Download Towards the Fabrication and Characterization of a Nanomechanical Electron Shuttle

Towards the Fabrication and Characterization
of a Nanomechanical Electron Shuttle
by
Benjamin B. C. Lucht
A thesis submitted to the
Department of Physics, Engineering Physics & Astronomy
in conformity with the requirements for
the degree of Master of Applied Science
Queen’s University
Kingston, Ontario, Canada
January, 2010
c Benjamin B. C. Lucht, 2010
Copyright Abstract
First proposed in the late 1990’s, a nanomechanical electron shuttle is a device where
an electrically isolated island moves a definite number of electrons between two leads,
producing a current that is directly related to the number of electrons moved in a
cycle and to the vibration frequency of the island. Since nanomechanical structures
can have very well defined vibration frequencies, a device of this type is useful as,
among other things, a current standard for metrology. The experimental shuttle
implementations to date have had large island-lead spacings, which has limited their
performance. The work presented here takes the first steps towards the fabrication of
a nanomechanical electron shuttle using the process of electromigration to define very
small lead-island gaps with conductivity on the order of the conductance quantum
G0 = 2e2 /h. These small gaps, coupled with the high vibration frequencies achievable
with nanostructures, will allow investigation deeper into the realm of quantum effects.
In this work, the fabrication steps for the creation of these devices were developed.
Electromigration of a single junction was successfully achieved to the 10–100 kΩ range.
The simultaneous and symmetric electromigration of two junctions, as required for the
shuttle, has not yet been achieved. The development of a fast electromigration cut-off
circuit, however, gives hope that double-breaking success will be achieved soon.
i
Acknowledgments
This thesis would not have been possible without the support and guidance of my
supervisor, Robert Knobel. His experience and insights helped me avoid problems,
kept me going, and both pushed and pulled me ever deeper into the realm of this
thing they call “science.” It is undeniably cliched, Rob, but I could not have done
this without you.
Equally valuable to me was the fine set of people that made up my group mates.
Jennifer, Josh, Scott, Devon, Saydur, Vincent: it has been a pleasure and a privilege
to work with you. I could always count on you to bounce ideas around with, set me
straight when I was confused, laugh with me when I did something stupid, and put
up with my terrible puns. I could not have asked for a better bunch of people to work
with. A special thank you goes to Josh for keeping me company through so many
late nights/early mornings in the final weeks of this thesis. I would say, Josh, that
you kept my head screwed on straight, but I suspect that most people would argue
that we just drove each other further into madness. Whatever the truth, it was good
times.
I would also like to tip my hat to the denizens of 154 and 511H who I was fortunate
enough to share offices with. Between YouTube Fridays, hospital lunches, and the
endless potential for interesting conversations, it is a wonder that I got any work
ii
done. You all made my time here rather enjoyable.
It is without hesitation that I say that I would not have been able to get anything
at all done if not for the excellent staff of Stirling Hall. There are so may of you, and
your contributions so varied and far-reaching, that to list you all here would take far
too much space (and I would inevitably forget someone!), but thank you all.
Finally, I would like to thank my family for all the love and support that they
have shown me throughout my education. A very special thank you goes to my wife
for putting up with countless absences, late lab nights, and having to share me with
physics. Claire, you’ve been wonderful; sometimes I don’t know how you put up with
me, but I’m grateful that you do.
iii
Statement of Contributions
There were two areas in which other people contributed directly to the work presented
in this thesis: single-junction electromigration and the cut-off circuit for doublejunction electromigration. Apart from the areas outlined below, all the work presented
in this thesis was my own.
The work on single-junction electromigration was shared equally between Jennifer
Campbell and myself, as this technique was also useful for her work. We worked
together in the creation of the patterns for electron beam lithography and both did
equal parts of the fabrication of the devices used to develop this technique. The
first version of the LabView control software was created by J. Campbell, although
we both discussed the logic behind it; further improvements and extensions to the
program were done by me, as well as the re-writing necessary to move from the source
meter to the SIM modules. We both contributed equally to the actual breaking trials,
both together and individually, as well as the subsequent imaging.
The cut-off circuit was designed in a group with Rob Knobel and Devon Stopps
and myself, although the circuit details and the specific parts to use were left to me. I
prototyped the board and evaluated its behaviour. J. Campbell assembled the circuit
board and got it box-ready. Troubleshooting and a handful of final modifications
were done by me in consultation with R. Knobel, D. Stopps, and Steve Gillen.
iv
Table of Contents
Abstract
i
Acknowledgments
ii
Statement of Contributions
iv
Table of Contents
v
List of Tables
ix
List of Figures
x
List of Symbols and Abbreviations
xiii
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
Brief Overview of the Field . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3
Statement of Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Chapter 2: Literature Review
2.1
. . . . . . . . . . . . . . . . . . . . . .
8
Single Electron Shuttle Theory . . . . . . . . . . . . . . . . . . . . .
8
2.1.1
Island Motion . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
10
2.2
2.3
2.1.2
The Arbitrary Assumptions . . . . . . . . . . . . . . . . . . .
14
2.1.3
Coulomb Blockade . . . . . . . . . . . . . . . . . . . . . . . .
17
2.1.4
Shuttle Accuracy . . . . . . . . . . . . . . . . . . . . . . . . .
20
Previous Shuttle Implementations . . . . . . . . . . . . . . . . . . . .
26
2.2.1
Externally Driven Shuttles . . . . . . . . . . . . . . . . . . . .
26
2.2.2
Instability-based Shuttles . . . . . . . . . . . . . . . . . . . . .
36
Electromigration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2.3.1
The Brief Theory of Electromigration . . . . . . . . . . . . . .
42
2.3.2
General Device Structure . . . . . . . . . . . . . . . . . . . . .
44
2.3.3
Controlling Electromigration . . . . . . . . . . . . . . . . . . .
45
2.3.4
Conductivity Regimes . . . . . . . . . . . . . . . . . . . . . .
48
2.3.5
Results and Uses . . . . . . . . . . . . . . . . . . . . . . . . .
50
2.3.6
Point Contact Displacement Detectors . . . . . . . . . . . . .
51
Chapter 3: Design and Fabrication
. . . . . . . . . . . . . . . . . . .
55
3.1
Relative Energy Scales . . . . . . . . . . . . . . . . . . . . . . . . . .
55
3.2
Device Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3.3
First Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.4
Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.4.1
Device Definition . . . . . . . . . . . . . . . . . . . . . . . . .
61
3.4.2
Electromigration . . . . . . . . . . . . . . . . . . . . . . . . .
71
3.4.3
Double-Junction Breaking . . . . . . . . . . . . . . . . . . . .
78
3.4.4
Device Suspension . . . . . . . . . . . . . . . . . . . . . . . .
83
Chapter 4: Experiments and Results
vi
. . . . . . . . . . . . . . . . . .
85
4.1
4.2
4.3
Single Junction Electromigration . . . . . . . . . . . . . . . . . . . .
85
4.1.1
Breaking Results . . . . . . . . . . . . . . . . . . . . . . . . .
86
4.1.2
Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
4.1.3
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
Double Junction Electromigration . . . . . . . . . . . . . . . . . . . .
93
4.2.1
1.5 K Cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.2.2
Four-Wire Electromigration Circuit . . . . . . . . . . . . . . .
95
4.2.3
The Breaking Cut-off Circuit . . . . . . . . . . . . . . . . . .
98
Device Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Chapter 5: Conclusions and Future Work
5.1
. . . . . . . . . . . . . . . 105
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.1.1
Device Suspension . . . . . . . . . . . . . . . . . . . . . . . . 106
5.1.2
Shuttle Fabrication . . . . . . . . . . . . . . . . . . . . . . . . 107
5.1.3
Other Experiments . . . . . . . . . . . . . . . . . . . . . . . . 108
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Appendix A: Detailed Fabrication Recipes
. . . . . . . . . . . . . . . 116
A.1 Spin Coating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
A.2 Electron Beam Lithography . . . . . . . . . . . . . . . . . . . . . . . 117
A.3 Chemical Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
A.3.1 Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
A.3.2 Liftoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
A.3.3 Wet Etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A.4 Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
vii
A.5 FIB Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Appendix B: Electromigration Per-Sample Results
Appendix C: 1.5 K Cryostat and Circuit Board
. . . . . . . . . . 124
. . . . . . . . . . . . 126
Appendix D: The Breaking Cut-off Circuit . . . . . . . . . . . . . . . 130
viii
List of Tables
3.1
Comparison of existing shuttle energy scales . . . . . . . . . . . . . .
56
3.2
Effect of bake temperature on resist development rate . . . . . . . . .
63
3.3
Electromigration program parameters . . . . . . . . . . . . . . . . . .
76
4.1
Optimal PID parameters . . . . . . . . . . . . . . . . . . . . . . . . .
97
A.1 Spin coating parameters . . . . . . . . . . . . . . . . . . . . . . . . . 116
A.2 SEM run-file parameters . . . . . . . . . . . . . . . . . . . . . . . . . 117
B.1 Single junction devices . . . . . . . . . . . . . . . . . . . . . . . . . . 125
B.2 Double junction devices . . . . . . . . . . . . . . . . . . . . . . . . . 125
ix
List of Figures
1.1
Feynman’s speech at the nanoscale . . . . . . . . . . . . . . . . . . .
3
2.1
Simple shuttle system proposed by Gorelik et al. . . . . . . . . . . . .
9
2.2
Shuttle charge-position profile . . . . . . . . . . . . . . . . . . . . . .
10
2.3
Schematic of the Coulomb blockade . . . . . . . . . . . . . . . . . . .
18
2.4
Numerical solutions to the shuttle system . . . . . . . . . . . . . . . .
20
2.5
Numerical and analytic results for shuttle accuracy at T = 0 . . . . .
23
2.6
Shuttle performance for increasing temperatures . . . . . . . . . . . .
25
2.7
SEM image of Erbe’s quantum bell . . . . . . . . . . . . . . . . . . .
27
2.8
“Quantum bell” results at room temperature . . . . . . . . . . . . . .
29
2.9
Small driving voltage giving shuttling of 7 ± 2 electrons at 30.5 MHz .
30
2.10 SEM image of the improved quantum bell . . . . . . . . . . . . . . .
31
2.11 Room temperature current spectrum of the improved quantum bell .
32
2.12 Quantum bell at 4.2 K . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.13 Schematic and results of the ultrasonically-driven electron shuttle . .
35
2.14 Nano-pillar and results from Kim et al. . . . . . . . . . . . . . . . . .
37
2.15 Using thiol thin films to hold a nanoparticle between electrodes . . .
38
2.16 Using an AFM to move a gold nanoparticle between two leads . . . .
39
2.17 Current-voltage results for the gold nanoparticle shuttles . . . . . . .
40
x
2.18 Current flow with and without the gold nanoparticle . . . . . . . . .
41
2.19 SEM image of a gap formed with electromigration . . . . . . . . . . .
45
2.20 Representative electromigration structure with 10 nm gap . . . . . . .
46
2.21 The few-atom regime of electromigration . . . . . . . . . . . . . . . .
47
2.22 Representative current traces for electromigration . . . . . . . . . . .
48
2.23 APC sensing of a vibrating beam . . . . . . . . . . . . . . . . . . . .
53
2.24 Two possible detector layouts for a cross correlation measurement . .
54
3.1
Schematic shape of a shuttle design . . . . . . . . . . . . . . . . . . .
57
3.2
Process flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
3.3
SEM pattern to define shuttle device . . . . . . . . . . . . . . . . . .
64
3.4
Schematic of the development process . . . . . . . . . . . . . . . . . .
66
3.5
Good and bad development results . . . . . . . . . . . . . . . . . . .
67
3.6
Schematic of normal and double-angle evaporation . . . . . . . . . . .
69
3.7
Schematic of the fabricated device . . . . . . . . . . . . . . . . . . . .
71
3.8
SEM images of fabricated device . . . . . . . . . . . . . . . . . . . . .
72
3.9
Single-junction device for electromigration . . . . . . . . . . . . . . .
72
3.10 Simple breaking circuit . . . . . . . . . . . . . . . . . . . . . . . . . .
73
3.11 Breaking software logic flow . . . . . . . . . . . . . . . . . . . . . . .
75
3.12 Janis ST-500 probe station . . . . . . . . . . . . . . . . . . . . . . . .
76
3.13 Single-junction electromigration result . . . . . . . . . . . . . . . . .
77
3.14 Result of uncontrolled electromigration . . . . . . . . . . . . . . . . .
78
3.15 FIB milling to define mechanical components . . . . . . . . . . . . . .
83
4.1
Single junction breaking data . . . . . . . . . . . . . . . . . . . . . .
87
4.2
I − V evolution for single-junction electromigration . . . . . . . . . .
88
xi
4.3
Normalized conductivity for single-junction electromigration . . . . .
90
4.4
Asymmetrical breaking of a double junction . . . . . . . . . . . . . .
94
4.5
Four-wire electromigration circuit . . . . . . . . . . . . . . . . . . . .
96
4.6
Overview of electromigration cut-off circuit . . . . . . . . . . . . . . .
99
4.7
Electromigration circuit with cut-off circuit box . . . . . . . . . . . . 100
4.8
Symmetric failure of a double-junction device . . . . . . . . . . . . . 102
4.9
Device after etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A.1 Example of poor focus during lithography . . . . . . . . . . . . . . . 118
A.2 Example of material left over after liftoff . . . . . . . . . . . . . . . . 120
A.3 FIB dose-depth profile . . . . . . . . . . . . . . . . . . . . . . . . . . 123
C.1 1.5 K cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
C.2 Cryostat circuit board . . . . . . . . . . . . . . . . . . . . . . . . . . 129
D.1 Cutoff circuit detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
xii
List of Symbols and Abbreviations
A
Island oscillation amplitude; also cross-sectional area
AC
Alternating current; a signal with frequency between DC and MW
AFM
Atomic Force Microscope
APC
Atomic Point Contact
C
Island capacitance
CΣ
Sum of island & junction capacitances
DC
Direct current; a signal with f = 0
e
Electron charge, 1.602×10−19 C
Ea
Activation energy
EBL
Electron Beam Lithography
E
Electric field magnitude; when bold, the vector field
ESD
Electrostatic Discharge
f, f0
Mechanical vibration frequency and eigenfrequency
xiii
F
Force
FIB
Focused Ion Beam
γ
Mechanical damping coefficient
G
Conductance
G0
Conductance quantum, ≈ 77.5 µS
Γ
Electron tunnelling rate
h
Planck’s constant, 6.626×10−34 J·s
~
Reduced Planck’s constant, h/2π = 1.055×10−34 J·s
I
Current
j
Current density
k
Mechanical spring constant of the island support
kB
Boltzmann constant, = 1.381×10−23 J/K
kF
Fermi wavevector
λ
Electron tunnelling length
l
Mean free path of an electron in a material
L
Length; as a subscript, the left lead or junction
m
Mass
MW
Microwave; a high-frequency signal
xiv
n
Number of extra (or missing) electrons on the shuttle island
N
Number of electrons transfered per shuttle cycle, = 2n
NPGS
Nanometre Pattern Generation System
ν
Charge fluctuation frequency, = 1/RC
Φ
Metal work function
PID
Proportional-Integral-Derivative feedback controller
P
Electrical power
Pn
Charge distribution function
q
Charge, usually of the island
Q
Quality factor, = f0 /∆f
RIE
Reactive Ion Etching
R
Resistance; as a subscript, the right lead or junction
R0
Equilibrium resistance
Rstop
Target (stop) resistance for electromigration
Rth
Threshold resistance; used as a feedback parameter in electromigration
SEM
Scanning Electron Microscope
σ
Electrical conductivity
t
Time
xv
t0
Shuttle contact time (as 2t0 )
τi
Transmission coefficient of the ith conduction channel
t¯f
Mean time to failure of a nanowire undergoing electromigration
T
Temperature; also shuttle oscillation period
V
Potential across device; often has subscripts to denote specifics
VC
Critical voltage before shuttle motion starts
Vdown
Voltage drop upon feedback threshold triggering in electromigration
Vdrive
Voltage used to drive shuttle motion
Vramp
Rate of voltage increase in electromigration
ω
Mechanical vibration frequency, = 2πf
ω0
Mechanical eigenfrequency of island vibration, = 2πf0
W
Work; often has a subscript to denote specifics
x
Displacement of island from equilibrium position
x0
Zero-point vibration amplitude
z∗
Atomic effective charge
S, D, G, C, I Source, drain, gate, clapper, island: shuttle components
xvi
Chapter 1
Introduction
In the year 2000, when they look back at this age, they will wonder why
it was not until the year 1960 that anybody began seriously to move in
this direction.
