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Progress Towards the Quantum Limit: High and Low Frequency Measurements of
Progress Towards the Quantum Limit:
High and Low Frequency Measurements of
Nanoscale Structures
by
Joshua Lewis Rideout
A thesis submitted to the
Department of Physics, Engineering Physics & Astronomy
in conformity with the requirements for
the degree of Master of Science
Queen’s University
Kingston, Ontario, Canada
March 2010
c Joshua Lewis Rideout, 2010
Copyright Abstract
In this thesis, I present the work performed towards a proposal to couple a piezoelectric, nanomechanical beam to a radio frequency single electron transistor (RF-SET).
Lumped element RF circuit theory is applied to 50 kΩ single electron transistors acting as the resistor in an RLC circuit. It is shown that for the expected inductances
and stray capacitances, at an operating frequency of 1.25 GHz, the RF-SET is expected to have a usable half-bandwidth of 175-200 MHz and a charge sensitivity on
√
the order of 10−5 e/ Hz. A fabricated RF-SET device is cryogenically cooled and
used to find experimental values of the stray capacitance.
A heterostructure made of gallium arsenide and aluminum gallium arsenide from
which piezoelectric beams can be made is designed to contain a 2-dimensional electron gas (2DEG). Quantum Hall effect samples are fabricated from the wafer, and
magnetoresistance measurements for each sample are presented. It is shown that the
2DEG has a high electron concentration of about 8 × 1011 cm−2 but a low mobility
of about 3.5 × 104 cm2 /(V·s) for this type of heterostructure.
i
Acknowledgments
I would like to extend my greatest thanks to my supervisor, Dr. Robert Knobel, for
guiding me through the M.Sc. degree at Queen’s. His patience, coupled with a strong
sense of dedication to his graduate students’ success, helped me get to where I am
today. I would also like to extend my thanks to Cindi and Sophie Knobel, for letting
Rob come to work at hours which no one outside the academic community would be
able to understand.
I extend great thanks to Zbig Wasilewski and Guy Austing at the National Research Council (NRC) for crystal growth services and initial ohmic contact recipes.
Without them the foundation of my research would not have taken place.
In the nanomechanics group, I wish to thank my fellow graduate student Scott
Pierobon for collaborating with me on the RF-SET portion of the project. I believe
that working together, we accomplished significant goals towards the fabrication of
complete devices, on which he his currently testing in the lab after my departure.
Also in this group, I thank Jennifer Campbell and Ben Lucht for many great
discussions about fabrication and their help in various stages of cool down procedures.
I appreciate the time of Devon Stopps, a crazy man who has more knowledge about
electric circuits and “how to fix stuff” than I could ever hope to acquire in my lifetime.
I wish to thank our postdoctoral fellow Saydur Rahman for his thoughtful comments
ii
and help with low temperature data interpretation.
In the department, I wish to acknowledge Gary Contant and Chuck Hearns for
helping me with their machine shop expertise, Kim MacKinder and John Odell for
running the helium plant, and Loanne Meldrum for answering questions about graduate studies, sending my thesis to the reviewers, and generally being an awesome
person.
Finally, I wish to dedicate this work to my wonderful girlfriend Nicole Rice, and
my family back home on The Rock. Without you all (Pat, Garry, Mark, Nan and
Pop Rideout, Bernie, Dee and all the rest), I would not have had the courage to live
the life upalong and do the things that give meaning to my life. Cheers, Love, and
Thanks!
iii
Table of Contents
Abstract
i
Acknowledgments
ii
Table of Contents
iv
List of Tables
viii
List of Figures
ix
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
Quantum Mechanics and You . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Contributions and Scope of Work . . . . . . . . . . . . . . . . . . . .
5
1.3.1
Scope of Work . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3.2
Specific Contributions . . . . . . . . . . . . . . . . . . . . . .
7
Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Chapter 2: Background and Theory . . . . . . . . . . . . . . . . . . .
10
1.4
2.1
The Single Electron Transistor (SET) . . . . . . . . . . . . . . . . . .
10
2.1.1
13
Electrostatics in the SET . . . . . . . . . . . . . . . . . . . . .
iv
2.1.2
Tunnelling in the SET . . . . . . . . . . . . . . . . . . . . . .
17
2.1.3
The Superconducting SET (SSET) . . . . . . . . . . . . . . .
19
2.1.4
Applications of the SET . . . . . . . . . . . . . . . . . . . . .
24
The Radio Frequency SET (RF-SET) . . . . . . . . . . . . . . . . . .
27
2.2.1
Theory of Operation . . . . . . . . . . . . . . . . . . . . . . .
28
2.2.2
Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.2.3
Noise and Sensitivity in the SET and RF-SET . . . . . . . . .
33
2.2.4
Applications of the RF-SET . . . . . . . . . . . . . . . . . . .
37
Two-Dimensional Electron Gases (2DEG) . . . . . . . . . . . . . . .
41
2.3.1
Inversion Layers and Doping . . . . . . . . . . . . . . . . . . .
41
2.3.2
Quantum Hall Effect Characterization . . . . . . . . . . . . .
45
2.3.3
Suspended 2DEGs and Previous Work . . . . . . . . . . . . .
52
A New Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
Chapter 3: Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
2.2
2.3
2.4
3.1
Design of the RF-SET . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.2
Determining the values of L and C . . . . . . . . . . . . . . . . . . .
63
3.3
Simulations of RF-SET Response . . . . . . . . . . . . . . . . . . . .
64
3.3.1
Perfect Matching . . . . . . . . . . . . . . . . . . . . . . . . .
64
3.3.2
Realistic Matching . . . . . . . . . . . . . . . . . . . . . . . .
66
Heterostructure Design and Simulation . . . . . . . . . . . . . . . . .
70
Chapter 4: Sample Fabrication . . . . . . . . . . . . . . . . . . . . . .
77
3.4
4.1
SET Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
4.1.1
78
Design of EBL Patterns . . . . . . . . . . . . . . . . . . . . .
v
4.2
4.1.2
Wafer Preparation . . . . . . . . . . . . . . . . . . . . . . . .
80
4.1.3
Exposure and Development . . . . . . . . . . . . . . . . . . .
81
4.1.4
Metallization and Oxidation . . . . . . . . . . . . . . . . . . .
84
4.1.5
Lift-Off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
2DEG Hall Bar Samples . . . . . . . . . . . . . . . . . . . . . . . . .
88
4.2.1
Photolithography . . . . . . . . . . . . . . . . . . . . . . . . .
88
4.2.2
Mask Design . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
4.2.3
Wafer Preparation . . . . . . . . . . . . . . . . . . . . . . . .
95
4.2.4
Exposure
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
4.2.5
Development . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
4.2.6
Ohmic Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2.7
Hall Bar Lithography and Etching . . . . . . . . . . . . . . . . 102
4.2.8
Hall Bar Suspension . . . . . . . . . . . . . . . . . . . . . . . 105
Chapter 5: Experimental Methods . . . . . . . . . . . . . . . . . . . . 109
5.1
5.2
Wiring up RF-SETs . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.1.1
Testing Fabricated SETs . . . . . . . . . . . . . . . . . . . . . 109
5.1.2
Sample Holder for the RF-SET . . . . . . . . . . . . . . . . . 111
5.1.3
1K-Cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.1.4
RF Test Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Wiring up 2DEGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2.1
Wirebonding Ohmic Contacts . . . . . . . . . . . . . . . . . . 118
5.2.2
He-3 Cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2.3
Circuit for Hall Bars . . . . . . . . . . . . . . . . . . . . . . . 123
5.2.4
Illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
vi
Chapter 6: Experimental Results and Discussion . . . . . . . . . . . 127
6.1
6.2
RF-SET Cooldown . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.1.1
SET Fabrication Results . . . . . . . . . . . . . . . . . . . . . 127
6.1.2
Temperature Dependence of CS . . . . . . . . . . . . . . . . . 128
2DEG Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2.1
First Cool Down - Unsuspended Samples . . . . . . . . . . . . 131
6.2.2
Ohmic Contact Tests . . . . . . . . . . . . . . . . . . . . . . . 139
6.2.3
Second Cool Down - Suspended Samples . . . . . . . . . . . . 143
6.2.4
Third Cool Down - Shifted Contacts . . . . . . . . . . . . . . 146
Chapter 7: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.1
7.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.1.1
Fabrication of 1 GHz Beams . . . . . . . . . . . . . . . . . . . 156
7.1.2
Coupled Nano-Electro-Mechanical System . . . . . . . . . . . 157
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Appendix A: Experimental Details . . . . . . . . . . . . . . . . . . . . 171
A.1 1K-Cryostat Cool Down Procedures . . . . . . . . . . . . . . . . . . . 171
A.2 He-3 Cryostat Cool Down Procedures . . . . . . . . . . . . . . . . . . 173
Appendix B: Mathematical Routines . . . . . . . . . . . . . . . . . . . 176
B.1 Cubic Spline Interpolation . . . . . . . . . . . . . . . . . . . . . . . . 176
B.2 Fast Fourier Transform (FFT) . . . . . . . . . . . . . . . . . . . . . . 178
Appendix C: List of Abbreviations and Symbols . . . . . . . . . . . . 180
vii
List of Tables
3.1
Experimentally obtained SET values from Greg’s Thesis. . . . . . . .
61
3.2
Perfect matching of a RF-SET to 50 Ω at f0 = 1.25 GHz. . . . . . . .
64
3.3
Good matching of a RF-SET to 50 Ω at f0 = 1.25 GHz. . . . . . . . .
66
4.1
Electron beam lithography parameters for pattern “rf-set15.RF6”. Patterning was performed using a 40 kV accelerating voltage.
1
Feature
size. 2 Magnification. 3 Electron charge dosage. 4 Beam current. 5 Centerto-center spacing. 6 Line spacing. 7 Pattern origin offset. . . . . . . . .
79
6.1
Experimentally obtained RF-SET properties from Figure 6.1. . . . . . 130
6.2
Experimentally obtained 2DEG properties from samples A and B. . . 138
6.3
Experimentally obtained 2DEG properties from samples D and E. . . 144
6.4
Filling factors i = h/(e2 RH ) from observable Hall plateaus of sample G. 150
6.5
Experimentally obtained 2DEG properties from sample G. . . . . . . 154
viii
List of Figures
2.1
Schematic of a SET. . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2
Electrostatics inside a SET. . . . . . . . . . . . . . . . . . . . . . . .
14
2.3
The “Coulomb diamonds” in the IV-curve of a SET.
. . . . . . . . .
16
2.4
Energy diagram of tunnelling through a thin barrier. . . . . . . . . .
18
2.5
Simulated IV-curve of a SET for (a) Ids vs. Vds and (b) Ids vs. Vg . .
20
2.6
The energy-band diagram of a Josephson junction.
. . . . . . . . . .
21
2.7
The IV-curve of a superconducting SET (SSET). . . . . . . . . . . .
23
2.8
Single electron box experiment. . . . . . . . . . . . . . . . . . . . . .
24
2.9
SET detection of a nanomechanical GaAs beam. . . . . . . . . . . . .
26
2.10 SET mixer output for a nanomechanical GaAs beam. . . . . . . . . .
27
2.11 Schematic circuit of a RF-SET. . . . . . . . . . . . . . . . . . . . . .
29
2.12 The reflected signal from an RF-SET. . . . . . . . . . . . . . . . . . .
31
2.13 The single electron trap. . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.14 Telegraph noise in the single electron trap. . . . . . . . . . . . . . . .
38
2.15 RF-SET coupled to SiN beam. . . . . . . . . . . . . . . . . . . . . . .
39
2.16 The MOSFET and its energy band diagram. . . . . . . . . . . . . . .
42
2.17 Energy gap and lattice constants for binary semiconductors. . . . . .
43
2.18 Energy bands for the two-layer GaAs/AlGaAs heterostructure. . . . .
44
2.19 Diagram of an GaAs/AlGaAs heterostructure. . . . . . . . . . . . . .
45
ix
2.20 Parabolic sub-bands in a 2DEG, and the step-wise density of states. .
47
2.21 Measurement of a Hall bar of material in a magnetic field. . . . . . .
48
2.22 Landau levels in a 2DEG . . . . . . . . . . . . . . . . . . . . . . . . .
50
2.23 Quantized Hall resistance in a 2DEG. . . . . . . . . . . . . . . . . . .
51
2.24 GaAs tuning fork. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
2.25 GaAs/AlGaAs cantilevers. . . . . . . . . . . . . . . . . . . . . . . . .
55
2.26 Suspended GaAs/AlGaAs Hall bar. . . . . . . . . . . . . . . . . . . .
56
2.27 A heterostructure beam coupled to quantum dots. . . . . . . . . . . .
57
2.28 QPC integrated with 2DEG resonator. . . . . . . . . . . . . . . . . .
58
2.29 RF-SET detection of a GaAs/AlGaAs nanomechanical beam. . . . . .
59
3.1
RF-SET matched to 50 Ω. . . . . . . . . . . . . . . . . . . . . . . . .
65
3.2
RF-SET mismatched to 50 Ω. . . . . . . . . . . . . . . . . . . . . . .
67
3.3
Change in the reflected signal versus SET resistance. . . . . . . . . .
68
3.4
Conductance of sample GregA. . . . . . . . . . . . . . . . . . . . . .
69
3.5
Heterostructure cross-section. . . . . . . . . . . . . . . . . . . . . . .
72
3.6
Conduction band of the unsuspended heterostructure. . . . . . . . . .
75
3.7
Conduction band of the suspended heterostructure. . . . . . . . . . .
76
4.1
SET pattern RF-SET15. . . . . . . . . . . . . . . . . . . . . . . . . .
80
4.2
SET fabrication steps. . . . . . . . . . . . . . . . . . . . . . . . . . .
82
4.3
Optical microscope images of developed SET patterns. . . . . . . . .
84
4.4
Bell jar deposition chamber. . . . . . . . . . . . . . . . . . . . . . . .
85
4.5
Tilting stage and double angle evaporation. . . . . . . . . . . . . . . .
86
4.6
Completed SET samples. . . . . . . . . . . . . . . . . . . . . . . . . .
88
4.7
The photolithographic process. . . . . . . . . . . . . . . . . . . . . . .
90
x
4.8
Mask patterns for large hall bars. . . . . . . . . . . . . . . . . . . . .
92
4.9
Mask patterns for small and suspended hall bars. . . . . . . . . . . .
93
4.10 Array of patterns on a photomask. . . . . . . . . . . . . . . . . . . .
94
4.11 Hall bar fabrication steps. . . . . . . . . . . . . . . . . . . . . . . . .
97
4.12 New photolithography base plate. . . . . . . . . . . . . . . . . . . . .
98
4.13 Example of 2DEG ohmic contact test data. . . . . . . . . . . . . . . . 103
4.14 Etched Hall bar shapes. . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.15 Suspended Hall bars. . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.1
Probe station circuit diagram. . . . . . . . . . . . . . . . . . . . . . . 111
5.2
New PCB for RF-SET experiment. . . . . . . . . . . . . . . . . . . . 112
5.3
1K-cryostat diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.4
1K-cryostat measurement circuit. . . . . . . . . . . . . . . . . . . . . 117
5.5
PCB for QHE experiment. . . . . . . . . . . . . . . . . . . . . . . . . 118
5.6
He-3 cryostat probe. . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.7
He-3 IVC contents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.8
Hall bar measurement circuit. . . . . . . . . . . . . . . . . . . . . . . 124
6.1
S11 versus frequency of an RF-SET. . . . . . . . . . . . . . . . . . . . 129
6.2
Sample A (a ∼100 µm wide, unsuspended Hall bar).
6.3
Resistance versus magnetic field for sample A. . . . . . . . . . . . . . 133
6.4
Classical carrier concentration for sample A. . . . . . . . . . . . . . . 134
6.5
FFT of SdH oscillations for sample A.
6.6
Sample B (a ∼1 µm wide, unsuspended Hall bar). . . . . . . . . . . . 136
6.7
Resistance versus magnetic field for sample B. . . . . . . . . . . . . . 137
6.8
Ohmic contacts to the wafer. . . . . . . . . . . . . . . . . . . . . . . . 140
xi
. . . . . . . . . 132
. . . . . . . . . . . . . . . . . 135
6.9
Annealing tests for ohmic contacts. . . . . . . . . . . . . . . . . . . . 142
6.10 Sample D (a ∼1 µm wide, suspended Hall bar). . . . . . . . . . . . . 144
6.11 Resistance versus magnetic field for sample D. . . . . . . . . . . . . . 145
6.12 Phase and VL,R versus magnetic field for sample A. . . . . . . . . . . 147
6.13 Edges states in the QHE. . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.14 Shifted Hall bar contacts.
. . . . . . . . . . . . . . . . . . . . . . . . 149
6.15 Sample G (a ∼2 µm wide, suspended Hall bar). . . . . . . . . . . . . 150
6.16 Resistance versus magnetic field for sample G (unilluminated). . . . . 151
6.17 Persistent photoconductivity in Hall bars. . . . . . . . . . . . . . . . 152
6.18 Resistance versus magnetic field for sample G (illuminated). . . . . . 153
6.19 Second subband filling. . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.20 FFT of SdH oscillations for sample G. . . . . . . . . . . . . . . . . . 155
B.1 Spline curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
B.2 Example of SdH spline curve. . . . . . . . . . . . . . . . . . . . . . . 178
xii
Chapter 1
Introduction
1.1
Quantum Mechanics and You
If there is but one anecdote undergraduate students learn from lessons in quantum
mechanics, it is typically Schrödinger’s Cat. In this scenario, which was first posed
in 1935 by Erwin Schrödinger, a cat is placed in a closed box with a vial of poison.
The vial is broken by a hammer if a connected Geiger counter detects a decay event.
If after one hour no counts have registered, the cat will still be alive, however if even
one count has been registered, the cat will surely be dead. If we cannot look in the
box, the cat is simultaneously alive and dead, and it is said to be in a superposition
of states. Opening the box destroys the superposition, as the cat is now either alive
or dead, and the so-called “collapse of the wavefunction” occurs [1].
Aside from wondering if Schrödinger had any pets at that time, an interesting
question perhaps to ask is: when does an object, such as a cat, dog, or other “macroscopic” object act as a coherent superposition of states, rather than a classical object?
1
CHAPTER 1. INTRODUCTION
2
This question has for over 30 years fueled the debate as to whether or not the “collapse of the wavefunction” is truly real, or just the appearance of purely quantum
mechanical systems interacting with their environment. The later is called quantum decoherence, and has gained much ground in describing the loss of quantum
information to the thermal bath surrounding a system of interest [1, 2].
1.2
Motivation
Nano-electro-mechanical systems (NEMS) in particular provide an opportunity to
study macroscopic and theoretically quantum mechanical systems up close. Cantilevers or doubly-clamped beams on the order of 100 − 1000 nm in size and weighing
10−15 − 10−17 g have already provided extremely sensitive tools in the areas of mass,
chemical, and biological sensing [3, 4]. If we assume for now that quantum mechanics holds for devices this large, at low enough excitation energies these beams can be
treated as quantum harmonic oscillators [5]. While it has yet to happen, there remains
an exciting possibility that one day we will observe a macroscopic, top-down fabricated device condensed into its lowest possible energy state, also called the quantum
ground state [5].
Attempting to observe a nanomechanical system in its ground state or a superposition of quantum mechanical states is a big challenge. Despite continuing advances in
photolithography, imprinting and scanning probe lithography, electron beam lithography (EBL) remains the most flexible system for nanoscale patterning. However, even
the highest quality EBL systems require significant user input. This means that for
every successfully released beam one makes, much time is spent fabricating samples
which may ultimately not work.
3
CHAPTER 1. INTRODUCTION
We should also consider: “how small is small?”; that is to say, how big are the
displacements nanomechanical beams make? Since we treat the beam as a quantum
harmonic oscillator, its Hamiltonian is of the form [5]:
Ĥ =
1 2 1
P̂ + mω0 X̂ 2
2m
2
(1.1)
where m is the mass of the beam, P̂ is the momentum operator, X̂ is the position
operator, and ω0 /2π is the mechanical resonant frequency of the beam. The energy
eigenvalues of the beam are of the form [2]:
1
E = (n + )h̄ω0
2
n = 0, 1, 2, ...
(1.2)
such that n represents the integer number of energy quanta the oscillator has. The
higher the value of n, the larger the beam displacement. When n = 0, E = 12 h̄ω0 , and
the oscillator is said to be in its ground state. It is well known the root-mean-square
(RMS) amplitude hx2 i1/2 = ∆x of an oscillator in its quantum ground state can be
found by applying Heisenberg’s uncertainty principle, and is given by [5]:
r
h̄
∆x = ∆xSQL =
2mω0
(1.3)
The value ∆xSQL is called the standard quantum limit (SQL) for displacement measurement [5]. For a ω0 /2π = 1 GHz beam having mass 10−15 kg, ∆xSQL ∼
= 10−15 m,
which is on the order of the width of an atomic nucleus [2]. Clearly, ultra-sensitive
detection methods must be employed to sense such small displacements.
To actually observe the quantum mechanical states, at least two general limits
must be satisfied. First, thermal fluctuations which give the resonator lots of energy
and push it out of the ground state must be low. In an environment with average
temperature T , the average energy of the beam is given by [2, 5]:
!
1
1
hEi = h̄ω0
+
= h̄ω0 hnth i
0
2 e kh̄ω
BT − 1
(1.4)
4
CHAPTER 1. INTRODUCTION
where hnth i is called the thermal occupation number, and the factor of 1/2 in the
brackets represents the ground state energy. When kB T ≪ h̄ω0 , hnth i < 1, and the
beam will be in its ground state. For a ω0 /2π = 1 GHz beam, a temperature of about
T ∼
= 50 mK is required, which is achievable inside a dilution refrigerator. To perform
these types of measurements, cryogenic lab equipment must be used.
Second, quantum fluctuations must be minimized. Unless the measurement for
beam detection commutes with Ĥ, the measurement invariably disturbs the value
of n in the oscillator. To measure displacement, which does not commute with Ĥ
(see Equation 1.1), even the best amplifier imparts a minimum amount of further
uncertainty in ∆x [5]. This uncertainty is called the backaction, and at T = 0
combining the backaction of a linear amplifier with the zero point motion of the
beam gives a revised ∆x of [6]:
∆xQL =
r
h̄
∼
= 1.35∆xSQL
ln 3mω0
(1.5)
where ∆xQL is called the quantum limit (QL) of displacement detection.
To attempt quantum limited displacement detection in our lab, we propose to
make a high frequency, nanomechanical resonator from a layered heterostructure of
GaAs and AlGaAs [7]. This wafer will be Si doped in particular layers to create a
2-dimensional electron gas (2DEG) in the suspended beam region. Since GaAs and
AlGaAs are piezoelectric materials, the beam will be driven by applying a voltage
between an electrode placed at one end of the resonator and the 2DEG. Due to the
beam motion, charge will build up on an electrode placed at the other end of the
beam [7].
A sensitive linear charge amplifier called a radio frequency single electron transistor (RF-SET) will be used to sense the charge buildup and thereby detect the
CHAPTER 1. INTRODUCTION
5
motion. Depending on the mechanical Q-factor of the beam and the displacement
sensitivity, a working design could lead to the closest approach ever to ∆xQL and allow
for fundamental studies of the transition between classical and quantum mechanical
states.
1.3
1.3.1
Contributions and Scope of Work
Scope of Work
This project was designed to push forward the fabrication steps for a device as outlined
in Reference [7]; a superconducting RF-SET coupled to a GaAs/AlGaAs piezoelectric
beam with an embedded 2DEG. Given below is the work completed towards the
project.
Using CAD software, patterns for single electron transistors (SETs) with two gate
electrodes were created and fabricated at Queen’s University using a scanning electron
microscope (SEM). Much time was spent optimizing the fabrication process, which
included varying the SEM beam currents, wet etching and oxidation times. The goal
of this work was to make reproducible SETs having a low total room temperature
resistance (RΣ = 50 − 100 kΩ), as devices in this resistance range give the best
sensitivity in RF-SET experiments. Specific details regarding SET fabrication recipes
are given in the thesis of Scott Pierobon [8], who spent significant time dealing with
these issues.
To perform the RF-SET experiment at T = 300 mK, a SET must be connected
to a printed circuit board (PCB) designed to fit onto our He-3 cryostat. It was determined that when a typical SET chip was connected to a tank circuit on a previously
CHAPTER 1. INTRODUCTION
6
fabricated PCB, the stray capacitance was too high. A new PCB was designed with
all copper traces removed under and around the SET chip. A 300 kΩ SET was
connected to 1/4 of this PCB and cooled to as low as T = 1.5 K, where the stray
capacitance was measured.
Aside from these experiments, a GaAs/AlGaAs wafer was grown by the NRC in
Ottawa. This wafer had been designed by Vincent Leduc (a previous M.Sc. student)
to contain a 2DEG. A photomask was created by the University of Alberta, and
ultraviolet photolithography was performed at Queen’s University to make Hall bars
and ohmic contacts on the 2DEG wafer. Photolithography was preferred at the time
to allow for parallel processing and high output.
Different recipes of Ni-Ge-Au contacts were deposited on the surface of the wafer
and annealed using a rapid thermal annealer. Current-voltage (IV) curves were collected using a probe station and data acquisition software. After the optimal recipe
was determined to give ohmic contact through the 2DEG, large (∼ 100 µm) and small
(∼ 2 µm) Hall bars were fabricated using a wet etch of hydrogen peroxide and citric
acid. Some of the small devices were completely suspended using a hydrofluoric acid
(HF) etch, which removed a sacrificial Al0.7 Ga0.3 As layer under the 2DEG layers.
The Hall bars were cooled to T = 300 mK, where the quantum Hall effect (QHE)
was used to find the carrier concentration, nc , and mobility, µ, of electrons in the
2DEG layer. It was determined that both unsuspended and suspended beams contained a 2DEG, which means piezoelectric beams can be fabricated from this wafer
for the proposed experiment. The original placement of ohmic contacts did not allow
for good high magnetic field ( 4 T) data to be collected. In the final sample studied,
the contacts were shifted to the mesa edges to contact edges states, and fully formed
CHAPTER 1. INTRODUCTION
7
quantum Hall plateaus were observed.
A doubly-clamped 2DEG beam coupled to a RF-SET was not realized at the end
of this project, as 100 kΩ SETs could not be made reproducibly in our lab as of late
2009. However once this obstacle has been overcome, a fully functional device (as
given in Reference [9]) should be obtainable.
1.3.2
Specific Contributions
The results in this thesis could not have been collected without the assistance of
several people. To distinguish their contributions from my own, listed below are the
specific items which they have worked on:
1. Scott Pierobon (a current member of the Knobel lab group) and I worked continuously on SET fabrication from October 2007 to January 2009. This included
performing lithography, aluminum deposition, oxidation and lift-off. While our
fabrications were performed independently, recipes were shared and modified in
order to achieve the goal of making reproducible 100 kΩ SETs. As discussed in
Chapter 6, Scott provided me with a 300 kΩ SET in October 2008 to test its
stray capacitance inside an RF-SET setup.
2. Vincent Leduc (who was a previous member of the Knobel lab group ) performed
the original Poisson-Schrödinger calculations for the design of the latest 2DEG
wafer. We received this wafer from the NRC in 2008. I have checked and
reproduced these calculations in Chapter 3, in order to show the design process.
3. Gary Contant (head instrument fabricator in the Department of Physics, Engineering Physics and Astronomy) fabricated the small sample lithography plate
CHAPTER 1. INTRODUCTION
8
for our Oriel mask aligner.
Specific items which were primarily my own work include, but are not limited to:
• Calculation of the expected RF-SET sensitivity from previously fabricated SETs
in our lab.
• Lithography and fabrication of SETs from Al/Alx Oy /Al.
• Design of the new printed circuit board (PCB) for RF-SET experiments in the
He-3 cryostat.
• Running the 1 K cryostat and programming LabView to collect data.
• Design of the Hall bar photomask using Coventor.
• Complete fabrication cycles of large, small, and suspended 2DEG Hall bars.
This includes lithography, wet chemical processing, deposition of Ni-Ge-Au contacts, annealing, lift off, and HF etching.
• Wirebonding ohmic contacts to the printed circuit board.
• Running the He-3 cryostat and programming MATLAB to collect data.
• RF-SET frequency sweep and Hall bar magnetoresistance data analysis and
graphing.
