the LC cir-
cuit were applied to the SET drain, and the reflected signal was recorded. Simulta-
neously, a 1 MHz sine wave bias of charge amplitude ?Q
g
(in units of electrons) was
applied to the gate of the SET, modulating the amplitude of the reflected carrier
signal, and producing sidebands at f
o
? nf,wheref = 1 MHz . The sidebands were
recovered using homodyne detection, mixing the reflected signal with the carrier,
and measured with a spectrum analyzer. The charge sensitivity was then calculated
from the ratio of the background noise power level P
Background
to the power in the
232
Figure B.10: Illustration of the experimental determination of the charge sensitivity
from amplitude-modulated reflectometry. The plot shows a 1 MHz sideband of
the measured reflected signal after recovery with homodyne mixing at the carrier
frequency.
233
1MHz sideband P
1MHz
(Fig. B.10):
?
S
Q
=
?Q
g
?
B
10
?SNR/10
, (B.23)
where SNR = P
1MHz
(dBM)-P
Background
(dBm) is the signal-to-noise ratio and B
is the resolution bandwidth of the spectrum analyzer.
In practice, due to losses in the sample lead and losses in the cabling and
filters inside the dilution refrigerator, the amplitude of the charge signal applied
to the SET gate at 1MHz was not known. However, from the dependence of the
reflected signal?s sideband amplitude on the amplitude of the 1 MHz modulation, I
can calculate the losses in the circuit and calibrate the charge sensitivity.
For a dc gate bias, V
g
? 0, the sideband power response can be approximated
by [153]
P = P
o
sin(2??Q
g
sin 2?ft), (B.24)
where f=1MHz. This can be expanded in terms of the Bessel functions J
n
(2??Q
g
)
[153]:
P =2P
o
?
summationdisplay
n=0
J
2n+1
(2??Q
g
)sin ((2n +1)?t). (B.25)
Using lock-in detection, we measured the amplitude of the fundamental of the
response, n = 0, as a function of the amplitude ?V
g
of the 1 MHz modulation at
the output of the waveform generator (Fig. B.11). I assume that the relationship
between the voltage modulation at the generator and the charge modulation at the
device is given by ?Q
g
= A?V
g
/e. Fitting the response to J
1
(A?V
g
), and knowing
that the first zero of J
1
(x) occurs at x =3.832, I extract A and determine the ratio
?Q
g
/?V
g
in electrons/volt. From the value of ?V
g
,Icalculate?Q
g
, and, using Eq.
234
B.23, I calculate the charge sensitivity at the operating points of the the Devices 1
- 4 (Table B.5).
For Devices 3 and 4, using the Bessel function fit, I find 5 - 7 dB attenuation in
the gate line. This is consistent with room temperature transmission measurements
of the same line (see Chapter 4). For Devices 1 and 2, we were not aware of the Bessel
function calibration technique. Thus the reported values of
?
S
Q
are estimated using
a value of 6 dB attenuation in the gate line.
Finally, the uncoupled energy sensitivity of the total detection scheme, SET
shot noise plus measurement circuit noise, is defined as [116]
?epsilon1 =
S
Q
2C
?
. (B.26)
235
0 0.5 1 1.5 2 2.5 3 3.5 4
?4
?2
0
2
4
6
8
x 10
?4
?V
g
  (V
rms
)
Sideband Amplitude (V
rm
s
)
Figure B.11: Bessel function fit to sideband response Device 3.
236
Table B.4: Charge modulation calibration Devices 1 - 4. Note that the attenuation
listed is the attenuation of the gate line. This does not include an additional 60 dB
of attenuation due to attenuators put in place at the top of the fridge.
Device A(
e
rms
V
) ?V
g
(V
rms
) ?Q
g
(e
rms
) Attenuation (dB)
1 - 0.50 ? .015 ? 6
2 - 0.50 ? .015 ? 6
3 .036 1.0 .036 7
4 .062 1.0 .062 5
Table B.5: Charge sensitivity Devices 1 - 4. (a) Measured charge sensitivity. (b)
Calculated charge sensitivity from curvature of I
SD
V
SD
V
g
map. Both at 1 MHz
Device SNR (dB)
?
S
Q
a
(?e
rms
/
?
Hz)
?
S
Q
b
?epsilon1(J/Hz) ?epsilon1
SET
(J/Hz)
1 56 20 30 85?h -
2 58 30 20 50?h 10?h
3 52 90 70 1300?h 600?h
4 50 200 150 3750?h 1100?h
237
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