— Richard Feynman, 1959
1.1
Brief Overview of the Field
Today’s field of nanotechnology is one that spans a wide range of disciplines — physics,
chemistry, materials science, biology, and countless others — and requires integration
and collaboration between these fields as no single approach or paradigm can succeed
on its own. Although the general perception is that nanotechnology is a new field,
this is not entirely true. The word “nanotechnology” is itself relatively recent. It was
first coined in 1974 by Norio Taniguchi; he used it to describe machining with tolerances of less than a micron [1]. The spirit of the field, however, goes back further, at
the very least to Richard Feynman’s 1959 talk titled “There’s Plenty of Room at the
1
CHAPTER 1. INTRODUCTION
2
Bottom” [2]. In this lecture at the American Physical Society’s annual meeting, he
started with the idea of writing the entire Encyclopaedia Brittanica on the head of a
pin and considered the difficulties involved in writing that information and reading it
back. He then discussed using small machines to build ever-smaller machines, talked
about some of the biological systems that already did this sort of thing, and even suggested the possibility of arranging atoms one-by-one and seeing what happens. In this
speech, Feynman not only anticipated many of the new research paths that are being
explored today, but he also captured part of the spirit of nanotechnology research
when he invoked many different research disciplines collaborating and exchanging
ideas. So visionary was Feynman’s speech that it is still quoted and honoured today.
For instance, Fig. 1.1 shows a passage from that speech written at the nanoscale.
It turns out that the use of nanotechnology goes back even further than Feynman’s speech. Further, in fact, than most of what we would consider to be modern
technology. Glass samples dating back to Rome in the 4th century A.D. show that
their stained glass got its colour from different sizes of metal nanoparticles [1, 4].
While it is almost certain that Roman glass blowers were unaware of the physical and
chemical processes they were harnessing in making this coloured glass, it is certainly
evidence that it is only the concept, and not the use, of nanotechnology that is new
to humanity. Finally, while people have been using nanotechnology of some form
for almost two thousand years, natural evolution has been working at the nanoscale
far longer than that. The abalone, a type of mollusk, builds its shell by organizing
nanostructured bricks made out of calcium carbonate. Because of this nanostructure,
any cracks that start on the outside of the shell cannot propagate through [4]. Just
consider the possibilities if we could figure out how to do this with modern materials!
CHAPTER 1. INTRODUCTION
Figure 1.1: Part of Feynman’s 1959 speech, written in letters
a couple of hundred nanometres in size, as patterned by the
Mirkin group at Northwestern University [3]. At no more than
several microns across, this paragraph would certainly fit on
the head of a pin.
3
CHAPTER 1. INTRODUCTION
4
While this discussion casts a bit of light on the long and varied history of nanotechnology, it offers very little insight into the nanotechnology of today. Whatever
else it may be, nanotechnology is a rapidly growing field. Some estimates say that,
since 1997, worldwide government funding of nanotechnology research has increased
five-fold and private investment is contributing at least as much [5]. It is not a small
amount of money, either. The U.S. government, for instance, invested more than $1.5
billion in nanotechnology in 2003 [1]. In keeping with our society’s love of defining
things, especially those things upon which we spend a great deal of money, the National Nanotechnology Initiative† has defined the main features of nanotechnology as
being [1, 5]
1. Research and technology development at the 1 nm to 100 nm range
2. Creation and use of structures that have novel properties because of their small
size
3. Building on the ability to control or manipulate at the atomic scale
In short, nanotechnology covers any structure, device, or idea where weird and wonderful things happen because it is small enough that dimensions can reasonably be
considered by counting the number of atoms involved. It is no wonder, then, that the
field is so wide and moving in so many different directions.
†
The NNI is a program of the U.S. government to coordinate their nanotechnology research and
development.
CHAPTER 1. INTRODUCTION
1.2
5
Motivation
Research in nanotechnology can be said to be proceeding along two parallel and often
complementary streams: that geared towards real-world (i.e. macro) applications, be
they commercial products such as nano-infused clothing [6] or environmental projects
like clean energy and pollution reduction [5], and that geared towards what could be
called pure science, the push to further our understanding of the basic properties of
our world. An example of the latter, and one which is particularly relevant to the work
presented in this thesis, is the development of ever-smaller nanomechanical systems
and ever more sensitive measurement schemes in the quest to reach the Standard
Quantum Limit, a measurement whose sensitivity is limited not by the equipment
used but by the Uncertainty Principle and the fundamental quantum nature of the
device [7, 8].
Rather like the drive to ever smaller devices in the semiconductor industry, there
is a continuing push towards smaller devices and greater sensitivity in the field of
nanomechanics and nanoelectronics [9]. Nanomechanical devices have a wide variety
of uses, such as force or magnetic field detectors. These types of devices have recently
been able to measure masses with zeptogram (10−21 g) resolution and the spin of a
single electron [7, 10]. Nanomechanical resonators may even find uses in quantum
computing before too long [10, 11].
In the theme of pushing the boundaries of mechanical devices ever smaller, Gorelik
and Isacson et al. [12, 13] in 1998 proposed a structure where a small mobile metal
island, electrically isolated from its surroundings, mechanically transports electrons
across a gap between two leads. This is intersting because, properly tuned and optimized, a nanomechanical electron “shuttle” could find a use in metrology as a current
CHAPTER 1. INTRODUCTION
6
standard based on the periodic transfer of electrons through the device [14] or in the
design of nanogenerators and nanoswitches [15]. With the addition of some form of
gate structure, an electron shuttle could also behave as a mechanical single electron
transistor (SET) [16], which is significant because SETs are the fastest and most
sensitive electrometers that exist today [17]. This is doubly interesting because traditional SET devices are limited by higher order tunnelling effects that are strongly
suppressed in mechanical devices [14, 17].
1.3
Statement of Purpose
The purpose of the project presented in this thesis is to design, fabricate, and test
a device that operates as a nanomechanical electron shuttle. The design portion
involves modelling the device to determine its electrical and mechanical properties
and to make sure that these properties are appropriate to the electron shuttle goal.
The bulk of the work will be the fabrication of the device, as this will require the
development and integration of three different fabrication recipes: the definition of
the overall layout of the device, the freeing of the mechanical structure, and the
creation of small gaps between the island and the leads. For this last part of the
fabrication, an electromigration technique will be used. This technique allows for
the controlled creation of nanometre-sized gaps in small metal wires. It is hoped
that this technique will allow for the creation of very small gaps, something that
would require only a small oscillation amplitude to function. This could allow the
fabrication of mechanically stiffer devices than has been seen to date, and thus allow
higher frequency electron shuttles to be explored in future work. As well, a small
force would be required to excite a small amplitude oscillation, allowing the use of
CHAPTER 1. INTRODUCTION
7
smaller driving voltages than has been possible to date. It is hoped that this will
allow for the observation of lower-energy quantum effects that have been predicted,
but not yet seen, in nanomechanical shuttle devices.
The scope of the work presented in this thesis is somewhat less than the project
purpose would suggest. Most of the work presented here pertains to the development
and testing of the nanolithography recipes necessary for the fabrication of these devices and to the use of electomigration to create the island-lead gaps. At this stage
of the project, the modelling necessary for the design of an optimal shuttle structure
is largely ignored. As well, some additional device features necessary for electromigration but detrimental to shuttle performance are introduced, but the later removal
of these structures for shuttle testing is not considered. These aspects of the project
will need to be dealt with in future work.
Chapter 2
Literature Review
I reject your reality and substitute my own.
— Adam Savage
2.1
Single Electron Shuttle Theory
In its purest form, a nanomechanical electron shuttle is a device where a small conducting island is mechanically attached to electrical leads by a deformable, elastic,
and highly resistive material. The island is made to oscillate between the leads and,
due to an electric field applied between the leads, electrons move onto the island when
close to one lead, travel on the island, and move onto the second lead at the far end of
the island’s motion [15]. This results in a flow of current. The concept of the electron
shuttle was first described by Gorelik et al. and Isacsson et al. in 1998 [12,13]. Their
proposed system is shown in Fig. 2.1; it consists of a 1–5 nm metallic grain vibrating
in a high resistance elastic medium between two electrodes.
In order for the island to act as a shuttle, two assumptions must be (for the
8
CHAPTER 2. LITERATURE REVIEW
9
Figure 2.1: (a) Simple model of the shuttle system proposed by
Gorelik et al., where a small metallic grain is elastically linked
to two electrodes. (b) Schematic of the loading and unloading
of electrons during the motion of the island [12]
moment) arbitrarily made: that the amplitude of the oscillation is much larger than
the electron tunnelling distance and that there is some limit to the number of electrons
on the island. When this is true, the island’s motion and charge fluctuation are
strongly coupled. For the case of a symmetric system, the charge-position profile is
shown in Fig. 2.2. In this system, a potential of V is placed across the leads; it is
referenced to the potential of the neutral island, so the island sees potentials of ±V /2
on the left and right leads. Once the island is moving,† the shuttling action can be
clearly seen. When the island is close to the negative lead, 2n electrons are loaded
onto the island. This replenishes the n electron deficit resulting from unloading at the
positive lead and also loads n extra‡ electrons onto the island. Thus, when combined
with the electric field between the leads, the island experiences a Coulomb force
†
‡
The question of how this motion starts will be dealt with later.
Compared to a neutral island
CHAPTER 2. LITERATURE REVIEW
10
Figure 2.2: Charge profile of an oscillating metallic grain. Electron transfer only occurs for |x| > L and, even then, only occurs
with the near lead [12]. Note that the negative lead is in the +x
direction from the shuttle; this sign convention is important in
the discussion of Eq. 2.3.
towards the other lead. When the island gets close to the positive lead 2n electrons
leave the island, resulting in an island that is missing n electrons. This causes the
electric field to accelerate the island back towards the negative lead and the whole
process repeats itself.
2.1.1
Island Motion
In the years since this type of structure was first proposed, there has been a great deal
of numerical and analytical examination of the mechanical and charge transport properties that could be exhibited by various implementations of the mechanical electron
shuttle concept. While most of this work treats electrons as semi-classical particles,
there is a pronounced divide when it comes to the treatment of the shuttle motion
itself. This depends entirely on the details of the shuttle structure being considered,
which can be classified as either classical [12–14,16,18] or quantum mechanical [19–22].
Although the nature of the quantum-classical transition is still poorly understood and
11
CHAPTER 2. LITERATURE REVIEW
a subject of much research [23], in this case the criterion is fairly straightforward: in
order to be able to treat the mechanical motion classically, either the amplitude of the
shuttle’s motion, x must be larger than its zero-point vibration amplitude, x0 , or x0
must be much less than the tunnelling length λ [15], where the zero-point amplitude
is given by
x0 =
r
~
.
mω0
(2.1)
Here, m is the mass of the oscillator and ω0 is its mechanical vibration frequency.
To put this requirement in perspective, using the value for gold of λ ≈ 0.84 ņ and
a reasonable frequency for a mechanical shuttle of 100 MHz [24], this means that the
mass of the shuttle must be much larger than ∼ 10−20 kg for the shuttle to behave
classically. That is on the order of merely 106 silicon atoms,‡ so it is safe to say that
a shuttle constructed using the techniques in this thesis will exhibit classical motion.
The shuttle’s motion in the classical regime can be described by Newton’s equation
of motion [12, 13, 18]
mẍ = Eq(t) − γ ẋ − kx,
(2.2)
where x(t) is the displacement of the island from rest, γ is the constant for the
damping force, k is the spring constant of the system,§ E = V /d is the electric
field between the two leads separated by distance d, and q(t) is the charge on the
island at time t. Once the shuttle motion has reached some steady state, it can be
characterized by an oscillation amplitude and frequency. To determine the amplitude
of the oscillation, consider what is driving it: the electric field acting on the shuttle.
†
‡
§
See Sec. 2.1.2 and Eq. 2.6
Using the mass of silicon
q of about 28 amu.
With, as usual, ω0 =
k
m
[13].
12
CHAPTER 2. LITERATURE REVIEW
The work done by this force on the shuttle is [12]
W =E
Z
ẋq(t)dt.
(2.3)
From Fig. 2.2, 2n electrons tunnel on or off the island when the shuttle is at the
limits of its motion, i.e. when the shuttle’s velocity changes direction. Obeying the
sign convention from that figure, this means that the product ẋq(t) is (almost) always
positive and so the electric field is always accelerating the grain. It follows, then, that
any small deviation of the island from its equilibrium position would lead to ever
larger oscillations. Since the shuttle will always have some thermal vibration, this
answers the question of how the shuttling motion begins. However, there is now an
electromechanical instability that is giving rise to an increasing oscillation amplitude.
The solution to this lies in the mechanical dissipation that must exist in any physical
system. This dissipation, as is seen in Eq. 2.2, scales with the speed of the island.
Each cycle, it dissipates energy equal to [12]
Wγ = πω0 γA2 ,
where A is the oscillation amplitude. Eventually, the oscillation amplitude will reach
a point where W = Wγ and an equilibrium oscillation will be reached. For a periodic
island motion x(t) = A sin(ω0 t), this amplitude is given by [12]
λΩV C
,
A= √
e γω0
r
where Ω ≡
e2
λCd
(2.4)
for an island with capcitance C; λ is the characteristic electron tunnelling length.
CHAPTER 2. LITERATURE REVIEW
13
As one might expect, the amplitude of oscillation depends directly on the voltage
applied between the leads and inversely on the strength of the mechanical damping.
This, however, can only be true for voltages greater than the critical voltage required
to move the island far enough to cause significant changes in the tunnelling rates.
This V > Vc condition is also seen in the shuttling simulation results. It should be
noted that these results were derived for the symmetric case of R0,L = R0,R . It has
been found, however, that this treatment still applies, with small corrections, for the
asymmetric case [12].
The vibration frequency is not as straightforward to determine, since analytic
solutions to Eq. 2.2 are extremely difficult to find for all but the simplest of geometries. Generally, numerical techniques are the only way to solve real systems [25];
commercial software packages exist to do this and the mathematical techniques and
approximations necessary are well known. One interesting feature to note, however, is
what happens if the elastic restoring force is much greater than either the damping or
the electric forces. In this case, the shuttle frequency is simply the natural mechanical
resonant frequency of the shuttle structure [12].†
The island excitation mechanism described here is, in fact, only one way to excite
motion resulting in electron shuttling. This self-excitation due to an electromechanical instability, the so-called “shuttle instability” [12, 13], is also conceptually the
simplest, which is why it was explored first. Beyond self-excitation, it is also possible
to drive the shuttle with a periodic voltage on one or more additional electrodes, capacitively coupled to the island and/or the island support, and to cause shuttling by
exciting the vibrational modes of a molecule bridging the gap between two leads [15].
†
In fact, there are many discrete mechanical eigenfrequencies, corresponding to different modes
and harmonics, but only one will be excited at a time.