1.4
Organization of Thesis
Chapter 2 presents an overview of the theory and previous work completed by other
researchers regarding SETs and 2DEGs. Inside it, the majority of equations and
concepts required for understanding this project are explained.
CHAPTER 1. INTRODUCTION
9
Chapter 3 gives the expected theoretical sensitivity for RF-SETs capable of being
made in our lab. It also presents the design and simulation results of the heterostructure wafer studied in this work.
Chapter 4 details the physical and chemical fabrication steps used to create SETs
and Hall bars, which were mostly performed in our Class 10000 cleanroom.
Chapter 5 discusses the experimental methods used to cool the RF-SET to T =
1.5 K, and the Hall bars to T = 300 mK, and the various electrical measurements
performed on each type of device.
Chapter 6 gives an analysis of the data that was collected from the reflected power
versus frequency sweep of the RF-SET, and the magnetoresistance measurements for
each Hall bar.
Finally, Chapter 7 discusses the major conclusions derived from this work, and
details what must be considered before the coupled RF-SET and piezoelectric beam
device can be made a reality.
Chapter 2
Background and Theory
The body of work completed for this project requires knowledge of the physics behind
single electron transistors and 2-dimensional electron gases. This chapter describes
most of the theoretical background behind these phenomena and extends the discussion of SETs to their high-speed counterpart, the radio-frequency single electron
transistor. Historically, significant calculations and experiments that refer to these
phenomena will be explored at the end of each section. Finally, Section 2.4 describes
the proposed experiment in more detail.
2.1
The Single Electron Transistor (SET)
A single electron transistor is a device which consists of a small conducting island separated from two conducting leads by insulating tunnel junctions and a gate electrode,
which is capacitively coupled to the island [10]. Typically, the leads and island are
composed of a metal while the insulating junctions are composed of an oxide of the
metal. A simple circuit diagram, as in Figure 2.1, shows that each tunnel junction can
10
CHAPTER 2. BACKGROUND AND THEORY
11
be thought of as a capacitor Ci operating in parallel with a resistor Ri . The device
is drain-source biased through the leads by a voltage Vds and is capacitively coupled
through Cg to the gate. The gate potential Vg is separately controlled. For certain
values of Vds and Vg , electrons can tunnel on and off the island through the junctions,
which manifests as a current. It will be shown that only a fraction of an electron
charge on the gate can drastically change the magnitude of this current, making the
SET the most sensitive electrometer [11].
Figure 2.1: Schematic diagram of a single electron transistor. Circuit component values
are labeled as in the text.
The idea of a SET was first conceived by Averin and Likharev in 1986 [12] while
the first practical device was fabricated in 1987 by Fulton and Dolan [10]. The useful
properties of the SET are most easily seen at low temperature, and electrostatic
analysis shows why this is the case. To begin, the energy to add one electron to the
CHAPTER 2. BACKGROUND AND THEORY
12
“capacitor” island of the SET through either tunnel junction is simply written [13,14]:
Ec =
e2
2CΣ
(2.1)
where CΣ = C1 + C2 + Cg + C0 , the sum of the individual tunnel junction capacitances, the gate capacitance Cg , and a background capacitance C0 . The background
capacitance accounts for a polarization charge due to trapped charges in the substrate
near the island [14, 15]. For small capacitances (∼fF), the act of adding one electron
to the island significantly changes its charging energy. Other electrons cannot tunnel
onto the island unless they have enough energy to overcome the potential barrier.
To keep a constant number of electrons on the island, thermal fluctuations must be
frozen out, and we require [11, 14]:
Ec >> kB T
(2.2)
as the first condition for SET operation. This condition is called Coulomb blockade
[11, 14].
The condition in Equation 2.2 automatically places requirements on the operating
temperature for a given device capacitance and size. The limits of electron beam
lithography allow typical islands about (100 nm × 1 µm) to be fabricated [16], with
tunnel junctions of Al:Alx Oy :Al being on the order of (100 × 100 nm). Values of CΣ
usually equal about 1 fF, which allows Ec /kB to be about 1 K. It is clear the SET
must be cooled well below liquid He-4 temperatures for any Coulomb blockade to
be observable [11]. This can be achieved using either a He-3 cryostat or a dilution
refrigerator, which are capable of reaching temperatures of 300 mK and below 50 mK
respectively [17].
A second condition for SET operation is that quantum fluctuations must remain
small enough that charges do not tunnel on or off the island while having less energy
13
CHAPTER 2. BACKGROUND AND THEORY
than Ec . These events are allowed by the finite quantum mechanical probability of
an electron tunnelling through a thin energy barrier. Following the approximation
found in Ferry and Goodnick [15], the energy-time uncertainty relationship:
∆E∆t > h
(2.3)
can be used to derive the second condition. The time for an electron to tunnel through
the lowest Ri tunnel junction is given by the circuit time constant, t ∼
= Ri CΣ , and
∆E ∼
= e2 /CΣ . This gives:
(e2 /CΣ )(Ri CΣ ) > h
(2.4)
Ideally the tunnel resistance of each junction should be:
Ri >>
h ∼
= 26 kΩ = RK
e2
(2.5)
where RK is known as the resistance quantum [11, 14]. Having each Ri > RK means
the tunnel barrier is sufficiently large to keep charge well localized on the island.
2.1.1
Electrostatics in the SET
Using electrostatics, we can find the values of Vds and Vg where tunnelling is allowed
[14,15]. On the other hand, using the basis of “orthodox” theory we can calculate the
actual values of the tunnelling current I(Vds , Vg ) for a given set of SET capacitances
at a finite temperature T [18–20].
Figure 2.2 presents a detailed electrostatic diagram of the SET. Each tunnel junction is represented as a capacitor of value Ci = C1,2 , such that charges qi = Ci Vi
build up on either side of the junctions. Due to the gate having capacitance Cg , using
Kirchoff’s Law on the right loop of Figure 2.2 shows that:
qg = Cg (Vg − V2 )
(2.6)
14
CHAPTER 2. BACKGROUND AND THEORY
where V2 is the voltage drop across C2 . Since qg is a polarization charge, it does not
need to be an integer number of electrons. The total capacitance of the island is
given by CΣ = C1 + C2 + Cg + C0 . The total charge on the island, q, is the sum of
the charges on the three halves of the capacitor plates the island is made of, plus the
background charge q0 . However this must be equal to a quantized amount of charge:
q = q2 − q1 − qg + q0 = ne
(2.7)
where n is the integer number of electrons on the island.
q
q1
-q1
q0
q2
-q2
V2
R1, C1
V ds
Cg
R2, C2
Vg
Figure 2.2: The electrostatic circuit diagram of a single electron transistor. The gate is
capacitively coupled through Cg to the island, and the tunnel junctions have resistance Ri
and capacitance Ci [15].
These equations allow us to find the free energy equation of the SET circuit for
a given number of charges n1 and n2 that have tunneled through junctions 1 and 2
respectively. The free energy, F = Es − Ws , is the electrostatic energy, Es , stored
15
CHAPTER 2. BACKGROUND AND THEORY
in the system, minus Ws , the work done by the voltage sources moving the discrete
electrons across the junctions as well as moving polarization charge in response to the
island potential changes. The change in the free energy when the island goes from a
state (n1 → n1 ± 1) is [14, 15]:
∆F1± (n) = F (n1 ± 1, n2 ) − F (n1 , n2 )
i
e he =
± (C2 + Cg )Vds − Cg Vg + ne − q0
CΣ 2
(2.8)
(2.9)
The change in the free energy when the island goes from a state (n2 → n2 ± 1) is:
∆F2± (n) = F (n1 , n2 ± 1) − F (n1 , n2 )
i
e he ± C1 Vds + Cg Vg − ne + q0
=
CΣ 2
(2.10)
(2.11)
The ± superscript denotes the island either gaining (+) or losing (-) an electron. At
T = 0, the only tunnelling events which are allowed are those which lower the overall
±
free energy in a response to changes of Vds or Vg . This means ∆F1,2
< 0 for tunnelling
to occur. This condition is not satisfied for a given number n electrons on the island
when [14, 15, 21]:
1
e n−
< (C2 + Cg ) Vds − Cg Vg − q0 < e n +
2
1
< C1 Vds + Cg Vg + q0 < e n +
e n−
2
1
2
1
2
(2.12)
(2.13)
Figure 2.3 below shows a graph of these inequalities in the Vg Vds -plane for various
integer number of electrons n on the island at T = 0. Here the background charge
on the island is q0 = 0 for simplicity. In the shaded region, tunnelling is suppressed
due to Coulomb blockade. When Vds = 0, each “Coulomb diamond” corresponds to
adding or removing another electron to the island with changes in Vg . Changing the
gate charge to half-integer values of an electron charge allows tunnelling and thus a
current to flow in the SET. The periodicity of of these points ∆Vg = e/Cg determines
CHAPTER 2. BACKGROUND AND THEORY
16
the gate capacitance. The slope of the lines formed from Equations 2.12 and 2.13
equal C1 /Cg and (C2 + Cg )/Cg respectively. A charging diagram such as Figure 2.3
can thus be used to find all SET capacitances and the charging energy.
Figure 2.3: The “Coulomb diamonds” of a normal state SET are shaded to show where
Ids = 0. In the other regions, the drain-source current is non-zero [21].
We can see from Figure 2.3 that for a small non-zero Vds around the half-integer
values of the gate charge, a large change in the drain-source current will occur. At
these areas, the SET can be operated as a highly sensitive electrometer, as changes
in the gate charge of only a fraction of an electron significantly change the output
current [14, 15, 21]. However the electrostatic model cannot predict the values of the
current nor Ri of each junction, as the tunnelling rate of electrons is fundamentally
quantum mechanical in nature.
CHAPTER 2. BACKGROUND AND THEORY
2.1.2
17
Tunnelling in the SET
The “orthodox” theory can be used to predict the current through a SET [18–20]. It
involves treating each oxide junction as a potential barrier which is a small perturbation of the larger metallic systems on either side. Using perturbation theory, the
transition rates of electrons in and out of each junction can be found and thus the
total current as a sum of these rates.
For a single metal-insulator-metal junction as shown in Figure 2.4, the tunnelling
rates of electrons through the insulator as allowed by quantum mechanics can be
found by considering the system Hamiltonian [20, 22]:
H = Hl + Hr + HT
(2.14)
where Hl and Hr are the single free particle Hamiltonians on either side of the barrier
and HT is called the transmission Hamiltonian. It is written as:
X
X
HT =
Tkl ,kr c†kr ckl +
Tkr ,kl c†kl ckr
kl ,kr
(2.15)
kl ,kr
The first term in the transmission Hamiltonian shows the process of annihilating a
particle of wavevector kl on the left side of the barrier and creating one of wavevector
kr on the right. Tkl ,kr is the tunnelling matrix element for the particular initial and
final k-states. The Hermitian conjugate in the second term gives tunnelling in the
opposite direction [20, 22].
If the barrier itself is thin, then it can be treated as a perturbation of both Hl
and Hr , so that first order perturbation theory applies. The tunnelling rate Γ for an
electron across the barrier reduces to [15, 20]:
Γ(∆F ) =
−∆F
− e∆F/kb T )
e2 RT (1
(2.16)
where ∆F is the change in the free energy due to tunnelling. The tunnelling resistance
18
CHAPTER 2. BACKGROUND AND THEORY
Figure 2.4: Energy diagram of an electron tunnelling through a thin barrier of width d
under a drain-source bias V . Here Hl and Hr are the left and right free particle Hamiltonians, while the shaded areas represent the electron states up to the left and right Fermi
levels [15].
RT is given by [15, 20]:
RT =
h̄
2πe2 |Tkl ,kr |2 Dl Dr
(2.17)
where Dl,r are the density of states on the left and right sides of the barrier.
It can be shown that for a system of N tunnel junctions in series, Equation 2.16
is valid for the individual junction tunnelling rates associated with changes in the
system free energy due to tunnelling through that particular junction [15, 20]. For a
two junction SET, ∆F → ∆Fi± (n), which are given by Equations 2.9 and 2.11 where
i = 1, 2 specifies the junction. The tunnelling rate through the i-th junction is [15,20]:
Γ±
i (n) =
−∆Fi± (n)
±
e2 Ri (1 − e∆Fi
(n)/kb T )
(2.18)
where Ri is the measured tunnel resistance of the i-th junction.
To calculate the IV characteristics of a SET requires knowing the current I as
a function of the circuit voltages and the charge state n of the island. Previously,
CHAPTER 2. BACKGROUND AND THEORY
19
it was shown the change in the system free energy associated with tunnelling events
at particular voltages allowed the tunnelling rates to be determined. However the
probability Pn of finding the island in a charge state n can change over time by one
electron charge tunnelling on or off the island. The rate of change of Pn is given by
the so-called “master equation” [15]. For stationary (average) values of the current,
we require dPn /dt = 0, and I is written as [20]:
∞
X
−
Ids = e
Pn [Γ+
1 (n) − Γ1 (n)]
(2.19)
+
Pn [Γ−
2 (n) − Γ2 (n)]
(2.20)
= e
n=−∞
∞
X
n=−∞
By calculating the tunnelling rates for particular Vds , Vg , and n using Equation
2.18, and then finding the relevant Pn values, the total current Ids through the SET
can be calculated. A calculated plot of Ids (Vds , Vg ) for a normal state SET is given in
Figure 2.5. We see that in Figure 2.5(b) for Vds = 0.1 mV, a small change in the gate
charge ng leads to a large change in the drain-source current. This is the expected
behaviour for the SET to act as a charge amplifier.
2.1.3
The Superconducting SET (SSET)
When aluminum is used as the electrodes and island in a SET, and the temperature
is reduced to below T = 1.3 K, the device is called a superconducting single electron
transistor [24]. Many additional charging effects can been seen in its IV-curve due to
superconductivity. Understanding why they occur requires a brief discussion on the
nature of superconductivity as it applies to charge tunnelling.
Superconductivity is the phenomenon of a material having perfect conductivity
(zero resistance) when it is cooled below a material specific transition temperature,
CHAPTER 2. BACKGROUND AND THEORY
20
Figure 2.5: The simulated IV-curve of a SET where R1 = R2 = 50 kΩ, C1 = C2 = 0.5 fF
and T = 100 mK. In (a) drain-source current vs. drain-source voltage for various ng , and
in (b) drain-source current vs. ng for Vds = 0.1 mV [23].
Tc . As described by the Bardeen-Cooper-Schrieffer (BCS) theory, when a Type I
superconductor is cooled below its Tc the electrons condense into a state of Bosonic
particles called Cooper pairs [22]. Each Cooper pair is made up of two electrons,
which have equal and opposite momentum and spin. The formation of the Cooper
pairs is due to an attractive electron-phonon interaction within the crystal lattice
of the material, and the Cooper pairs are bound by a certain energy specified by a
quantity called the energy gap ∆(T ) [22]. At T = 0, the amount of energy required
to break Cooper pairs back into individual electrons (called quasi-particles) is [22]:
Eg (0) = 2∆(0) = 3.528kb Tc
(2.21)
Below Tc , the Cooper pairs in a superconductor are the majority charge carriers,
which propagate with zero resistance and give rise to the supercurrent.
When two superconductors are separated by a thin insulating barrier, the resulting
structure is called a Josephson junction, and its energy-band diagram is pictured in
CHAPTER 2. BACKGROUND AND THEORY
21
Figure 2.6 [22, 25]. For a superconductor using the BCS theory, at T = 0, quasiparticles completely occupy a valence band and an empty conduction band is above
it, each separated from the Fermi energy by ∆. In order for quasi-particles to tunnel
from one side of the junction to the other, unoccupied quasi-particle states must
be available at the same or lower energy than the tunnelling electrons. As such, a
bias voltage Vds = 2∆/e is required to start quasi-particle tunnelling, which is also
called Giaever tunnelling [24, 25]. At E = ∆, where the density of states goes to
infinity [22] and there are many free states for electrons to tunnel into, there is a
large discontinuous jump in the current Ids at eV = 2∆.
a)
b)
Conduction bands
eV
2
2
Valence Bands
Figure 2.6: The energy-band diagram of a Josephson junction. The density of states goes
to infinity in the valence and conduction bands at a value of E = ∆ from the Fermi energy.
(a) Quasi-particle tunnelling can occur at T = 0 if the applied potential satisfies eV > 2∆.
(b) Cooper pair tunnelling is possible when the Fermi energies (dashed lines) of the two
superconductors are equal [25].
The other type of tunnelling which occurs in a single Josephson junction is called
CHAPTER 2. BACKGROUND AND THEORY
22
Cooper pair tunnelling [22, 26]. This type of tunnelling occurs at Vds = 0, and is a
result of the phase difference in the superconducting wave functions of each electrode.
Cooper pair tunnelling can be thought of as a pair of electrons moving from the Fermi
energy of one superconducting electrode across the barrier to the Fermi energy of the
other electrode.
A SSET can be thought of as two low capacitance and highly resistive Josephson
junctions in series. The IV-curve for a SSET displays additional charging effects
due to quasi-particle and Cooper pair tunnelling [24, 26–28]. An example curve is
given in Figure 2.7. For instance, quasi-particle current does not begin to flow until
Vds = 4∆/e, where the voltage drop is 2∆/e over each junction. For Vds > 4∆/e,
Coulomb blockade for electrons is still relevant so long as Ec >> kb T , so that “gaprise”
of quasi-particle current is modulated by the gate voltage to be between 4∆/e < Vds <
(4∆+2Ec )/e [21]. When a system is coupled to changes in the gate charge qg = Cg Vg ,
gaprise is a very sensitive point to bias a SSET at for detecting the changes, in that
its gain or modulated dI/dqg is large.
For Vds < 4∆/e, no pure quasi-particle current can flow due to the gap in the
superconducting density of states. Nevertheless in this region a number of non-zero
current peaks are evident. One such peak is due to the Josephson quasi-particle
(JQP) cycle, in which it is energetically favourable for a Cooper pair to tunnel off (or
on) the island through one junction followed by successive quasi-particle tunnelling
on (or off) the island through the other junction [26–28]. With an island initially in
the n = 0 state, the corresponding transitions would be n = 0 → 2 → 1 → 0.
CHAPTER 2. BACKGROUND AND THEORY
23
Figure 2.7: The IV-curve of a superconducting SET shows a number of charging features.
In this device, Ec /e > 2∆/3e, so that DJQP peaks are visible [29]. Using electrostatic
arguments similar to that in Section 2.1.1, the values for threshold and peak JQP and
DJQP effects can be derived as given in the figure [25].
Another cycle called the double Josephson quasi-particle (DJQP) or 3e cycle involves a two Cooper pairs and two quasi-particles tunnelling through opposite junctions sequentially [24,27]. With an island initially in the n = 0 state, the corresponding transitions would be n = 0 → 2 → 1 → −1 → 0. As seen in Figure 2.7, both JQP
and DJQP peaks have large gain dI/dqg , however these peaks are used less often in
sensing schemes than the gaprise because large gate coupling capacitances can make
Ec too small [21].
CHAPTER 2. BACKGROUND AND THEORY
2.1.4
24
Applications of the SET
After Fulton and Dolan created the first SET in 1987, the SET was put to good use in
fundamental and applied studies. For instance, Lafarge in 1991 used a SET coupled
to a single electron box to clearly show charge quantization [30]. A single electron
box is simply a metallic island coupled to one side of a voltage source through a
tunnel junction capacitance Cj and to the other side through a gate capacitor Cg .
The charging energy of the box is given by Ec = e/2CΣ where CΣ = Cg + Cj . The
total electrostatic energy E = Ec (n − ng )2 of the single electron box is governed
by the number of electrons on the island n and the continuous applied gate charge
ng = Cg Vg /e. For increasing gate charge ng , the box will seek the lowest energy state
and therefore single electrons will tunnel on and off the island over a range of gate
charge values [25, 30].
Figure 2.8: The single electron box coupled to a single-electron transistor [30]. (a)
Schematic diagram of the experiment, where a single-electron box with two tunnel junctions is capactively coupled to the SET through Cc . (b) Measured average charge hni on
the box versus applied gate charge ng = Q̃g /e. The increasing values of θ = (0.01, 0.1, 10)
correspond to increasing temperature T , which are set by the capacitance values in Ref. [30].
As seen in Figure 2.8(a) above, by capacitively coupling a single electron box to
CHAPTER 2. BACKGROUND AND THEORY
25
the island of a SET through Cc , and measuring the current through the transistor,
changes in the average charge hni on the island were sensitively measured. Figure
2.8(b) shows the Coulomb staircase achieved for the measurement. It should be noted
that at T > 0, the average charge on the box hni is not discretely quantized and is
described by a Boltzmann weighted average of charge states including a temperature
parameter [30]. However, the true number of electrons n on the box is quantized [25].
Thus when ng = 1/2 and hni = 1/2 was measured, the true charge on the box was
rapidly changing between n = 0 and n = 1. The tunnelling rate Γn=0→1 was on the
order of 500 MHz at T = 150 mK, which was too high to measure with the limited
bandwidth of the SET [25].
A SET was also used by Knobel and Cleland in 2003 to measure the displacement
of a nanomechanical beam [9]. As shown in Figure 2.9, a SET was capacitively
coupled to a GaAs doubly-clamped beam having f0 = 116.7 MHz, which was driven
in-plane using the magnetomotive technique [9].
To detect the motion of the beam, the SET was used as an electrical mixer [31].
The mixer technique involves applying a DC voltage Vbeam to a wire on the beam
which acts as a SET gate, and then applying a small driving voltage at frequency
f0 to the same wire. As the beam moves, the charge on the SET island changes by
∆q = Vbeam ∆C, where ∆C is the change in the capacitance between the island and
the beam. A local oscillator voltage VLO (fLO ) is applied to a second gate. Due to the
non-linear current response of the SET to changes in gate voltage, the device acts as a
mixer such that the output current Ids (fIF ) at intermediate frequency fIF = |fLO −f0 |
gives a measurement of the displacement amplitude ∆x of the beam [31].
As shown in Figure 2.10, for a driving signal power of P = −125 dBm with a
CHAPTER 2. BACKGROUND AND THEORY
26
Figure 2.9: (a) A 3 µm long × 250 nm wide × 200 nm thick GaAs doubly clamped
beam is capacitively coupled to a SET. The scale bar in the lower right corner is 1 µm.
(b) Schematic diagram of the circuit used for the measurement [9]. The measurement of
Ids (fIF ) at intermediate frequency fIF = |fLO − f0 | gives a measurement of the beam
displacement ∆x.
beam of mechanical Q-factor Q ∼
= 1700, the maximum amplitude of the beam was
measured to be ∆x = 2.3 × 10−14 m. This is on the order of the diameter of an
atomic nucleus [32]. The displacement sensitivity of the measurement was found to
√
√
be Sx = 2.0 × 10−15 m/ Hz, about a factor of only 100 larger than ∆xQL for this
device [9].
SETs have also previously been used to investigate simultaneous tunnelling of
Cooper pairs and photons into a 2DEG microstrip [33], and have been placed inside
XOR gates for logic circuits [34].
CHAPTER 2. BACKGROUND AND THEORY
27
Figure 2.10: (a) The SET mixer output voltage (Vds = Ids R) versus driving frequency of
the beam gives a resonant frequency of f0 = 116.7 MHz. (b) Mixer output versus frequency
changes rapidly as the SET gate voltage is moved off the sensitive bias point. This is due
to interference with background signals, as the phase of the mixer output varies with the
gate voltage [9].
2.2
The Radio Frequency SET (RF-SET)
The RF-SET is a single electron transistor designed to operate at much higher frequencies than a SET and was first demonstrated by Robert Schoelkopf et al. in
1998 [35]. There are a number of reasons why operating at higher frequency is advantageous.
First, for a fabricated SET it is well known that stray capacitance to ground, CS ,
due to the bond pads of the device and nearby grounded metal parts can be anywhere
CHAPTER 2. BACKGROUND AND THEORY
28
from 0.1 to 5 nF [36, 37]. For a total SET resistance of RP = R1 + R2 = 50 kΩ,
a cutoff frequency of f = 1/2πRP CS = 500 − 30000 Hz appears at the start of an
output roll-off due to the RC-filter formed by the device. Thus if the gate charge on
the SET changes at high frequency (i.e. greater than 1 MHz), the change in the DC
current Ids will be attenuated severely. This means a standard SET cannot typically
be utilized for fast measurements, such as the detection of vibrating nanomechanical
beams with resonant frequencies from 0.2 − 1 GHz.
Second, 1/f noise caused by the motion of trapped background charges in the
substrate near a real device lowers the measured charge sensitivity of a SET operating
at low frequencies [38]. As will be discussed more extensively later, the sensitivity
can be improved by operating the SET in the MHz to GHz frequency range.
2.2.1
Theory of Operation
An RF-SET is created by putting an inductor in series with a single electron transistor. The inductance, L, is chosen to create a tank circuit with CS in order to transform
the impedance of the SET to as close to 50 Ω as possible [35–37]. Since the dynamic
resistance of the SET, RS (qg ) =
∂Ids
|
,
∂Vds q=qg
is highly sensitive to the gate charge, the
impedance of the RLC circuit is also. By sending out a low power RF-signal, and
measuring the reflected portion of it, the charge on the island can be measured in
a non-dissipative manner. Further, if qg is modified by a modulated system such as
a vibrating beam or charge trap, it will cause RS , and thus the reflected signal to
change.
Figure 2.11 shows a schematic diagram of the inductance L, stray capacitance CS ,
SET dynamic resistance RS and RF-cable input impedance Z0 = 50 Ω in an RF-SET.
CHAPTER 2. BACKGROUND AND THEORY
29
Figure 2.11: Schematic circuit of an RF-SET. The SET dynamic resistance RS (qg ) =
∂Ids
∂Vds |q=qg is a strong function of the gate charge qg .
The impedance “looking in” from the RF-cable, where we assume the bias tee has a
50 Ω impedance, is the combination of the RS , L, and CS components. The input
impedance is therefore [36, 39]:
Zin (ω) = iωL +
RS [1 − iωRS CS ]
1 + ω 2 RS2 CS2
(2.22)
where ω is the angular frequency of an incoming signal. On resonance, ω = ω0 and
the imaginary part of Zin goes to zero. This gives the resonant frequency as:
s
1
1
ω0 =
− 2 2
(2.23)
LCS
RS CS
r
p
1
∼
, when RS ≫ L/C
(2.24)
=
LCS
where since RS = RP is a minimum of tens of kΩ, the approximation always holds
true. On resonance, Zin can be expressed as:
Zin (ω0 ) =
L
RS CS
(2.25)
30
CHAPTER 2. BACKGROUND AND THEORY
The reflection coefficient, Γ(ω), defines the ratio of the reflected RF-signal voltage
to the input voltage at any frequency ω. The quantity Γ(ω) is a complex number
defined as [36, 39]:
Γ(ω) =
Zin (ω) − Z0
Vout
=
Zin (ω) + Z0
Vin
(2.26)
On resonance, changes in the the SET resistance RS due to changes in Vds and
Vg can be measured by collecting the magnitude and phase of Γ(ω0 ). This allows
the entire Γ − qg curve to be mapped out. For instance when a normal state SET
is biased such that it is Coulomb blockaded, RS ∼
= ∞ and |Γ| ∼
= 1, while above the
blockade RS = RP and |Γ| < 1. This is shown in Figure 2.12 below, where a SET
has been biased in both the Coulomb blockade and normal regions of the IV-curve.
On the y-axis, S11 = 20 log |Γ| is the reflection coefficient in logarithmic units [39].
The signal has been amplified (S11 > 0 dB), however we observe maximum reflection
in the Coulomb blockade and minimum reflection above blockade.
The depth of modulation,
∂Γ
,
∂RS (qg )
when there is no Coulomb blockade is a max-
imum when the RLC circuit of the RF-SET is impedance matched to the RF-cable
impedance. At resonance, when the input impedance equals the 50 Ω line impedance,
we require [36, 39]:
L
= Z0
RS CS
(2.27)
and thus
ZLC =
p
RS Z0
where ZLC =
r
1
L
= ω0 L =
CS
ω0 CS
(2.28)
One can use the Q-factor, which gives the ratio of the stored to dissipated electromagnetic energy in the circuit, to simplify Equation 2.28 and get the circuit bandwidth.