CHAPTER 2. LITERATURE REVIEW
14
Systems exploiting this latter mechanism have been fabricated and tested [26] as well
as theoretically analysed [27]; it is an interesting phenomenon, but that type of system
and the corresponding physics are beyond the scope of this thesis. Electrode-driven
shuttles have also been fabricated and measured [24, 28] and are mechanically very
similar to self-excited shuttles. There are two main differences. The first is that, in
addition to the potential VSD applied across the source and drain leads, there is a
periodic voltage Vdrive applied at the electrode(s) to drive the shuttle. Ideally, VSD
would drive the electron tunnelling and Coulomb blockade while Vdrive would only
affect the motion of the island. In practice, however, there is cross-coupling between
the driving electrode and the leads which changes the tunnelling [29]. The second
difference is that, since the shuttle’s motion is being driven by an external field, motion is not limited to the mechanical eigenfrequencies. The shuttle can move at any
frequency, albeit at a greatly reduced amplitude (and thus reduced source-drain current). Considering the greatly enhanced motion on resonance, being able to move the
shuttle at any frequency does not initially seem like a significant advantage. However,
usual shuttle structures have several eigenfrequencies, and driving with an external
field means that the shuttle can be operated at any of those frequencies instead of just
the frequency which happens to get excited in the shuttle instability scheme. This
could be useful for designing a shuttle with a specific mode spectrum, something that
could be used for a filter, and that minimizes leakage currents and 1/f noise [24].
2.1.2
The Arbitrary Assumptions
At the beginning of this chapter, two assumptions were stated regarding the operation
of the electron shuttle: that the amplitude of that mechanical oscillation was much
15
CHAPTER 2. LITERATURE REVIEW
higher than the electron tunnelling length and that the number of extra electrons on
the island was limited. These assumptions were not justified at the time and they will
now be revisited, starting with the oscillation amplitude requirement. As is shown in
Fig. 2.2, charge transfer between the island and the leads only occurs when the island
is closest to the lead and not when the island is travelling from one lead to the other.
This is because the tunnelling resistance between the island and a lead increases
exponentially with the distance between them, a relationship quantified as [12]
RL,R (x) = R0 e±x/λ ,
(2.5)
where the L and R subscripts denote the left and right leads, x is the displacement of
the island from its rest position, and λ is the characteristic electron tunnelling length
given by†
√
λ
−1
=
2me Φ
.
~
(2.6)
Depending on the materials in question, values for λ can fall anywhere between 0.05–
3 Å [13]. For gold, which has a work function Φ ≈ 5.38 eV [31], this works out to
λ ≈ 0.84 Å.
An experimentally- or engineering-minded person might be inclined to stop here,
as having a motion that is larger than an ångström is hardly a tall order. In the
interests of completeness, however, consider the case where the vibration amplitude
is similar to the tunnelling length. In this case, x ≈ λ always. This means two
things: that tunnelling can occur between the island and both leads simultaneously
and that tunnelling can occur from one lead to the other, albeit at a smaller rate
†
This can be derived by looking at the barrier potential problem in quantum mechanics [30].
CHAPTER 2. LITERATURE REVIEW
16
than to the island. Since the flow of current through the device in this state is not
dependent on the location of the island at any given time, electron transport does
not occur in discrete packets and any shuttling is incidental. As this is clearly not the
desired result, this leads to the requirement that A λ for proper shuttling. This
feature is also one of the significant advantages to nanomechanical single electron
devices. In standard SETs, higher-order tunnelling processes, such as co-tunnelling
and simultaneous tunnelling between the island and both electrodes, are a significant
factor [14]. Since these processes are suppressed in the nanomechanical shuttle, they
are capable of better current accuracy without the need for complicated designs.
The second assumption was that the number of electrons on the island was limited.
There are two ways of achieving this, using the motion of the island to limit the charge
or having a Coulomb blockade. In the first case, the vibration frequency of the island
must be larger than the charge fluctuation frequency, ω > ν = 1/RC. This means
that the island will move away from the lead before the tunnelling process is complete,
i.e. before the island is “full” of electrons. This leads to a “chopping” of the electron
tunnelling and thus limits the number of electrons on the island [24]. While this
method will certainly work for limiting the number of electrons on the island, the
number of electrons shuttled per cycle will change with the mechanical frequency,
thus making this type of operation sensitive to a variety of factors. Also, operating in
this regime means that the contact time is small. As will be shown in Sec. 2.1.4, the
best electron shuttle accuracy is obtained in the limit of long contact times. Clearly,
these two requirements are in conflict.
The other way of limiting the size of the shuttled packets is to have a Coulomb
blockade in the system. In this regime, the number of electrons shuttled is limited
CHAPTER 2. LITERATURE REVIEW
17
by the physics of the relative energy levels of the leads and the shuttle as well as the
available electronic states on the island. As such, the device performance is limited
in a well-understood way by the potential on the device and the temperature rather
than the vibration frequency.
2.1.3
Coulomb Blockade
The charge transfer from one lead to the other depends entirely on the shuttling
motion of the island, so it is no surprise that the current through this system depends
on the frequency of the motion and the “shuttling capacity” of the island. This
current is given by [12]
I = 2nef,
(2.7)
where f = ω0 /2π is the mechanical shuttling frequency of the island, e is the electron
charge, and 2n, as above, is the number of electrons shuttled per cycle. Nanomechanical systems can have very high quality factors [9], which means that f is precisely
set and thus I is well known for a certain n.† At this point, the observant reader will
notice that the quantity n has appeared several times without really being addressed.
Indeed, much of the discussion of electron shuttling to this point has revolved around
the assumption that the island can only support some definite number of extra electrons. For operation frequencies that do not induce strong “chopping”, the validity
of this assumption depends on the shuttle island demonstrating Coulomb blockade
behaviour [14]. A Coulomb blockade occurs when the Fermi energy levels of the leads
lie between the n and (n + 1) energy levels of the island [32]. In this state, no additional electrons can move onto or off the island unless the relative energies change. If
†
Or vice versa
18
CHAPTER 2. LITERATURE REVIEW
a potential difference is applied between a lead and the island, this shifts the Fermi
energy of the lead relative to the island energy levels, allowing electron flow until all
the island states are filled below the Fermi energy and empty above. This process is
illustrated in Fig. 2.3. The surplus (or deficit) of electrons is given by [12]
1.
2.
3.
4.
∆E
(a)
(b)
Figure 2.3: (a) Electrons are energetically blocked from tunnelling onto the island (centre) as there is no empty state accessible. (b) A relative change in the energy levels allows an
electron to tunnel from the source into a state on the island (1–
2) and then off the island to the drain (3–4). However, while
the electron is on the island, the accessible state is filled and
no more electrons can tunnel onto the island [33].
CV
1
n=
+
,
e
2
(2.8)
where V is the potential across the leads and C is the capacitance of the island. In
addition to energy gained from the electric field, electrons also have thermal energy.
In order for the Coulomb blockade to exist, the electrons in the system must not have
enough thermal energy to overcome the barrier onto the island in the absence of an
electric field, i.e. they must have less thermal energy than the charging energy of the
19
CHAPTER 2. LITERATURE REVIEW
island. This means that, for the Coulomb blockade to exist and thus for shuttling to
occur, the temperature of the system must be such that [12, 32]
kB T e2
.
C
(2.9)
For a room-temperature Coulomb blockade, this results in a required capacitance of
10−18 –10−19 F. This improves, however, to ∼ 10−16 F at liquid helium temperatures.
Another condition for the Coulomb blockade is that [13]
~
e2
,
RC
C
(2.10)
which works out to R & 4 kΩ.
Gorelik et al. found numerical solutions to the equation of motion of the island,
described in Sec. 2.1.1 and given in Eq. 2.2, and to the current through the system,
given by [12]
ν
I=
2T
Z
0
T
e−x/λ
X
Γ(n, n + 1)Pn dt,
n
where ν = 1/RC is the charge fluctuation frequency, T is the oscillation period of
the shuttle, Γ(n, n + 1) is the dimensionless tunnelling rate, and Pn is the charge
distribution function. The resulting normalized I − V curves are shown in Fig. 2.4
for different amounts of damping. There are several important features shown in this
result. The first is the appearance of “steps” in the current for increasing voltage.
This is known as a “Coulomb staircase”; it is a feature of the Coulomb blockade and
shows that the current flow is due to a discrete number of electrons. It is indicative
of the relative energy levels shifting to allow one more electron to tunnel onto the
island as it oscillates; this interpretation is also suggested by looking at Eq. 2.7. The
CHAPTER 2. LITERATURE REVIEW
20
Figure 2.4: Numerical solutions of the normalized currentvoltage relationship for different amounts of damping in the
electron shuttle system. Note the characteristic Coulomb
blockade steps that appear above a critical voltage. Ω is defined
with Eq. 2.4 [12].
second feature is that the Coulomb staircase, and thus the shuttling of electrons,
only exists above some critical voltage Vc . This is explained in Sec. 2.1.1 and is a
feature of a shuttle mechanism with some mechanical stiffness and damping. The
third feature of interest is that Vc increases with higher damping (γ) until, for large
damping, there is no shuttling. The behaviour of Vc with the damping suggests that
mechanical damping should be minimized in the system to obtain optimal shuttling
behaviour.
2.1.4
Shuttle Accuracy
If nanomechanical electron shuttles are to be of any use at all, especially as precision
current sources, some thought must be given to the level of accuracy and precision
CHAPTER 2. LITERATURE REVIEW
21
possible with these devices. Soon after the proposal and description of the shuttle,
Weiss & Zwerger [14] worked to extend Gorelik and Isacsson’s analysis with an eye
towards predicting shuttle accuracy and examining the limits of Eqs. 2.7 and 2.8. In
this case, a useful metric is the root mean square deviation of the number of electrons
transferred. It is given by
∆N ≡
p
hN 2 i − hN i2 ,
where hN i is the average number of electrons transferred per cycle and is expected
to be equal to 2n.
To characterize the behaviour of the current through the shuttle, Weiss & Zwerger
started with a master equation for the probability p(m, t) of finding m extra electrons
on the island at time t [14], given by
h
i
d
(+)
(+)
(−)
(−)
p(m, t) = − ΓL (m, t) + ΓR (m, t) + ΓL (m, t) + ΓR (m, t) p(m, t)+
dt
h
i
(+)
(+)
ΓL (m − 1, t) + ΓR (m − 1, t) p(m − 1, t)+
h
i
(−)
(−)
ΓL (m + 1, t) + ΓR (m + 1, t) p(m + 1, t).
(2.11)
Equation 2.11 may look intimidating at first glance, but it is simply the combination
of the electron gain and loss terms as given by the tunnelling rates Γ(m, t). These
rates are given by [14]
Γ=
where ∆E ∝
e2
.
C
1
e2 R
∆E
h
i,
∆E
1 − exp − kb T
(2.12)
Both R and C are time dependent, however the exponential dis-
placement (and thus, time) dependence in R dominates.† As such, the tunnelling
†
For the sake of clarity, it should be noted that the subscript R, as in ΓR , refers to the right lead
(contrast with ΓL ) and must not be confused with the resistance R.
22
CHAPTER 2. LITERATURE REVIEW
rates can be factored into
(∓)
(∓)
ΓR (m, t) = gR (t)ΓR,m (t).
In this form, the strong time dependence of R is contained in the gR (t) term and
(∓)
ΓR,m (t) depends solely on the linearly-varying capacitance. This second term is given
by
(∓)
ΓR,m (t)
± m+
1
h =
τ 1 − exp − ± m +
CL (t)V
e
− 12
i,
CL (t)V
1
e2
− 2 CΣ (t)kB T
e
with the time constant τ = RR (tmax )CR (tmax ) and CΣ = CR + CL . The “max”
subscript is used to indicate the time (tmax ) or displacement (xmax ) when the island
is closest to the lead. An equivalent expression exists for the left electrode, with the
indices R and L exchanged, V → −V and x(t) → −x(t). The function g(t) is of
a much simpler form, provided the capacitance is assumed to vary linearly with the
shuttle motion, i.e. CL (t) ∝ (x0 + xmax + x(t))−1 . In this case,
x0 + xmax − x(t)
xmax − x(t)
gR (t) =
exp −
.
x0
λ
This function is strongly peaked at tmax . This makes sense, since the tunnelling rate
to an electrode is highest when the island is closest to that electrode.
The theory to this point is sufficient for a numerical solution to show how the
shuttle packet size and root mean square fluctuations develop with bias voltage, temperature, and shuttle frequency. However, another interesting parameter falls out if
one final substitution is made: the replacement of gR (t) with g̃R (t), a step function
23
CHAPTER 2. LITERATURE REVIEW
having the same height and area,
g̃R (t) ≡



1 : tmax − t0 < t < tmax + t0 ,
.


0 : otherwise
Written this way, 2 t0 corresponds to an effective contact time in which tunnelling
can occur. It is given by
1
t0 ≡
ω
r
πλ
2xmax
λ
1+
.
2x0
(2.13)
As one might expect, the contact time goes down as the vibration frequency increases.
As well as introducing t0 , the step function approximation allows the master equation
to be solved analytically.
Weiss & Zwerger first looked at the low-temperature limit to see the effect of
contact time. Their results are shown in Fig. 2.5 for both the analytical and numerical
solutions (which can be seen to agree). A few features are evident. First, the Coulomb
(a) Average number of electrons transferred
(b) Root mean square fluctuations
Figure 2.5: Shuttle performance at T = 0 for increasing contact
times. Analytical solutions are lines, numerical solutions are
crosses [14].
CHAPTER 2. LITERATURE REVIEW
24
blockade effect can clearly be seen for long contact times, t0 τ , and no current flows
for very short contact times, t0 τ . Second, these results agree with Fig. 2.4 in that
there is a critical voltage below which current does not flow. Finally, the current
fluctuations also depend on the contact time. They disappear for both long and short
contact times and are at a maximum of ∆N ≈ 0.7 for t0 /τ ≈ 1. At low temperature
and for a maximum of one extra electron on the island, it was found that the number
of electrons transferred is given by [14]
1 − a3
,
[1 + a][1 + 12 a + a2 ]
6 + 9a + 22a2 + 13a3 + 10a4
(∆N )2 = 2a(1 − a)
,
[2a2 + a + 2]2 [a + 1]2
hN i = 2
(2.14)
where a ≡ exp − tτ0 . Note that a → 0 in the limit of long contact times and a → 1
for short contact times. Equation 2.14 reduces to Eq. 2.8 in the limit of t0 τ and no
electrons are transferred for t0 τ . These results suggest that operating the shuttle
with long contact times will give the best performance.
The shuttle’s calculated behaviour at non-zero temperatures is shown in Fig. 2.6.
As expected, the Coulomb staircase disappears with increasing temperature. What is
interesting is that the current fluctuations are strongly suppressed at low temperature
and at the midpoint of a Coulomb step. The mid-step improvement in accuracy was
also seen in the T = 0 case. This suggests that the shuttle should be operated in the
middle of a Coulomb step as opposed to on a step edge for optimal accuracy.
In this analysis, then, Weiss & Zwerger confirmed the expression expected by
Gorelik et al. for the number of electrons transferred per shuttle period, subject
to the condition of long contact time. They also confirmed the expected Coulomb
25
CHAPTER 2. LITERATURE REVIEW
(a) Average number of electrons transferred
(b) Root mean fluctuations
Figure 2.6: Shuttle performance for t0 τ and increasing
e2
temperature. Energy scale is Ec = 2C
[14].
Σ
blockade behaviour for low temperature and large amplitude. Most importantly,
however, they showed how the behaviour of the shuttle changes as the system gets
farther from the requirement of kB T e2 /2C and they predicted the behaviour of
the root mean fluctuations in the average number of electrons transferred. This latter
point is especially important in order to push the limits of accuracy.
The master equation-based treatment of Weiss & Zwerger was focused on here
because it is instructive in examining the properties of the electron shuttle while
still being conceptually straightforward, much in the spirit of the original proposal
by Gorelik et al.. Other approaches exist in the literature, however. For instance,
Chtchelkatchev et al. [16] looks at the electron shuttle in the context of single electron
transistor performance. Armour & MacKinnon [19] and Fedorets et al. [20] both look
at the electron shuttle in the quantum limits of motion and charge. However, these
approaches all predict similar qualitative characteristics and so they are not examined
here. These different approaches are still quite interesting, and they are mentioned
here in case the interested reader wishes to pursue them further.
CHAPTER 2. LITERATURE REVIEW
2.2
26
Previous Shuttle Implementations
The nanomechanical electron shuttle does not exist solely in the realm of theory. Various research groups have managed to realize devices exhibiting this type of behaviour
that make use of all three excitation methods: shuttle instability, external driving,
and molecular vibrations [15]. Molecular vibration-based shuttles, as was mentioned
earlier, are different enough from the rest and from the work in this thesis that they
will not be explored here. Of the remaining shuttles, the majority to date have been
externally driven.
2.2.1
Externally Driven Shuttles
Externally driven shuttles are, arguably, the easiest to achieve as they require nothing
more than the application of the shuttle ideas to the reasonably well-understood area
of nanomechanical beams. With the inclusion of an external driving signal, this type
of shuttle also has the potential to be used as a nanomechanical SET. These shuttles
are very well represented by the work of Erbe et al. [24, 29], who essentially scaled
a classical bell down to the nanoscale. An SEM image of their first device is shown
in Fig. 2.7. This device consists of an island (C, for “clapper” – their term, which
flows from the bell comparison) at the end of a singly-clamped suspended beam, 1 µm
long × 150 nm wide × 190 nm thick. The beam is driven by an oscillating voltage
on two gates, G1 and G2 with a π phase shift between them, at frequencies up to
100 MHz. This causes the island to move between the source, S, and the drain, D.
The clapper is biased with some DC voltage VCD and the source is grounded. The
resultant clapper-drain current is amplified and measured. The device was patterned
using optical and electron beam lithography on a silicon-on-insulator (SOI) wafer.
CHAPTER 2. LITERATURE REVIEW
Figure 2.7: SEM image of the first “quantum bell” structure
produced by Erbe et al.. The image is approximately 11 µm
across. The inset shows the equivalent circuit schematic of this
device.
27
CHAPTER 2. LITERATURE REVIEW
28
This wafer consists of 190 nm of Si (the mechanical layer) on 390 nm of SiO2 (the
sacrificial layer) on a silicon substrate. After lithography, they deposited a 1.5 nm
thick NiCr adhesion layer, 50 nm of Au, and a 30 nm thick Al etch mask. They then
used reactive ion etching (RIE) to etch down 600 nm and finally used an HF etch
to remove the SiO2 and thus mechanically free the clapper. Their 300 K results are
shown in Fig. 2.8(a); the 4.2 K results were qualitatively similar, but a larger VCD
was required before shuttling began and the overall current was smaller.
The current-frequency profile in the figures show strong peaks due to the mechanical resonances of the beam, indicating that shuttling is occuring. These peaks have
small quality factors, ranging from Qη1 ≈ 100 to Qη3 ≈ 15.
The inset of Fig. 2.8(a) shows the number of electrons shuttled per cycle, N =
I/ef , as a function of frequency. Below ∼20 MHz, 103 -104 electrons were being shuttled. On the 73 MHz peak, this was reduced to ∼ 130 electrons. As there is no
Coulomb blockade at work here,† this is showing the clipping effect discussed in
Sec. 2.1.2. In Fig. 2.8(b) results are shown for different clapper-drain biases. The
inset shows the current at 73 MHz as a function of VCD with an exponential fit overlaid. This shows that the current increases with the bias, which is not unexpected
since increasing bias lowers the relative energy level of the island and allows more
electrons onto the island, and confirms that the resistance of the tunnelling gaps has
the exponential form given in Eq. 2.5. From this exponential relationship, Erbe et al.
calculated a peak displacement of xmax ≈ 5 nm.
As a final measure of this device’s performance, Erbe et al. looked at the effect
of reducing the driving voltage VGpp . Again at room temperature, they aimed for a
†
A variety of factors prevent Coulomb blockade in this structure, most importantly the lack of
an electrically isolated island.
CHAPTER 2. LITERATURE REVIEW
(a) Clapper-drain current for a range of gate driving frequencies with VGpp = ±5 V. The peaks in the data show
enhanced shuttling due to the mechanical resonances of the
beam. Inset: Number of electrons shuttled per cycle. Fewer
electrons are shuttled at higher frequencies.
(b) The effect on the clapper-drain current of increasing
VCD . A stronger bias increases both the overall current and
the noise. Inset: Amplitude of the 73 MHz peak with increasing VCD with an exponential fit.
Figure 2.8: “Quantum bell” results at room temperature from
Erbe et al. [24].
29
CHAPTER 2. LITERATURE REVIEW
30
signal-to-noise ratio of S/N > 3 and reduced the driving voltage until they reached
this goal. They achieved this when they measured a shuttling of 7 ± 2 electrons at
30.5 MHz. This result is shown in Fig. 2.9
Figure 2.9: Current fluctuations around the 30.5 MHz peak at
low gate voltage. The precision is ±2 electrons and the peak
current is 7 electrons. Inset: Current fluctuations in time of
±2 electrons [24].
One issue with this device is that the beam and the island that make up the
clapper are covered in a continuous coating of metal. This means that there is no
real island of the type envisioned by Goelik et al., just a larger structure on the end
of a metal beam. This shortfall was addressed in later work by the group, in which
they used largely the same fabrication procedure as before but with slightly different
mask patterns [29]. This second device is shown in Fig. 2.10; its room temperature
current spectrum is in Fig. 2.11. Like the first device, this one shows a spectrum of
peaks superimposed on a non-zero background current. Since there was no measured
current through the system when the beam was immobilized (Fig. 2.11 top inset),
CHAPTER 2. LITERATURE REVIEW
Figure 2.10: SEM image of a newer quantum bell. Note that the
island on the end of the clapper beam is not metallically connected to the body of the clapper. Because of this, the clapper
is grounded and the tunnelling bias applied across the source
and the drain. The inset shows the circuit schematic [29].
31
CHAPTER 2. LITERATURE REVIEW
Figure 2.11: Room-temperature current spectrum for the improved quantum bell driven at VGpp = ±3 V. Note complicated
peak structure superimposed on a non-zero background. Insets: Top: Current for an immobile beam. Bottom: Amplitude
of a peak for varying driving voltage [29].
32
CHAPTER 2. LITERATURE REVIEW
33
it is safe to say that this signal is a result of beam motion. At lower temperatures,
Erbe et al. found that the background current was suppressed, confirming that it was
due to the thermal motion of the beam, and that the increased stiffness of the beam
resulted in a much simpler peak spectrum. The one remaining peak, at 120 MHz, is
shown in Fig. 2.12. This result shows shuttling of an average of 0.11 ± 0.001 electrons
at 4.2 K. This is a current of 2.3±0.02 pA. At this sensitivity, Erbe et al. predict being
able to resolve the Coulomb blockade structure once they achieve lower temperature.
They estimate needing to be below 600 mK to see the Coulomb blockade in these
devices.
Figure 2.12: Plot of the only resonance peak appearing at
4.2 K. Inset: Electric field interactions between the gate and
island [29].
One final interesting point on these results is that, with the improved shuttle,
Erbe et al. did not see the dependence of current on source-drain bias predicted by
CHAPTER 2. LITERATURE REVIEW
34
Weiss & Zwerger (Sec. 2.1.4). Instead they saw a dependence on the driving voltage.
Considering, however, that the source-drain voltages were smaller than the driving
voltages by a factor of 103 , this makes sense. This is further borne out by finite
element calculations, shown in the inset of Fig. 2.12, that show significant interaction
between the gate and the island. This was taken into account in the analysis of
Chichalkatchev et al. on a gated, oscillating carbon nanotube electron shuttle [16].
Their analysis parallels that of Weiss & Zwerger save that the bias voltage term, ±V /2,
is simply replaced by the difference between the gate and lead voltages, (VR,L − VG ).
One way of retaining the control over the shuttle operation given by driving with
a gate while avoiding the gate voltage dominating over the source-drain voltage is to
change how the shuttle is driven. One approach to this was presented by Koenig et al.
Instead of capacitively driving the shuttle, they used ultrasonic transducers to excite
the shuttle [17]. They accomplished this by placing their shuttle (actually an array of
shuttles) inside a Faraday cage with a piezoelectric transducer attached to the outer
wall. A schematic of this system is shown in Fig. 2.13(a). The Faraday cage isolates
their system from any external electric fields, such as those used to actuate the piezoelectric base. They found that the mechanical motion caused by vibrating the piezo
was transmitted through the cage wall and excited vibrations in the shuttles. The
results of this device are shown in Fig. 2.13(b) for a temperature of 20 K. The current
shows peaks at certain frequencies where there is increased shuttling. These different
peaks correspond to the eigenfrequencies of different shuttles in the array. These results show two things. The first is that Koenig et al. were able to excite vibrations in
the shuttles without the application of an oscillating electric field, meaning that they
do not lose sensitivity to the source-drain voltage the way that Erbe et al. did. The
CHAPTER 2. LITERATURE REVIEW
(a) Schematic of the ultrasonically-driven array of electron shuttles. The piezo is actuated to excite the shuttle
motion. The Faraday cage ensures that no unwanted
electric fields interfere with the shuttles.
(b) Results of the shuttle at 20 K. The different peaks
correspond to the eigenfrequencies of different shuttles
in the array.
Figure 2.13: Schematic and results for the ultrasonically-driven
electron shuttle [17].
35
CHAPTER 2. LITERATURE REVIEW
36
second is that, as one might expect, the frequency response of an array of shuttles has
features corresponding to the mechanical eigenfrequencies of each of those shuttles.
This means that a device could be designed to have a specific shuttling frequency
spectrum without relying on designing a single shuttle with a complicated system of
resonances.
Although they performed the experiment at 20 K, Koenig et al. still did not see a
Coulomb blockade. Their simulations, which corresponded well to the experimental
results, show that a temperature of less than 200 mK is necessary to see a Coulomb
blockade in their devices.
2.2.2
Instability-based Shuttles
The realization of a mechanical electron shuttle that operated due to a shuttle instability was first accomplished by Tuominen et al. [34], who used a 2.06 mm brass
ball suspended between two brass plates as the shuttle and drove it with source-drain
voltages up to 500 V. Although this is clearly not a nano-scale system, it is worth
mentioning because they noticed some of the same voltage, frequency, and charge
relationships predicted by Gorelik et al. [12] and Isacsson et al. [13]. As such, they
felt that this type of macro-scale system could provide general understanding and
insight into the current-voltage behaviour and the interactions with the vibrational
modes of similar nano-scale systems.
One implementation of an instability-based nanoscale shuttle was achieved by
Kim et al. [35]. As shown in Fig. 2.14(a), their device consisted of a 60 nm wide ×
250 nm tall pillar with a 45 nm gold layer deposited on top. This pillar is nominally
in the middle of a 110 nm gap between two electrodes. To make the pillar move,
37
CHAPTER 2. LITERATURE REVIEW
(a) SEM image of the nano-pillar between
two electrodes
(b) Room-temperature results with an exciting AC signal of 19 dBm.
Figure 2.14: Nano-pillar device and results from Kim et al. [35].
they used a bias-tee to combine a small RF waveform with a DC bias voltage across
the source and drain leads. The RF signal they used was not large enough to move
the pillar on its own but, when combined with a large DC signal, resulted in mixing
of the RF signal with the mechanical motion of the excited pillar. This gave a
signal at the difference frequency ∆ω = ωRF − ω0 , which they were able to measure
more accurately than the eigenfrequency itself. The results for a 19 dBm AC signal
superimposed on DC biases from 0 to 16 V are shown in Fig. 2.14(b). They found a
mechanical resonance at 10.5 MHz and, at 16 V, approximately 100 electrons shuttled
per oscillation.
The closest experimental realization so far of Gorelik’s proposed system has been
simulated [36] and fabricated [36, 37] by Moskalenko et al. Using an AFM tip, they
CHAPTER 2. LITERATURE REVIEW
38
placed a 20 nm gold nanoparticle in a gap between two gold electrodes. The nanoparticle was attached to the electrodes by a coating of organic molecules, either a multilayer of 1-octanethiol or a single layer of 1,8-octanedithiol. These thin films serve
as both an elastic medium to keep the nanoparticle in place and as a high-resistance
tunnel barrier between the electrodes and the nanoparticle island. Schematics for the
multilayer (“Type A”) and monolayer (“Type B”) systems are shown in Fig. 2.15.
In the figure, note the similarities between this system and Gorelik’s proposal from
Fig. 2.1: a small nanoparticle is held between two leads by an elastic, highly resistive
medium.
Figure 2.15: Schematic of the 1-octanethiol multilayer (Type A)
and 1,8-octanedithiol monolayer (Type B) systems for holding
a gold nanoparticle between two electrodes [37].
The leads were patterned using electron beam lithography and then 30 nm of gold
was deposited onto a 5 nm titanium sticking layer. Using this technique, they were
able to achieve curved electrodes separated by 10–30 nm. The electrodes were then
CHAPTER 2. LITERATURE REVIEW
39
cleaned with solvents and a ultraviolet/ozone cleaner and a solution of nanoparticles
and free thiol chains was placed on the surface and allowed to dry. This coated the
surface with the thiol thin film and left several nanoparticles close to the gap. After
imaging with an AFM, they used the AFM tip to move one of the nanoparticles onto
the gap. Figure 2.16 shows AFM images of this stage of the process.
Figure 2.16: AFM images showing the moving of a 20 nm gold
nanoparticle onto the gap between two gold leads. Top figures
show images taken at stages during the move while the bottom image shows the final device and the path taken by the
nanoparticle [37].
The responses of the Type A and Type B shuttles are shown in Fig. 2.17 for
an applied DC voltage. As predicted, there is a critical voltage above which the
current through the device shows a marked increase; this is interpreted as the start
CHAPTER 2. LITERATURE REVIEW
Figure 2.17: Current-voltage results for the Type A (top) and
Type B (bottom) shuttles. In each case, the devices show shuttling above a critical voltage, although this voltage is much
higher for the multilayer (A) case. The leakage current is indicated by the dashed line [37].
40
CHAPTER 2. LITERATURE REVIEW
41
of mechanical motion and thus shuttling. To prove this, Moskalenko et al. tested the
current-voltage response without the nanoparticle as well. These results are shown in
Fig. 2.18. The current shows a significant increase when the nanoparticle is added,
suggesting that shuttling may be taking place. More compelling evidence of shuttling
would involve some direct measurements of the oscillations. However, due to the small
size (10−10 m) and high frequency (108 Hz), they found it difficult to obtain reliable
measurements. They propose studying the frequency dependence of the current using
an AC excitation method similar to that used by Kim et al.
Figure 2.18: Current flow with (curve 1) and without (2) the
gold nanoparticle held between the leads. The current shows a
significant increase when the nanoparticle is added, suggesting
that shuttling is taking place [36].
CHAPTER 2. LITERATURE REVIEW
2.3
42
Electromigration
One of the significant fabrication difficulties involved in the creation of an electron
shuttle with good performance is the high precision needed to properly locate the
shuttling island in a small gap between electrodes [36]. Moskalenko et al. used an
AFM to position the island after the gap had been formed. Erbe et al. formed the
gap and island at the same time, but had a large gap and thus had to drive the shuttle
at high amplitude. Another possibility is to form the gaps in-place once the leads and
island have been fabricated. A technique that could do this, called electromigration,
has already been used to form small gaps in metal nanowires.
It should be noted that there is a lot of interesting physics involved electromigration and atomic point contacts. Much of this physics goes above and beyond the background necessary to understand the techniques and results presented in this thesis,
but there are excellent reviews on the topic by Ho & Kwok [38] and Agraı̈t et al. [39]
in case the reader is interested.
2.3.1
The Brief Theory of Electromigration
Electromigration is the motion of atoms in a metal under the influence of a large
current density [40]. It has been studied in some way or other since the mid-1800s
and has long been known as an important failure mechanism in electronic circuitry [38,
41]. The main component of the electromigration force on an atom is the result of
interactions between the atoms of the metal and the flowing electrons associated with
the current. In these collisions, momentum is conserved and so some momentum from
the electrons is transferred to the atoms. Although energy imparted to the atoms in
each individual collision is small, the density of flowing electrons is high, on the order
CHAPTER 2. LITERATURE REVIEW
43
of 105 − 108 A/cm2 , and they are travelling with a large velocity. As a result, the net
effect of all the electrons acts similarly to pressure exerted by a gas and the atoms feel
a force in the direction of current representative of the time average of the electron
interactions [41,42]. Since this force is dependent on many electron-atom interactions,
it follows that the magnitude of the force increases with increased current density.