CHAPTER 2. BACKGROUND AND THEORY
31
Figure 2.12: Amplified, reflected signal for an RF-SET with inductance L = 660 nH,
ω0 = 354 MHz and RΣ = 454 kΩ. When the SET is Coulomb blockaded S11 is a maximum,
while above the blockade S11 is a minimum. The variation in S11 at other frequencies is due
to cabling standing waves which have not been zeroed in the network analyzer calibration
[36].
The unloaded-Q of an RLC circuit when not attached to an input impedance is defined as [40]:
Q=
ω0
R
=
∆ω
ω0 L
(2.29)
where ∆ω is the bandwidth, which is the difference in frequency between the fullwidth at half-maximum points of |Γ|2 . Once connected to the 50 Ω input impedance,
the loaded-Q of the circuit is the given by [40]:
1
1
1
=
+
QL
Qe Q
(2.30)
CHAPTER 2. BACKGROUND AND THEORY
32
where the external-Q factor is [40]:
ω0 L
Z0
Qe =
(2.31)
When L and CS are chosen for a perfect match to 50 Ω, Equation 2.31 can be rewritten
as:
Q2e =
RS
Z0
(2.32)
which shows the tank circuit acts to transform the SET impedance downwards. The
bandwidth ∆f in Hz of the circuit and the whether it is perfectly matched or not, is
given using the loaded-Q [40]:
∆f =
ω0
2πQL
(2.33)
With perfect matching, 2QL = Qe , and we have:
∆f =
ω0
πQe
(2.34)
From Equation 2.33, the bandwidth is proportional to the resonant frequency and
inversely proportional to the loaded-Q. Thus to impedance match to larger device
impedances, there is a decrease in the bandwidth that can be offset in principle by
operating at higher resonant frequency. Maintaining good bandwidth is important as
any modulation of qg at a frequency of ωm << ω0 will modulate the reflected signal
such that if fm > ∆f /2, the changes in reflection will be severely attenuated.
2.2.2
Noise
Noise in a system can be defined as random fluctuations in an output signal with
time [39, 40]. The quantity of noise in a system determines the smallest signal level
that can be detected with instrumentation such as an amplifier. Since the SET is an
amplifier, in order to determine the smallest quantity of charge one can measure, we
CHAPTER 2. BACKGROUND AND THEORY
33
must consider its noise characteristics.
In electrical circuits there are at least three different types of noise due to different
physical processes. These are the Johnson-Nyquist or thermal noise (caused by random thermal motions of electrons), shot noise (caused by the discreet nature of charge
carriers), and 1/f noise [39, 40]. Single electron transistors are highly affected by 1/f
noise, with the largest contribution due to trapped charges in the substrate moving
about near the junctions, metal island, and electrodes [38]. It is therefore advantageous to move away from low frequency measurements and improve the signal-to-noise
ratio by using the RF-SET.
2.2.3
Noise and Sensitivity in the SET and RF-SET
Intrinsic noise in the SET and RF-SET has been computed theoretically in the literature [41–44].
The intrinsic noise in a DC-SET operating in the normal state has been described
by Korotkov [41]. This model uses the orthodox theory, in the limits where kb T ≪ Ec
and the frequency range of interest is above all the 1/f noise (∼ 10 kHz), but below
the SET tunnelling rate (RP CP )−1 ∼ (1 − 100) GHz. There are two distinct
contributions to the noise spectral density: (1) the shot noise due to the tunnelling
current Ids , and (2) the fluctuations in the island potential, φ. Since the current Ids is
a stochastic (probabilistic) process, counting statistics apply and lead to shot noise,
while the island potential changes by an amount e/CP each time a single electron
tunnels on or off the island [41]. The potential fluctuations lead to what is known
as the SET backaction; fluctuations in the island potential φ cause changes in the
system coupled to the island which is being measured [41].
34
CHAPTER 2. BACKGROUND AND THEORY
In the limits given near the Coulomb-blockade threshold, the shot noise spectral
density, SII (ω), can be approximated as [41]:
SII (ω) ∼
= ηI 2eIds
(2.35)
Γ21 + Γ22
(Γ1 + Γ2 )2
(2.36)
where:
ηI =
The factor ηI is like a Fano factor in that it accounts for correlations between electron
tunnelling events using the tunnelling rates Γ1,2 of each junction. It follows the noise
in the detected gate charge, Sqg (ω), is related to the noise in the drain-source current
by [41]:
∂Ids
Sqg (ω) = SII (ω)
∂qg
−2
(2.37)
Similarly, the potential noise spectral density, Sφφ (ω), can be approximated by [41]:
e2
Sφφ (ω) ∼
= ηφ P
C
(2.38)
4Γ1 Γ2
(Γ1 + Γ2 )2
(2.39)
where:
ηφ =
The factor ηI similarly accounts for correlations in the stepwise change in island
potential φ. The quantities SII (ω) and Sφφ (ω) can be correlated so that another
noise source SIφ (ω) can describe these correlations, however in practice these can
be minimized by choosing a correct bias point for a given temperature and source
impedance [29].
Korotkov was able to show that the intrinsic SET sensitivity due to classical noise
√
should be about 1.2 × 10−5 e/ Hz [41], while other authors have pegged the best
√
DC-SET sensitivity to a slightly higher value of ∼ 1.7 × 10−6 e/ Hz [11, 42]. In each
35
CHAPTER 2. BACKGROUND AND THEORY
case, the authors have noted the 1/f noise due to charge traps in the substrate of real
devices causes the most major decrease in experimental sensitivity from theoretical
values [11, 41, 42].
Noise and sensitivity in the RF-SET have been modeled theoretically and studied experimentally [43, 45–47]. The groundwork for the shot noise limited RF-SET
sensitivity was laid by Korotkov and Paalanen [43], where using the orthodox theory
in the frequency limit ω ≪ Ids /e they showed the optimized charge sensitivity of the
RF-SET is only 1.4 times worse than a conventional SET. Numerical analysis of RFSET operation later showed that the best response ∂Γ/∂qg and sensitivity are found
when zero DC bias is used or the SET is DC biased within the Coulomb blockade
regime [45]. From the theory, predicted RF-SET sensitivities for a typical device are
√
on the order of ∼ 1 − 2 × 10−6 e/ Hz [45]. However, in real experiments, the normal
√
state RF-SET sensitivity can vary between ∼ 9 − 10 × 10−6 e/ Hz [35, 46], while the
√
superconducting RF-SET sensitivity ranges between ∼ 3 − 6 × 10−6 e/ Hz [46, 47],
with the decrease from the predicted range due primarily to preamplifier noise.
In the scope of this work, I wish to show that to maximize the RF-SET sensitivity
in a circuit dominated by preamplifier noise, the charge gain ∂Γ/∂qg should be maximized while maintaining the desired sensing bandwidth. Using the analysis presented
by Roschier [44], we can show that for an RF-SET with an incoming RF carrier signal
v0 cos ω0 t, the reflected signal coming back from the impedance matching circuit is:
vr (t) = v0 Γ cos ω0 t + n(t)
(2.40)
where Γ = Γ(ω0 ) is the reflection coefficient given by Equation 2.26, and n(t) is the
noise voltage component of the reflected signal. If a time-dependent voltage is applied
to the gate such that the reflection coefficient is modulated sinusoidally by the gate
CHAPTER 2. BACKGROUND AND THEORY
36
charge, the reflected signal can be written [44]:
vr (t) = v0 (Γ0 + ∆Γ cos ωm t) cos ω0 t + n(t)
(2.41)
where ∆Γ cos ωm t is the change in the reflected signal at the operating point Γ0 due
to the change in the gate charge. It is assumed the modulation frequency ωm is small
enough so that ωm ≪ ω0 , and we see that the reflected signal is amplitude modulated.
After homodyne detection, the signal has an RMS amplitude of [44]:
∆Γ
vs,RM S = v0 √
8
(2.42)
It can be shown that if the noise source n(t) is taken as a white noise source, the
√
√
noise power spectrum density, SN , in units of [V/ Hz] can be written as [44]:
r
p
kB T0 Z0
SN =
(2.43)
2
where T0 is the equivalent noise temperature of the measurement circuit. The signalto-noise ratio (S/N) is therefore:
2
2
vS,RM
v0 ∆Γ
S
S ∼ = 2
= √
N
vN,RM S
4kB T0 Z0 ∆B (2.44)
such that ∆B is the measurement bandwidth. The sensitivity is defined when S/N =
1, and by using:
∆Γ =
we find the sensitivity
p
√
2∆ΓRM S =
∂|Γ| √
2δqRM S
∂q
√
Sq in units of [e/ Hz] is:
√
p
2kB T0 Z0
δqRM S
Sq = √
=
e ∆B
ev0 ∂|Γ|
∂q
(2.45)
(2.46)
For a RF-SET circuit dominated by the noise temperature T0 , the best sensitivity is found by maximizing v0 ∂|Γ|
. The value of v0 cannot be made too large as the
∂q
RF-signal will begin to probe part of the SET IV-curve that does not change with qg .
37
CHAPTER 2. BACKGROUND AND THEORY
The value
∂|Γ|
∂q
will change drastically depending on Vds , whether the SET is super-
conducting or not, and the temperature. Thus to get the best sensitivity, Equation
2.46 states that we must maximize the change in signal
2.2.4
∂|Γ|
.
∂q
Applications of the RF-SET
Figure 2.13: (a) Circuit schematic of the single electron trap. (b) The energy band diagram of the single electron trap for the four lowest energy states in (nIsl , nT rap ). Changing
the applied trap charge to ngT = 0.5 allows an equal probability for charge to tunnel on
and off the trap island. (c) One of two allowed single electron pathways to get onto the
trap island [36].
The RF-SET has been applied as a wide bandwidth sensor for numerous measurements since 1998. For instance, it was used by Julie Love in 2007 for detection of
telegraph noise in charge traps similar to a single electron box [36]. As given in Figure
2.13, the single electron trap consists of an intermediate island containing a charge
nIsl and a trap island containing a charge nT rap . Each island has its own gate voltage
for bias control, and the energy level diagram for a system with ECIsl /kB = 2.5 K
and ECT rap /kB = 0.7 K is shown. To add one single electron to the trap island, the
CHAPTER 2. BACKGROUND AND THEORY
38
two pathways for charge state changes are:
(nIsl , nT rap ) = (0, 0) → (1, 0) → (0, 1)
(2.47)
(nIsl , nT rap ) = (0, 0) → (−1, 1) → (0, 1)
(2.48)
and:
At an applied trap gate charge of ngT = CgT VgT /e = 1/2, these pathways cause
the electron to tunnel into higher energy bands before reaching the next lowest energy
state. Because of this, the transition rates at T = 50 mK are Γn=0↔1 ∼
= 35 kHz, which
are significantly slower than in the single electron box [36].
Figure 2.14: “Telegraph noise” in the single electron trap. At the degeneracy point of
ngT = 0.5, Γn=0→1 = Γn=1→0 and the trap island spends about the same amount of time
in the nT rap = 0 state as the nT rap = 1 state. The blue curve is the collected data, while
the green curve is an “on-off” fit [36].
To observe charge tunnelling in real time (charge dynamics), the SET reflected
power was first calibrated against the number of electrons nT rap on the trap island.
By setting the trap gate to ngT = 0.5 and taking a fast time-domain measurement
of the reflected power, real-time tunnelling events of single electrons on and off the
island were able to be measured and are given in Figure 2.14. These measurements,
as quoted by Love, were the very first “time-resolved measurements of non-thermal
CHAPTER 2. BACKGROUND AND THEORY
39
tunnelling in a metallic system or quantum dot” [36].
RF-SETs coupled to nanomechanical beams for displacement detection have also
been documented in the literature [29, 48]. In one example, LaHaye et al. used an
RF-SET capacitively coupled to a SiN doubly-clamped beam to measure its thermal
motion without driving it [48]. An SEM image of the device and schematic diagram
are given in Figure 2.15.
Figure 2.15: (a) SEM image of the tank circuit coupled to the RF-SET. The inductor
(inset) is a spiral on-chip type, made of 100 nm thick aluminum with L = 46 nH. The stray
capacitance is small enough that an aluminum inter-digitated capacitor having C = 290 fF
is required for a tank resonant frequency of f0 = 1.37 GHz. (b) Close up of the SET, SET
gate and 8 µm long SiN beam. (c) Schematic circuit of the measurement. VN R applied to
the resonator couples its thermal motion to the SET as a gate [48].
The goal of the experiment was to measure the thermal motion of a nanomechanical beam capacitively coupled to the SET, using the bias voltage VN R applied to the
CHAPTER 2. BACKGROUND AND THEORY
40
gold electrode on the beam. The detection scheme is carried out by monitoring the
reflected RF-signal from the SET impedance given by:
vr (t) = Γv0 (t)
(2.49)
where v0 (t) = v0 cos ω0 t. A change in the reflected signal ∆vr (t) due to a change in
the reflection coefficient given by ∆Γ(t) happens over time as the beam moves closer
and father away from the SET. The change is specified as [29]:
∆vr (t) = ∆Γ(t)vc (t)
(2.50)
It can be shown for a resonator having an average fundamental mode displacement of
ym (t), the change in the reflected signal coming back from the SET is approximately
[29]:
∆vr (t) ∼
=
CN R VN R
−2Q2e Z0 b
dN R
∂ 2 Ids
ym (t)v0 (t)
∂Vds ∂Qg
(2.51)
where CN R is the nanoresonator coupling capacitance, dN R is the static distance from
the beam to the SET island, and b is a numerically calculated, geometrical capacitance
coefficient of order unity [29].
LaHaye was able to use the high-gain sensing point near a DJQP peak to conduct
the nanoresonator measurements. For the f0 = 19.65 MHz beam, ym (t) = 50 pmRMS
was measured at the dilution fridge base temperature of T = 35 mK [48]. The best
√
√
displacement sensitivity achieved was Sx = 3.8 × 10−15 m/ Hz, which is the same
order of magnitude as that of Knobel et al. in Ref. [9]. However, this measurement
is only a factor of 4.3 from ∆xQL [48].
The RF-SET has also previously been used to observe single electron tunnelling
in a quantum dot [49], and to cool a vibrating nanomechanical beam [50]. The later
experiment produced a measurement of ∆x = 3.9∆xQL which to date remains the
CHAPTER 2. BACKGROUND AND THEORY
41
closest approach to ∆xQL achieved [50].
2.3
Two-Dimensional Electron Gases (2DEG)
The experiment proposal in Chapter 1 uses a nanomechanical beam which has an
embedded layer known as a 2-dimensional electron gas. In order to understand the
motion detection scheme of this beam, an understanding of the phenomena present
in the 2DEG is required. This section presents a brief overview of what a 2DEG is,
how they are created and characterized, and finally a review of some previous work
using suspended beams with integrated 2DEG layers.
2.3.1
Inversion Layers and Doping
A two-dimensional electron gas is a thin layer in a material where electrons are confined in a two-dimensional plane [51]. A common example of this effect happens inside
a metal-oxide-semiconductor field effect transistor (MOSFET). As shown in Figure
2.16, a doped semiconductor is covered by a thin layer of oxide followed by a thin
layer of metal which acts as a gate. If the semiconductor is p-type, and a positive
potential is applied to the gate metal, electrons are attracted to the gate and create a
negatively charged depletion region in the semiconductor. With more electrons near
the oxide the surface is more n-type and the Fermi level, EF , is closer to the conduction band. By using Poisson’s equation, it is easy to show that having a net charge
density in the surface region causes the energy bands to bend downward towards the
metal EF [32].
With a large applied gate potential, the energy bands can bend so much that the
CHAPTER 2. BACKGROUND AND THEORY
42
Figure 2.16: (a) A standard MOSFET. Using a positive potential on the aluminum gate,
an inversion layer of electrons is formed at the interface of the p-type silicon and the
gate insulating silicon dioxide. The n-type source and drain regions provide the charge
carriers [51]. (b) The energy band diagram under positive gate bias. The electric field on
the gate causes band bending in the p-type silicon, such that for large enough bias the
conduction band will fall below EF . Electrons will fill the hole states first, and then start
filling the conduction band [51].
conduction band will lie below the Fermi level over a very small depth in the p-type
semiconductor. If a voltage is applied along the plane in which the electrons are
sitting, there is no energy gap and the electrons can gain kinetic energy, such that the
material will be conductive and a current will flow. This region is known as a 2DEG
because the potential energy barriers on either side keep the electrons in a very thin,
nearly two-dimensional quantum well [51].
Using a 2DEG like that found in a MOSFET can be problematic for devices which
need high conductivity because ionized dopants in the 2DEG layer scatter the charge
carriers significantly [15]. As well, the oxide is not typically a single crystal material
and thus lattice dislocations and other defects can trap and scatter charge carriers.
CHAPTER 2. BACKGROUND AND THEORY
43
One class of materials that overcomes these problems is the AlGaAs/GaAs heterostructure. A heterostructure is a multilayered, epitaxially grown semiconductor [52]. As shown in Figure 2.18 below, the materials GaAs and AlAs are closely
latticed matched materials at a ∼
= 5.65 Å and can be grown on top of one another as
single crystals with minimum defects [52]. Moreover, GaAs can be alloyed with Al to
produce a continuous variation in the material composition such that Alx Ga1−x As,
in the range of x = 0 → 1, has a corresponding change in the band gap from 1.4
to 2.2 eV. Using the crystal growing method of molecular beam epitaxy (MBE), the
changes in materials of different x values can be made in essentially one atomic layer
with minimum defects [52].
Figure 2.17: Energy gap (band gap) versus lattice constant of several common binary
semiconductors [52].
When two semiconductors of different band gaps are placed in contact, electrons
in the conduction band of the larger band gap material will tend to seek lower energy
CHAPTER 2. BACKGROUND AND THEORY
44
states in the smaller band gap material [52]. If the larger gap material is highly
n-doped, a large number of electrons will fall into the conduction band of the other
material near the surface leaving behind ionized donors in a spatially separate region.
EF in both materials at thermal equilibrium is the same, and band bending at the
junction for high enough doping concentration allows the now n-type surface of the
small band gap material to dip below EF . As shown in Figure 2.18, a 2DEG can be
observed to form at the surface of the smaller gap material. The technique of doping
only one side of a heterostructure is known as “modulation doping” [52].
Figure 2.18: The modulation doping technique used in a GaAs/AlGaAs heterostructure
produces a 2DEG at the interface. Electrons from the donor atoms occupy hole states in
the GaAs, and then begin to fill the conduction band, forming the 2DEG. The AlGaAs is
heavily n-doped, and the undoped GaAs is weakly p-type [52].
In a typical AlGaAs/GaAs heterostructure, AlGaAs is doped using very thin
“donor layers” of Si which are typically only one atomic layer thick. This thin layer
is referred to as “delta doping” and is situated about 100 Å away from the undoped
CHAPTER 2. BACKGROUND AND THEORY
45
GaAs layer. In this way the ionized donor atoms are spatially separated from the
2DEG region by “buffer layers” of intrinsic AlGaAs. Since the AlGaAs is crystalline,
the density of defects is low, and the scattering processes due to ionized impurities
near the carriers and defects near the carriers are minimized [51, 52].
2.3.2
Quantum Hall Effect Characterization
A 2DEG will show what is known as the quantum Hall effect under applied magnetic
fields. This effect was first presented in the literature by von Klitzing, Dorda and
Pepper in 1980 [53], and has since become a standard method for determining the
mobility, µ, and carrier concentration, nc , of 2DEG samples. The quantum Hall effect
will be briefly discussed, with emphasis on why it occurs and its experimental usage.
Quantization of 2DEG Energy
Figure 2.19: The GaAs/AlGaAs heterostructure of Blick et al. is designed so that the
2DEG sits inside an approximately square well potential. Illumination changes the band
gap by ionizing more donor atoms [54].
As shown in Figure 2.19 above, if an AlGaAs/GaAs heterostructure is created
CHAPTER 2. BACKGROUND AND THEORY
46
such that its layers are mirror images about the GaAs layer, the band bending in the
materials produces an rough finite square well potential in the intrinsic GaAs region.
From quantum mechanics, the electron wavefunctions in the 2DEG must satisfy the
time-independent Schrödinger equation with a potential term V (r) = V (z), where
the z-direction is perpendicular to the plane of the 2DEG [15]. The solution leads to
the electron energy eigenvalues of [52]:
h̄2 2
E=
k + En
2m∗xy xy
(2.52)
where kxy is the wavevector component and m∗xy is the effective mass in the 2DEG
plane. The first term in Equation 2.52 states that in plane the electrons behave as
free particles, while the second term En is a spectrum of discrete values due to the
electrons being in a quantum well. If the quantum well is very deep, the En are
approximately the eigenvalues of a particle in an infinite square well potential [52]:
h̄2 π 2 n2
2m∗z dz
En =
n = 1, 2, 3, ...
(2.53)
where m∗z is the effective mass component perpendicular to 2DEG plane, and dz is
the width of the quantum well. One can obtain the density of states in k-space for
these 2D electrons the same way it can be done for 3D electrons. The number of
states dN in a ring of thickness dk and radius k is given by [52]:
dN = 2 ×
2πk dk
(2π)2
(2.54)
To take into account spin degeneracy of the electrons, an extra factor of 2 was used
in Equation 2.54. By using dE = h̄2 k dk/m∗xy , the density of states per unit area is:
D=
dN
m∗
=
= Const.
dE
πh̄2
(2.55)
The parabolic bands in k-space are called sub-bands, and as shown in Figure 2.20,
the density of states is a step function due to the addition of the constant values as
CHAPTER 2. BACKGROUND AND THEORY
47
n increases [52].
Figure 2.20: (a) The energy sub-bands of a 2DEG are parabolic in shape versus the wavevector kx , where ky = 0, in the plane of free propagation perpendicular to the ẑ direction.
(b) The density of states is a step function reflecting the quantization of sub-bands in the
2DEG [52].
Classical Hall Effect
In a standard Hall effect measurement a shown in Figure 2.21 below, a bar of material
is placed in a magnetic field Bz perpendicular to the direction of a current density
Jx = Ix /(wd) = −enc vx flowing through it. Here e is the unit charge of the electron,
nc is the carrier concentration, vx is the drift velocity of the carriers, Ix is the current
and W and d are the width and thickness of the bar respectively.
The charge carriers feel a Lorentz force of magnitude |Fm | = evx Bz toward the
sides of the bar, and the building charge produces an electric force |Fe | = eEy in the
opposite direction. When the two forces balance, the equilibrium voltage measured
between the sides of the bar is called the Hall voltage, VH , and the quantity known
CHAPTER 2. BACKGROUND AND THEORY
48
Figure 2.21: A typical Hall bar measurement. The magnetic field of magnitude Bz points
perpendicular to the surface of the Hall bar, while a drain-source current density Jx flows
in the longitudinal direction. The measured Hall voltage, VH , and longitudinal voltage, VL ,
are referenced for the quantum Hall effect as well [52].
as the Hall resistance takes the value [32]:
RH =
VH
Bz
=−
Ix
enc d
(2.56)
where the sign in Equation 2.56 simply accounts for the polarity of the Hall voltage.
For a two-dimensional current we define the current density Jx = Ix /w, so the
classical Hall resistance of a 2DEG takes the value:
RH =
VH
Bz
=−
Ix
enc
(2.57)
Quantum Hall Effect
The quantum Hall effect (QHE), for which von Klitzing won the 1985 Nobel prize [53],
is a phenomenon in which the Hall resistance is dependent only on fundamental
constants at certain values of the magnetic field. While the single particle energies in
Equation 2.52 are valid for zero magnetic field, in a non-zero field the electrons take
CHAPTER 2. BACKGROUND AND THEORY
49
up cyclotron orbits around the direction of the magnetic field. The orbital frequency
or cyclotron frequency ωc is [51, 52]:
ωc =
−eB
m∗xy
(2.58)
Circular orbits such as angular momentum or cyclotron motion can be decomposed
into simple harmonic motion, and therefore the energy eigenvalues of the single electrons are given by [51, 52]:
1
E = (j + )h̄ωc + En
2
j = 0, 1, 2, ...
(2.59)
where only a term involving the spin splitting of the energies at very high field has
been omitted. Instead of parabolas, the energy is split into singular points called
Landau levels [51, 52]. The density of states is no longer constant but consists of
δ-function-like peaks separated by h̄ωc in energy. Since no single particle states are
lost, the number of states per unit area in each Landau level is NL = h̄ωc D0 = eB/h,
where a factor of 2 has been removed due to the lifting of spin degeneracy by the
magnetic field. The Landau levels are pictured in Figure 2.22 below.
At very low temperatures (T ∼
= 0), electrons in a 2DEG all lie in the single particle
states below EF and each Landau level contains exactly NL electrons. The Landau
levels are not perfect δ-functions and are broadened by phonon scattering, finite
sample size, and crystal defects [15,51,52]. With an increase in the magnetic field Bz ,
the number of states available in each level increases linearly and the energy splitting
h̄ωc moves each level to a higher energy. As a Landau level crosses EF , it is emptied
of electrons which fall into the lower energy empty states of the next lowest Landau
level. This has an effect on the conductivity of the 2DEG as conductivity is related
to the ability of electrons to take up the smallest amounts of energy by moving into
higher energy states under an electric field. When the middle of a Landau level crosses
50
CHAPTER 2. BACKGROUND AND THEORY
D(E)
hωc
E
Figure 2.22: The Landau levels in a 2DEG. The dashed line is the density of states in
the first sub-band of the 2DEG for B = 0. The peaks each contain NL = eB/h states for a
non-zero B-field [52].
the Fermi level, the conductivity is a maximum as there are many states immediately
available for electrons to scatter into, while when the Fermi level is between Landau
levels the conductivity is a minimum [15, 51, 52].
These oscillations in the conductivity are called Shubnikov-de Haas (SdH) oscillations and are an indication of the existence of a 2DEG in a medium [15, 51, 52]. The
conductivity is the inverse of the resistivity tensor in two dimensions σ = 1/ρ, such
that the longitudinal resistivity is [15, 51]:
ρxx =
σxx
2
+ σxy
2
σxx
(2.60)
When the longitudinal conductivity σxx = 0, the longitudinal resistivity ρxx = 0,
provided σxy does not vanish. Since σxy ∼ RH , the changes in the conductivity at
high magnetic field are mirrored in the changes in the resistivity.
The 2DEG electron density which carries the current can only change in multiples
of the degeneracy NL of a Landau level, and this leads to quantized Hall resistance.
51
CHAPTER 2. BACKGROUND AND THEORY
In Figure 2.23 below, typical magnetoresistance measurements are shown. With a
source current Ix moving through the sample, the Hall resistance measured in the
y-direction is given by Equation 2.57. At values of the magnetic field where nc charge
carriers per unit area completely fill iNL states per unit area, the Hall resistance is
given by [51, 52]:
|RH | =
B
h1
= 2
iNL e
e i
i = 1, 2, 3, ...
(2.61)
Figure 2.23: Magnetoresistance measurements after illumination of an GaAs/AlGaAs
heterostructure show SdH oscillations in the longitudinal resistance and quantization in the
Hall resistance, revealing the presence of a 2DEG [55].
Magnetoresistance measurements can be used to experimentally determine the
carrier concentration nc and mobility µ of a 2DEG. From Equation 2.57, if the current
and Hall voltage are measured for a 2DEG while sweeping the magnetic field, a plot
of IBz versus eVH for low B-field will give the classical carrier concentration nc as
the slope. At higher magnetic fields when the gap between Landau levels is large and
the Fermi level sits between them, the spacing between the levels is periodic in
1
B
and
CHAPTER 2. BACKGROUND AND THEORY
thus SdH oscillations show similar periodicity such that [15, 51]:
1
ge
∆
=
B
hnc
52
(2.62)
The factor of g = 2 in Equation 2.62 is the spin degeneracy factor, and thus must be
replaced by the factor 1 when spin-splitting of SdH oscillations occurs at very high
field [15, 51].
The mobility µ of the 2DEG is a quantity which describes how easily the charge
carriers can move under the influence an electric field with zero applied magnetic field.