The current density depends not only on the magnitude of the current, but also on
the geometry of the metal structure in question.
The electromigration force, F, on an atom is usually expressed in terms of an
effective charge z ∗ [38]:
F = ez ∗ E,
(2.15)
where E is the electric field in the material. The effective charge of the atom is a
function of both the valence of the atom and the coupling between the atom and
the moving electrons. Experimentally, it has been found that z ∗ is negative and
large compared to the atomic valence [41]. In addition to the electromigration effect,
current flow through a material with some non-zero resistance will result in Joule
heating. This also has the result of increasing the movement of the atoms in the
metal. However, this motion is not correlated with the current direction and so
the net effect is just that atoms move away from areas with high current density
(i.e. hotter areas). This means that local heating is also an important effect in the
failure of metallic leads [42, 43]. This latter effect, however, can be reduced by the
use of cryogenic temperatures, to the point of making the contribution of heating
almost negligible at the lowest temperatures. The effect of local temperature on
44
CHAPTER 2. LITERATURE REVIEW
electromigration is shown in Black’s equation [44, 45],
A
t¯f = n exp
j
Ea
kB T
,
(2.16)
which gives the mean time to failure, t¯f , of a nanowire with cross-sectional area A and
current density j. The quantity Ea is the activation energy of the metal undergoing
electromigration. Empirical results show that n is usually between 1 and 3, with
n = 2 being the usual approximation [44, 45].
Much of the work in the past several decades has been to reduce the effect of electromigration, with the goal of increasing device reliability and lifetime [38,41]. Lately,
however, electromigration has been used by researchers to controllably break metal
nanowires. This technique can be likened to the slow, controlled failing of a fuse and
the motivation is to push the limits of top-down fabrication. Before electromigration,
feature sizes were limited by the resolution of lithography techniques, ∼ 10 nm for
electron beams [40]. Using electromigration, however, groups have achieved repeatable electrode gaps on the order of a few nanometers [40, 42, 43, 46–52]. An example
of these gaps, from the original paper describing this as a fabrication technique, is
shown in Fig. 2.19.
2.3.2
General Device Structure
The formation of electromigration break junctions makes use of the fact that atoms
feel more force in regions where the current density is higher. This is done by creating
a structure such that the region where the break is desired has a much smaller crosssectional area than the rest of the structure. This means that atoms in this “neck”
feel a very high force and thus move away from the neck due to the higher current
CHAPTER 2. LITERATURE REVIEW
45
Figure 2.19: SEM image of electrodes with nanometre separation formed using electromigration. The left image is before
electromigration, the right image is after electromigration [40].
density in this region. Once out of the neck, the force felt by the atom is reduced.
Consequently there is a net movement of metal atoms away from the neck, resulting
in the eventual formation of a gap in that region [48]. An example of this type
of structure is shown in Fig. 2.20. Most of the devices described in literature use
gold leads deposited on a sticking layer of chromium. The patterning is done using
a combination of electron beam and photolithography [40, 42, 43, 46–50, 52]. The
techniques involved in controlling the electromigration vary, and are discussed in the
following section.
2.3.3
Controlling Electromigration
As described by Strachan et al. [43], there are three regions in the electromigration
process for break junction formation. At high conductance, Joule heating and electromigration cause atoms to move away from the neck once the local temperature
CHAPTER 2. LITERATURE REVIEW
46
Figure 2.20: A representitive structure for the formation of
nanogaps using electromigration. (a) The narrow section of the
devices causes a very high local current density. This ensures
that (b) the nanogap is formed where is is wanted. In this case,
a gap of 10 nm has been achieved [47].
exceeds some critical value (typically around 400 K [42]). This causes the conductance to slowly fall. Once the neck has narrowed to a few atoms, the conductance
shows plateaus and jumps as it falls. These are near multiples of G0 = 2e2 /h, the
conductance quantum. This effect is shown in Fig. 2.21 In this regime, the critical
temperature also falls. Finally, the nanogap is formed when the conductance falls
below G0 and current flow enters the tunnelling regime.
Each group has a slightly different way of controlling their electromigration process. They all, however, consist of sourcing one of voltage or current and measuring
the other to observe how the resistance of the neck changes as the sourced quantity is varied. Most of the groups source voltage and measure current, however
Park et al. used a notably different approach and sourced current instead [40]. The
CHAPTER 2. LITERATURE REVIEW
47
Figure 2.21: Jumps and plateaus in the conductance, normalized to G0 , of a metallic neck in the few-atom regime [43].
voltage-sourcing groups can be further categorized as either having a feedback circuit [43, 46, 49–51, 53] or not [42, 47, 48, 52]. Park et al., also, did not use feedback on
their current control. Representative current-voltage traces with and without feedback are shown in Fig. 2.22. The effect of the feedback is to reduce the voltage once
the resistance has increased by a certain threshold value, which is increased incrementally, and to use this to control the electromigration rate. The groups who do this feel
that it increases their control over the breaking process. In both the feedback and
non-feedback cases, the time to “failure”† is on the order of a few minutes. Although
their methods differ, all the groups agree that the electromigration process must be
controlled in some way. Uncontrolled electromigration results in catastrophic failure
of the structure, giving gaps 100 nm.
†
In this case, “failure” is used to describe the point where the nanogap forms. As such, the failure
of the nanowire marks the success of the electromigration process.
48
CHAPTER 2. LITERATURE REVIEW
(a) Feedback controlled [43]
(b) No feedback [42]
Figure 2.22: Representative current-voltage traces, (a) with and
(b) without feedback, for the breaking of metallic necks using
electromigration. Without feedback, the source voltage simply
ramps up until failure. With feedback, the voltage ramps up
until the conductance changes by a threshold value, at which
point the voltage is reduced and ramping resumes. This repeats
until the neck fails.
2.3.4
Conductivity Regimes
The three electromigration regimes described by Strachan et al. correspond to the
three conduction regimes: macroscopic, mesoscopic, and tunnelling. In the macroscopic regime, the the device structure is large enough that conduction obeys a simple
Ohm’s law relationship [54]
A
G=σ ,
L
where σ is is conductivity of the material and A and L are the cross-sectional area
and length of the conductor, respectively. As the conductor becomes smaller, Ohm’s
law begins to break down and the conductance enters the mesoscopic regime. This
occurs when the characteristic length of the conductor, L, becomes less than the
49
CHAPTER 2. LITERATURE REVIEW
phase coherence length,† Lφ [39]. This regime is also interesting because it tends to
be where electron transport shifts from diffusive (l L, where l is the mean free path
of electrons in the material) to ballistic (l > L) and where a characteristic length of
the conductor becomes comparable to the Fermi wavelength, λF and semi-classical
treatments break down. In the mesoscopic regime, conductivity is described by the
Landauer formula [39, 54]. It has the form
N
2e2 X
τi ,
G=
h i=1
(2.17)
where τi is the transmission coefficient of the ith conduction channel [54]. For a
conductor that is only a single atom wide, the number of conduction channels, N , is
given by [39]
N=
kF a
2
2
,
where kF is the Fermi wavevector and a is the contact radius. For metals, N is usually
between 1 and 3; its exact value depends on the atom’s valence orbital structure. It is
the mesoscopic regime that gives rise to the quantized conductivity seen in Fig. 2.21.
Since the quantization is in near-integer multiples of G0 = 2e2 /h, it suggests that
τi ≈ 1 in this device and the progressing electromigration is removing conduction
channels.
Once the last conduction channel is broken and the conductivity drops below G0 ,
a potential barrier forms across the conductor. This is the tunnelling regime, where
the conductivity drops exponentially with the width of the barrier and the resistance
†
The phase coherence length is the distance over which quantum coherence is preserved.
CHAPTER 2. LITERATURE REVIEW
50
is thus given by Eq. 2.5:
R(x) = R0 e±x/λ .
(2.5)
It is here that a “true,” physical gap is formed in the wire and most electromigration
experiments to date have achieved devices in this regime, although some have achieved
devices in the mesoscopic regime [55].
The fact that electromigration can be used to create devices that fall into two
conductivity regimes raises an interesting question of nomenclature: what to call
the structure created by electromigration? In the tunnelling regime, a gap results
from electromigration and much of the literature describes devices that fall into this
category. On the other hand, electromigration ending in the mesoscopic regime creates
devices that are perhaps better described as thin atomic chains and that may have a
spring-like behaviour when stretched [55]. Indeed, the work presented in this thesis
aims at creating junctions at the low-conductivity end of the mesoscopic regime. In
the discussion of electromigration, however, references to “nanogaps or thin chains
of atoms” are unwieldy and would quickly become tiresome. As a result, this work
will continue to use the gap nomenclature, with the understanding that the actual
structure and behaviour of the final device depends on the resistance.†
2.3.5
Results and Uses
Regardless of whether or not feedback was used or whether current or voltage was
sourced, electromigration has proved capable of forming electrode gaps smaller than is
possible with traditional lithography. The groups that used feedback report that they
†
To provide a numerical point of comparison, the transition from the mesoscopic to tunnelling
regime occurs at about 1/G0 = 12.9 kΩ.
CHAPTER 2. LITERATURE REVIEW
51
were able to form gaps ∼ 5 nm while the non-feedback groups report gaps of 1 − 2 nm.
Caution should be made comparing these two sets of results, however, as there are
more procedural and apparatus variations between groups than simply whether or not
they used feedback and so this difference could be due to other factors. All groups
report being able to repeatably form gaps close to a target width (or resistance), but
few provide numbers to back that up. Strachan et al. [43] report being able to form
gaps within a factor of three from targets in the 0.5 MΩ–1 TΩ range. These results
represent electromigration both at room temperature [42, 43, 48, 51, 53] and cryogenic
temperatures [40, 42, 47].
Electromigration break junctions are useful for any application that would benefit from having electrode spacings less than what is achievable by lithography alone.
These break junctions have been used to define the electrodes for molecular singleelectron transistors [40, 47] and could be useful for a variety of other molecular electronics [43]. In an interesting tie back to the electron shuttle discussion, the group
who made the C60 -based shuttle used electromigration to define a gap in their electrode in which to place the molecule [26]. As well, studying the formation of break
junctions under different conditions has been used to gain a better understanding of
the electromigration process itself [42, 49, 51, 53]
2.3.6
Point Contact Displacement Detectors
A gap formed via controlled electromigration forms as a result of thinning of the neck
until it finally thins to the point of failure. This means that the break in the wire
tends to be relatively large except for at the atomically sharp point where it finally
failed. At this location, a so-called atomic point contact (APC) is formed. Due
CHAPTER 2. LITERATURE REVIEW
52
to the exponential relationship between resistance and distance, this small distance
determines the behaviour of the gap as a whole, and it is this distance that is usually
referred to as the gap width. Atomic point contacts are interesting in part because
they can be used as transducers and amplifiers of a mechanical motion changing the
with of the gap between them [55] and because they can, theoretically, be a quantum
mechanically ideal amplifier [55, 56]. An electron shuttle where the island-lead gaps
are formed by electromigration will, in essence, have a pair of APCs responding to
the displacement of the island, so it is useful to understand how these APC structures
are used as displacement detectors.
As was explored in Sec. 2.1.2, the tunnelling resistance of a small gap depends on
the size of that gap (Eq. 2.5). The problem with this is that it requires knowledge of
both the size of the gap, x, and the x = 0 resistance, R0 . When looking at mechanical
motion, however, it is useful to look at displacement. For an APC with voltage V
across it, resulting in an average current hIi, the current change in response to a small
displacement is given by [55]
dhIi =
1 ∂R
hIi dx.
R ∂x
(2.18)
The displacement sensing properties of atomic point contacts has been explored by
Flowers-Jacobs et al. [55], who used a point contact formed via electromigration to
measure the vibration of a electromagnetically-actuated beam. Shown in Fig. 2.23,
the beam was 5.6 µm long × 200 nm wide × 100 nm thick and had a total mass of
2.3×10−15 kg. Using electromigration at 4 K, they fabricated their APC detector
to have an equilibrium resistance of R0 = 33 kΩ. In testing, they found that their
detector had a displacement response of (1/R)(∂R/∂x) = 0.4 nm−1 , a displacement
CHAPTER 2. LITERATURE REVIEW
53
Figure 2.23: SEM image of the beam and APC used to sense
its motion. The APC has not yet been formed in this image.
Electromigration will cause the triangle attached to the beam
to thin out until the APC is formed [55].
√
√
sensitivity of 2.3±0.1 fm/ Hz, and backaction of 78±20 aN/ Hz. This corresponds
to a total displacement uncertainty of 42 times the standard quantum limit. Using
finite element modelling for small displacements from equilibrium, they estimated the
APC has having the same effect on the beam’s motion as a 180 N/m spring attached
to the beam at the location of the APC.
In order to approach a quantum-limited displacement measurement, the backaction† of the measurement transducer on the system must be minimized. Recent
theoretical work by Doiron et al. suggests that backaction can be reduced or even
eliminated by using cross correlations between two point contact detectors [11]. Two
possible configurations, in phase and out of phase, are shown in Fig. 2.24. The out
of phase detection scheme is particularly interesting from the perspective of an electron shuttle-type device created using electromigration, which essentially has point
contact displacement detectors positioned on either side of a vibrating island. This
suggests an possible additional research path requiring only a few modifications to
the shuttle structure discussed in this thesis.
†
In quantum mechanics, any measurement made on a system has an effect on that system. This
is termed backaction.
CHAPTER 2. LITERATURE REVIEW
a)
b)
Figure 2.24: (a) In phase and (b) out of phase detector layouts
for a cross correlation displacement measurement [11].
54
Chapter 3
Design and Fabrication
[Nanofabrication] is building at the ultimate level of finesse.
— Richard Smalley
3.1
Relative Energy Scales
The operation of nanomechanical shuttles involves the interaction of phenomena on
several energy scales, notably thermal, mechanical, and electrical. Many interesting
effects, for instance, are observable when the thermal energy, given by kB T , is below the others. For this reason, it is useful to compare the energy scales of some
of the existing shuttle implementations and to compare them to the device design
described in this thesis. This comparison is shown in Table 3.1. For the electrical
energies, both the charging energy of the island (required to be larger than kB T to see
Coulomb blockade), e2 /2C, and the energy supplied by the source-drain bias, eVDS ,
are of interest. Where sufficient information was available, the mechanical energy was
calculated from the elastic potential energy stored at the limit of the island’s motion,
55
56
CHAPTER 3. DESIGN AND FABRICATION
1
kx2max ;
2
the quantum mechanical energy of the motion, hfmech , is also given.
Table 3.1: Comparison of the energy scales of some electron
shuttle implementations. All energy values are given in electronvolts (eV).
Erbe
Koenig
Kim
Moskalenko
Thesis device
∗
Thermal
3.6×10−3
1.7×10−3
2.5×10−2
2.5×10−2
10−5 –10−4
Elastic
3600
—∗
—∗
1.6×10−3
19
QM
4×10−7
2×10−8
4×10−8
4×10−8
6×10−7
Blockade
Bias
5.4×10−5 5.0×10−3
1.7×10−4
0.8
−2
1.6×10
16
1.6×10−2
3
−3
3×10
< 0.25
Not enough information
The energy scales for the devices described in this thesis were calculated as follows.
The thermal energy is based on the cryogenic capabilities of the available equipment,
with temperatures from liquid helium (4×10−4 eV) down to 300 mK (2×10−5 eV)
being achievable. The Coulomb blockade energy was calculated using the device
capacitance of 60 aF that was determined with finite-element modelling (Sec. 3.2).
The elastic and quantum mechanical energies are based on a few estimates, since
the details of the mechanical design were only loosely considered over the course
of this work. The vibration frequency of the beam was estimated to be roughly
200 MHz. Based on previous simulations of a beam of similar scale, composition, and
frequency, k = 30 N/m was used as the spring constant of the beam [57]. When this is
combined in series with two point contact junctions, each with kAPC ≈ 180 N/m [55],
it gives a spring constant of k ≈ 23 N/m for the overall system. Finally, the bias
was chosen based on the requirement that electromigration not happen during device
operation. This number reflects experimental observations that electromigration only
occurs above a particular bias voltage.
CHAPTER 3. DESIGN AND FABRICATION
3.2
57
Device Design
Distilled down to the simplest level, the design for the nanomechanical electron shuttle
presented in this thesis must satisfy three necessary requirements. The first, is that
the device consist of a movable metallic island between source and drain leads. In
order for the island to be able to move, it should be patterned on a multilayer substrate
that lends itself to the creation of mechanical structures. The schematic device shown
in Fig. 3.1 satisfies this overall shape requirement, with the understanding that the
island will be separated from the leads via electromigration at the locations marked
Left Gap and Right Gap. In this design, the island is 1 µm across. If this were to be
Figure 3.1: Schematic layout that satisfies the general structural requirements of a nanomechanical electron shuttle. The
source (S), island (I), and drain (D) are marked along with
the two locations where electromigration will be used to define
nanogaps. The vibrating beam, marked in grey, is electrically
insulating.
patterned on a substrate such as silicon nitride on silicon (SiN† wafers) or silicon-oninsulator (SOI‡ wafers), then this pattern could also be used to define a mechanical
structure where the island sits on a non-conducting nanomechanical beam vibrating in
†
Stoichiometric silicon nitride is Si3 N4 . However, the silicon nitride layer on these wafers is
seldom stoichiometric so this notation will be used for simplicity.
‡
Silicon on silicon dioxide on silicon, where the SiO2 acts as a sacrificial etch layer.
CHAPTER 3. DESIGN AND FABRICATION
58
the plane of the page. This beam could either be doubly-clamped, like the one shown
in Fig. 3.1, or clamped at only one end with the island at the other (e.g. remove
the lower half of the beam in Fig. 3.1). The mechanical motion would be highly
dependent on the size and shape of this beam, so this will have to be considered in
the final device design. For the purpose of this general requirement, however, Fig. 3.1
patterned on SiN or SOI wafers is sufficient for the moment.
The second requirement of the shuttle design is that the number of electrons
on the island must be limited. This can be accomplished one of two ways, either
with a Coulomb blockade or with fast island movement. For a Coulomb blockade
at 4.2 K, the island capacitance must be less than 10−16 F and the gap resistance
must be greater than 4 kΩ. The R > 4 kΩ requirement is not difficult to achieve with
electromigration; groups have made gaps orders of magnitude more resistive without
issue.† The capacitance is more difficult. For the design in Fig. 3.1, simulation using
CoventorWare 2008 [58] gives a capacitance of 6×10−17 F (60 aF), which is much larger
than that required for Coulomb blockade at room temperature. However, it is within
the range where a Coulomb blockade should exist at liquid helium temperatures.
This capacitance scales with the volume of the island [59], so it should be possible to
improve this value by reducing the island size once all the fabrication steps are well
characterized and optimized.
For limiting the number of electrons by having the island vibrate faster than
electron tunnelling can fill it up, the frequency of the island must be such that
fmech > 1/RC. For Erbe et al., who had gap resistances ∼1 GΩ and a capacitance
of 25 aF [24], this worked out to fmech > 40 MHz. Their beam had resonances up
†
In fact, achieving resistances as small as 4 kΩ with electromigration is difficult.
CHAPTER 3. DESIGN AND FABRICATION
59
to 400 MHz so this was easily achievable. Other beams have been designed and fabricated in the tens [55] and hundreds [57] of megahertz range, suggesting that this
is a reasonable frequency range for nanomechanical structures on the scale of a few
hundred nanometres to a micron. For the shuttles presented in this project, the aim
was junction resistances on the order of 10–100 kΩ. This gives 1/RC ≈ 200 GHz–
2 THz. Fabricating a structure to vibrate faster than this is not possible with current
techniques for and understanding of nanomechanical structures.† One way to achieve
clipping of the shuttled electrons would be to increase the resistance and capacitance
of the device. Although this would reduce the magnitude of the overall signal, it is
the best option for realizing shuttling if a Coulomb blockade is not achievable. Increasing the capacitance requires increasing the size of the island, something that is
not possible in an already-fabricated device. Resistance, however, can be increased
using electromigration. As such, should an unsuccessful attempt be made at achieving Coulomb blockade, increasing the junction resistances is an option for in situ
adjustment of a device to achieve shuttling in the clipping regime.
3.3
First Steps
The creation of a device like that in Fig. 3.1, which pulls together a variety of different
fabrication steps and techniques, must generally be approached via some intermediate
stages. For this project, the intermediate stage was due to the requirements of forming
two junctions with electromigration. In the ultimate shuttle design, the island is
electrically isolated from its surroundings. However, this means that the two junctions
†
On the other hand, shuttling with a Coulomb blockade requires long contact times. This is
easily achievable if 1/RC is hundreds of gigahertz.
CHAPTER 3. DESIGN AND FABRICATION
60
are connected in series, which is a problem for electromigration. Any asymmetry
between the gaps, whether it exists initially or develops through electromigration,
would result in more power being dissipated in the thinner neck and thus only one
gap would be formed.
The solution to this problem was to deposit metal on the mechanical beam supporting the island and use it as a third electrical contact. The island contact allows
the junctions to be connected in parallel, which allows for greater control of the process and better symmetry between the resulting gaps. Initially the intention was to
remove the third contact prior to device testing, thus maintaining an electrically isolated island for shuttling. It was discovered, however, that the gaps are quite fragile
when formed and removing the island contact became impractical. As the island is
not isolated in this scheme, it will not be possible to achieve a Coulomb blockade.
As demonstrated by Erbe et al. [24], however, it is still possible to see shuttling if
the island can be made to move quickly enough. There are also other interesting
experiments that can be done with two tunnelling junctions to a metal beam [11], so
this is an interesting structure in its own right as well.
To achieve a Coulomb blockade shuttle from this intermediate design, it will be
necessary to remove the electrical contact to the island while maintaining the integrity
of the mechanical beam and the junctions. Doing this is beyond the scope of this
work, but can be explored in future work.
3.4
Fabrication
The fabrication of these devices was divided into three sections: device definition,
device suspension, and electromigration. The device definition steps involved the
CHAPTER 3. DESIGN AND FABRICATION
61
lithography and deposition processes that actually put material on the substrate in
the desired shape. In the device suspension steps, sacrificial material was removed
from around the mechanical portions of the device, freeing the island to move. The
final step, electromigration, formed the nanogaps that resulted in the tunnelling junctions on either side of the shuttle island. Using devices produced in the first step,
development and fine-tuning of the final two steps proceeded independently and in
parallel and then were finally integrated once both were working individually. To
foreshadow the discussion to come, the fabrication process used is shown in Fig. 3.2.
Where possible, the discussion in the following sections will merely be an overview of
the fabrication process focusing more on the what and why rather than the how of
fabrication. Additional details, such as specific procedures and recipes, are included
in Appendix A.
3.4.1
Device Definition
Lithography
The first step in the fabrication was to coat the wafer in electron beam resist. For
this project, a wafer consisting of 50 nm of silicon nitride on a silicon substrate was
used. With this type of wafer, the SiN is used as a mechanical layer and the Si is both
the sacrificial and substrate material. After the wafer was cleaned, a ∼600 nm thin
film of LOR 7A [60] was spun onto the wafer using a spin coater. After a 5 minute
bake at 200 ◦ C, a second layer of resist, 120 nm of PMMA A3† [61], was spun onto
the wafer. The wafer was then baked for 40 minutes at 180 ◦ C. Baking the PMMA
for such a long time ensures that it will be stiff enough to support itself once it has
†
The A3 designation indicates that the PMMA is diluted to 3% in anisole.
62
CHAPTER 3. DESIGN AND FABRICATION
Spin coat
LOR & PMMA
?
EBL
?
Resist development
?
Evap Cr/Au
(Double Angle)
?
Lift-off
?
FIB to expose
Si & SiN
around beam
and junctions
?
Wet etch
HF + KOH†
?
Electromigration
?
Device testing
Figure 3.2: Fabrication process used in the creation of the
electron shuttles.
†
The HF + KOH order of etching is for SiN wafers. For SOI, the order would be reversed.
63
CHAPTER 3. DESIGN AND FABRICATION
been undercut. The LOR bake temperature was chosen so make it develop more
slowly; this helps ensure better control over the amount of undercut. The effect of
baking temperature on development rate is shown in Table 3.2. This resist structure
of PMMA on LOR on the wafer is an example of a bilayer resist. The bilayer structure
allows for undercutting the pattern defined in the top resist, which in turn allows for
the use of double-angle evaporation and helps improve lift-off results.
Table 3.2: Effect of bake temperature on resist development
rate for three common lower-layer resists used in the group.
Data were obtained using AFM measurement of the thickness
of the resist layer, measured on the edge of a scratch, before
and after 5 s of development in MF-319. The bake time was 5
minutes in each case.
Resist \
BakeTemp
LOR 7B
LOR 7A
PMGI‡
180 ◦ C
85 nm/s
60 nm/s
S
200 ◦ C
65 nm/s
S†
–§
250 ◦ C
6 nm/s
S
–
†
Thickness change too small to measure, so development rate is slow.
Not exposed prior to development, which may be the cause of the slow rate.
§
Rate not measured since 180 ◦ C result was so slow.
‡
Once the resist layers had been applied to the sample, electron beam lithography (EBL) was performed with a JEOL JSM-6400 scanning electron microscope
(SEM) [62] to write the device pattern, shown in Fig. 3.3, into the top layer of the
resist. The pattern was created and written using the Nanometer Pattern Generation
System (NPGS) lithography control software and hardware [63].
Development
With the lithography complete, the next step in the fabrication process was development of the resist. The first development step is to immerse the sample in a 1:3
CHAPTER 3. DESIGN AND FABRICATION
200�µm
Figure 3.3: SEM pattern used to define the device shape. This
pattern contains two devices, one attached to the top and right
centre leads and the other attached to the bottom and left
centre leads. The section in pink is the interesting part of the
device; the rest is simply the leads and bond pads necessary
to allow electrical hookups to the device. Note that there is a
200 nm gap between the point of each lead and the island.
64
CHAPTER 3. DESIGN AND FABRICATION
65
solution of MIBK developer and isopropyl alcohol (IPA) [61]. A chemical reaction
dissolves the regions of the PMMA that were exposed by the electron beam, leaving
holes through to the LOR layer in the shape of the pattern written.† Schematics of
the resist developments are shown in Fig. 3.4. The development of the LOR is done
in the MF-319 developer [60, 64]. LOR resists have two very interesting features:
that they do not require exposure to be developed and that they are removed at an
equal rate in every direction. This means that the LOR is removed sideways as well
as down during development. Since the MF-319 can only gain access to the LOR
through the holes developed in the PMMA, this has the effect of removing a region
of LOR that is the same shape as the EBL pattern but expanded in every direction.
Because the PMMA holes are not affected by this process, the PMMA gets undercut.
Care was taken at this stage of the development because too much undercut would
have suspended more of the PMMA than it could support by itself and resulted in
the collapse of portions of the PMMA. Beyond destroying the shape of a portion of
the pattern, PMMA collapse can result in very poor liftoff and the loss of the device.
Examples of good and bad development are shown in Fig. 3.5.
Evaporation
Once development was completed, the sample was ready for evaporation. This is a
process of physical vapour deposition where a metal is heated with an electron beam,
causing some of the metal to evaporate. The free metal atoms then propagate through
a vacuum chamber and come to rest wherever they happen to land. Evaporation is
a line-of-sight process, meaning that the atoms propagate in a straight line from
†
It is actually merely close to the shape of the pattern written. Due to improper doses and the
proximity effect, the developed pattern can sometimes be very different from the desired pattern.
66
CHAPTER 3. DESIGN AND FABRICATION
PMMA
EBL-Defined
Pattern
LOR
Substrate
Larger
Undercut
Region
Figure 3.4: Schematic of the development process, with the
leads in the pattern omitted for clarity. Top: Pattern defined
by EBL is developed in PMMA. Bottom: The LOR is developed through the holes in the PMMA. Note that the developed
region retains the shape of the EBL-defined pattern, but expands in every direction. This creates an undercut under the
PMMA layer.
CHAPTER 3. DESIGN AND FABRICATION
(a) Good lithography & development
(b) Collapse starting – Note the darker regions
Figure 3.5: Good and bad development results. The device
in the lower figure shows partial collapse of the PMMA layer
(dark regions). In both figures, note the light area around the
edge of the pattern. This is the undercut region.
67
CHAPTER 3. DESIGN AND FABRICATION
68
the source. The propagation directions are, to at least a reasonable approximation,
evenly distributed through the solid angle subtended by the sample. In this stage,
the holes in the PMMA act as a mask, allowing metal atoms to land on the surface
of the substrate in the shape of the pattern and blocking the rest. The evaporation
process for the electron shuttle was slightly more complicated in that it involved doing
double-angle evaporation. Unlike most evaporation, which is done with the sample
normal pointing towards the metal source, a double-angle evaporation involves metal
incident to the sample at an angle of +θ to the normal and then again at −θ. Because
the PMMA mask is raised from the substrate surface by the 600 nm height of the LOR
layer, it shadows the incident metal atoms and the result is deposition on the substrate
in the shape of the pattern, but offset by d = (600 nm) tan θ. Because the pattern
is offset compared to the openings in the PMMA, it is important that the undercut
in the LOR is greater than d. If this is not the case, some of the deposited metal
will fall on the sidewalls of the LOR and be removed with liftoff. When deposition
is performed at two different angles, sharp features on the pattern can be made to
overlap by a much smaller area than is possible to define with EBL. With the electron
shuttle geometry, this meant that the necks for electromigration could be made very
small. A schematic of evaporation, showing both normal and double-angle deposition,
is shown in Fig. 3.6.
For the electron shuttle devices, four evaporations were done. The first two, 300 Å
of Au on a 50 Å Cr sticking layer, were done normal to the samples. This created a
base for the device that adhered well to the substrate. The final two evaporations,
both 300 Å of Au, were done at angles of ±15◦ from sample normal for a total throw
on the surface (sum of the two offsets) of approximately 350 nm. This distance was
69
CHAPTER 3. DESIGN AND FABRICATION
Au
Cr
Au
Au
Au
Cr
Au
Figure 3.6: Schematic of normal (top) and double-angle (bottom) evaporation for the deposition of metal for the electron
shuttle. Note the small and narrow overlapping regions where
the nanogaps will be formed.
CHAPTER 3. DESIGN AND FABRICATION
70
chosen based on the need to cover the distance between the island and lead points as
developed in the PMMA. Atomic force microscope (AFM) measurements indicated
that this distance was slightly less than 350 nm. The difference between this distance
and the 200 nm in the NPGS pattern can be explained by imperfect focus and the
lack of proximity effect around the points in the pattern. This caused the points to
be slightly rounded and thus shorter than the perfectly sharp points in the pattern.
The pattern fidelity could be improved by giving more dose to the tips of the points,
but developing and testing this would be a significant time investment for little to no
gain in device performance, so this was not explored.
Liftoff
The final step in the device definition process was liftoff, a process where the LOR on
the sample is removed, causing the PMMA and unwanted metal on top of the LOR
to lift off the sample and leave only the device behind. This is done by immersing the
sample in NanoTM Remover PG [65], heated to 78 ◦ C, for approximately five minutes.
While this is sufficient for many devices, the electron shuttle samples needed a slightly
more forceful approach – after five minutes, a small pipette was used to squirt the
immersed sample with Remover. When done with care, this removed the extra gold.