Qualitatively, lower mobilities signal there are more defects and charge trapping states
in the 2DEG layer (it is “dirty”), and the larger the magnetic field required for SdH
oscillations to begin showing [15, 51]. The mobility at zero magnetic field is given by:
µ=
1
ρxx nc e
(2.63)
where the longitudinal resistivity is given by:
ρxx =
w VL
l Ix
(2.64)
In Equation 2.64, VL is the longitudinal voltage drop as displayed in Figure 2.21, l is
the length between the longitudinal contacts, and w is the width of the Hall bar. It
may be noticed that, in two dimensions, resistivity has units of Ω which is the same as
that of resistance, so that to distinguish between them the units of “ohms-per-square”
are used [51].
2.3.3
Suspended 2DEGs and Previous Work
In NEMS devices, a 2DEG can act as a built-in electrode and therefore several mechanical devices have incorporated them into designs. Of particular interest are devices which are made of the GaAs/AlGaAs heterostructure, as both of these materials
CHAPTER 2. BACKGROUND AND THEORY
53
display piezoelectric properties. Put simply, piezoelectricity is the phenomenon where
applying stress to a material generates an electric field, and conversely applying an
electric field generates stress [56]. Appling stress moves electric dipoles in the material, and the resulting bound surface and volume charges generate the field. The
coupled equations which describe piezoelectricity are [56]:
S = sE=0 T + dT E
(2.65)
D = dT + ǫT =0 E
(2.66)
In the first equation, S is the strain matrix , sE=0 is the compliance matrix at zero
applied electric field, T is the stress matrix, and E is the electric field vector. In the
second equation, D is the charge density displacement vector and ǫT =0 is the electric
permittivity matrix at zero applied stress [56].
The history of using GaAs and AlGaAs as piezoelectric materials can be traced
to about the early 1990’s, when Frike, Soderkvist and Hjort published a series of
papers detailing the basic piezoelectric properties of these materials [56–58]. These
publications include derivations of the components of the piezoelectric tensor d for
GaAs wafers oriented in the (e1 , e2 , e3 ) = ([01̄1], [01̄1̄], [100]) direction [56], and show
that for a thin beam oriented along e1 an electric field along e3 will produce elongation
of the beam in e1 and e2 directions [57]. If the field is not uniform over the thickness
of the beam, the stress will couple to out-of-plane flexing of the beam. In Figure 2.24,
by choosing the correct placement of metal electrodes on the sides of the beam, the
authors were able to demonstrate in-plane tuning forks with resonant frequencies of
32.8 kHz and mechanical Q-factors of (1 − 2.7) × 104 in air at room temperature and
atmospheric pressure [58].
CHAPTER 2. BACKGROUND AND THEORY
54
Figure 2.24: A GaAs tuning fork with integrated metal electrodes for in-plane flexural
vibrations [58].
Using GaAs/AlGaAs heterostructures with embedded 2DEGs for mechanical devices has been utilized since about 1998, when Beck et al. developed self-sensing AFM
cantilevers based on previous work of field-effect transistor (FET) strain sensors (see
Figure 2.25) [59]. Their heterostructure consists of GaAs/Al0.3 Ga0.7 As mirrored layers which contain a 2DEG generated by Si δ-doping, and on top of the cantilever rests
an integrated FET which senses the surface charge developed on cantilever bending.
At T = 2.2 K, the sensing scheme allowed for imaging a mica sheet with FET gate
√
√
p
p
voltage noise of SVg = 1 µV/ Hz and charge noise Sq = 0.1 e/ Hz at 100 Hz.
√
√
This gave a displacement noise of Sx = 10 Å/ Hz [59].
In 1998, Blick and Roukes created a 100 nm thick GaAs/AlGaAs Hall bar shape
and suspended it by etching a 400 nm thick Al0.8 Ga0.2 As sacrificial layer grown underneath it using hydrofluoric acid (HF) wet etchant [54]. At T = 4.2 K, the unsuspended 15 nm wide GaAs 2DEG layer was found to have a carrier concentration
of 2.64 × 1011 cm−2 and a mobility of 5.1 × 105 cm2 /V·s. A similar device was later
used to characterize the transport of charge in 2DEGs and quantum dots [60]. However, the investigators found that when suspended, the 2DEG only had a mobility of
(1.4 − 1.6) × 104 cm2 /V·s, which they attribute to damage from the reactive ion etch
CHAPTER 2. BACKGROUND AND THEORY
55
Figure 2.25: (a) SEM image of an GaAs/AlGaAs cantilever, having dimensions 65 × 12 ×
0.25 µm. (b) A smaller cantilever with an integrated strain-sensing FET and electron-beam
deposited AFM tip. (c) A reference FET near the small cantilever can be used for differential
strain measurement. (d) Schematic diagram of the heterostructure in the cantilever [59].
step used to create the Hall bar shape and non-uniform surface tension along the device [60]. The suspended device and its longitudinal magnetoresistance measurements
are given in Figure 2.26 below.
The first true nanomechanical GaAs/AlGaAs resonators were created by Hohberger et al. [61] as well as Cleland et al. [62] in 2002. In Hohberger et al. [61],
the heterostructure is nearly identical to that found in Blick et al. [54]. Magnetoresistance measurements of the suspended heterostructure gave a carrier concentration
of 3.19 × 1011 cm−2 and mobility of 5.7 × 104 cm2 /V·s. The overall goal of the project
was to couple NEMs with 2DEGs to quantum dots, and thereby sense the motion capacitively [61]. In the next iteration, a fully realized GaAs/AlGaAs suspended beam
was driven using the magnetomotive technique [63]. Figure 2.27 below shows that by
CHAPTER 2. BACKGROUND AND THEORY
56
Figure 2.26: (a) SEM image a 2 × 1 µm GaAs/AlGaAs Hall bar. (b) Magnetoresistance
measurements after illumination show SdH oscillations in the longitudinal resistance [60].
monitoring the reflected signal of a network analyzer, the resonant frequency of the
beam was determined for various incident microwave powers. A displacement sensi√
√
tivity of Sx = 2.9 × 10−12 m/ Hz was obtained using the changes in the quantum
dot IV-curves with beam motion [63].
By magnetomotively driving a similar beam with a surface mounted gold wire,
Cleland et al. used the 2DEG as an electrode to sense the beam motion piezoelectrically [62]. As in Figure 2.27, the motion of the beam was set out of plane and
therefore modulated the potential of a quantum point contact (QPC) on top of the
beam. The current through the quantum point contact is modulated by the outof-plane piezoelectric fields which develop due to the out-of-plane stress similar to a
top gate, and therefore a lock-in measurement of the current will reflect the resonant
frequency of the beam. The device and data collected for the motion are shown in
Figure 2.28 below. At the resonant frequencies studied, Cleland et al. obtained a
√
√
displacement sensitivity of ∼ Sx = 3 × 10−12 m/ Hz [62].
CHAPTER 2. BACKGROUND AND THEORY
57
Figure 2.27: (a) SEM image of a doubly-clamped heterostructure beam coupled to quantum dots. (b) Using magnetomotive drive, the resonant frequency of the beam was determined for various incidence drive power [63]
2.4
A New Experiment
A proposal by Robert Knobel and Andrew Cleland put forward in 2002 would see
an RF-SET used to measure the displacement of a piezoelectric beam made from a
GaAs/AlGaAs heterostructure [7]. The mechanism of sensing is displayed schematically in Figure 2.29. A large island electrode of the RF-SET would be positioned
over one end of the top side of the beam, while another electrode connected to a RF
driving signal would be positioned at the other. Both the RF-SET carrier signal and
beam driving signal would use contacts to the embedded 2DEG as a ground plane.
By applying the time varying voltage to the drive electrode along ẑ, the piezoelectric
CHAPTER 2. BACKGROUND AND THEORY
58
Figure 2.28: (a)SEM image of the QPC on top of the piezoelectric beam. Here (1) is the
wire providing the driving Lorentz force, (2) and (5) are source-drain ohmic contacts for
the 2DEG, (3) and (4) are QPCs patterned on top of the beam. (b) Circuit diagram of the
experiment. A magnetic field B moves the beam out-of-plane at a frequency determined by
the local oscialltor (LO). Current through the QPC is lock-in measured at the intermediate
frequency (IF). (c) The current response of the QPC is a maximum at the mechanical
resonance frequency of the beam [62].
beam will have induced strain in the xy-plane and this will couple to out-of-plane
vibrations of the beam [7]. At the RF-SET island, an induced charge due to the
reverse piezoelectric effect will build up on the island, which can be sensed by the
SET.
By solving the coupled electrical and mechanical equations for the response of the
beam, the effect of the detection circuit on the motion of the beam was studied [7].
The modelled piezoelectric beam was chosen to have a resonant frequency of ωm /2π =
1 GHz, so that h̄ω ≥ kB T for T ∼
= 10 mK, which is achievable in a dilution refrigerator.
The displacement noise SX (ω) as a function of the resonant frequency of the beam is:
R
B
SX (ω) = SX
(ω) + SX
(ω)
(2.67)
R
where SX
(ω) is the readout noise of the RF-SET due to the tunnelling current and
B
SX
(ω) is the backaction noise of potential fluctuations on the SET island causing the
beam to move. For SET junction parameters of R1 = R2 = 100 kΩ, C1 = C2 = 0.15
CHAPTER 2. BACKGROUND AND THEORY
59
Figure 2.29: (a) The proposed heterostructure of GaAs/AlGaAs includes a 1 µm sacrificial
layer for HF etching and suspension of the beam. (b) Schematic diagram of the circuit used
for the RF-SET coupled measurement of the nanomechanical beam. The RF-SET is drawn
schematically in the top left corner [7].
fF, and gate capacitances (to the beam and a biasing electrode) of Cg1 = Cg2 = 65 aF a
√
√
displacement sensitivity of ∼ Sx = (1 − 5) × 10−17 m/ Hz was found. This is about
one order of magnitude better than the best and most recent capacitively coupled
nanomechanical beam measurement [50]. Depending on the mechanical Q-factor of
the beam, it could lead to a very close approach to ∆xQL and is the experimental
goal for which the current thesis has been carried out.
Chapter 3
Modelling
In this project, steps were made towards fabricating and testing the first RF-SETs
in our lab. In order to come up with the RF-circuit required for measuring high
frequency piezoelectric beams as outlined in Section 2.4, I have used the theory to
determine expected device bandwidth and depth of modulation. An initial prediction
of the required circuit is made in the first half of this chapter. In the second half, I
discuss how the conduction band energy of the designed heterostructure is predicted
using a one dimensional Poisson-Schrödinger solver.
3.1
Design of the RF-SET
To measure the flexing of a piezoelectric beam with an RF-SET, there are at least four
experimental goals which must be individually reached: (1) Demonstrate the ability
to fabricate a working RF-SET, (2) develop a heterostructure with an embedded
2DEG that can be suspended, (3) fabricate a suspended heterostructure beam with
a high enough mechanical resonant frequency fm = ωm /2π such that h̄ωm > kB T
60
61
CHAPTER 3. MODELLING
at cryogenic temperatures, and (4) fabricate the RF-SET on the heterostructure and
couple it to the beam for measurement.
Single electron transistors have previously been fabricated in our research group
[64], and therefore goal (1) has been partially reached. We can use the DC-SET
parameters as obtained by Greg Dubejsky as a starting point for what one would
expect of a SET that our current fabrication facilities can produce. The parameters
of two SETs cooled to a refrigerator temperature of T = 300 mK are given in Table
3.1 [64].
Table 3.1: Experimentally obtained SET values from Greg’s Thesis.
Device
GregA
GregB
RΣ (kΩ)
432 ± 14
170 ± 8
CΣ (aF)
479 ± 46
670 ± 47
Cg (aF)
13.48 ± 0.01
13.8 ± 0.5
Ec
e
(µV)
170 ± 10
119 ± 4
∆
e
(µV)
350 ± 10
360 ± 10
The RΣ values in Table 3.1 are large enough such that RΣ > 2RK as in Equation
2.5 and quantum fluctuations are highly suppressed.
We can examine how the reflection coefficient and sensitivity of a RF-SET depends on RS . From Equations 2.25 and 2.26, on resonance the change in the voltage
reflection coefficient with change in transistor resistance,
∂|Γ|
2Z0 LCS
=
∂RS
(L + Z0 RS CS )2
Using partial derivatives, the value
∂|Γ|
∂qg
∂|Γ|
,
∂RS
takes the form:
(3.1)
takes the form:
∂|Γ|
∂|Γ| ∂RS
=
∂qg
∂RS ∂qg
(3.2)
Thus in Equation 3.1 for increasing values of RS the derivative becomes less negative
and closer to zero, while in Equation 2.26 the reflection coefficient becomes closer
to negative one. The changes in the reflected signal with qg are not as great if we
62
CHAPTER 3. MODELLING
have a large RS to begin with, and therefore we lose sensitivity as explicitly given
in Equations 3.2 and 2.46. To get the best sensitivity, we should try to make the
normal state resistance RS = RΣ as low as possible without going below RS ∼
= 2RK ,
as this is the smallest magnitude resistance that can be obtained with the SET IVcurve. In principle the partial derivative
∂RS
∂qg
depends on where one biases the SET,
the operating temperature and weather the device is superconducting or not, and
can be maximized by trial and error, or advanced knowledge of the SET and system
characteristics [44].
We can now state some RF-SET parameters required for a sensitive detector,
based on our previous fabrication and other past results. The normal state resistance
RΣ should be made as small as possible, with a lower bound of about RΣ = 50 kΩ. A
minimum stray capacitance CS = 0.2 − 0.5 pF should be expected when using surface
mounted inductors wirebonded to the SET bond pads [35,36,65,66]. When an on-chip
spiral inductor is used, the stray capacitance of the inductor alone is usually between
10-90 fF [67, 68], and better matching can occur. In the literature, experimentalists
have typically fabricated interdigitated on-chip capacitors with these inductors to
achieve the desired resonant frequency, and to date total stray capacitances of CS =
0.19 − 0.34 pF have been achieved [23, 29, 37]. Finally, the inductance L should be
chosen such that the resonant frequency of the RLC circuit will be f0 = 1.25 ±
0.25 GHz. This f0 is within the bandwidth of our QuinStar model QCA-L1 HEMT
amplifier [69], which will be used to amplify the reflected RF-SET signal at low
temperature.
CHAPTER 3. MODELLING
3.2
63
Determining the values of L and C
In order to maximize the change in reflected signal and maintain good bandwidth, one
must carefully consider how to pick the inductor for the tank circuit. Experimental
limitations on CS and f0 prevent perfect matching most of the time. It is well known
that CS is mostly due to the bond pads and wirebonds to the SET, as well as grounded
circuit board traces and mounting parts [35–37]. CS is also a strong function of
temperature, as the electrical properties of the substrate the SET is fabricated on
will change upon cooling down. It is usually necessary to cool down a test device
below the “freeze out” point of the substrate charge carriers, which for Si is about 40
K [32], in order to estimate CS and change L to get the correct resonant frequency.
As well, if the inductor is poorly chosen relative to CS , f0 could be too high or too
low to be in the bandwidth of the HEMT amplifier [23, 36].
Choosing an inductor based on an expected SET resistance was done theoretically
√
and experimentally. To find the best possible impedance match (ZLC = RΣ Z0 ), we
used the method outlined in Ref. [23]: First, we found the external-Q using Equation
2.32 which gives the best match for the SET resistance. Next we found L from
Equation 2.31 and CS from the desired f0 = 1.25 GHz. The bandwidth, ∆f , was
determined by using Equations 2.30 and 2.33.
In a real experiment (see Chapter 6), one has to tweak these values based on
the minimum stray capacitance measured at low temperature. As given in Ref. [36]:
one can begin to minimize CS by removing as much grounded metal near the SET
as possible. We can then estimate the real stray capacitance, and choose L using
Equation 2.23 to get the closest impedance match for the center frequency of the
HEMT amplifier. The ∆f and the minimum reflection can be estimated using these
64
CHAPTER 3. MODELLING
values, and once the SET is cold can be measured using a network analyzer. A new
value for the inductor can be chosen if ∆f and f0 are unfavourable for the experiment.
This type of search was performed on a fabricated SET as discussed in Chapters 5
and 6.
3.3
Simulations of RF-SET Response
3.3.1
Perfect Matching
Here we consider the case of a RF-SET having normal state RΣ = 50 kΩ, which we
want to have a resonant frequency of f0 = 1.25 GHz. Table 3.2 gives the calculated
parameters of the external-Q factor, inductance, stray capacitance, loaded-Q factor,
bandwidth and -0.2 dB bandwith (∆f−0.2 dB ) expected for perfect impedance matching to Z0 = 50 Ω. ∆f0.2 dB denotes the bandwidth over which the change in S11 from
RS = 50 to 1000 kΩ is larger than 0.2 dB, which we arbitrarily set as an expected
detection limit in the system due to noise.
Since CS = 0.2−0.5 pF >> C1,2 in these experiments, we neglect the effect of SET
junction capacitances in the circuit [35, 36, 64]. Using Matlab [70] and the equations
developed in Section 2.2, the reflection coefficient S11 = 20 log |Γ| is modelled over
the range of 1-1.5 GHz for RS = 50 and 1000 kΩ and is shown in Figure 3.1.
Table 3.2: Perfect matching of a RF-SET to 50 Ω at f0 = 1.25 GHz.
Qe
31.6
L (nH)
201.3
CS (fF)
80.5
ZLC (Ω)
1581
QL
15.8
∆f (MHz)
79.1
∆f−0.2 dB (MHz)
396
The large change in the reflected signal when RS changes by hundreds of kΩ is the
65
CHAPTER 3. MODELLING
0
-5
-10
0
S11 [dB]
S11 [dB]
-15
-20
-0.5
-25
-1
-30
1
1.25
f [GHz]
1.5
-35
-40
-45
RS = 50 kΩ
RS = 1000 kΩ
1
1.05
1.1
1.15
1.2
1.25
f [GHz]
1.3
1.35
1.4
1.45
1.5
Figure 3.1: Reflection coefficient for a perfectly matched RF-SET at f0 = 1.25 GHz. The
inset shows that at RS = 1000 kΩ, more of the signal is reflected.
fast sensing technique used to determine RS (qg ) =
∂Ids
|
∂Vsd q=qg
and the charge state of
the SET [35]. It should be noted in Figure 3.1 that on resonance for perfect matching
S11 → −∞ since |Γ| = 0, which has been cut off due to the point spacing of δf = 1
MHz in the MATLAB plot.
The ∆f /2 = 39.6 MHz denotes the half-width at half-maximum point of |Γ|2 in
Figure 3.1 at RΣ = 50 kΩ. ∆f−0.2 dB /2 = 198 MHz is a more practical measure of
the half-bandwidth. It denotes a point where we can no longer detect fast changes
in the modulation of |Γ| by RS (qg ), at sidebands f = fc ± fm where fm is the charge
modulation frequency.
66
CHAPTER 3. MODELLING
3.3.2
Realistic Matching
While perfect impedance matching gives large gain and good bandwidth, it is rarely
possible to obtain such a good match. This is because realistically the value of the
stray capacitance is about CS = 0.3 pF when a surface-mounted inductor is used in
the circuit [35, 36, 65, 66]. To reach goal (1) of making a working RF-SET, we have
used Coilcraft surface-mounted inductors in our first circuits [71]. For this estimated
stray capacitance value, the calculated inductance required to have f0 = 1.25 GHz is
L = 54.0 nH. The associated circuit parameters are given in Table 3.3 for RΣ = 50
kΩ. The value of S11 is modelled over the range of 1-1.5 GHz for RS = 50 and 1000
kΩ and is shown in Figure 3.2.
Table 3.3: Good matching of a RF-SET to 50 Ω at f0 = 1.25 GHz.
CS (fF)
300
L (nH)
54.0
ZLC (Ω)
424
Zin (Ω)
3.6
Qe
8.5
QL
7.9
∆f (MHz)
157.8
∆f−0.2 dB (MHz)
374
In this case, the calculated value ∆f /2 = 78.9 MHz is double that of the perfectly
matched SET. While its practical bandwidth ∆f−0.2 dB /2 = 187 MHz is only slightly
worse than the perfectly matched SET, there is large reduction in the minimum
reflected signal S11 at the lowest SET resistance. The sensitivity is reduced, and to
visualize the effect a plot of
∂|Γ|
∂RS
is given in Figure 3.1. We see that at RS = 50 kΩ,
the derivative of the realistically matched SET is about 4 times less than that of the
perfectly matched SET; the sensitivity is lowered by the same factor.
The sensitivity of a realistically matched RF-SET can be estimated using the
analysis found in Roschier et al [44]. The maximum sensitivity is given by Equation
2.46, however we must make some slight modifications. To begin, the actual peak
67
CHAPTER 3. MODELLING
0
-0.2
0
-0.4
S11 [dB]
S11 [dB]
-0.02
-0.6
-0.04
-0.06
-0.8
-0.08
-1
1
1.25
f [GHz]
1.5
-1.2
RS = 50 kΩ
RS = 1000 kΩ
-1.4
1
1.05
1.1
1.15
1.2
1.25
f [GHz]
1.3
1.35
1.4
1.45
1.5
Figure 3.2: Reflection coefficient for a realistically matched RF-SET at f0 = 1.25 GHz.
The inset shows that at RS = 1000 kΩ, more of the signal is reflected.
voltage over the SET, which we call vSET , is not the same as the peak voltage v0
of the incoming high-frequency signal due to power dissipation in the LC section of
the matching circuit. The power of the incoming wave is given by v02 /2Z0, while the
2
power dissipated in the SET is given by vSET
/2RS , and we let the coefficient K be
the ratio of the dissipated power to the incoming power. We obtain [44]:
r
Z0
v0 = vSET
RS K
such that Equation 2.46 can be rewritten as:
√
p
2kB T0 RS K
Sq =
evSET ∂|Γ|
∂q
(3.3)
(3.4)
The K factor can be formally determined using an S-matrix representation of the
68
CHAPTER 3. MODELLING
0
0
100
200
300
400
500
RS [kΩ]
600
700
800
900
0.5
d|Γ|
dRS
|Γ|
|Γ| Perfect
|Γ| Realistic
d|Γ|/dR Perfect
d|Γ|/dR Realistic
0.5
[S]
1×10−5
1
0
1000
Figure 3.3: Resonant change in the reflection coefficient with SET resistance at f0 = 1.25
GHz.
LC-impedance transformer, and the equations for this representation are explicitly
stated in Roschier for practical use [44]. The K factor used below has been calculated
based on the source and load impedances, which are 50 Ω and RS respectively, as well
as the CS and L values expected. Parasitic resistance and capacitance of the inductor
L have been accounted for in the S-matrix using parameters specified for Coilcraft
model 0402CS surface mount inductors [71].
If we were to take the previously mentioned sample GregA and use it at the heart
of a realistically matched RF-SET experiment, we can now estimate the minimum
69
CHAPTER 3. MODELLING
(best) sensitivity expected for a modulated gate charge. For our purposes, we take
the SET as being in its normal state and DC biased at the center of its IV-curve. We
can assume a noise temperature of T = 1.5 K, as our HEMT amplifier is heatsinked
to a pumped He-4 1 K pot (see Chapter 5). As shown in Figure 3.4, at the center
of the IV-curve the maximum change in SET conductance is from 450.5 kΩ to 403.2
kΩ, such that ∆R = 47.3 kΩ. Using the approximation given by Roschier [44]:
∂RS
1
= 3.44∆R
∂q
e
(3.5)
Figure 3.4: The conductance in Siemens of sample GregA versus the gate voltage, where
Vsd is -0.95 µV to offset the background charge and hold the SET at the center of its IVcurve [64]. One period in the oscillation of conductance corresponds to a change of charge
by 1e− on the island. Maximum and minimum values of RS are given in the figure.
we obtain ∂RS /∂q = 162.7 kΩ/e = 2.603 × 10−14 Ω/C. The voltage across the
SET should be maximized over the region where the current changes, so we let
vSET = Ec /e = 170 µV. From the previous simulations of the reflection coefficient
with resistance using Equation 3.1, we obtain ∂|Γ|/∂RS = 0.44 µS at the approximate operating point of RS = 426.9 kΩ. This corresponds to the point on the
CHAPTER 3. MODELLING
70
transfer function in Figure 3.4 with the highest slope. Using the S-matrix method,
the calculated K gives a value of 0.0124, such that the final charge sensitivity is
√
p
Sq = 3.9 × 10−5 e/ Hz. For a higher noise temperature T0 = 4.2 K, which would
√
p
be the temperature using a liquid He-4 heatsink, we obtain Sq = 6.5 × 10−5 e/ Hz.
It should be noted that this calculation does not take into account (i) the 1/f noise,
(ii) the shot noise of the SET, and (iii) that the noise temperature can vary significantly with the impedance at the input of HEMT amplifier, which clearly changes
with time as it is the SET impedance [44]. Regarding point (i), if the gate charge is
modulated at a frequency greater than a few MHz, it has been shown the 1/f -noise
essentially falls to the level of SET shot noise [23, 72]. Regarding point (ii), the shot
√
noise limit of the RF-SET has been predicted to be 1.4 × 10−6 e/ Hz [45], so we
can ignore this small contribution in our estimate. Regarding point (iii), LaHaye has
shown that the true circuit noise temperature T0 for a RF-SET experiment conducted
with a helium-4 heatsinked HEMT amplifier is between ∼12-20 K [29], which increases
(worsens) the best sensitivity by an equivalent factor of ∼3.5-4.5. For this reason,
fabrication attempts should aim to minimize the SET resistance, and an operating
point in the superconducting mode could be used to obtain a higher ∂RS /∂q. Nevertheless this analysis shows a charge sensitivity at or better than the DC-SET could
be obtained with a realistically matched RF-SET which we are able to fabricate.
3.4
Heterostructure Design and Simulation
Design of the GaAs/AlGaAs heterostructure was primarily completed by Vincent
Leduc in our lab group prior to my beginning this part of the project. I present an
updated simulation of the conduction band energy with depth, using a 1D-Poisson
CHAPTER 3. MODELLING
71
simulation program.
The heterostructure in the proposal of Ref. [7] requires at least three properties:
(1) it must be able to be suspended as a nanomechanical beam, (2) it must contain a
2DEG which resides in both the suspended and unsuspended parts of the wafer, and
(3) it must be made of piezoelectric materials to couple charge buildup upon motion
to the island of the RF-SET.
Since the GaAs/Alx Ga1−x As heterostructure materials are all piezoelectric, this
type of heterostructure satisfies condition (3). The 2DEG formed in this system has
been shown to exist in all sections of the wafer [59, 60], which satisfies condition (2).
However to suspend a beam made of the layers responsible for producing the 2DEG, a
material must be included underneath them which is sensitive to a particular etching
process. Since hydrofluoric acid (HF) selectively attacks high aluminum percentage
Alx Ga1−x As much more readily that lower percentage alloys [73], a GaAs/Al0.3 Ga0.7As
heterostructure with a Al0.7 Ga0.3 As sacrificial layer was decided upon.
Figure 3.5 shows a schematic cross section of the heterostructure designed. The
substrate is undoped GaAs on which the subsequent materials are deposited using
molecular beam epitaxy (MBE). On top of the substrate is a 400 nm thick Al0.7 Ga0.3As
sacrificial layer. Working from the outside towards the center of the layers which
create the 2DEG, 5 nm GaAs cap layers prevent oxidation to the 30 nm thick undoped
Al0.3 Ga0.7As spacer layers. The δ-doping which provides charge carriers is contained
in single Si monolayers in the middle of 2 nm thick GaAs layers on either side of
the quantum well. The barrier layers are 40 nm thick undoped Al0.3 Ga0.7 As, and the
2DEG is contained in the 20 nm thick GaAs layer. Since there is identical doping
on either side of the 2DEG, a symmetric quantum well would be expected as in the
CHAPTER 3. MODELLING
72
heterostructure of Blick et al. in Figure 2.19 in Section 2.3.2 [54].
Figure 3.5: The cross-section of the designed 2DEG heterostructure. The δ-doped layers
have an equivalent doping level of 2 × 1019 cm−3 .
The conduction band energy in the heterostructure can be calculated using the
materials specified by solving the Poisson and Schrödinger equations self-consistently.
This was done using the 1DPoisson program written by Greg Snider at the University
of Indiana. Exact details of the self-consistent solver can be found in Refs. [74, 75],
but a summary is presented here. The 1DPoisson program solves the one dimensional
Schrödinger equation [74, 75]:
2
1 d
h̄ d
+ V (z) ψk (z) = Ek ψk (z)
−
2 dz m∗ (z) dz
(3.6)
73
CHAPTER 3. MODELLING
and one dimensional Poisson equation [74, 75]:
d
d
q[ND (z) − n(z)]
ǫs (z)
φ(x) = −
dz
dz
ǫ0
(3.7)
simultaneously. In Equation 3.6, ψk (z) is the electron wavefunction as function of
depth z into the heterostructure, m∗ (z) is the effective electron mass, and Ek is the
eigenenergy. In Equation 3.7, ǫs (z) is the dielectric constant, φ(z) is the electrostatic
potential, ND (z) is the ionized donor concentration, and n(z) is the electron concentration. We set the potential energy V (z) to be the conduction band energy, but in a
quantum well of arbitrary shape the potential is set to be V (z) = −qφ(z) + ∆Ec (z),
so that ∆Ec (z) is an offset energy to correct for the “jump” in band offset at material
interfaces [74, 75].
To get the electron concentration in the conduction band, we relate where the
electron wavefunction probability is non-zero to the occupation of each state. The
occupation of each state nk (z) is given by the Fermi-Dirac distribution [74]:
Z
1
m∗ (z) ∞
dE
nk (z) =
2
(E−E
F )/kB T
πh̄
Ek 1 + e
(3.8)
where EF is the Fermi level, and the total density of electrons is given by [74]:
s
X
n(z) =
φ∗k (z)φk (z)nk (z)
(3.9)
k=1
The charge density is the density of ionized donors minus the density of electrons,
such that ρ(z) = q(ND − nz ) in Poisson’s equation (Equation 3.7).
The 1DPoisson program meshes the total thickness of the heterostructure into
segments of varying width, and starting with a trial potential φi (z) and potential
energy Vi (z) solves the for the wavefunctions and eigenenergies using Equation 3.6.
It then calculates the electron density ni using Equations 3.8 and 3.9. When the
next iteration occurs, the computed ni (z) and a given donor concentration ND (z)
CHAPTER 3. MODELLING
74
is used to calculate φi+1 (z) using Equation 3.7. A new potential energy Vi+1 (z) is
obtained, and the process repeats itself until φi+n+1(z) − φi+n (z) < a, where a is an
error criteria [74].
This process does not always converge, and the total number of mesh segments
cannot exceed 1000 in the program. Boundary conditions are required at exposed
surfaces of the heterostructure and at the interface to an infinite substrate. For GaAs
exposed to air as in the top and bottom surfaces of a suspended structure, a Schottky
barrier of 0.6 eV is sufficient to account for the effect of surface states [76]. For an
unsuspended structure, the slope of the conduction band must be zero at the interface
to the bulk GaAs wafer [74].
To simulate the designed heterostructure, the materials, thicknesses, doping type
and concentration for each layer were entered into a ASCII text run file, along with
the mesh size, temperature and boundary conditions. In this work the temperature
used was 1 K, with a mesh size of 4 Å. In the design phase the doping was adjusted so
that a 2DEG having an integrated electron density of ∼ 5×1011 cm−2 was formed [55].
This doping level was found to be 2 × 1019 cm−3 in each δ-doping layer.
Figure 3.6 shows the simulation results for an unsuspended heterostructure, while
Figure 3.7 shows that of a suspended one. The conduction band energy is plotted
versus depth in the heterostructure, and we see a clear region where the conduction
band dips below EF in the undoped GaAs region at the center of the heterostructure.
However in the unsuspended case, a second 2DEG forms at the interface of the GaAs
cap layer with the sacrificial layer. This would not be desirable if the wafer was to be
used “as-is,” due to transport effects between the 2DEGs. Since the end goal was to
create suspended structures, the second 2DEG was not treated with great concern;
75
CHAPTER 3. MODELLING
it has also been previously shown that attempting to get rid of this second 2DEG
with a low-temperature GaAs cap layer led to depletion of the primary 2DEG in the
center [55]. Finally, in the unsuspended heterostructure a sheet electron density of
5.526 ×1011 cm−2 is calculated in the 2DEG layer, and 5.532 ×1011 cm−2 is calculated
for the suspended case.
0.8
0.7
0.6
Ec [eV]
0.5
0.4
0.3
0.2
0.1
0
-0.1
0
100
200
300
z [nm]
400
500
600
Figure 3.6: The conduction band of the unsuspended heterostructure. At 77 nm below
the surface, the conduction band dips below the Fermi level (at Ec = 0 eV), and a 2DEG
forms. A second 2DEG is seen to form at the interface of the heterostructure with the
sacrificial Al0.7 Ga0.3 As layer.
76
CHAPTER 3. MODELLING
0.8
0.7
0.6
Ec [eV]
0.5
0.4
0.3
0.2
0.1
0
-0.1
0
20
40
60
80
100
120
140
160
180
z [nm]
Figure 3.7: The conduction band of the suspended heterostructure. The Fermi level is at
Ec = 0 eV. With the sacrificial layer removed, only the central 2DEG remains.
Chapter 4
Sample Fabrication
In this chapter, I outline the steps performed to fabricate SET samples on silicon
wafers, and Hall bar shapes from the GaAs/AlGaAs wafer. These descriptions cover
the design of patterns for lithography, evaporation of electrode materials, etching of
Hall bar shapes, and annealing of ohmic contacts to the 2DEG. The first section
covers SETs, which were made in one electron beam lithography step. The second
section covers Hall bars, which required two separate lithography steps to deposit
ohmic contact pads, and then to make the etched Hall bar shapes.
Since the fabrication of nanoscale devices is as much of an art as it is a science, in
particular areas I outline problems with certain fabrication steps and provide evidence
of their solutions.
4.1
SET Samples
Many hours in this project were devoted to the fabrication of low resistance SETs to
be used in an RF-SET experiment. This section outlines the steps required to take a
77
CHAPTER 4. SAMPLE FABRICATION
78
bare piece of silicon through our cleanroom fabrication facilities and turn it into a SET
sample. Several optimizations of pattern designs, resist development and deposition
parameters were performed during this work, to attempt to make a RΣ = 50 kΩ SET.
The search for low resistance SETs continues in our group, and as such I present the
techniques we have used in the device fabrication.
4.1.1
Design of EBL Patterns
In order to produce SET tunnel junction with areas of 100 × 100 nm or less, electron
beam lithography was used as the primary pattern transfer tool. This process uses
a finely focused beam of electrons produced inside a scanning electron microscope
(SEM) to pattern the surface of resist-coated substrates [16]. The resists are organic
chemicals which become either more or less sensitive to a developer chemical when
exposed to the electron beam. Using a positive resist, exposed areas are chemically
removed and a positive of the pattern is left on the wafer surface. With a negative
resist, they are preserved and a negative of the pattern remains [16].
To make the patterns for the SET island, leads, and wirebond pads, the commercial software DesignCAD LT was used [77]. This is a CAD program which allows one
to draw boxes, circles and other polygons, and can sort these polygons into layers.
The completed drawing is called a pattern file. Another program used to control
the SEM during writing is used to set the layer doses, which is the charge per unit
area the resist receives. This software is called the Nano Pattern Generation System
(NPGS) [78], and a specified run file for a particular pattern file is used to associate
a dose to a layer. The NPGS run file also sets the magnification and beam current
required to pattern the elements in each layer, as well as other exposure parameters.
79
CHAPTER 4. SAMPLE FABRICATION
The only parameter not capable of being set in NPGS is the proximity effect, which is
the tendency for forward scattered electrons in the resist to expose the area between
two closely patterned features. In this project, proximity effect was compensated for
in the designed pattern dimensions.
The pattern used most frequently in this project for SET fabrication was “rfset15.RF6,” and an image of this pattern is given in Figure 4.1. It is composed
of 4 separate layers, one each for the wirebond pads and large leads, small leads,
SET leads, and SET island. Table 4.1 below gives a summary of the EBL settings
in the run file “RF-SET15.DC2”. Each layer had a separate dose, and in general
higher doses require less patterning time. The magnification used for the smallest
features was set to a power of 10, as these values activate less amplification circuits
in the SEM, leading to less noise. The origin offset, center-to-center spacing and line
spacing are displacement parameters optimized to maintain the alignment of features
when changing the magnification, and to ensure even dosage of exposure points during
scanning.
Table 4.1: Electron beam lithography parameters for pattern “rf-set15.RF6”. Patterning
was performed using a 40 kV accelerating voltage. 1 Feature size. 2 Magnification. 3 Electron
charge dosage. 4 Beam current. 5 Center-to-center spacing. 6 Line spacing. 7 Pattern origin
offset.
Size (µm)1
10-200 µm
1-10 µm
0.2-1 µm
<0.2 µm
M.2
85
1000
1000
1000
µC 3
D. ( cm
2)
750
900
14800
22075
I (pA)4
2100
10
10
10
C-t-C (nm)5
16.16
9.61
4.12
4.12
L. Sp. (nm)6
16.16
9.61
4.12
4.12
Off. (µm)7
-4.8,-6.6
0,0
0,0
0,0
CHAPTER 4. SAMPLE FABRICATION
80
Figure 4.1: The SET pattern “rf-set15.RF6” as made using DesignCAD LT. (a) The bond
pads and large leads for the drain, source, and both gates are shown. (b) The design for
the island and source/drain leads requires single linewrites using the electron beam.
4.1.2
Wafer Preparation
All SET samples were prepared on single-side polished 4” silicon wafers, with a thermally grown 5000 Å oxide layer on the polished surface. These wafers were supplied
by Virginia Semiconductors and had a primary flat in the [100] direction [79]. Before
spin-coating, the wafers were cleaned by rinsing with deionized water, isopropanol
and then blown dry with N2 gas. A dehydration bake on a hot plate at 190◦ C for five
minutes was performed to drive off any adsorbed water vapour.
The wafer was then held on a vacuum chuck inside a Laurel spin-coater, at which
point liquid Lift-Off-Resist 7B (LOR7B) from MicroChem was dropped across the
entire surface of the wafer. This resist was spun at 500 RPM for five seconds to
CHAPTER 4. SAMPLE FABRICATION
81
spread the resist, followed by a spin at 4100 RPM for 40 seconds, which leaves a
nominal film thickness of about 6000 Å as per the LOR7B spin-speed curve [80]. The
wafer was spun down to 3000 RPM for five seconds and then stopped. After removal
of edge bead and thin “whiskers” of resist using edge bead remover (EBR), the wafer
was baked on a hot plate at 190◦C for 10 minutes.
The positive electron beam resist used for pattern transfer was 3% polymethylmethacrylate in anisole (PMMA) supplied by MicroChem. This resist was spun on
top of the LOR7B using a spreading spin of 500 RPM for five seconds, followed by a
spin at 4000 RPM for 45 seconds, which leaves a nominal film thickness of about 1500
Å as per the PMMA spin-speed curve [81]. The wafer was spun down to 3000 RPM
for five seconds and then stopped. Following more edge bead removal, the wafer was
again baked on the hot plate at 180◦ C for 10 minutes.
For the rest of this section, Figure 4.2 provides a reference to the SET fabrication
process.
4.1.3
Exposure and Development
After spincoating, the wafers were cleaved into ∼ 5 × 5 mm2 chips using a diamond
scribe so they could be loaded into the JEOL model JSM 6400 SEM for patterning [82]. A LaB6 filament, which is a thermionic emission source, was used in the
microscope, allowing for a minimum linewidth of about 50-70 nm. Focusing on the
plane of the resist without exposing the entire chip to the electron beam was allowed by dipping a fine metal point into a jar of pure silver dust, and transferring a
small amount of dust to the edge of the resist coated chip. After placing the chip on
the sample stage and loading it into the microscope, focus, astigmatism and wobble
CHAPTER 4. SAMPLE FABRICATION
82
Figure 4.2: The SET fabrication steps performed in this project. (a) The resist bi-layer
was spun on the wafer surface. (b) EBL exposure of the resist was performed in the SEM.
(c) Resist development was performed in the cleanroom. (d) Double-angle evaporation of
aluminum with an oxidation step in between each deposition formed the tunnel junctions.
(e) Lift-off of the bi-layer leaves the deposited SET on the wafer surface.
CHAPTER 4. SAMPLE FABRICATION
83
corrections were adjusted using a gold nanoparticle standard glued the sample stage.
Using a ScanService Model 890 beam blanker [83], the beam was blanked and the
sample was moved using a translation stage to an area where patterning was to take
place. NPGS was run using the pattern settings in Table 4.1 above, and since NPGS
can control the beam blanker automatically, pauses between layers allowed different
magnification and current settings to be adjusted manually at the SEM controls.
When the patterning was finished, the chip was removed from the SEM and brought
to the cleanroom for development.
During development, the two-layer or “bi-layer” stack of PMMA on top of LOR7B
allowed us to create what is known as an undercut profile into the resist. The PMMA
is a positive photoresist, such that when exposed to the electron beam it becomes
soluble in a developer chemical, which in this case is a 1:3 mixture of MIBK:ISO
(methyl isobutyl ketone:isopropanol). On the other hand the LOR7B is unchanged
upon electron beam exposure, and is instead removed by the developer MF-319 [84].
When the exposed PMMA layer is removed, the chip can be emersed in MF-319 for
a different amount of time. This will subsequently dissolve LOR7B outward from
the center of the pattern, but not attack the unexposed PMMA. In this manner, an
undercut pattern such as that shown in Figure 4.2(c) is obtained in the developed
areas.
The development step for the PMMA layer in pattern “RF-SET15.rf6” was a two
minute dip in 1:3-MIBK:ISO, followed by a rinse in pure isopropanol and a N2 dry. If
when examined under an optical microscope the patterns looked well developed, the
LOR7B development step was carried out by performing a six to ten second dip in
MF-319, followed by a rinse in deionized water, isopropanol and a N2 dry. In Figure
CHAPTER 4. SAMPLE FABRICATION
84
4.3(a), a well developed bi-layer with clear undercut into the LOR7B is shown, while
in Figure 4.3(b), a collapsed resist bridge around the island and junction areas is
pictured. Fine tuning of the bake times and temperatures as well as development
times for both resists was carried out over the course of the project, to obtain good
undercut without having the top PMMA film collapse.
Figure 4.3: Optical microscope images of developed SET patterns. (a) Clear undercut of
1 µm into the LOR7B layer is observed after sample development using MF-319. (b) The
same amount of undercut for a sample with a different PMMA bake time has caused the
resist bridge required to make the SET island to collapse.
4.1.4
Metallization and Oxidation
To deposit the SET leads and make the tunnel junctions, double-angle thermal evaporation was used. Figure 4.4 gives an image of the bell jar used in our lab to deposit
aluminum films which formed the SET leads and island.
As shown in Figure 4.5(a), inside the bell jar developed SET samples were loaded
onto a tilting stage connected to a stepper motor. The stage rested above a thermal
evaporation source, which was a molybdenum boat containing 99.999% Al shot from
CHAPTER 4. SAMPLE FABRICATION
85
Figure 4.4: (a) Image of the bell jar used to deposit the metal coatings in this experiment.
(b) On the thermal evaporation side, molybdenum boats filled with evaporation materials
were heated using high current leads. On the electron beam side, graphite crucibles were
used to hold the materials for Hall bar ohmic contacts, which is discussed in Section 4.2.
Kurt J. Lesker Co. [85]. Once the bell jar was pumped down to ∼ 2 × 10−6 Torr using
a roughing pump and a cryopump, deposition began.
A two step method called double-angle evaporation was used to create the SET
junctions [10], and is schematically given in Figure 4.5(b). The usefulness of the
undercut created in the LOR7B layer now becomes apparent; a suspended resist
bridge of PMMA is used to create two distinct overlapping layers of oxidized Al
and pure Al, forming the tunnel junctions. The first step in this method involved
determining the actual thicknesses of the resist layers spun on the wafer, well before
CHAPTER 4. SAMPLE FABRICATION
86
deposition occurred. Using a Veeco model CPII atomic force microscope [86], it was
determined during development that the thickness of the LOR7B layer was 550 ± 20
nm, while the thickness of the PMMA was found to be 120 ± 10 nm. Since the distance
in the SET pattern from the island hole to the tip of the lead holes was 268 nm, and
the width of the island hole was typically 100 nm, from simple trigonometry the
minimum angle needed to begin overlap was arctan (268 − 100)/(2 ∗ 600) = θ = 8.0◦ .
After several trials, the angle for the best overlap with minimal extension of the leads
past the island material was found to be θ = 11.7◦.
Figure 4.5: (a) The tilting stage used for double-angle evaporation was attached to a
stepper motor having steps of 0.9◦ . (b) Schematic scale diagram of double-angle evaporation.
The angle θ used in the fabrication does not give an ideal overlap in the scale diagram. This
is because simple trigonometry does not account for the spreading of the Al vapour from
the non-ideal point source, or variations in the resist thickness due to spin-coating.
Each deposition was conducted by flowing high current through copper leads into
the boat, raising its temperature and causing the Al shot to vapourize. By changing
the current flow the deposition rate of the Al vapour was controlled. The rate was
monitored using a Inficon XTM/2 quartz crystal monitor, programmed with the final
thickness, density and Z-ratio of aluminum [87]. The rate was typically maintained
CHAPTER 4. SAMPLE FABRICATION
87
at 2-3 Å/s. After the first deposition of 25 nm of Al at an angle of θ = 11.7◦ , the
Inficon caused a blocking shutter between the sample and source to close, and the
sample was allowed to cool for 15 minutes.
To oxidize the deposited Al, pumping on the bell jar was stopped. A 9”×3/8”
copper tube connected to the jar was evacuated and back-filled with a 90%:10%
mixture of Ar:O2 , such that when emptied into the bell jar a pressure of 33 mTorr was
measured. The gas was allowed to oxidize the freshly deposited Al for five minutes,
and then pumped out with the cryopump. A second Al deposition was performed to
deposit a further 35 nm of Al at an angle of −θ = −11.7◦ , thereby creating the island
and tunnel junctions. After letting the samples cool for 15 minutes, the bell jar was
vented to atmospheric pressure with N2 gas, and the samples were removed.
4.1.5
Lift-Off
When evaporation had been completed, the resist was removed by immersing the
sample in NanoRemover PG (NRPG) solution from MicroChem [88]. The NRPG attacks both the LOR7B and PMMA layers, such that any metal clinging to the resist is
left floating above the sample, hence the metal and resist “lift-off” the surface. Areas
where resist had been removed due to pattern exposure and development remained
attached to the wafer, and the completed SET was formed. After lift-off the samples
were rinsed in deionized water, followed by isopropanol and blown dry with N2 gas.
Figure 4.6 gives two SEM images of SETs fabricated in our laboratory.
CHAPTER 4. SAMPLE FABRICATION
88
Figure 4.6: (a) SEM image of a completed SET sample, showing the island and its
overlap with the leads, forming tunnel junctions. (b) SEM image of a SET fabricated by
Scott Pierobon, with smaller junctions.
4.2
2DEG Hall Bar Samples
The section outlines the process for making Hall bars using a GaAs/AlGaAs heterostructure wafer. I will discuss the second lithography technique used to produce
samples in this project, namely photolithography, and give the recipe for the Hall
bars fabricated in our laboratory.
4.2.1
Photolithography
In previous work towards this project, electron beam lithography was used to make
ohmic contacts to the 2DEG and define the Hall bar shape [55]. However in this
work, Hall bar fabrication was exclusively performed with photolithography, so that
many devices could be made in parallel.
Photolithography is a pattern transfer technique which uses ultraviolet (UV) light
to make patterns on a resist coated wafer [16]. It continues to be the standard
CHAPTER 4. SAMPLE FABRICATION
89
technique used in the fabrication of all major integrated circuits (ICs) and large scale
integration of transistors for computer processors [89]. Although EBL is capable of
producing linewidths as small as 10 nm, it is a raster or serial process and so the
electron beam must scan across the wafer surface line-by-line. Photolithography on
the other hand is a parallel process, which means an entire wafer with many different
features or individual devices can be exposed all at once, which is required for the
high volume production of commercial devices.
The basic diagram of an optical lithographic process is given in Figure 4.7(a)
below. A sample, usually a full wafer of silicon or other semiconductor, is cleaned
and coated with a UV-sensitive photoresist polymer. The resist-coated wafer is placed
in a mask aligner printing system under a UV lamp and a photomask containing the
pattern to be transferred. When the UV lamp is turned on and the output intensity
has stabilized, a shutter between the lamp and mask is opened and UV light is
allowed to pass through the mask to expose the photoresist. After a set time period,
the shutter is closed and the lamp turned off.
Photomasks are usually written with SEM or a small spot size laser, and in final
form contain the pattern to be transferred in an etched chrome layer on UV transparent glass [89]. A common light source for UV optical lithography is the high pressure
arc lamp seen in Figure 4.7(b), filled with either Xe or Hg gas. A strike voltage in the
kV range is used to ignite the vapour into a plasma which produces a glow discharge.
The discharge is the result of two optical sources in the lamp, namely high temperature electrons and the atoms themselves, which undergo excitation to higher energy
states and relaxation which produces photons [90]. The hot electrons only produce
deep-UV light in the 75 nm wavelength range, which is mostly absorbed by the silica
CHAPTER 4. SAMPLE FABRICATION
90
Figure 4.7: (a) The photolithographic process uses a UV lamp to expose photoresist
underneath a photomask containing the pattern to be transferred [90]. The polarity of the
resist determines the polarity of the pattern. (b) Arc lamps such as these which contain Xe
and Hg vapours produce the UV light necessary for photolithography [91].
tube containing the vapour. Thus the line spectra from the hot atoms are used to
expose the photoresist; for mecury lamps either the g-line (436 nm) or i-line (365 nm)
can be used, while for xenon lamps wavelengths between 200-300 nm are used [90].
4.2.2
Mask Design
To make the Hall bar shapes and deposit ohmic contacts with photolithography, a
photomask was designed using the software Coventor produced by CoventorWare [92].
CoventorWare is a finite element modelling tool which can be used to simulate the
electromechanical behaviour of MEMS systems such as beams and cantilevers, but
can also be used as CAD software for the production of photomasks. Specifically,
CHAPTER 4. SAMPLE FABRICATION
91
Coventor can output CAD drawings in the GDSII file format, which is a standardized
format for drawings of IC layouts.
Design of a photomask using the GDSII file format is layer and cell based. Layers
are used to define the materials in a foundry process when making a multilayered
mask for ICs. For a photomask, only a thin layer of chrome will contain patterns and
thus only one layer besides the substrate must be included in the design schematic.
Another layer for measurements and layout positioning can be included as a guide,
but is not submitted in the final process.
Cells are used to contain the polygons which make up one layer of a complete
device. Figure 4.8 displays the two cells used to make 100 µm wide Hall bars. The
first cell in Figure 4.8(a) contains the design of mask holes used to transfer a pattern
for ohmic contacts to the resist, and the cell in Figure 4.8(b) contains the design for
the mesa to be etched into the wafer. The pads were designed to be 200 × 200 µm
in area, which is smaller than the 250 × 250 µm mesa areas around them. This was
so the ohmic contacts would be completely covered by the photoresist during etching
and not be undercut, which could lead to loss of contact with the 2DEG layer [55].
Similarly Figure 4.9 shows the cells designed to make 2 µm wide Hall bars. The
first cell in Figure 4.9(a) contains the design of mask holes for the ohmic contacts,
and the cell in Figure 4.9(b) contains the design for the mesa to be etched into the
wafer. The mesas are again larger than the pads so the photoresist can protect them
during etching. Also in this case, the length-to-width ratio of the small Hall bars is
significantly smaller than for the large Hall bars. This was required for suspended
beams, as long, thin beams are more likely to collapse upon release from the substrate,
due to being less stiff in the direction perpendicular to the plane of the wafer.
CHAPTER 4. SAMPLE FABRICATION
92
Figure 4.8: Photomask patterns for 100 µm Hall bars. (a) The ohmic contacts and (b)
Hall bar mesas. The crosses and L-shapes are for alignment purposes, while the T-shapes
provide antisymmetry to label the contact pads.
CHAPTER 4. SAMPLE FABRICATION
93
Figure 4.9: Photomask patterns for 2 µm Hall bars. (a) The ohmic contacts and (b) Hall
bar mesas. The inset shows that the contact leads must narrow to form the small hall bar.
CHAPTER 4. SAMPLE FABRICATION
94
Once the patterns for large and small Hall bars had been finalized and saved as
cells, an array of identical cells was created using the commands in Coventor. Figure
4.10 shows red squares which are 1 cm2 and were used to layout the cells evenly on
the photomask as large arrays. In this way, parallel exposure of multiple devices on
the same chip was able to be carried out in one fabrication step.
Figure 4.10: Using Coventor, an array of cells can be made so that over a small area,
many individual devices can be exposed in the mask aligner. In the center of this figure,
the 1 cm2 red square contains a 3 × 4 array of large Hall bar mesa designs.
The completed 5”×5” inch photomask design was sent to the University of Alberta
NanoFab facility for writing using their Heidelberg DWL-200 pattern generator [93].
This machine is a commercial quality laser pattern generator, which is capable of
writing 1 µm minimum feature sizes on a photomask. In submitting this mask design,
the minimum linewrite of 1 µm was used over the entire surface of the mask to ensure
the 2 µm wide Hall bars were exposed and etched with the best possible resolution.
CHAPTER 4. SAMPLE FABRICATION
4.2.3
95
Wafer Preparation
The heterostructure wafer was grown at the National Research Council facilities in
Ottawa to the design specifications as given in Chapter 3, with production number
V0653. Upon obtaining the photomask, the wafer processing began with cleaving,
cleaning and resist spinning. Since the epitaxially grown wafer is expensive, the wafer
was cleaved into ∼ 1 cm2 pieces (chips) using a diamond-tipped scribe tool. By using
the (01̄1) wafer flats of the GaAs substrate, perpendicular sides of the pieces were
aligned to within about 5◦ of the (01̄1) and (01̄1̄) directions in the heterostructure.
These are also natural cleavage planes in GaAs [56], and so the quality of cleaving
was a good indication of how close the sample sides aligned to these crystallographic
directions.
The first lithography step in making these Hall bars was carried out to deposit
metals for ohmic contacts. The wafer was initially cleaned using a rinse in isopropanol,
a rinse in deionized water, and blown dry with N2 gas. A dehydration bake on a hot
plate at 112◦ C for ten minutes was performed to drive off any adsorbed water vapour.
The wafer was then held on a vacuum chuck inside a Laurel spin-coater, at which
point LOR7B was dropped across the entire surface of the chip. This resist was spun
at 4100 RPM with 10000 RPM per minute acceleration for 45 seconds, which left a
nominal film thickness of 5000 Å [80]. After removal of edge bead and thin “whiskers”
of resist using edge bead remover, the wafer was baked on a hot plate at 150◦C for
three minutes. A resist layer of Shipley 1813 (S1813) was then spun on top of the
LOR7B at 4100 RPM with 10000 RPM per minute acceleration for 45 seconds. This
left a nominal film thickness of 13000 Å as per the spin speed curve [94]. Following
more edge bead removal, the wafer was baked again on the hot plate at 112◦ C for 65
CHAPTER 4. SAMPLE FABRICATION
96
seconds.
As in the SET fabrication, we employed a bi-layer stack of S1813 on top of LOR7B
to enable undercut. In this case, the S1813 is a positive photoresist which is sensitive
to the same MF-319 developer that removes the LOR7B [94]. Thus additional fine
tuning of the bake times and temperatures for both resists was required to obtain
good undercut while leaving the S1813 well suspended.
For the rest of this section, Figure 4.11 provides a reference to the Hall bar fabrication process.
4.2.4
Exposure
Exposure of resist coated samples was carried out using an Oriel Mask Aligner system
and UV lamp [95]. The lamp inside the exposure fixture was a 1kW Hg/Xe arc bulb,
which produced 8.4 mW/cm2 of intensity at the surface of the substrate. Dichroic
mirrors inside the lamp housing limited the exposure wavelength to the 436 nm gline [95].
The mask aligner fixture contained a baseplate with a hole connected to a vacuum
line, which was used to hold large substrates in place during mask alignment and
exposure. This fixture also held the 5”×5” photomask above the substrate, and when
both were properly aligned, the substrate was raised in 1 µm steps into soft contact
with the mask. Soft contact was achieved when the mask began to lift off its resting
posts and when Newton’s rings were seen through the largest holes in the mask [16].