Another option was to use a sonicator to shake the material loose, but this resulted in
the destruction of the fine features of the sample as often as not and so this technique
was not used. A schematic of the final device is shown in Fig. 3.7. Notice that there
is no Cr sticking layer below the junction regions. This is a side-effect of the doubleangle evaporation and it ensures both that there is nothing extra holding the gold
atoms in place during electromigration and that, once the nanogap is formed in the
71
CHAPTER 3. DESIGN AND FABRICATION
gold, there is no electrical conduction path through the Cr layer. SEM images of the
device after liftoff are shown in Fig. 3.8. The three-fold repetition that can be seen
in Fig. 3.8(a) is due to the double-angle evaporation.
Junctions
Figure 3.7: Isometric and cutaway views of the device after
liftoff. Note the lack of Cr sticking layer under the junctions.
3.4.2
Electromigration
The fabrication of suspended structures is something that is well-understood in the
group [57, 66, 67], so the bulk of the work for this project was focused on forming
the two nanogaps with electromigration. Although there is a great deal of published
research on this topic, no one involved in this project had first-hand experience with
electromigration. To get a feel for the process, then, the initial work on electromigration involved single-junction devices.† A representative device is shown in Fig. 3.9.
†
Single-junction breaking was also needed for another project [57], so the initial electromigration
work was a collaboration with J. Campbell. She is due thanks and recognition for her contributions.
CHAPTER 3. DESIGN AND FABRICATION
72
(a) SEM image of fabricated shuttle region show- (b) SEM image of overall pattern. The damage
ing necks of ∼120 nm thick.
to the bond pads was caused by a DC probe.
Figure 3.8: SEM images of the fabricated devices. Compare
with the written pattern in Fig. 3.3.
Figure 3.9: SEM image of a single-junction device for the development of electromigration recipes.
73
CHAPTER 3. DESIGN AND FABRICATION
Feedback and Control
As was discussed back in Sec. 2.3, getting good results from electromigration depends
in part on an effective feedback system to control the breaking. Ignoring implementation details for the moment, the required system controls the voltage across the device
and measures the resulting current through it. A simple Ohm’s law calculation gives
the device resistance and feedback is performed to keep this resistance from increasing
too much or too quickly. Figure 3.10 shows the circuit for this simple scheme. The
source measure unit (SMU) used was a Keithley 2400LV [68] controlled via GPIB
by a LabView 8.5 [69] program. All logic and instrument control at this stage was
done in software. This consisted of setting a source voltage V , which was slowly
SMU
-�+
A
Device
Figure 3.10: Simple breaking circuit. The SMU used was a
Keithley 2400LV controlled via GPIB.
increased at a ramp rate of Vramp , and measuring the resulting current. Ramping,
in this case, was implemented as a small voltage increase proportional to Vramp after
CHAPTER 3. DESIGN AND FABRICATION
74
a configurable time step. The resistance at that time was then calculated and compared to the target resistance, Rstop , and to the feedback threshold resistance, Rth .
The program returned the voltage to zero and stopped once Rstop was reached. The
feedback threshold resistance was set as a percentage above the initial resistance for
that feedback cycle. Once that threshold was crossed, the voltage was reduced by a
large step (usually Vdown > 100 mV), and the new threshold resistance was calculated
for the next cycle. This mechanism was intended to quench the electromigration process after it had progressed a small amount. In this way, the target resistance could
be approached slowly and controllably in small steps rather than trying to stop a
single electromigration event at the desired resistance. The logic of this program is
presented more clearly in the flowchart of Fig. 3.11.
Testing of this electromigration scheme took place in a Janis ST-500-1 micromanipulated cryogenic probe station [70] like the one shown in Fig. 3.12. This probe
station has two DC probes, one microwave (MW) probe, and an optical fibre probe
connected to micrometer positioning stages. It is capable of operating down to 4.2 K
with liquid helium, but it was only used from room temperature to liquid nitrogen
temperature in this work.
Through trial and error experimentation, it was determined that the most consistent results were given by the parameters given in Table 3.3. Through all these trials,
the target resistance was 10 kΩ per junction, approximately the resistance where the
tunnelling regime begins (R0 = 1/G0 ≈ 12.9 kΩ). It was anticipated that the resistance would overshoot this target somewhat and produce a device in the 10–100 kΩ
range. Figure 3.13 shows the result of a successful† electromigration. The device
†
“Successful” here refers to electromigration that ends with a device in the 10–100 kΩ range.
75
CHAPTER 3. DESIGN AND FABRICATION
Start
?
Initialize
?
-
V → V + Vramp
-
Measure I
?
?
Calc. R =
V
I
?
@
@
@
@
Yes
R > Rstop ? @
@
@
@
@
@
- Stop
No
?
@
@
@
@
No
Yes
R > Rth ? @
@
@
@
?
@
@
@
Vramp @
time
step @
@
@exceeded?
No
@
@
@
@
@
?
Large V step down
?
Calc. new Rth
Yes
Figure 3.11: Logic flow of the electromigration control software.
CHAPTER 3. DESIGN AND FABRICATION
Figure 3.12: Janis ST-500 micromanipulated probe station
Table 3.3: Parameters for the electromigration program that
gave optimal results
Parameter
Value
Rstop
10 kΩ
Vramp
4 mV/s
Vdown
100–200 mV
Rth
R + 1%
Time step
0.1 s
76
CHAPTER 3. DESIGN AND FABRICATION
77
shown here started with a resistance of 33.7 Ω. After electromigration, the final resistance was ∼20 kΩ. A test was also done without feedback to see what uncontrolled
electromigration looked like. Figure 3.14 shows the rather catastrophic failure that resulted. This indicated that an effective feedback control system was indeed necessary
for successful electromigtion.
(a) Before electromigration
Gap
(b) After breaking
Figure 3.13:
device.
Successful electromigration on a single-junction
CHAPTER 3. DESIGN AND FABRICATION
78
Figure 3.14: Catastrophic failure of a device that underwent
uncontrolled electromigration.
3.4.3
Double-Junction Breaking
The topic of performing electromigration in multiple junctions simultaneously was
explored by Johnston et al. in 2007 [50]. Their finding was that breaking junctions in
series resulted in runaway electromigration in one junction, leaving the others practically untouched. They also found that electromigration on junctions in parallel tended
to form relatively uniform gaps across all the junctions. To explore why this is, recall
Eq. 2.15, which states that the electromigration force in a junction is proportional
to the local electric field. Assuming the junctions act as resistors, this means that
the electromigration force is dependent on the voltage that falls across the junction.
Consider, then, two junctions that are to undergo electromigration simultaneously.
They act as two resistors and can either be connected in series or in parallel. Which
configuration will better promote symmetric evolution of the nanogaps? In order to
CHAPTER 3. DESIGN AND FABRICATION
79
answer this question, the stability of the system must be considered.
As electromigration progresses, the resistance of each junction will slowly increase.
In general, this increase will be unique to a particular junction. In a stable system, an
increase in one junction will cause the relative electromigration rates to shift to the
other junction and, in this way, promote symmetric evolution. Instability would be if
a resistance increase in one junction caused the electromigration rate in that junction
to increase, causing one nanogap to form preferentially. It is important to note that
stability, in this sense, is merely the promotion of symmetric electromigration between
the two junctions rather than the overall behaviour of the electromigration as viewed
from an external circuit. As such, a stable-symmetric system could still undergo
“runaway” electromigration, with the result being very large, but symmetric, gaps.
Thus, external feedback and control is still necessary. For a voltage-controlled system
where the resistance changes are happening faster than the feedback system, the
approximation can be made that the voltage, V , across this whole system is constant.
At this point, the power dissipated in each junction becomes a useful indicator of the
electromigration force. In order for the system to be stable, an increase in resistance
in junction 1 must result in a larger change in power across junction 2 than across
junction 1. That is [50],
∂P2
∂P1
<
.
∂R1
∂R1
(3.1)
CHAPTER 3. DESIGN AND FABRICATION
80
Junctions in Series
In this case, the equivalent resistance of the system is Req = (R1 + R2 + RL ), where
RL is some lead resistance. The current through the system is
I=
V
Req
and the voltage across one junction is
Vi = IRi = V
Ri
.
Req
This means that the power dissipated in one junction is
Pi = V 2
Ri
.
(Req )2
Thus,
∂P1
V2
2V 2 R1
=
−
∂R1
(Req )2
(Req )3
2V 2 R2
∂P2
=−
.
∂R1
(Req )3
Working through from Eq. 3.1 and substituting the expression for Req gives
R1 + R2 + RL
< R1 − R2 .
2
(3.2)
For R{1,2,L} > 0 and R1 ≈ R2 , there is no solution that satisfies this. Therefore, a
series configuration will not promote symmetric evolution of the two junctions.
81
CHAPTER 3. DESIGN AND FABRICATION
Junctions in Parallel
For this case, consider R1 ||R2 in series with a lead resistance RL and with V across
all of them. Now, the equivalent resistance is Req = RL + R|| with R|| =
R1 R2
.
R1 +R2
Thus,
the voltage across each junction is
V0 =
R||
RL + R||
and the power dissipated is
Pi =
(R|| )2
V 02
=
.
Ri
Ri (RL + R|| )2
At this point, one need only look at the partial derivative
∂R||
R2
R1 R2
=
−
∂R1
R1 + R2 (R1 + R2 )2
to know that expressions are going to start getting messy. The end result is that, for
the symmetric electromigration of two junctions in parallel,
2RL (R2 − R1 ) <
R1
R2
2
.
(3.3)
Since the initial definition of this problem had R1 > R2 , this is always true. The
parallel configuration is thus the best option for symmetric nanogaps (the optimal
configuration for a shuttle).
As originally designed, the shuttle device had two junctions in series. In order
to perform electromigration in parallel, electrical contact to the island was necessary.
CHAPTER 3. DESIGN AND FABRICATION
82
Initially, using an AFM tip to contact the island was considered. This initially looked
feasible as the Veeco CP-II AFM has an external connection to the tip bias. It was
decided, however that this solution would be too difficult to implement efficiently for
frequent experiments, as was required for double-junction electromigration, and that
it carried the risk of causing electromigration at the tip-electrode interface instead
of at the necks. For these reasons, the beam supporting the island was instead fully
metallized and thus a third electrode was added to the design. This allowed an
electrical connection to be made directly to the island using the same methods as to
the left and right leads.
Because of this third contact, electromigration in the probe station is slightly more
complicated, since there are only two DC probes. This issue was avoided by the use
of the microwave probe. The bond pads for the left and right leads were fabricated to
be separated by the pitch of the MW probe leads so that the MW probe bridged these
pads. The signal and ground leads of the probe were then shorted with a shorting
cap on a T-connector and this formed the lead for one output of the source meter.
A DC probe was then used to contact the island bond pad, completing the circuit.
Apart from this, the electromigration circuit, software, and parameters stayed largely
the same as for the single junction. The one additional difference is that the target
resistance was set to Rstop = 5 kΩ. This is the equivalent resistance, seen by the
external circuit, of two 10 kΩ junctions in parallel. Detailed double-junction breaking
results, along with some changes to the breaking process, are explored in the next
chapter.
CHAPTER 3. DESIGN AND FABRICATION
3.4.4
83
Device Suspension
The suspension of the device involved three processing steps: focused ion beam (FIB)
milling and two steps of wet etching. FIB milling involves a focused beam of gallium
ions incident on the surface of the sample. Like the SEM, this beam can be steered.
The main difference is that, unlike electrons, gallium ion have significant mass. When
they impact the surface, their momentum is transferred to the surface atoms and some
of these atoms are knocked free. This results in the removal of material in the path
of the ion beam. A Micrion 2500 [71] FIB with FIB Assist software from Fibics [72]
was used to mill away the material around the mechanical components of the shuttle.
To prevent charging effects, the sample was first coated with 70 Å of Al. A dose of
0.4 nC/µm2 was used for a mill depth of 70 nm. A milled device is shown in Fig. 3.15.
Figure 3.15: SEM image of a device with the mechanical components defined by FIB milling. This particular device had
poor SEM lithography, resulting in a large lead-island gap, but
the milled shape in the nitride can still be seen.
CHAPTER 3. DESIGN AND FABRICATION
84
Once the FIB milling was completed, two wet etches were performed, an HF
etch which isotropically etches silicon nitride and a KOH etch which anisotropically
attacks silicon [73]. Both of these etchants also attack aluminium, so the anti-charging
layer applied for FIB milling was also removed in these step. The KOH was used to
remove the silicon underneath the nitride. It could only attack the silicon exposed
by the FIB milling, so the beam was undercut and freed without etching away the
substrate underneath the entire sample. In most suspended devices, milling and a
KOH etch would be sufficient to free the structure. However, the APC junctions
create a complication in freeing the island: the nitride connections between the leads
and the island must also be removed. They cannot be milled away, however, because
the gold above them must remain intact. To try to remove this nitride, the KOH
etch was preceded by an HF etch. This also removed nitride from the sides of the
beam, so care was taken to not etch for too long to avoid the removal of the entire
mechanical structure.
Chapter 4
Experiments and Results
We may sometimes feel that nature is loath to give up her secrets without
a considerable expenditure of effort on our part...
— Philip R. Bevington
4.1
Single Junction Electromigration
In order to develop procedures and experience with electromigration, the initial work
was done on single-junction devices.† This stage of the experimentation was done
using the two DC probes in the Janis probe station to make contact to the device.
Sourcing voltage across the device and measuring the resulting current were both
done with a Keithley 2400LV source meter controlled by LabView via GPIB. This
circuit was shown in Fig. 3.10, reproduced below.
After some testing at room temperature, it was determined that lower temperature would be necessary for successful electromigration to the 10–100 kΩ range. Recall
†
As opposed to double-junction devices like the shuttles.
85
CHAPTER 4. EXPERIMENTS AND RESULTS
86
SMU
-�+
A
Device
Figure 3.10: Single junction breaking circuit
that there is a component of electromigration that is dependent on the Joule heating of the device. Recent work has indicated that this heating will not affect the
electromigration until a critical local temperature is reached [42]. This means that
a lower overall temperature will reduce the effect of heating on the electromigration
process, which should render it slower and thus more controllable. The combination
of liquid nitrogen and a Cryo-Con Model 32B [74] temperature controller connected
to the probe station was used to maintain a temperature of 78.000 ± 0.005 K† during
breaking.
4.1.1
Breaking Results
The recorded data for a sample representative of successful breaking and the I − V
curve of this data are shown in Figs. 4.1 and 4.2. The final resistance value recorded by
†
Measured at the base of the probe station, not on the sample
87
CHAPTER 4. EXPERIMENTS AND RESULTS
the software was 20 kΩ; at this point, the resistance was above the target resistance
and the software shut down the voltage to stop electromigration. Measurements†
taken immediately afterwards showed a resistance of ∼165 kΩ, indicating that migration continued briefly after the last measurement.
Voltage
(V)
3
2
1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
×104
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
×104
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
×104
Current
(mA)
20
15
10
5
0
Resistance
(Ω)
105
100
Time
(s)
Figure 4.1: Data recorded from a successful single junction
electromigration.
One interesting feature of these data is that there seems to be some initial threshold that must be crossed before electromigration starts. This can be seen in the
current and voltage plots of Fig. 4.1 and slightly more clearly as the long straight
†
Taken at low voltage so as to not cause further electromigration.
88
CHAPTER 4. EXPERIMENTS AND RESULTS
18
16
14
Current
(mA)
12
10
8
6
4
2
0
0
0.5
1.0
Voltage
(V)
1.5
2.0
Figure 4.2: I − V curve from a single-junction electromigration
with the progression in time indicated by arrows. The large
initial section builds up to the threshold for the start of electromigration, after which point less power is required. Each
step in the curved, falling section is due to the software stepping the voltage back after a small resistance increase.
2.5
CHAPTER 4. EXPERIMENTS AND RESULTS
89
line in Fig. 4.2 before the current starts to fall. This same shape was seen in the
literature (Sec. 2.3). In the successfully fabricated devices, this critical power was in
the 5–10 mW range.