Alignment was conducted using an integrated stereo-microscope suspended above
the fixture. Cross and L-shaped alignment marks were positioned over corresponding
marks already etched or deposited on the substrate surface. In this manner, multiple
CHAPTER 4. SAMPLE FABRICATION
97
Figure 4.11: The Hall bar fabrication steps. (a) The resist bi-layer was spun on the
wafer surface. (b) UV exposure of the resist in the mask aligner. (c) Resist development
using MF-319. (d) Evaporation of Ni-Ge-Au films for ohmic contacts in the electron beam
evaporator. (e) Lift-off of the bi-layer left the deposited material for ohmics on the wafer
surface. (f) Annealing of contacts in the Heatpulse RTA. (g) A second spincoating step to
make the Hall bars. (h) Alignment to the contacts and lithography was again conducted.
(i) Development of the negative resist left the Hall bar shape protected. (j) Hall bars were
etched with citric acid/H2 O2 . (k) Lift-off of the negative photoresist left finished Hall bars.
(l) HF suspension was performed on the 2 µm wide structures.
CHAPTER 4. SAMPLE FABRICATION
98
aligned layers of photolithography were be performed.
Since the ∼ 1 cm2 pieces of wafer were too small to completely cover the vacuum
hole in the original baseplate used for 4” wide wafers, a new baseplate with many 0.1”
holes across its surface was designed using SolidEdge V20 [96]. The baseplate was
fabricated out of aluminum by Gary Contant in the Department of Physics’ machine
shop, and is pictured in Figure 4.12. By blocking off uncovered holes in the plate
with pieces of plastic transparency and Scotch tape, a small chip of wafer could be
placed anywhere on its surface and be held down using vacuum.
Figure 4.12: The designed small sample substrate holder for the mask aligner was fabricated from aluminum. The top and bottom halves of the baseplate were mated with four
countersunk screws, and a series of o-rings sealed the two together. In this manner, a cavity
connecting all the small holes to the main vacuum line was formed.
CHAPTER 4. SAMPLE FABRICATION
99
To perform photolithography the small sample plate was installed and a sample
was placed under the photomask while unused vacuum holes were blocked off. The
substrate vacuum was engaged and increased to reach 7 − 10 inHg. An old mask was
placed in the aligner, and it was checked that the sample vacuum did not leak into
the space between it and the sample. With only the substrate vacuum engaged, the
photomask to be used for exposure was placed in the aligner micrometer positioning
knobs were used to align the mask to the substrate. The substrate was then raised
into soft contact with the mask, and the leak valve was slowly closed, until just over
1 inHg of vacuum was measured between the mask and substrate. The lamp was
powered up and the shutter timer was set to 15 seconds. Once the lamp had warmed
up, the aligner fixture was slid under the lamp lens. The leak valve was closed until
2.5 inHg of vacuum was measured between the mask and substrate, the shutter was
opened and the sample was exposed for 15 seconds. When the shutter closed, the
previous steps were reversed to release the sample.
4.2.5
Development
The development of the Shipley 1813/LOR7B bi-layer was optimized over the course
of this project using the chemical MF-319 from MicroChem [84]. Both exposed Shipley and unexposed LOR7B are sensitive to MF-319, and so only one development
step was required. The undercut in LOR7B was optimized to facilitate easier liftoff
after the metal deposition in the next fabrication step.
After exposure, the samples were dipped in MF-319 for at least 10 seconds, but
the average time required to obtain good undercut without collapse was about 12
seconds. The exact time required changed based on the age of the resist, temperature,
CHAPTER 4. SAMPLE FABRICATION
100
and humidity in the cleanroom, such that it was always necessary to take the sample
out of MF-319 at five second intervals and check the progress under a microscope.
When removed from MF-319, the samples were rinsed in deionized water and blown
dry with N2 gas.
4.2.6
Ohmic Contacts
Recipe
The general recipe used to make ohmic contacts to a 2DEG made of GaAs/AlGaAs
layers involves the deposition and annealing of three metals, namely nickel, germanium and gold. To make the contacts, one must first decide on the thickness and
order of each metal to be deposited on the wafer. After deposition and liftoff has
taken place, the samples are placed in a rapid thermal annealer (RTA) which quickly
heats the wafer to temperatures > 400◦ C in only a few seconds. Upon cooling, if
the contact recipe is sound, “spikes” of the alloyed materials should penetrate down
many nanometers into the 2DEG layer [97, 98]. The resulting IV-curve between any
two contact pads on the wafer surface should be linear.
The recipe used for ohmic contacts to the 2DEG was suggested by Guy Austing
at the NRC and was previously used by Vincent Leduc in our lab in the production
of similar Hall bar samples [55]. The recipe is to evaporate 25 nm of Ni, 55 nm of
Ge and 80 nm of Au on the wafer surface, and anneal in the RTA at 415◦ C for 15
seconds, under an atmosphere of 95%:5% N2 :H2 forming gas. It was found during
trial runs that 15 seconds of annealing at this temperature did not produced ohmic
contacts to wafer V0653. Thus the temperature was increased to 425◦ C and the time
to 60 seconds, which was found to produce ohmic contacts.
CHAPTER 4. SAMPLE FABRICATION
101
Evaporation
The electron beam evaporation side of our bell jar contains a 3kW Thermionics RCseries, 3-crucible linear electron beam source [99]. This apparatus was used to deposit
the materials for ohmic contacts to the 2DEG layer. The metals Ni and Ge, each
having purity of 99.99% were placed in Fabmate crucible liners [99], while 99.999%
pure Au was placed in a coated graphite crucible liner inside the water-cooled copper
source tray. Evaporation of materials took place when the bell jar was evacuated to
2 × 10−6 Torr, and a sweeper system was used to control the position and flux of
the electron beam. This ensured uniform heating of only the deposition material and
not its containing crucible. Before starting the electron beam, the final thickness,
density and Z-ratio of the material to be deposited were programmed into the Inficon
quartz crystal monitor [87]. Once a material was melted and evaporation had begun,
the blocking shutter was opened and the evaporation rate was controlled by varying
the current to the electron beam filament. For Ni and Ge, the evaporation rate
was typically 2-4 Å/s, while for Au the rate was typically 5 Å/s. When the desired
thickness had been reached, the Inficon automatically closed the shutter.
Lift-off
After evaporation, the resist covered areas were cleaned by immersing the samples in
78◦ C NRPG solution. Once the surface of the sample began to look cracked and the
films began to float away, a quick swirl of the solution was typically enough to dislodge
any areas which were still partially clinging to the sample. Ultrasonic cleaning was
not required in this lift-off process. After lift-off the samples were rinsed in deionized
water, followed by isopropanol, and blown dry with N2 gas.
CHAPTER 4. SAMPLE FABRICATION
102
Annealing and Contact Testing
Contact annealing took place in a 410 Heatpluse rapid thermal annealer made by
AGA Associates.
To test if the contacts displayed ohmic, rather than Schottky behaviour, a Wentworth probe station and test circuit as shown in Figure 5.1 (see Chapter 5) were used
to collect voltage and current data. A LabView computer program was used to control a current source and voltmeter using GPIB connections. Typically, the current
through the circuit was varied from -20 to 20 µA in 1 µA steps while the voltage
was read at each step. If the contacts were ohmic, a displayed real-time IV-curve was
observed to be linear, and a least squares fit gave the measured resistance. If the
contacts behaved as Schottky barriers, the current increased exponentially with the
voltage measured [32]. Example IV-curves for non-annealed (Schottky) and properly
annealed (ohmic) contacts are given in Figure 4.13.
Typically the resistance between two ohmic contacts on an etched Hall bar shape
ranged from 10-20 kΩ for large Hall bars, and from 30-50 kΩ for small Hall bars.
4.2.7
Hall Bar Lithography and Etching
Defining Hall bar shapes requires a second step of photolithography to protect areas
of the surface we do not wish to etch. In this work, ohmic contact pads were always
prepared before the mesa was etched, thus an applied etch mask must also protect
the contacts from damage when the etch is conducted.
For the etch mask, we chose to apply the negative photoresist ma-N 405 made by
MicroResist Technology [100]. Since it is a negative resist, areas exposed to UV-light
become more resistant to developer chemicals and remain on the surface, protecting
103
CHAPTER 4. SAMPLE FABRICATION
IV-Curve of Ni/Ge/Au Contacts After Annealing
2
1.5
Voltage [mV]
1
0.5
0
Before Annealing
Data
Fit
4
2
V [V]
-0.5
-1
-2
-1.5
-2
-25
0
-4
-1
-20
-15
-10
-5
0
5
Current [µA]
-0.5
10
0 0.5
I [µA]
15
20
1
25
Figure 4.13: Voltage versus current data for the central pair of annealed 400 µm ohmic
contacts shown in the inset. The linear fit gives a contact resistance of 82±1 Ω. The second
inset shows that before annealing, the IV-curve was Schottky type.
the wafer. The spin-coating method of applying the resist allows the ohmic contacts
to be protected as equally well as the wafer surface. This is not the case for etch
masks such as Ni or Ti films, which have been demonstrated to leave etch holes in the
mesa during a wet etch process, while at the same time leave the wafer unprotected
near the ohmic contact edges [55]. As per the ma-N 405 datasheet, this photoresist
displays good etch resistance to both wet and dry processes for most semiconductor
materials [100].
To begin mesa etching, each sample was spin-coated with ma-N 405 at 3000 RPM
CHAPTER 4. SAMPLE FABRICATION
104
with 10000 RPM per minute acceleration for 30 seconds. Edge bead was removed,
and the samples were hard baked at 100◦ C for one minute before being placed in the
mask aligner. The mask patterns as shown in Figures 4.8(b) and 4.9(b) were used to
make large and small Hall bar mesas. The samples were UV-exposed for two seconds,
and developed in ma-D 331 developer chemical from MicroResist [100] for three to
four minutes until all unexposed resist was gone.
Wet etching of the Hall bar shapes is the process of immersing the wafer pieces
into a solution of prepared etchant and controlling the immersion time. The materials
that must be etched in the heterostructure include pure GaAs, Al0.3 Ga0.7 As, and
Al0.7 Ga0.3As. One particular etchant that has been previously used to etch all three
of these materials is a solution of citric acid and hydrogen peroxide [101, 102]. This
etchant is of particular interest because: (1) there are several references giving etch
rates of these materials along common crystal directions, (2) the etch rate of GaAs
can be made nearly the same as that of Al0.3 Ga0.7As by controlling the ratio of citric
acid to H2 02 , and (3) the solution is inexpensive to produce.
With 50% by mass citric acid and 30% by mass H2 02 solutions, a prepared solution
having volumetric ratio of
VCitric Acid
VH2 O2
= κ = 20 gives a GaAs etch rate of 762 ± 31
Å/min and a Al0.3 Ga0.7As etch rate of 918 ± 96 Å/min along the (100) direction at
room temperature [102]. Thus the selectivity of GaAs to Al0.3 Ga0.7 As is about 0.83,
which is the closest value to unity possible with this etchant. Etching is significantly
slower along the (01̄1̄) and (011̄) directions, with each direction having sidewalls at
132◦ and 129◦ from the etched (100) plane respectively [101].
To etch the samples in this project, a 50% solution of citric acid was made from
CHAPTER 4. SAMPLE FABRICATION
105
deionized water and citric acid monohydrate (C6 H8 O7 · H2 O) supplied by SigmaAldrich Corporation [103]. Once the temperature had stabilized (the dissolution is
endothermic) and immediately before etching, the citric acid was measured and mixed
with 30% H2 O2 to give a κ = 20 solution. The wafers to be etched were placed in the
solution for four and a half minutes, ensuring the heterostructure would be sufficiently
etched to reach the sacrificial layer. Since the etchant is water-soluble, etch stop was
performed by rinsing the samples in deionized water, followed by a blow dry using N2
gas.
After etching, the ma-N 405 resist was removed using a 10-20 minute dip in rem660 remover supplied by MicroResist [100], then rinsed in deionized water, isopropanol
and blown dry with N2 . The results of Hall bar etching for large, small, and poorly
shaped Hall bars are given in Figure 4.14. Under an optical microscope, there is a
clearly defined edge between the mesas and etched areas, and the exposed Ni-Ge-Au
alignment marks are tarnished by the acid etching. In Figure 4.14(d), a poorly etched
Hall bar can result if the photolithography is not conducted with the highest possible
resolution. With our Oriel mask aligner, using medium vacuum pressure and good
soft contact between the mask and substrate, the resolution in ma-N 405 resist is
limited to about 1 µm. Since the smallest Hall bars are only 2 µm wide, extremely
good lithographic control was required to make these samples.
4.2.8
Hall Bar Suspension
To make suspended Hall bars from the 2 µm wide samples, the chips were placed in
a 5% by volume hydrofluoric acid solution, made by diluting 50% stock HF(aq) with
deionized water. The sacrificial layer under a 2 µm wide beam took 40 seconds to
CHAPTER 4. SAMPLE FABRICATION
106
Figure 4.14: Etched Hall bars which are: (a) a 100 µm wide, (b) 2 µm wide, (c) 2 µm
wide with more resolved features and (d) poorly defined due to poor lithography.
undercut in this concentration of HF, which is slightly longer than is expected [73].
Initially, samples were immediately rinsed in deionized water, isopropanol and N2
dried. However, it was found this caused a majority of suspended samples to collapse,
as shown in Figure 4.15.
The reason for collapse is that as the samples are dried in air, the beam is pulled
down by the surface tension of shrinking solvent droplets as they evaporate [89]. To
fix this problem, critical point drying (CPD) was used on the next batch of suspended
CHAPTER 4. SAMPLE FABRICATION
107
Figure 4.15: (a) Optical image of a suspended Hall bar, with its corresponding SEM
micrograph in (b). (c) Optical image of a suspended Hall bar which has collapsed, with its
corresponding SEM micrograph in (d). The green color in the optical microscope images
clearly shows the undercut of the heterostructure after HF suspension, while in (c) the
collapsed Hall bar shows thin film interference fringes as its displacement changes. Neither
of these samples were tested at low temperature due to their poorly defined shapes.
beams. CPD is a technique in which the sample is immersed in liquid CO2 , and the
bath is brought to its critical point of 31◦ C and 1072 psi [89]. By raising the pressure
and temperature above the critical point, the liquid-gas boundary fades away, and
the supercritical gas can be slowly vented so that the sample is dried without having
to pass through the boundary at all.
After rinsing but before being blown dry, the next batch of HF suspended samples
were immersed in a boat full of isopropanol, which was then placed inside a SPI
CHAPTER 4. SAMPLE FABRICATION
108
Supplies jumbo SPI-DRY CPD apparatus [104]. The dryer was flushed with liquid
CO2 from a commercial cannister to remove the isopropanol, filled with liquid CO2 ,
and closed off. The apparatus was then heated to 35◦ C with warm water from a
faucet. After the pressure reached above 1100 psi and the liquid-gas interface was
seen to “vanish,” the CPD tank was slowly vented back to atmospheric pressure
and the dried samples were removed. As shown later in Chapter 6, CPD allowed
suspended 2 µm wide Hall bars to be made without collapse.
Chapter 5
Experimental Methods
In this chapter I explain the experimental methods used to obtain the resonant frequency of an RF-SET and the electrical properties of quantum Hall effect samples,
namely the carrier concentration and carrier mobility in the 2DEG. To reach the
low temperatures required for these experiments, a 1K-cryostat was used for the RFSET sample, while a closed system He-3 cryostat was used for the Hall bar samples.
The base (lowest) temperature for the 1K-cryostat was around 1.5 K, while the base
temperature of the He-3 system was around 300 mK.
5.1
5.1.1
Wiring up RF-SETs
Testing Fabricated SETs
The first measurements conducted on fabricated SETs were the collection of IV-curves
to collect the normal state, room-temperature resistance RΣ of each device. Chips
were placed under a stereomicroscope inside a Wentworth probe station [105], and
109
110
CHAPTER 5. EXPERIMENTAL METHODS
needle probes with tips 25 µm in diameter were lowered onto bond pads. These
probes allowed the SET to be connected to a Keithley model 220 programmable
current source [106] and HP model 34401A digital multimeter [107]. This is shown
in the circuit schematic of Figure 5.1 below. By sourcing a current I and measuring
the voltage V across the device, using Ohm’s law RΣ was determined. Typically a
current of 5 nA was used to measure the resistance.
Since aluminum oxide has a dielectric breakdown voltage of about 11 MV/cm
[108], only about 1 V across a nanometer thick junction can cause it to either blow
open or short the metal island to ground. Thinner or narrower regions of a device
could allow this voltage to be much lower. To avoid damaging the devices from electrostatic discharge (ESD), proper grounding techniques were utilized. A grounding
wrist strap was worn when working at the probe station and the probes were shorted
to ground before touching the SET. Also since the Keithley 220 current source has
a minimum compliance voltage of 1 V, a variable 0 to 920 kΩ resistor was placed in
parallel with the test device. This allowed the probes to be shorted when lowered
onto the bond pads, but when turned up to 920 kΩ would limit the maximum voltage
across the SET even if it was high resistance.
As an example measurement, with the parallel resistor turned to 920 kΩ and
output current to I = 5 nA, a voltage V = 0.454 mV would be measured for a SET
having RΣ = 100 kΩ. In general the SET resistance can be found from the measured
voltage and sourced current using:
V
RΣ =
I
920 × 103 Ω
920 × 103 Ω −
V
I
!
(5.1)
which is a good estimate of SET resistance for values at or below 1 MΩ. Equation 5.1
is non-linear with the voltage, and above 1 MΩ small changes in the SET resistance
CHAPTER 5. EXPERIMENTAL METHODS
111
Figure 5.1: The probe station circuit. After the probes had been lowered onto the SET
pads and the variable resistor set to 920 kΩ, a current I was sourced and the voltage V was
measured to find RΣ .
are harder to measure accurately.
5.1.2
Sample Holder for the RF-SET
In Section 3.2, it was discussed that the best way to minimize the stray capacitance in
an RF-SET experiment is to remove as much grounded metal as possible around the
SET. However, to reach mK temperatures, the SET must be mounted to a cryostat
fitted with RF amplifiers, coaxial cables and filters, all of which must use the main
body of the cryostat as signal ground. The best way of mounting the RF-SET components was therefore a printed circuit board (PCB). The PCB was custom designed
to fit onto the bottom of our He-3 cryostat and further isolate the chip from ground.
A PCB for the RF-SET experiment had been previously designed by Greg Dubejsky [64] using the software ExpressPCB [109]. This board was fabricated by ExpressPCB using 62 mil thick FR-4 epoxy glass, with 1.7 mil copper on the front
and back for traces between components. It contained coplanar waveguides to bring
RF-signals from MMCX connectors towards the SET chip while maintaining a 50 Ω
CHAPTER 5. EXPERIMENTAL METHODS
112
cable input impedance, as well as bias tees (consisting of an inductor and capacitor
electrically connected in the shape of a “T”) to allow for DC biasing of the SET
through the inductor while blocking RF.
Figure 5.2: The new, 1.3” diameter PCB for the RF-SET experiment. The blue square
in the center schematically represents a SET. The lettered traces are connected to MMCX
connectors on the back of the board. When a real SET is wirebonded up, traces B and C
are for DC biasing while allowing RF signals to reach the device, while traces A and D are
for either DC or RF gate biasing. The white square denotes the 1/4 of the PCB used in
the cool down as described in Chapter 6.
Upon testing Greg’s PCB at room temperature with a RΣ = 220 kΩ SET electrically connected to a test circuit (see Section 5.1.4), no resonance peak was observed.
The stray capacitance was so high that even with the smallest available tank circuit
inductors (10-20 nH), no peak was observed above the 10 MHz lower limit of our
network analyzer. It was for this reason that a new circuit board was designed and
fabricated using the ExpressPCB software and manufacturing process.
The new PCB is given in Figure 5.2, and was made with the same thicknesses of
CHAPTER 5. EXPERIMENTAL METHODS
113
FR-4 and copper as the previous one. To reduce the stray capacitance, all copper
underneath and immediately to the sides of the chip mounting area on the PCB was
removed. A SET chip was glued to the center of the board using General Electric
(GE) 7031 varnish [110], and was electrically connected to DC and RF traces using
a West-Bond ultrasonic wire bonder [111]. A Coilcraft model 0402CS-R10 surface
mount inductor [71] with L = 100 nH was secured upside-down on top of a teflon
block using GE varnish, and the block was glued to the board using the same varnish.
This kept the inductor far away from the metal traces on the board. To electrically
connect the drain of the SET to the inductor, a wire was ultrasonically bonded to
the SET bond pad and then to the wound copper wire that formed the inductor and
ended in a solder pad. In this manner soldering the inductor and adding further metal
near the SET was avoided.
On the new PCB, the removal of the ground planes meant the RF traces could
not be kept as 50 Ω cable impedances all the way up to the tank circuit inductor.
However as a previous RF-SET experiment has shown, this method works quite well
to minimize the stray capacitance [36], it was assumed that any impedance mismatch
immediately before the SET would be small enough that it would not significantly
reduce the total RF power reaching the tank circuit. Finally, bias tees were added to
the RF traces by soldering 220 pF surface mount capacitors and 0.53 µH Coilcraft
model BCC-531JL broadband conical inductors [71] to the board.
5.1.3
1K-Cryostat
The PCB was designed to perform measurements on an RF-SET in the He-3 cryostat, however to find the resonant frequency of the loaded tank circuit we used the
CHAPTER 5. EXPERIMENTAL METHODS
114
1K-cryostat. This system was originally designed by Cory Dean, a previous undergraduate student, and was constructed by Gary Contant and Cory Dean. It requires
less liquid helium to cool to base temperature than the He-3 system, and is therefore
better suited to preliminary tests of measurement circuitry. A scale diagram of the
1K-cryostat from the program SolidEdge is given in Figure 5.3.
The most important components of the 1K-cryostat include the inner vacuum can
(IVC), which contains the sample and electrical connections held under high vacuum,
and the 1 K pot, which is located in the IVC space as shown in Figure 5.3. When the
cryostat probe is completely immersed in liquid He-4, the probe temperature drops
to 4.2 K by conductive cooling of the cold liquid. However, the temperature of the
sample can be lowered further by pumping on the 1 K pot, which is connected to the
He-4 bath by a thin siphon. The pumping action draws liquid He-4 into the brass
pot, and by having chosen a siphon with the correct diameter, the 1 K pot can be
kept full of liquid while pumping away its vapor as the liquid evaporates. This action
lowers the vapor pressure and causes to the pot and the attached sample to cool to
1.5 K [17].
In order to cool down, the PCB mounted sample was installed at the bottom of
the cryostat. Since the new PCB had a diameter which was too big to fit inside
the 1K-cryostat sample can, as illustrated in Figure 5.2 we used only 1/4 of this
board to mount a chip. The sample mount was removed and the PCB was screwed
to the sample can mount with non-magnetic stainless steel 4-40 screws. One coaxial
line made of 0.085” outer diameter, semi-rigid stainless steel and filled with teflon
dielectric was used to launch the RF signal from the top of the cryostat down to
the MMCX connector on the board. This cabling has low electrical and thermal
CHAPTER 5. EXPERIMENTAL METHODS
115
Figure 5.3: Scale diagram of the 1K-cryostat. The colors indicate the materials which the
components are made of; yellow is for brass, orange is for copper, light gray is for stainless
steel and dark grey is for aluminum.
CHAPTER 5. EXPERIMENTAL METHODS
116
conductivity due to inner and outer conductors being made of stainless steel, and
its cost is relatively high (∼ $45/ft) [36]. The low thermal conductivity allowed the
semi-rigid cable to cross the steep thermal gradient between room temperature and
1.5 K, and minimize the heat load experienced by the 1 K pot.
To measure the temperature, a nominally 100 Ω Allen-Bradley carbon resistance
thermometer (CRT) was attached to the sample can mount. Its 4-wire resistance
was continuously monitored using a Lakeshore 340 temperature controller [112] and a
LabView measurement program [113]. DC lines used for this purpose were made from
40 AWG Evanohm wire and ran from a 19-pin connector at the top of the cryostat,
through the IVC pumpout tube to copper traces on top of electrically insulating paper.
This paper was GE varnished to the 1 K pot as a heatsink, and more Evanohm wire
was used to connect the CRT to the traces. Full details of the 1K-cryostat cool down
procedure are summarized in Appendix A.
5.1.4
RF Test Circuit
The circuit used to measure the tank circuit resonant frequency when loaded by the
SET is given in Figure 5.4. The sample was connected through the bias tee to the
semi-rigid coaxial cable, and at the top of the cryostat SMA feedthroughs allowed the
RF signal to be accessed by room temperature electronics.
To scan for the resonant frequency, the output port of an Anritsu model MS4623B
vector network analyzer (VNA) [114] was connected to the coupled line of a directional
coupler attached to the coaxial line. The coupling factor of the directional coupler
was 10 dB. The input port of the directional coupler was connected to the top of the
cryostat, and the output port was connected to the input of the VNA. At each of
CHAPTER 5. EXPERIMENTAL METHODS
117
Figure 5.4: RF-SET measurement circuit on the 1K-cryostat.
the four stable temperatures of T = 300 K, 77 K, 4.2 K and 1.5 K, the VNA was
operated in frequency sweep mode. In this mode, -15 dBm sinusoidal signals over the
range of 10 MHz to 1.5 GHz were sent down the coaxial line and the reflected power
S11 was recorded. Over the full range a frequency step size of 7.475 MHz was used.
The temperature was monitored during the entire experiment, to ensure that heating
of the cold stage from power dissipated in the RF circuit was minimized.
CHAPTER 5. EXPERIMENTAL METHODS
5.2
5.2.1
118
Wiring up 2DEGs
Wirebonding Ohmic Contacts
As described in Section 4.2, Ni-Ge-Au contacts deposited on the surface of the heterostructure wafer had to be annealed to reach the 2DEG layer. Once the contacts
were determined to be working, the samples were GE varnished to the PCB shown
in Figure 5.5, and the contacts were wire bonded to copper traces for DC lines.
Figure 5.5: The PCB for the QHE experiment. Side (a) faces towards the lab floor and
is where the Hall bar sample is GE varnished, while side (b) faces towards the lab ceiling.
Black areas are copper trace, while in white areas there is no copper. This PCB was designed
and previously used by Vincent Leduc [55].
5.2.2
He-3 Cryostat
The quantum Hall effect measurements on the 2DEG wafer were performed at T =
300 mK using a Janis model He-3 SSV cryostat [115]. The major structural components of this cryostat are shown in Figure 5.6, while the contents of the inner vacuum
CHAPTER 5. EXPERIMENTAL METHODS
119
can are shown in Figure 5.7. As can be seen in Figure 5.7, the IVC surrounds the
sample, low temperature thermometers, filters and most cooling components.
On this probe, the cooling components which allow one to reach T = 300 mK
include the 1 K pot, the charcoal sorption pump, and the He-3 pot. The functional
operation of these together is briefly given as follows: At room temperature on the
top of the cryostat, a small can of He-3 gas is connected to a thin walled stainless
steel tube which runs down the center of the probe to the charcoal sorption pump
and He-3 pot. Once the cryostat is at a temperature of 4.2 K and is surrounded by a
bath of liquid He-4, the 1 K pot can be brought to a temperature of 1.5 K by filling
the pot with He-4 using a siphon in the bath and simultaneously pumping on the pot.
Another siphon is used to flow He-4 around the charcoal sorption pump to cool
the activated charcoal inside it and thereby adsorb the He-3 gas. Once all the He-3
gas has been adsorbed it is liquified by simultaneously heating the charcoal to 44 K
and driving off all the gas while keeping the 1 K pot cold. Since He-3 liquifies at a
temperature of 1.8 K, over a few hours the He-3 gas will condense and run down into
the He-3 pot. Finally, pumping on the He-3 pot lowers the vapor pressure and cools
the pot and attached samples to a temperature nearing 300 mK [17]. Full details of
the cool down procedure are given in the Janis He-3 cryostat manual [115], and are
summarized in Appendix A.
Wiring and Filters
The DC cabling used to connect the top of the cryostat at room temperature to the
T = 300 mK sample was made of a number of different sections. This allowed us to
dump as much heat as possible into heatsinks at the 4.2 K and 1.5 K sections of the
CHAPTER 5. EXPERIMENTAL METHODS
Figure 5.6: A labelled diagram of the He-3 cryostat [64].
120
CHAPTER 5. EXPERIMENTAL METHODS
Figure 5.7: A labelled diagram of the He-3 cryostat IVC contents.
121
CHAPTER 5. EXPERIMENTAL METHODS
122
cryostat, and to reduce noise. To start this cable run, a 19 pin breakout box of BNC
connectors was attached to a 19 pin connector at the top of the cryostat. This was
done using a cable of 30 AWG copper wires with stainless steel shielding.
To go from 300 K to 1.5 K, a group of 5 mil insulated manganin wires ran from
the 19 pin connector to a wire plug post on the 1 K plate (which is connected to the
1 K pot). On the other side of the post, teflon shielded, 34 AWG copper wire ran to
a OFHC copper box containing powder filters. These filters were was also heatsunk
to the 1 K plate [64].
To go from 1.5 K to 300 mK, a cryocable made from 40 AWG Evanohm wire,
teflon insulation and stainless steel shielding connected the powder filters to a micro
miniature D-sub connector, which was heatsunk beside the He-3 pot [64]. Color coded
copper wires ran from the D-sub connector on the sample stage to pins on the PCB
attached to the copper traces.
For this experiment, the only filtering used on the DC cabling was the powder filter
box attached to the 1 K plate inside the IVC. This filter box consisted of homemade
inductors fabricated from 34 AWG copper magnet wire, wound to form coils 2 mm
in diameter and 50 mm long. Each inductor was connected to an individual DC line,
and the copper box was filled with 304 stainless steel powder (#325 mesh). When
tested with a network analyzer, this filter lowered unwanted signals in the frequency
range > 1 GHz to the noise floor of the instrument (-130 dB). Further details about
the cryostat wiring and filtering can be found in Greg Dubejsky’s thesis [64, 116].
Temperature sensors on the cryostat included silicon diode thermometers attached
to the 1 K pot and charcoal adsorption pump, and a ruthenium oxide (“RuOx”) thermometer attached to the He-3 pot. All temperatures on the cryostat were measured
CHAPTER 5. EXPERIMENTAL METHODS
123
using a Lakeshore model 340 temperature controller with standard calibration curves
for silicon diode and RuOx thermometers [112]. Cabling for the temperature sensors
and wire round heaters ran through a Janis cryocable to a separate 19 pin connector
at the top of the cryostat [115].
Magnet
Inside the dewar and centered on the sample section of the IVC, a Janis C-1082-M
superconducting magnet was used to apply magnetic fields as high as 9 Tesla to the
Hall bars [115]. To charge the magnet coil with current, a 45.8 Ω persistent switch
heater supplied by Janis was attached to the piece of superconducting wire which
shorts the coil across its charging leads. This heater is required as a voltage applied
by the leads to opposite ends of the coil would only cause a supercurrent to flow
through the shorting wire between them. By turning on the persistent switch heater,
the shorting wire is switched to an ohmic state, and current will only flow through the
non-resistive superconducting coil. Once a desired field value has been reached the
heater can be turned off, causing the wire to become superconducting again and loop
the current. In this work, persistent current mode was not used as the field needed
to be swept over its full range to collect the QHE data. The voltage for charging the
magnet was supplied by a Cryomagnetics model CS4-10V power supply [117].
5.2.3
Circuit for Hall Bars
The electrical properties of the 2DEG were studied at T = 300 mK using quantum
Hall effect measurements and quantum well theory as given in Chapter 2. For each
Hall bar an excitation voltage Vds was applied between contacts at opposite ends of
CHAPTER 5. EXPERIMENTAL METHODS
124
the bar. The magnetic field Bz was swept over a range of 0 to 9 Tesla and as in
Figure 2.21, the longitudinal voltage VL and Hall voltage VH were measured. These
measurements allowed the carrier concentration to be calculated from: (i) the nearzero field data using Equation 2.57, and (ii) the low field SdH oscillations using
Equation 2.62. The length and width of the Hall bar were measured using a calibrated
Zeiss optical microscope [118], and with these dimensions the mobility was calculated
using Equations 2.63 and 2.64.
Figure 5.8: Hall bar measurement circuit on the He-3 cryostat.
A schematic diagram of the data collection circuit is given in Figure 5.8. A
Vds = 0.500 Vrms sinusoidal signal with a frequency of 17 Hz was applied to the
series combination of the Hall bar and a 10.00 MΩ resistor. This excitation voltage
came from the sine output of a Stanford Research Systems model SR 830 lock-in
CHAPTER 5. EXPERIMENTAL METHODS
125
amplifier [119]. The voltage VL was measured using the difference inputs A-B attached
to the generator lock-in, while the voltage VH was measured using the A-B inputs on
a separate SR 850 lock-in. The lock-ins were phase locked at 17 Hz using a reference
TTL signal output from the voltage generating SR 830. For the first cool down, the
current through the Hall bar was measured continuously using another available SR
830. For the second and third cool down, the current was only measured at zero
magnetic field and the Hall bars were grounded at one end during the field sweeps.
Though the longitudinal Hall resistance and therefore the current changed with
applied field during each sweep, we can approximate the current through the bar
as being a constant. The highest resistance Hall bars are 30-50 kΩ and increase by
no more than about 10 kΩ at 9 T. The expected current through the Hall bar was
therefore between Ix = 49.70 and 49.71 nArms depending on the applied field. This
fluctuation is on the order of the experimental uncertainty in the measured current
due to noise and is a negligible effect.
To collect the QHE data, all instruments were connected to a desktop computer
using a National Instruments GPIB interface [113]. A Matlab [70] program was used
to sweep the charging current at a constant rate of 0.005 A/s, which changed the
magnetic field at 0.001 T/s. The field was swept from 0 to 9 T or vice versa, and
every two seconds data was read from the lock-ins, the temperature controller, and
the magnet supply.
5.2.4
Illumination
Illumination of each sample was first attempted using a 5 mm red LED. At room
temperature, applying 1.8 V to the LED alone produced a current of 1 mA and a
CHAPTER 5. EXPERIMENTAL METHODS
126
dim light. On the cryostat the LED was connected in series with a 1.0 kΩ resistor,
HP 34401A ammeter and a variable DC voltage supply. Using a LabView program,
the temperature, current through the LED, VL and VH were measured while the DC
voltage was manually controlled.
In the first cool down the illumination procedure did not work. No current could
be observed to flow in the LED circuit even with the application of 20 V from a
Keithley model 2400 sourcemeter [106]. When the probe was warmed back to room
temperature, the LED was observed to function normally. It was hypothesized that
higher voltage was required to get current to flow in the nearly (if not completely)
frozen out p-n junction. In the second cool down 50 V was applied to the LED circuit
using a general purpose bipolar operational amplifier. Again no current was observed
to flow in the LED circuit. We did not have access to a DC voltage source capable of
producing more than 50 V, so for the third cool down the LED was replaced. Various
LEDs were tested using a dip stick in liquid He-4, and a 3 mm red LED which turned
on at around 35 V was selected. In the third cool down, illumination was successfully
performed. After illumination of this sample, another magnetoresistance data set was
collected.
Chapter 6
Experimental Results and
Discussion
The single RF-SET cool down and three Hall bar cool downs gave much experimental
data for analysis. In this chapter I present the results from both sets of experiments,
with some discussion and rationalization. References to specific phenomenon in the
results which have not been previously discussed in Chapter 2 will be given in the
text.
6.1
6.1.1
RF-SET Cooldown
SET Fabrication Results
Fabrication of SETs was mainly performed from October 2007 to January 2009, in
attempts to get devices suitable for cooling down in our He-3 cryostat. These devices
were to ideally to have RΣ ∼
= 50 − 100 kΩ, and with island lead overlaps of ∼ 75 × 75
127
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
128
nm2 or lower, so that C1 and C2 would be small and EC would be large.
During this time, about 20 complete fabrication cycles were performed, with only
12 of these cycles giving chips with fully formed SETs. Of these samples, only sample
JRB05 contained a SET with RΣ = 106 ± 3 kΩ as measured on the probe station.
Other samples had resistances which ranged from 7-500 kΩ, but once wirebonded to
the PCB and connected to the He-3 cryostat, all of these samples were measured as
being open or shorted. Thus for the purposes of finding f0 and CS of an RF-SET
setup using 1/4 of our PCB, a RΣ ∼
= 300 kΩ SET was provided by Scott Pierobon.
Much more detail regarding oxidation control and ESD protection of SETs to get
working RΣ ∼
= 100 kΩ devices can be found in Scott’s thesis [8].
6.1.2
Temperature Dependence of CS
Figure 6.1 shows the temperature dependent reflected power, S11 , versus sweep frequency, f , for the connected RF-SET. Since the impedance of the semi-rigid stainless
steel coaxial cables changes by a large amount with temperature, a network analyzer
calibration performed at T = 300 K does not completely remove cable oscillations at
T = 1.5 K. For this reason, we did not to calibrate at room temperature and observed
many cable oscillations in S11 .
At T = 300 K, with the bias tee inductor grounded and no SET connected to the
tank inductor, the red curve shows there were no resonance peaks other than cable
oscillations. Next, the cyan curve shows that when the bias tee was floating a large
peak appeared at fb = 279 MHz, which we denote the bias tee resonant frequency.
After the SET was wirebonded to the tank circuit, the peak at fb remained. Also
a broad, low frequency peak at fs = 240 MHz, which we denote the tank circuit
129
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
Determination of RF-SET Resonance Frequency
-15
-20
S11 [dB]
-25
-30
-35
300 K (Grounded, no SET)
300 K (Floating, no SET)
300 K
77 K
4.2 K
1.5 K
-40
-45
0
0.5
1
1.5
f [GHz]
Figure 6.1: Temperature dependence of the reflected power S11 versus probe frequency f
for a RΣ ∼
= 300 kΩ SET. The “Grounded” and “Floating” refer to the DC (inductor) side
of the bias tee on the PCB. The peak at fb = 294 MHz is attributed to a floating bias tee,
which does not change significantly with temperature.
resonant frequency, was observed.
During liquid nitrogen cooling to T = 77 K, the tank circuit peak was directly
observed on the VNA screen to move higher in frequency to a stable value of fs = 473
MHz and lose many dB in power. The bias tee resonance moved to fb = 294 MHz
and remained relatively constant in power.
When the SET was placed in a liquid helium bath at T = 4.2 K, the tank circuit
resonance lost a few dB in power, and moved to a value of fs = 1.17 GHz. At T = 1.5
K, fs did not change, while the peak shape became non-ideal due to changes in cable
130
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
resonances. From T = 77 K to 1.5 K, the peak at fb did not change significantly, nor
did its reflected power.
We surmise that the peak associated with fs is f0 , the resonance frequency of
the SET connection and the temperature dependent CS . This peak: (i) increased in
frequency with decreasing temperature, which is expected due to freeze out of the
charge carriers in Si [32], and (ii) decreased in reflected power, which is predicted
from Equations 2.25 and 2.26 for decreasing CS . The peak associated with fb is a
resonance between the inductance of the ungrounded bias tee and a stray capacitance
which has not been accounted for.
For each temperature T , the estimated f0 and CS ∼
=
1
(2πf )2 L
for L = 100 nH
are given in Table 6.1. Assuming that CS does not change significantly for a full
Table 6.1: Experimentally obtained RF-SET properties from Figure 6.1.
Temperature [K]
300
77
4.2
1.5
f0 [GHz]
0.240
0.473
1.17
1.17
CS [pF]
4.40
1.13
0.185
0.185
PCB in the He-3 cryostat, an f0 = 1.17 GHz would be an ideal operating frequency
for the RF-SET experiment, as it falls in the flat bandwidth region of our HEMT
amplifier [69].
The resistance of the RΣ ∼
= 300 kΩ SET was not directly measured during the
cool down. At low temperature: (i) it would take many cool downs to accurately
calibrate out the RF-cables, and (ii) we do not know the precise reactance of the tank
circuit and how it changed from room temperature. Thus it is difficult to calculate
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
131
RΣ from S11 alone. However in a full RF-SET experiment, only the real impedance
RS changes with Vds and Vg . In this manner values of Γ can be mapped to accurately
measured values of RS , and the total Γ − qg curve acquired.
6.2
6.2.1
2DEG Samples
First Cool Down - Unsuspended Samples
The first 2DEG samples studied at 300 mK in this project were ∼100 µm and ∼1
µm wide unsuspended Hall bars. These samples, known as sample A and sample
B respectively, were measured individually using the lock-in technique described in
Chapter 5.
Sample A - Large Hall Bar
Figure 6.2 below shows an image of sample A. After etching, sample A had dimensions
very close to that of its design. This was expected as its smallest linewidths are 10
µm, which is well above the limit of our mask aligner. Two Ni-Ge-Au contacts on
the same side of the Hall bar were connected to make the measurements of VL . One
of these leads and its counterpart on the opposite side of the Hall bar were connected
to measure VH . Optical measurements of the Hall bar width and length between
longitudinal contacts are given later in Table 6.2. Errors in the optical measurements
are the estimated error of the scaling calibration and the estimated standard deviation
of the dimensions (i.e. the non-uniform width of some Hall bars) added in quadrature.
Figure 6.3 shows a plot of the longitudinal and Hall resistances versus an applied
magnetic field from 0 to 9 T. The data was collected without illumination. Each
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
132
Figure 6.2: Optical microscope image of sample A, which is a ∼100 µm wide, unsuspended
Hall bar.
resistance value was calculated from the in-phase voltage divided by the in-phase
current as measured by the lock-ins. At low fields, RH increases linearly as per the
classical Hall effect. At Bz ∼
= 0.5 T, RL begins to oscillate periodically with the field;
these are the Shubnikov de Haas (SdH) oscillations. The SdH oscillations continue
to increase in amplitude all the way up into the high field region.
Quantum Hall plateaus begin to form at the same time as the SdH oscillations,
however there are some unusual features in the high field (> 4 T) region. It can be
seen that as the plateaus begin to fully form, RH spikes to large positive or negative
(not shown) values instead of remaining constant. RL exhibits similar behaviour,
although it should fall to zero in these regions. These features are not expected for
typical QHE data, and are discussed with the results for the third cool down.
Although precise values are difficult to obtain, filling factors i = h/(e2 RH ) can be
estimated from Figure 6.3. At Bz = 2.405 T, the value RH = 1861 Ω corresponds to
133
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
Magnetoresistance Measurements of Sample A
10
9
RLongitudinal
RHall
8
Resistance [kΩ]
7
6
5
4
3
2
1
0
0
1
2
3
4
5
Magnetic Field [T]
6
7
8
9
Figure 6.3: The longitudinal and Hall resistances versus applied magnetic field for sample
A.
i∼
= 14. Near Bz = 6.657 T, spin splitting begins to occur, and the small plateau at
RH = 5258 Ω corresponds to i ∼
= 5. The wide region of data between B = 5.2 − 6
T should correspond to RH (i = 6) = 4300 Ω, and given that a few points near the
beginning and end of this region give RH = 4336 Ω, the plateaus are in their correct
positions.
Figure 6.4 shows the plot of the low field data used to find the classical carrier
concentration. Equation 2.57 was used to determine nc . To get the low field carrier
concentration from the SdH oscillations and Equation 2.62, either a plot of RL versus
1/Bz or a Fourier transform of the SdH oscillations will work. The Fourier transform method was chosen as it was used to analyze the data for the third cool down
134
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
2.2
×10−24
2
Calculation of Classical nc for Sample A
Experimental Data
Fit
1.8
eVH [CV]
1.6
1.4
1.2
1
0.8
0.6
0.4
0.4
0.6
0.8
1
1.2
Ix Bz [AT]
1.4
1.6
1.8
2
×10−8
Figure 6.4: A plot of eVH versus Ix Bz for sample A. The inverse of the fit slope is
equal to the classical carrier concentration displayed in Table 6.2. The fit y-intercept is
non-zero due to magnetic field remnants in the superconducting magnet. From this data,
nc = (7.89 ± 0.02) × 1011 cm−2 .
experiment.
To take the Fourier transform of discreet data, a fast Fourier transform (FFT)
algorithm can be utilized [120]. To use this type of algorithm, the data must be
sampled at a constant rate in the time domain, which in this case is 1/Bz , to give the
power spectrum in the frequency domain, Bz [120]. In these experiments, the field
was ideally ramped at a constant rate while data was collected in two second intervals
by the computer. However the intervals are not exactly two seconds apart, due to
communication delays with the instruments over the GPIB connection. Thus to FFT
the data, a “spline” technique was implemented using Matlab to interpolate RL at
135
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
evenly spaced points in 1/Bz [121]. The results of the FFT on the interpolated SdH
data are given in Figure 6.5. Further reading about these techniques can be found in
Appendix B.
FFT Amplitude [Arb. Units]
Calculation of nc from SdH Oscillations for Sample A
0.2
0.15
0.1
0.05
0
0
10
20
30
50
60
40
SdH Frequency [T]
70
80
90
100
Figure 6.5: An FFT plot of the longitudinal resistance SdH oscillations for sample A. The
peak at fSdH = 16.6 ± 0.4 T corresponds to the growing SdH oscillations observed in Figure
6.3. The spline curve interpolation range was between 0.31 ≤ 1/B ≤ 2.5 T−1 only, to remove
distortions in the FFT from the high field data. From this data, nc = (8.0 ± 0.2) × 1011
cm−2 .
Using the classical nc value and the measured ρxx , µ was calculated at zero field
from Equation 2.63. The summarized results for sample A are given in Table 6.2.
Sample B - Small Hall Bar
Figure 6.6 below shows an image of sample B. After etching, sample B had a width
smaller than the designed 2 µm, while the voltage measurement leads were wider. The
bar also does not have exceptionally straight sidewalls. These effects are attributed to
imperfect mask contact with the ∼ 1 cm2 substrate chip during Hall bar lithography,
as well as the resolution of our mask aligner being at best 2 − 3 µm. The dimensions of sample B were measured optically and are given in Table 6.2. These values
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
136
have larger estimated relative uncertainties, due to their larger non-uniformities and
smaller dimensions.
Figure 6.6: Optical microscope image of sample B, which is a ∼1 µm wide, unsuspended
Hall bar.
The RL and RH data is given in Figure 6.7. The data was collected without
illumination. The zero-field RL is larger for the small Hall bar, which is expected by
Equation 2.64, assuming ρxx is approximately constant for any size Hall bar. The
amplitudes of the SdH oscillations are noticeably larger for sample B, while in the
high field region resistance spikes are observed. The summarized results for sample
B are given in Table 6.2.
General Discussion
The first cool down revealed the measured classical and low field carrier concentration
in each of samples A and B matched within experimental error. There was a small
increase in nc in sample B compared to the sample A. With nc ∼
= 8 × 1011 cm−2 in
137
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
Magnetoresistance Measurements of Sample B
10
9
RLongitudinal
RHall
8
Resistance [kΩ]
7
6
5
4
3
2
1
0
0
1
2
3
4
5
Magnetic Field [T]
6
7
8
9
Figure 6.7: The longitudinal and Hall resistances versus applied magnetic field for sample
B.
each sample, the wafer carrier concentration exceeds that of the designed 5 × 1011
cm−2 .
It was also observed that the measured mobility for sample A matches that of
sample B within experimental error. However with µ ∼
= 3.7 × 104 cm2 /(V·s) in
each sample, the measured value is approximately one 10-100 times lower than some
previous GaAs/AlGaAs heterostructures [15, 54]. Other heterostructures have been
measured to have a similar or lower mobility of µ ∼
= (1 − 3) × 104 cm2 /(V·s) [60, 61],
but these values were attributed to reactive ion etching damage during the mesa etch.
It is hypothesized that a poor growth recipe could have caused the mobility to
decrease. One idea is that too much δ-doping could have caused large amounts of
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
138
ionized impurity scattering, even with thick spacer layers. In this wafer, a Si sheet
density of 3.5×1012 cm−2 was used by our crystal grower Zbig Wasilewski at the NRC
to create the δ-doped regions. Since Si is amphoteric in GaAs, placing too many Si
atoms next to one another in the same plane can cause the Si to autocompensate [122].
Above a critical sheet density, free electrons produced by donor Si atoms in Ga sites
begin to be taken up by acceptor Si atoms in As sites. In this way, ionized scattering
centers can increase without an increase in carrier concentration and with a possible
decrease in mobility.
It is known that the critical sheet density for Si dopants in (100) GaAs is about
1.3 × 1013 cm−2 [122], so in our wafer all the Si dopants should be sitting in Ga sites
and autocompensation should not be a problem. Other problems with the growth,
such as excessive impurities or dislocation defects, could account for the lower mobility
observed [15], although these effects are harder to determine accurately.
The first QHE experiment also revealed no evidence of a permanent depletion
width near the edges of the small Hall bar. This width, known as wd , is an effect of
surface states at the sidewalls of the Hall bar shape [123]. Many of these states can
trap charge carriers and partially deplete the 2DEG, effectively lowering the electrical
width we by the relation we = w − 2wd and increasing ρxx . If wd ∼
= w, the total width
Table 6.2: Experimentally obtained 2DEG properties from samples A and B.
Property
l [µm]
w [µm]
nc (classical, zero field) [cm−2 ]
nc (SdH, low field) [cm−2 ]
µ [cm2 /(V·s)]
Sample A
699 ± 11
97 ± 1
(7.89 ± 0.02) × 1011
(8.0 ± 0.2) × 1011
(3.63 ± 0.07) × 104
Sample B
16 ± 2
0.9 ± 0.1
(8.45 ± 0.03) × 1011
(9.0 ± 0.2) × 1011
(3.8 ± 0.6) × 104
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
139
of the Hall bar, the 2DEG will be totally depleted and no conduction will occur.
For GaAs/AlGaAs heterostructures, previous theoretical predictions and experiments have estimated the depletion width to range from 175-800 nm [61,123] depending on the sample’s carrier concentration and mobility. In our sample A, w ≫ wd for
any of these estimates and depletion was not observed. The resistivity of sample A
was found to be ρxx = 219 ± 4 Ω/Square. In sample B, some ohmic contacts were
observed to be high resistance to one another during initial tests. However, after a
few hours at T = 300 mK, all of these contacts were observed to become ohmic again
with low resistance. The resistivity was measured to be ρxx = 200 ± 30 Ω/Square,
which matches that of Sample A within the experimental uncertainty.
6.2.2
Ohmic Contact Tests
After initial experiments on samples A and B had been conducted at low temperature,
there was some suspicion that the recipe for ohmic contacts did not reach into the
2DEG as well as suggested, and a new recipe was formulated. Upon review of the
literature it was found that gold-germanium in the weight ratio of 88%:12% Au:Ge
behaves as a eutectic with a melting point of 361◦ C, and the best thickness of nickel
was determined to be one quarter the total gold-germanium thickness [124]. Thus
a recipe of 40 nm of Ni, 53 nm of Ge and 107 nm of Au would be ideal. However,
our RTA cannot reproduce the exact characteristics of pressure control as found in
Ref. [124], and there was concern over the depth of penetration of the ohmic contact
spikes if too much Ni was used. Thus the revised, second recipe used was 25 nm of
Ni, 53 nm of Ge and 107 nm of Au. This recipe was used for all other devices in this
project (samples C-G).
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
140
An additional problem was that some wafers which were annealed before samples
A and B displayed non-ideal surface morphologies. For example as shown in Figure
6.8(b), one common problem was for the Ni-Ge-Au films to bubble and lift-off the
surface of the wafers during annealing.
Figure 6.8: (a) Image of good ohmic contacts to the heterostructure wafer after annealing.
The contact surface looks rough, as melting and recrystallization has occurred. (b) Image
of poorly annealed contacts; the surface was not properly cleaned and gas phase products
have bubbled up the contacts during the high temperature annealing.
It was hypothesized that unremoved resist and/or oxide layers which had not been
removed before metal deposition were decomposing into gas phase byproducts during
annealing. For this reason, extra cleaning steps including oxygen plasma cleaning and
hydrochloric acid etching were added into the recipe immediately after development
but before metal deposition.
To plasma clean the wafers after development, they were placed inside a PlasmaPreen oxygen plasma cleaner made by Terra Universal [125].
This cleaner is a
standard household microwave outfitted with vacuum controls to pump out a glassenclosed sample space, and flow controls to let in a precise flow rate of O2 gas when
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
141
running. The 2.45 GHz microwaves are used to turn the oxygen flowing around the
sample into a plasma of ionized atoms and free radicals, which react strongly with
the carbon-containing resist residue stuck to the wafer surface. The residue is then
turned into CO2 and other gaseous products which can be pumped away [89].
It was found that by using a flow rate of three SCFH of O2 and 100% microwave
power, the etching rate of LOR7B was about 40 nm/min as measured by a Veeco
model Dektak 150 surface profiler [86]. By cleaning the developed wafers for two
minutes in the plasma, it was assured that at least 80 nm of LOR7B residue would
have been removed if present in the developed areas, a thickness of which would
clearly present as a significant color change in optical microscope images. Thus it was
assumed after plasma cleaning all resist in the developed areas had been removed.
The plasma cleaning did not significantly alter the edges of the developed S1813 which
defines the resolution of the pattern.
To further clean the developed areas, hydrochloric acid was used on the wafers
after plasma cleaning. It is known that after sitting in air for a while, the GaAs cap
layers on the heterostructure develop a native oxide containing both Ga-O bonds and
As-O bonds [126]. However it has been shown in the literature that after a one minute
etch in concentric HCl(aq) (35% by mass), the oxide is completely removed [126, 127].
Samples C-G in this work were subjected to a two minute O2 plasma clean and
a one minute HCl(aq) etch immediately before being placed in the bell jar for metal
deposition.
Due to the bubbling Ni-Ge-Au contacts and poor high field magnetoresistance
data, tests were performed after the previous experiment to check that ohmic contacts were being fabricated properly. First, it was found that performing O2 plasma
142
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
cleaning and HCl acid etching prior to evaporation of the metals decreased the number of bubbled contacts after annealing to near zero. Second, annealing time tests
were performed on ∼ 1 µm wide Hall bar samples C, D, E and F. To avoid variability in the processes preceding annealing, these samples were fabricated with the
revised cleaning steps on the same wafer chip and diced using a wafer saw. Figure
6.9 presents the collected IV-curves of each sample after annealing at T = 425◦ C for
various times.
Determination of Ohmic Anneal Time @ 425◦ C
5
Voltage [V]
2.5
C - Unannealed
C - Annealed 15 s
D - Annealed 30 s
E - Annealed 60 s
F - Annealed 120 s
0
-2.5
-5
-50
-25
0
Current [µA]
25
50
Figure 6.9: A plot of the resistance of Ni-Ge-Au contacts for various annealing times. For
each sample C-F, a ∼ 1 µm wide Hall bar was fabricated on wafer V0653, and the contacts
at opposite ends of each Hall bar were probed. At 425◦ C, after about 60 s, the contacts are
clearly ohmic and further annealing does not significantly reduce the resistance.
From the graph it is clear that after 30 s the contacts became ohmic, and after
60 s further annealing did not significantly decrease the contact resistance. From this
data, it was surmised that the wafer cleaning steps and contact recipes could not
be improved further. Samples D and E, which were the best etched samples, were
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
143
selected for HF suspension and used as suspended Hall bars in the next cool down.
6.2.3
Second Cool Down - Suspended Samples
The second set of Hall bars studied, samples D and E, were annealled for 30 and 60 s
respectively. They were completely suspended along their lengths using HF etching.
Graphical results from sample D only are presented here, while sample E behaved in
a similar manner and its properties are recorded for completeness.
Sample D and Sample E - Suspended Hall Bars
Figure 6.10 below shows an image of sample D after HF etching. While its width is
smaller than 2 µm, after mesa etching it had straighter sidewalls than sample B due
to better lithography. Optical measurements of the Hall bar width and length are
given in Table 6.3.
Magnetoresistance data is plotted in Figure 6.11. The data was collected without
illumination. For samples D and E, the resistances were calculated using the measured in-phase voltage divided by a constant current of about Ix ∼
= 48 nA. The SdH
oscillations are observed to form at Bz ∼
= 0.7 T, and increase in amplitude all the
way up to 9 T. In the high field region, spikes in both resistances are observed where
RL heads towards 0 Ω and Hall plateaus would be expected. In this manner sample
D shares magnetoresistance characteristics with samples A and B. The summarized
results for samples D and E are given in Table 6.3.
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
144
Figure 6.10: Optical microscope image of sample D, which is a ∼1 µm wide, suspended
Hall bar. The central suspended beam has not collapsed, as there are no color (interference)
fringes to indicate a change in height.
General Discussion
From the second cool down it was observed that suspended heterostructure beams
fabricated from our wafer V0653 contained a 2DEG. Although the beams were less
than 1 µm wide, SdH oscillations and low field Hall effect were clearly observed.
This is in contrast to previous experiments performed in our lab on a different
Table 6.3: Experimentally obtained 2DEG properties from samples D and E.
Property
l [µm]
w [µm]
nc (classical, zero field) [cm−2 ]
nc (SdH, low field) [cm−2 ]
µ [cm2 /(V·s)]
Sample D
16 ± 1
0.84 ± 0.09
(7.87 ± 0.05) × 1011
(8.0 ± 0.2) × 1011
(2.9 ± 0.4) × 104
Sample E
16 ± 1
0.8 ± 0.1
(6.78 ± 0.02) × 1011
(6.5 ± 0.2) × 1011
(3.7 ± 0.5) × 104
145
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
Magnetoresistance Measurements of Sample D
15
RLongitudinal
RHall
Resistance [kΩ]
10
5
0
0
1
2
3
4
5
Magnetic Field [T]
6
7
8
9
Figure 6.11: The longitudinal and Hall resistances versus applied magnetic field for sample
D.
GaAs/AlGaAs wafer with suspended beams. In Ref. [55], 0.3 − 1 µm wide, wet
and dry mesa etched, HF suspended Hall bar samples did not show any conduction
unless illuminated with an LED. At T = 300 mK, no classical Hall effect or SdH
oscillations were observed with varying field. We surmise that: (i) the removal of
a low-temperature GaAs (LT-GaAs) layer from wafer V0653 that was originally in
the previous wafer, and (ii) the use of photoresist rather than Ni mesa etch masks
helped to solve this problem. The removal of the LT-GaAs likely decreased the defect
density near the 2DEG layer, while the photoresist masks kept ∼ 100 nm wide etch
pits from forming at the surface of the wafer [55].
Finally, the second QHE experiment revealed the sheet densities and mobilities
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
146
of samples D and E closely matched those of samples A and B. In the case of the
mobilities, all are equal within experimental errors.
6.2.4
Third Cool Down - Shifted Contacts
The third and final cool down was used to study sample G, a ∼ 2 µm wide, suspended
Hall bar.
High Field Regions
The first and second cool downs revealed a high field spiking pattern in the magnetoresistance data. It was determined these spikes were a result of the contacts
becoming highly resistive at the Hall plateaus. As plotted in Figure 6.12, for sample
A the phase of VL versus Bz in these regions becomes becomes non-zero, indicating
capacitive and/or inductive impedances are contributing to the circuit. The total
q
2
2
VL,R = VL,x
+ VL,y
increases, and we surmised that the behaviour was due to the
Ni-Ge-Au contacts becoming non-ohmic and capacitively coupling to one another
through the 2DEG.
We hypothesized the differences between “bulk” and “edge state” 2DEG theory,
when applied to our samples, explained why contacts not touching the mesa edges
became non-ohmic in the plateau regions [51, 128]. As discussed in Chapter 2, electrons which carry the current in the bulk wafer are confined to Landau levels below
EF . When Bz is increased and the m + 1 occupied Landau level empties, EF does not
discontinually jump to the m Landau level because of localized states between them.
Since these states are caused by localized impurities and disorder, they can carry no
current, and only the extended states at the core of the Landau levels contribute to
147
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
Deviation of Longitudinal Voltage Phase
Longitudinal Voltage [mV]
Phase [Deg.]
100
50
0
-50
-100
1.5
1
0.5
0
0
1
2
3
4
5
Magnetic Field [T]
6
7
8
9
q
2 + V 2 for sample A at high field. At
Figure 6.12: Phase variations in VL,R = VL,x
L,y
select points, the voltage exceeded the measurement range of the SR 830 lock-in, causing
the measurements to look flat.
the Hall conductivity [51, 128].
Once the sample has been etched, infinite potential barriers at the mesa edges add
further confinement to the 2DEG. If the width of the 2DEG is in the y-direction, with
its length along the x-direction, without derivation we can state that the energy levels
given by Equation 2.59 (Landau levels in a single ith subband) must go to infinity
at y1 = −w/2 and y2 = w/2. This is schematically shown in Figure 6.13(a). A term
dependent on the wavevector ky is introduced, and the new energy levels are given
by [15]:
h̄2 ky2
1
E = (j + )h̄ωc + En +
2
2M
j = 0, 1, 2, ...
(6.1)
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
148
where M is an effective mass. As in Figure 6.13(b), the levels become steeply sloped
near the edges, such that empty states in an occupied Landau level can be pushed
over EF near the edge. These are called edge states, and can carry a current in the
Hall plateau region [128].
Figure 6.13: Schematic diagrams of edge states in the quantum Hall effect. (a) Landau
levels of the first subband head to infinity near the “hard-wall” potential barriers at y1
and y2 , the Hall bar edges. The value y0 is called the center coordinate of the magnetic
state, which is the average position of the wavefunction. (b) Single particle electron states
close to the edges. Inside the blue oval, an occupied Landau level crosses EF . In this case,
electrons just below EF can take up infinitesimal amounts of energy and carry a current at
the edge [128].
Using a multi-probe method, Büttiker [129] has previously shown that the absence
of backscattering in the current carrying edge states leads to the same quantization of
the Hall resistance as in Equation 2.61 and vanishing longitudinal resistance. However
to access the edge states, the ohmic contacts must be within a few tens of nanometers
from the mesa edge. On samples A-F, as pictured in Figure 6.14(a) all ohmic contacts
were ∼ 25 µm away from edges, and at high field only a diode-like current would be
expected through the semi-insulating GaAs to the edge states. Thus on sample G, as
pictured in Figure 6.14(b), the contacts were fabricated directly over the mesa edges,
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
149
to receive the edge state currents.
Figure 6.14: (a) On sample D, the ohmic contacts were fabricated ∼ 25 µm away from
the mesa edges. (b) On sample G, the contacts were shifted upwards to lay over the mesa
edges, and connect to the edge states.
Sample G - Suspended Hall Bar
Figure 6.15 below shows an image of sample G after HF etching. This image shows
extra fringes near the mesa edges, which could be an indication of anisotropic sidewall
etching. Also, while it is clear that areas under the voltage probes are not completely
undercut, the central part of the Hall bar appears to be. The colors in the image are
darker due to changes in the camera settings on the microscope. Optical measurements of the Hall bar width and length between longitudinal contacts are given in
Table 6.5.
Magnetoresistance measurements are given in Figure 6.16. In this sample, the Hall
plateaus fully form in the high field region, and the SdH oscillations go completely
to zero. There are no spikes observed in either of the resistances. Observed filling
factors were calculated by averaging RH over the width of each plateau, and values
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
150
Figure 6.15: Optical microscope image of sample G, which is a ∼2 µm wide, suspended
Hall bar. The central suspended beam has not collapsed, as there are no color (interference)
fringes to indicate a change in height. However, the suspension is incomplete near the
voltage leads.
including the standard deviations are given in Table 6.4.
Table 6.4: Filling factors i = h/(e2 RH ) from observable Hall plateaus of sample G.
Bz [T]
7.750
5.342
4.117
3.301
2.764
i (Theoretical)
4
6
8
10
12
i (Experimental)
3.97 ± 0.02
5.99 ± 0.01
7.95 ± 0.02
9.87 ± 0.04
11.75 ± 0.01
After collecting the data in Figure 6.16, at zero magnetic field illumination was
performed by manually increasing the LED voltage to about 35 V. When RL was
observed to fall and the He-3 pot temperature increased, the voltage was quickly
lowered back to zero. Figure 6.17 shows the illumination current and RL with time.
151
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
Magnetoresistance Measurements of Sample G (Unilluminated)
7
RLongitudinal
RHall
6
Resistance [kΩ]
5
4
3
2
1
0
0
1
2
3
4
5
Magnetic Field [T]
6
7
8
9
Figure 6.16: The longitudinal and Hall resistances versus applied magnetic field for sample
sample G (unilluminated).
Magnetoresistance data collected after illumination are presented in Figure 6.18.
The SdH oscillations show evidence of a second frequency or “beat” pattern with
increasing Bz , and there many Hall plateaus. The summarized results for sample G
before and after illumination are given in Table 6.5.
Sample G confirmed our hypothesis about the edge state picture, as fully formed
quantum Hall plateaus were observed. At low field all samples behaved well, which
is expected as the broad Landau levels have not fully separated and the bulk 2DEG
formalism applies [129].
152
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
20
1.6
15
1.4
10
1.2
5
1
0
0.8
0
50
100
150
Time [s]
200
250
LED Current [mA]
Long. Resistance [kΩ]
Persistent Photoconductivity of Sample G
1.8
-5
300
Figure 6.17: Illumination current and longitudinal resistance of sample G versus time.
The peak illumination current reached was 18.4 mA, and the He-3 pot momentarily spiked
to T = 815 mK. After illumination, the longitudinal resistance dropped and relaxed to
about a factor of two lower than its original value.
Second Sub-Band Filling
After illumination of sample G we observed second subband filling in the 2DEG. This
phenomenon is schematically illustrated in Figure 6.19. As the carrier density n0
in the first subband increases such that Landau levels are filled up past the square
well energy E1 , the second subband will begin to populate. This effect has been
previously used in similar GaAs/AlGaAs heterostructures to determine the effective
mass in each subband and study nonparabolic band shapes [130, 131].
In a 2DEG with more than one occupied subband, the conductivity ρxx is the sum
of their oscillatory conductivities. The carrier density ni in the i-th subband is given
by the frequency of the low field SdH oscillations [131]:
fiSdH =
hni
hni
=
ge
2e
(6.2)
153
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
Magnetoresistance Measurements of Sample G (Illuminated)
7
RLongitudinal
RHall
6
Resistance [kΩ]
5
4
3
2
1
0
0
1
2
3
4
5
Magnetic Field [T]
6
7
8
9
Figure 6.18: The longitudinal and Hall resistances versus applied magnetic field for sample
sample G (illuminated).
which can easily be calculated. Figure 6.20 presents an FFT of the SdH data for
sample G. From Table 6.5:
n0 + n1 = (1.10 ± 0.04) × 1012 cm−2
(6.3)
which matches the total, classical nc = (1.085 ± 0.001) × 1012 cm−2 within experimental error. We propose that further photoconductivity tests of the 2DEG should
be computer controlled, to avoid manually overexposing the sample with such a high
LED current.
154
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
Table 6.5: Experimentally obtained 2DEG properties from sample G.
Property
l [µm]
w [µm]
nc (classical, zero field) [cm−2 ]
n0 (SdH, low field) [cm−2 ]
n1 (SdH, low field) [cm−2 ]
µ [cm2 /(V·s)]
Before Illum.
16 ± 4
2.1 ± 0.1
(7.729 ± 0.008) × 1011
(7.5 ± 0.2) × 1011
—
(4 ± 1) × 104
After Illum.
16 ± 4
2.1 ± 0.1
(1.085 ± 0.001) × 1012
(9.7 ± 0.2) × 1011
(1.3 ± 0.2) × 1011
(5 ± 1) × 104
D(E)
hωc
hωc
E
E0
E1
EF
Figure 6.19: Schematic diagram of second subband filling. E0 is the square well energy
eigenvalue for the first subband, while E1 is the eigenvalue for the second subband. For a
large enough nc , at energies below EF the second subband will begin to populate. In this
diagram, EF is between both sets of Landau levels, and a Hall plateau would be observed.
155
CHAPTER 6. EXPERIMENTAL RESULTS AND DISCUSSION
FFT Amplitude [Arb. Units]
Calculation of nc from SdH Oscillations for Sample G
0.2
Illuminated
Unilluminated
0.15
0.1
0.05
0
0
10
20
30
40
60
50
SdH Frequency [T]
70
80
90
100
Figure 6.20: An FFT plot of the longitudinal resistance SdH oscillations for sample
G, before and after illumination. After illumination, the peak at fSdH = 20.0 ± 0.4 T
corresponds to n0 , while the peak at fSdH = 2.7 ± 0.4 T corresponds to n1 . The spline curve
interpolation range was between 0.25 ≤ 1/B ≤ 2.5 T−1 only, to remove distortions in the
FFT from the high field data.
Chapter 7
Conclusions
7.1
7.1.1
Future Work
Fabrication of 1 GHz Beams
Given that there exists a 2DEG in wafer V0653 for suspended structures having
widths down to at least w = 0.8 ± 0.1 µm, the next logical step is to use electron
beam lithography in conjunction with reactive ion etching (RIE) to form beams which
meet the requirements of the proposed nanomechanical experiment. Using a Ni etch
mask, RIE with BCl3 plasma has been performed on a similar GaAs/AlGaAs wafer
to make 0.3 − 1 µm wide beams having extremely straight sidewalls [55].
It has been previously shown that a suspended GaAs/AlGaAs beam having a
thickness of 164 nm, length of 836 nm and width of 250 nm would have a fundamental, out-of-plane resonant frequency of 925.6 MHz [55]. Magnetomotive drive
could be used to determine the mechanical Q-factor of these beams [2], which have
an assumed Q = 104 in Ref. [7]. For these beam widths additional magnetoresistance
156
CHAPTER 7. CONCLUSIONS
157
measurements should be performed, to confirm that total depletion of the 2DEG does
not occur at low temperature.
7.1.2
Coupled Nano-Electro-Mechanical System
Once control of the SET oxidation process has reached a sufficient level in our electron
beam evaporator, we should be able to create many SET devices having RΣ = 100±30
kΩ just as in Refs. [21, 29, 36]. A complete RF-SET Γ − qg data curve should be
collected for various Vds to determine its electrical properties (i.e. Ri , Ci , Cg , Ec , etc.)
at T = 300 mK. Once working amplifiers are demonstrated, movement towards a
coupled RF-SET and piezoelectric beam device should progress rapidly.
One major issue that poses unanswered questions is the utilization of the 2DEG
layer, which will sit only 77 nm below any fabricated SETs on the wafer surface, as a
ground plane for the actuation of the piezoelectric beam. If design considerations are
not furthered beyond the proposal of Ref. [7], the 2DEG will introduce a very large
stray capacitance to ground through the bond pads, pulling the RF-SET resonance
frequency down. There has been some consideration to etching the 2DEG away
underneath the area where the SET device will sit, however it remains an open
question if this will work in a fully realized device.
7.2
Summary
RF circuit theory was applied to RΣ = 50 kΩ SETs in an tank circuit. The reflected
power was modelled for perfect and imperfect impedance matches to a Z0 = 50 Ω
cable input impedance, over the 1-1.5 GHz frequency range. It was determined that
CHAPTER 7. CONCLUSIONS
158
for reasonable expected values of the stray capacitance to ground, the usable halfbandwidth of an imperfectly matched RF-SET was as high as 175-200 MHz. This
analysis assumed a noise threshold of 0.2 dB between the maximally and minimally
(Coulomb blockaded) conductive states of the SET. For a SET having RΣ = 432 ± 14
kΩ that was previously fabricated in our lab, for a noise temperature of T = 1.5 K
√
p
the best calculated charge sensitivity was Sq = 3.9 × 10−5 e/ Hz.
Fabrication recipes were developed in our lab to create SETs with RΣ ∼
= 100 kΩ.
Many attempts at cooling a SET down to T = 300 mK and measuring its properties were performed, however each time the sample was destroyed from electrostatic
discharge. To find the resonance frequency and stray capacitance of a RF-SET on a
PCB, Scott Pierobon provided a RΣ ∼
= 300 kΩ device which was cooled to T = 1.5 K.
At this temperature, the device was found to have a resonance frequency of f0 = 1.2
GHz and a stray capacitance of CS = 0.2 pF.
A GaAs/AlGaAs 2DEG heterostructure was simulated using Poisson-Schrödinger
self-consistent theory. For a Si doping density of 2 × 1019 cm−3 in δ-doped layers
above and below a GaAs quantum well region, an electron density of nc ∼
= 5 × 1011
cm−2 was calculated to condense in the well at T = 1 K. This carrier concentration
was not expected to change significantly between suspended and unsuspended parts
of the wafer.
The designed heterostructure wafer was grown by the NRC in Ottawa. In our
lab, several 100 µm wide and 1 − 2 µm wide Hall bar samples were fabricated using
photolithography and wet chemical etching. Some of the narrower Hall bars were
suspended from the substrate using a HF dip. Magnetoresistance measurements at
T = 300 mK revealed a 2DEG existed in both suspended and unsuspended parts of
CHAPTER 7. CONCLUSIONS
159
the wafer. The 2DEG had a carrier concentration of about nc = 8 × 1011 cm−2 and
a mobility of about µ = 3.5 × 104 cm2 /(V·s). Evidence of edge states and second
subband conduction were observed in the Hall bars studied.
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[132] J. R. Clement and E. H. Quinnelll. The low temperature characteristics of
carbon composition thermometers. Rev. Sci. Instru., 23(5):213216, 1952.
Appendix A
Experimental Details
A.1
1K-Cryostat Cool Down Procedures
Sample Mounting and Leak Checking
To cool the 1K-cryostat once a sample was connected and checked, as shown in Figure
5.3 the IVC was placed on the tapered brass plug and sealed with Ceralow, a low
temperature solder alloy which melts at about 58◦ C [17]. The IVC was then pumped
out from the top of the cryostat and leak checked with an Alcatel model ASM 110
Turbo CL leak detector. Leak checking involved pumping the IVC space down to
10−5 Torr and monitoring the He-4 background while spraying dry He-4 around all
solder joints and o-ring seals to the IVC space. The 1 K pot was also leak checked
by backfilling it with dry He-4 gas and monitoring the IVC background.
171
APPENDIX A. EXPERIMENTAL DETAILS
172
Transfer to Liquid N2 Bath
Once the cryostat was found to be leak tight, the IVC space was valved off while still
under vacuum, and placed in a dewar of liquid nitrogen. A constant stream of dry
He-4 gas was flowed through the 1 K pot line to ensure the attached capillary did not
become blocked by ice. About 200 Torr of dry N2 gas was introduced to the IVC to
help transfer heat to the cold bath. Once the temperature was read to be stable for
some time, the IVC was pumped out to make sure it still held high vacuum (this is
because leaks which are not evident at room temperature may become more severe
as the cryostat cools). After this step about 200 Torr of dry He gas was placed in
the IVC, the IVC pumpout was valved off, and the probe was lowered slowly into the
He-4 dewar.
Transfer to Liquid He-4 Bath
The probe was immersed in a liquid He-4 dewar and the temperature was monitored
until stable. The IVC was again leaked checked, the flowing He-4 gas was stopped, and
the IVC was evacuated. To reach base temperature of 1.5 K, the 1 K pot was pumped
using a roughing pump. The CRT resistance was observed to stabilize at a value that
against a typically calibration curve [132] gave a temperature of T = 1.5 ± 0.3 K
(there was no approximate curve for this particular CRT).
APPENDIX A. EXPERIMENTAL DETAILS
A.2
173
He-3 Cryostat Cool Down Procedures
Sample Mounting
To begin a He-3 cryostat cool down, as shown in Figure 5.7 the Hall bar PCB was
mounted on an oxygen free, high conductivity (OFHC) copper sample mount connected to the He-3 pot. This sample mount maintained the PCB in good thermal
contact with the He-3 pot, and also kept it parallel to the floor and perpendicular to
the direction of the applied magnetic field. A “light tight,” 50 mil thick stainless steel
can was placed around the sample to shield the sample from electromagnetic noise.
Once all electrical connections to the device were checked by measuring the contact
resistances from the top of the dewar, the IVC was installed on the probe. This
involved placing a 1 mm diameter piece of pure indium wire in a o-ring groove at the
opening of the IVC. Twenty-two stainless steel bolts were used to attach the IVC to
the probe flange. The bolts were tightened in a torque pattern to evenly crush the
ring of indium and thus provide a gas tight helium seal down to 4.2 K.
Leak Checking
The cryostat was leak checked by pumping out the IVC and spraying He-4 gas around
the indium seal, solder joints and all other vacuum seals at the top of the cryostat.
The 1 K pot system was also leak checked by back-filling the 1 K pot with 3 psi of He4 gas and maintaining vacuum on the IVC. When these checks were completed, the
cryostat probe was raised using a hoist and slowly lowered into the dewar. Aluminum
tape was used to keep the 1 K pot and charcoal cooling siphons close to the IVC
while lowering took place.
APPENDIX A. EXPERIMENTAL DETAILS
174
Cryogenic Liquid Transfers
Once the probe was inside the dewar, about 200 Torr of dry N2 gas was introduced
into the IVC, and dry He-4 was flowed through the charcoal cooling pump siphon.
The dewar was filled to the top with liquid N2 , while pumping on the 1 K pot with
a closed-cycle helium roughing pump. Once the entire probe reached a temperature
of about 77 K, the IVC was evacuated and the dry He-4 flow was stopped.
At this point, the liquid N2 was blown out of the dewar by pressurizing it with
dry He-4 gas. By flushing and pumping out the dewar with dry He-4 a few times, it
was ensured no liquid N2 remained. While leak checking the IVC, and again flowing
He-4 gas through the charcoal cooling siphon, liquid He-4 was slowly transferred to
the dewar using a transfer tube. The 1 K pot was pumped out during this step.
When the transfer was finished, a needle valve to the 1 K pot siphon was slightly
opened to allow filling of the 1 K pot with liquid He-4. The 1 K pot was pumped
out simultaneously to reach 1.5 K. Positive pressure in the dewar was used to flow
cold gas through the charcoal cooling capillary, causing the charcoal pump to drop
to 5 K. The pressure in the He-3 storage vessel was observed to drop to -30 psi, and
the temperatures of the charcoal pump and 1 K pot eventually dropped to 5 K and
1.5 K respectively. After a few hours the charcoal cooling was stopped. The pump
was heated to 44 K using 1.5 W of power on a 25 Ω wire wound heater attached to
it, while the temperature of the 1 K pot was kept as low as possible using the needle
valve at the top of the cryostat.
APPENDIX A. EXPERIMENTAL DETAILS
175
Cool to Base Temperature
After the 1 K pot was maintained below 1.8 K for about two hours, causing most of
the He-3 gas to liquify and run into the He-3 pot, the charcoal heating was stopped
and cooling with the cold He-4 gas began once more. By cooling the charcoal a
pumping action was engaged on the He-3 pot, lowering its temperature to about 300
mK after an hour and a half. With a full dewar of liquid He-4 and the 1 K pot kept at
1.5 K, this temperature was continuously maintained for many days at a time, until
the He-4 level dropped below the ends of the siphons.
Appendix B
Mathematical Routines
B.1
Cubic Spline Interpolation
Given a set of n “knot” points x0 , ..., xn that are known and satisfy a function
f (x0 ), ..., f (xn ), cubic spline interpolation is a method which allows one to interpolate the value f (x) for any value of x. To minimize the interpolation error, we
seek (n − 1) piece-wise continuous polynomials Pi (x) = ai x3 + bi x2 + ci x + di between
points xi and xi+1 that satisfy the conditions [121]:
Pi (xi+1 ) = Pi+1 (xi+1 )
(B.1)
′
Pi′ (xi+1 ) = Pi+1
(xi+1 )
(B.2)
′′
Pi′′ (xi+1 ) = Pi+1
(xi+1 )
(B.3)
Equation B.1 states the condition that at the knot points, the values of adjacent
polynomials must be equal, while Equations B.2 and B.3 state the conditions that the
first and second derivatives of adjacent polynomials must also be equal at the knots
respectively.
176
APPENDIX B. MATHEMATICAL ROUTINES
177
The coefficients (ai , bi , ci , di ) are all unknown, but can be found using Equations
B.1-B.3, plus two extra conditions that are typically [121]:
f ′′ (x0 ) = 0 and f ′′ (xn ) = 0
(B.4)
These conditions lead to what is called the natural spline curve [121]. A diagram
illustrating a fitted spline curve is given in Figure B.1.
Figure B.1: Cubic spine curves between four points xi = x0 , ..., x3 . Knowing the functions
Pi (x) = ai x3 + bi x2 + ci x + di in each interval i = 1, 2, 3 allows us to calculate any value of
f (x), up to a definable error.
For Shubnikov de Haas (SdH) oscillations, we used Matlab [70] to perform cubic spline interpolation on the RH data along 1/Bz (“time” domain). This allowed
us to estimate RH values for evenly spaced 1/Bz data, and perform a fast Fourier
transform (FFT). The FFT gave the frequency, amplitude and phase of the RH oscillations against Bz (“frequency” domain). Figure B.2 shows an example of how spline
interpolation allowed for evenly spaced points along 1/Bz to be found. In Matlab,
178
APPENDIX B. MATHEMATICAL ROUTINES
we used the command “yy = spline(x,Y,xx)” to find the interpolated values (yy) for
a vector of equally spaced points (xx), given the array of known knot points (x,Y).
1.8
Resistance [kΩ]
1.7
1.6
1.5
1.4
1.3
Points Given
Points Found
Spline Curve
1.2
0.8
0.9
1
1.1
1/Bz [1/T]
1.2
1.3
1.4
Figure B.2: An example of the spline curve applied to RH data (SdH oscillations). The
interpolation spacing is ∆(1/Bz ) = 0.001 T−1
B.2
Fast Fourier Transform (FFT)
The fast Fourier transform is a fast algorithm which computes the discrete Fourier
transform (DFT) of a set of (N − 1) points [120]. The DFT of a set of N − 1 points
x0 , ..., xN −1 collected at a sampling rate fs is computed in Matlab using [70]:
Xk =
N
−1
X
n=0
n
e−2jπk N , 0 ≤ k ≤ N − 1
(B.5)
where Xk = ak + jbk are complex numbers. The ak and bk represent the amplitudes
and phases of respective cosine and sine waves which add to give the approximate
APPENDIX B. MATHEMATICAL ROUTINES
179
time domain signal. The frequencies are given by fk = kfs /N. To find the wave
p
magnitude for a given frequency fk , we use Rk = a2k + b2k .
In Matlab, we used the command “y=fft(x)” to find the complex values (y) of the
waves which make up the time domain signal (x).
Appendix C
List of Abbreviations and Symbols
Abbreviations
2DEG
2-dimensional electron gas
EBL
Electron beam lithography
MOSFET
Metal-oxide-semiconductor field effect transistor
IVC
Inner vacuum can
LOR7B
Lift-off resist 7B
MEMS(NEMS)
Micro-electro-mechanical systems (Nano-electro-mechanical systems)
MF-319
A solvent
QHE
Quantum Hall effect
RF-SET
Radio frequency single electron transistor
RTA
Rapid thermal annealler
S1813
Shipley 1813 photoresist
SQL
Standard quantum limit
SET
Single electron transistor
SSET
Superconducting single electron transistor
180
APPENDIX C. LIST OF ABBREVIATIONS AND SYMBOLS
Symbols
Bz
Magnetic field in z-direction
C0
Background capacitance
Ci
Junction capacitance, i = 1, 2
Cg
Gate capacitance
CΣ
Total SET capacitance
CS
Stray capacitance
e
Electron charge
EC
Charging energy
EF
Fermi energy
f0 , ω0
Resonance frequency and angular frequency
fm , ωm
Modulation frequency and angular frequency
Ix
Current in the x-direction
Ids
Drain-source current
kB
Boltzmann constant
L
Inductance
µ
Mobility
hnth i
Thermal occupation number
nc
Carrier concentration
ni
Carrier concentration in the i-th subband
Q
Quality factor
Qe
External quality factor
QL
Loaded quality factor
181
APPENDIX C. LIST OF ABBREVIATIONS AND SYMBOLS
Symbols cont.
q0
Background charge
qi
Junction charge, i = 1, 2
qg
Gate charge
ρ
Resistivity
Ri
Junction resistance, i = 1, 2
RH
Hall resistance
RK
Resistance quantum
RL
Longitudinal resistance
RΣ
Total SET resistance
RS
Dynamic SET resistance
σ
p
Sq
√
Sx
Conductivity
Charge sensitivity
T
Temperature
Vds
Drain-source voltage
Vg
Gate voltage
VH
Hall voltage
VL
Longitudinal voltage
∆xQL
Quantum limit of displacement
∆xSQL
Standard quantum limit of displacement
Displacement sensitivity
182
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