Finally, the normalized conductivity is shown in Fig. 4.3 for the final segment of
the gap formation. Plateaus can be seen at 2G0 and 1.5G0 as well as less distinct
plateaus at 3.5G0 and 5G0 . Recall that the quantization of conductivity in G0 is a
feature of the mesoscopic conduction regime. The presence of the half-integer plateaus
suggests that there may be conduction modes in the device at this stage that do not
have perfect transmission. They may also indicate the presence of some kind of
contamination on the junction. It is also possible that the plateaus seen here are
artifacts of the measurement technique, which involved slowly increasing the voltage
across the device and suddenly dropping it after a large resistance change. Moving
from one plateau to another was enough to trigger these voltage drops, although there
were also voltage drops that do not correspond to a conductance jump to the next
lower plateau. More convincing evidence that these plateaus are due to mesoscopic
conduction would require measurements in this region that are taken at a constant
voltage, one just high enough to produce electromigration but not so high that the
process proceeds too rapidly. This measurement, however, has not been done.
4.1.2
Stability
An important metric for junctions formed via electromigration is the time and temperature stability of the resistance once the junction is formed. Immediately after
electromigration had completed, the device was measured for several minutes at low
90
CHAPTER 4. EXPERIMENTS AND RESULTS
5.5
5
4.5
4
G
G0
3.5
3
2.5
2
1.5
1
0.5
1.4
1.5
1.6
1.7
1.8
1.9
Time
(s)
Figure 4.3: Normalized conductivity plot for the final section
of the electromigration process. There is some evidence of the
conductivity plateaus predicted for the mesoscopic conductivity regime.
2
×104
CHAPTER 4. EXPERIMENTS AND RESULTS
91
voltage. Over this time, a resistance of 164±10 kΩ was recorded. This was repeated after approximately 20 minutes, when the resistance was measured as 160±3 kΩ. These
measurements were taken while the device was still at 78 K. After the probe station
has warmed up, the resistance was checked again. At room temperature, the device
had a resistance of 50±1 kΩ. This suggests that thermal expansion may play a role
in the geometry of electromigration junctions, an important fact if there is a desire
to operate the shuttles at temperatures other than that at which the junctions were
formed. Another possible explanation is that, as this device was in the mesoscopic
conductivity regime, the changes in resistance were due to small rearrangements of
the gold atoms that formed the junction.
4.1.3
Problems
Across many electromigration trials, some common problems appeared. First and
foremost was unreliable contact to the pads using the DC probes. Care was taken
while positioning the probes to make sure they were pushing down on the sample with
sufficient force but, despite this, it was not uncommon for the probes to shift over the
course of an experiment. If they shifted so that they were in weaker contact with the
pads, or shifted off the pads altogether, this created a false increase in resistance to be
recorded. It was thought that this shifting may have been due to thermal contraction
in the probes or to vibrations being transmitted into the probe station. As a result,
a long time was allowed to elapse after heating or cooling the probe station to allow
the probe temperature to stabilize and changes in temperature were always done with
the probes not touching the sample. Also, care was taken to not knock the table or
probe station and to walk softly while an experiment was in progress. None of these
CHAPTER 4. EXPERIMENTS AND RESULTS
92
measures, however, improved the reliability of the probe contact and the root cause
of the shifting is as yet unknown.
Another problem with this system was that it was not very good at catching
the final electromigration step in time. That is, the feedback was not fast enough.
Data were measured at approximately 78 ms intervals and the resistance would often
increase by several orders of magnitude, or “blow open,” in the span of a single time
step, resulting in a very high resistance device. The chances of catching this sudden
resistance increase were increased by slowing the voltage ramp rate and lowering the
percentage by which the resistance threshold was raised for each feedback cycle, as
well as by breaking at low temperature, but these were Band-Aid solutions to the
underlying issue of speed.
Finally, these tunnelling junctions proved to be quite fragile once formed. One
thought for removing the electrical contact to the island after electromigration had
been to cover the beam in aluminium, using a second lithography sequence, while
having the rest of the device still made from gold. Aluminium is removed quickly
in MF-319, the resist developer used already in the fabrication to develop the LOR
layer. Because MF-319 does not attack gold or silicon, this seemed a convenient way to
remove the contact along the beam. In testing, this process very effectively removed
the aluminium contact. However, the wet processing also destroyed the junctions.
Electrostatic discharge (ESD) also proved to be a problem for the junctions, as it
caused a very high field in the region of the junction. This resulted in the same kind
of catastrophic failure seen in the testing of electromigration without feedback.
All in all, then, single junction electromigration was successfully achieved. It
was not a completely reliable process, however, in large part to the issues discussed
CHAPTER 4. EXPERIMENTS AND RESULTS
93
above. The per-device results are included in Appendix B; all told, less than half of
the single-junction electromigration trials resulted in devices with a resistance in the
desired range of 10–100 kΩ.
4.2
Double Junction Electromigration
Initial attempts at breaking two junctions in parallel were done in the probe station in
the same way as the single junctions. A parallel configuration was achieved by using
the microwave probe to contact the bond bads for both the left and right leads; the
signal and ground contacts of the MW probe were then shorted. The island lead was
contacted with a DC probe. It was quickly discovered that the microwave probe was
even more susceptible to shifting over time than the DC probes and, despite efforts,
this could not be improved. As such, double breaking with the probe station was
not successfully achieved.† Although electromigration was achieved, the MW probe
would often shift off of one pad, putting a sudden burst of power through the other
junction and causing a large gap. An example of this is shown in Fig. 4.4.
4.2.1
1.5 K Cryostat
Because of the problems with probe contact, it was decided to move away from the
probe station. It had been found, however, that the controllability of the electromigration improved at lower temperatures. To accommodate this low temperature
requirement, a circuit board was created to allow these devices to be mounted in a
†
As an aside, a double-junction device was successfully created on the probe station. However,
this was done by breaking the two junctions individually and thus is not directly relevant to this
section.
CHAPTER 4. EXPERIMENTS AND RESULTS
94
Figure 4.4: SEM image of a double-junction device. Sudden
failure of the right junction during a double-break electromigration attempt caused the asymmetry and large gap.
home-built 1.5 K cryostat. Further details of the cryostat and circuit board are included in Appendix C. Here, it suffices to say that the cryostat had 15 DC lines that
could be used for devices. These lines were connected to quick-connect style header
blocks inside the cryostat for easy connection and removal of the circuit board. The
samples with the devices were attached to the circuit board with GE varnish. Electrical connections between the device pads and the contacts on the circuit board were
made using a West·Bond model 7476E [75] wire bonder with thin aluminium wire.
A Note on ESD
As was mentioned earlier, these devices were vulnerable to ESD. They were particularly vulnerable when they were wire bonded to the circuit board. This is because
the devices were electrically attached to a much larger area of metal, increasing the
CHAPTER 4. EXPERIMENTS AND RESULTS
95
chance of part of the device coming into contact with something at a different potential. To minimize the risks, a grounding wrist strap was worn whenever the devices
were on the circuit board. As well, the pins on the circuit board were shorted together
when it was not in the cryostat. The board was also stored and transported in an
ESD-safe plastic container.
The devices were also at risk while the board was mounted in the cryostat. The
DC connections for the cryostat came out to a breakout box with grounding switches.
These switches were kept in the grounded position for all connections that were not
actively being used. This ensured that the only potential differences across the devices
were those being purposefully applied.
Resistive Wire
One consequence of moving to the cryostat was that the wires to the sample had a
resistance of approximately 170 Ω per line. This is non-trivial compared to the device
resistance, especially at the beginning when it is only a few tens of ohms. A fourterminal measurement, in the style of Wu et al. [51], was needed to correct for the
lead resistances. With the 15 DC lines on the cryostat, this meant that three devices
could be connected per cool-down.
4.2.2
Four-Wire Electromigration Circuit
The switch to a four-wire measurement necessitated the design of a new breaking
circuit, shown in Fig. 4.5. The basic principle of this circuit is the same as with the
source meter: source voltage and measure current. The current measurement was
done with an Agilent 34401A multimeter [76] measuring the current passing through
96
CHAPTER 4. EXPERIMENTS AND RESULTS
PID
G≈3.6
+
+
A
_
-
Figure 4.5: Circuit for four-wire electromigration. The lead
resistances are RL ≈ 170 Ω.
to the device. The voltage sourcing was more interesting. Instead of using a simple
voltage source, a PID controller was used. This is a device that changes its Output†
to make a Measurement match a Setpoint value. A Signal Recovery Model 5113 preamp [77], used as a unity gain differential amplifier, was connected across two leads to
measure the voltage solely across the device. With this providing the Measure signal
for the PID, it allowed the voltage across the device to be Set directly, with the PID
adjusting the voltage across the whole system as required. Using this set voltage and
the measured current allowed the resistance of the device to be calculated without
the lead resistances. The fourth lead was connected to ground to provide a current
return path.
There was another benefit to setting the voltage across the device as opposed
to the voltage across the series combination of the device and the leads: improved
†
Capitalization is meant to differentiate PID inputs from generic quantities.
CHAPTER 4. EXPERIMENTS AND RESULTS
97
stability compared to a two-wire connection with highly resistive wires.† If the voltage
across the system as a whole were kept constant, then any increased resistance in the
junctions would cause more of the total voltage to fall across the junctions and thus
increase the rate of electromigration. On the other hand, with the PID controller
working to set the voltage across the device, the same increase in resistance will
cause the PID to decrease the voltage across the whole system by whatever amount
is necessary to keep the device voltage constant. The PID controller used was a
SIM960, part of a series of Small Instrumentation Modules [78]. This controller has a
bandwidth of 100 kHz, which allowed it to make small corrections to the voltage far
faster than LabView on a GPIB interface could hope to. The P, I, and D feedback
coefficients were tuned, using a 50 kHz square wave, to have the fastest rise and
settling times possible. These parameters are given in Table 4.1. The Set voltage was
sourced from a SIM928 isolated voltage source in a SIM900 mainframe (with the PID
module). This voltage source was controlled by LabView via the GPIB connection
on the mainframe.
Table 4.1: Optimal feedback parameters for the SIM960 PID
controller in the circuit shown in Fig. 4.5. The proportional
gain, P , is negative due to the inverting configuration of the
BOP amplifier.
Value
Parameter
P
-0.2
I
15.0×104
D
1.2×10−5
†
Although there should be improved stability compared to the highly resistive two-wire measurement, the stability gains from a four-wire measurement will not improve it beyond the stability of a
two-wire system with negligible lead resistance.
CHAPTER 4. EXPERIMENTS AND RESULTS
98
One final note needs to be made on the sourcing of voltage in this circuit. The
high resistance of the DC leads meant that, during the initial breaking stages when
the device resistance was low, most of the applied voltage fell across the leads. For a
typical device starting resistance of 40 Ω, only about 10% of the Output voltage was
actually applied to the device. Early measurements in the cryostat indicated that, as
with the single break junctions, the double junctions had a critical voltage of slightly
over 1 V before electomigration started to occur.† The PID has a maximum output of
10 V, which meant that it could not source enough voltage to begin electromigration
with the 170 Ω leads. To increase the sourced voltage, the PID output was fed through
a Kepco BOP 36-6M amplifier [79]. In its default configuration, the BOP is set so
that input voltages in the ±10 V range address the entire range of the BOP’s output
voltage. Since the 36-6M has a maximum output voltage of ±36 V, this means that
the gain of the BOP was approximately -3.6, with the negative sign indicating that
it is an inverting amplifier.
4.2.3
The Breaking Cut-off Circuit
Electromigration attempts with the PID circuit showed improvements over breaking
in the probe station. Contacts to the devices were much more reliable‡ and the fourwire measurement meant that the effect of lead resistances on the data was eliminated.
The speed of the measurement was slightly improved, but the software was still taking
tens of milliseconds between points and the stabilizing effect of the PID feedback did
not have as significant an effect on the evolution of the junction as had been hoped. To
†
These measurements were done with lower resistance wire, ∼ 65 Ω per lead. The DC lines were
replaced with the higher resistance wire midway through the project.
‡
After some practice on the wire bonder
99
CHAPTER 4. EXPERIMENTS AND RESULTS
this end, an “emergency brake” of sorts was designed and constructed using purely
hardware components, elements of which were inspired by Roch et al. [80]. This
circuit, shown in Fig. 4.6 and discussed in more detail in Appendix D, was inserted
into the existing connections as shown in Fig. 4.7.
Rset
In
External�Box
Vset
1�kΩ
1�kΩ
R
5.09�kΩ
From
DUT
OP27
J
Vdd
MC14027
K
S
R
Q
Q
LF411
LF411
R
Vi
Output
Vdd
LM311
LF411
PID
In
TC4425
Bypass
Switch
To
DUT
Figure 4.6: Overview of the electromigration cut-off circuit
The function of this circuit is to ground the input to the device, effectively removing the potential driving electromigration, once the resistance has passed the
target value. This is the same as the stop point programmed into the software, but
the advantage of doing this in hardware is that it can operate much more quickly.
Bench-top testing showed that a potential across a test resistor was grounded within
150 ns of being triggered, much faster than the tens of milliseconds response of the
CHAPTER 4. EXPERIMENTS AND RESULTS
PID
+
G≈3.6
+
100
A
_
-
Figure 4.7: The electromigration circuit with the cut-off circuit
box included.
LabView program. This should allow the termination of electromigration very close
to the target value, since the event that forms the junction has been estimated as
occurring over a 100 µs timescale [81].
For the cut-off box to function properly, electrical noise had to be well controlled. This was especially true for the transimpedance amplifier† and the comparator.‡ When the comparator seemed to be consistently triggering too soon, it was
discovered that the 5113 pre-amp, used to measure the voltage across the device,
injected ∼ 30 mV of noise onto its inputs and thus onto the input (and consequently
the output) of the transimpedance amplifier. This was a significant fraction of the
∼ 150 mV level representing the target resistance, and explained the early triggering
of the comparator. As a result, the 5113 pre-amp was replaced by a home-built unity
†
Used to convert the current through the device into a voltage.
Compares the output of the transimpedance amplifier to an input voltage representing the
current through a device of the target resistance and triggers the switching circuitry to stop
electromigration.
‡
CHAPTER 4. EXPERIMENTS AND RESULTS
101
gain difference amplifier constructed around a LF411 operational amplifier. A side effect of this was that, to keep a high amplifier speed while reducing overshoot, the DC
input impedance of the difference amplifier was only 200 kΩ. This meant that a simple R = V /I calculation of the set voltage and measured current through the system
would no longer give the resistance of the device. Instead, it would give the resistance
of the device in parallel with ∼ 200.3 kΩ (200 kΩ + 2RL ). A calculation correcting
for this was done by LabView during data collection. Testing showed that, in order
to keep the cut-off set voltage above the noise in the circuit, a minimum voltage of
50 mV across the device was needed.
Getting the cut-off circuit working properly involved a significant amount of troubleshooting and many devices were destroyed while discovering the bugs in the circuit
that were not found during testing. By the end of it, there were only two devices
remaining for testing. Unfortunately, these devices did not survive cooling down†
and there was insufficient time to fabricate more. This means that, despite positive
indicators along the way, parallel double-junction electromigration has not yet been
achieved.‡ There is a bit of a silver lining, however. SEM images of the devices blown
when the cut-off circuit was not triggering properly, an example of which is shown
in Fig. 4.8, showed that the two junctions failed symmetrically. This is in contrast
to the probe station results, Fig. 4.4, where one junction failed before the other and
resulted in an asymmetric device. This symmetric failure suggests that the parallel
configuration used here does indeed promote symmetric evolution of the junctions
during electromigration, and that this holds true even during the fast, large-scale
failure shown in these results. This would seem to confirm the analysis of Sec. 3.4.3
†
At this stage of the experiment, electromigration testing was happening at liquid helium temperatures, ∼ 4.2 K.
‡
The per-device results are in Appendix B.
CHAPTER 4. EXPERIMENTS AND RESULTS
Figure 4.8: SEM image of a double-junction device that failed
catastrophically when undergoing electromigration in the parallel configuration. The junctions have failed symmetrically.
102
CHAPTER 4. EXPERIMENTS AND RESULTS
103
and means that the author remains hopeful about the viability of this technique once
the bugs have been worked out of the cut-off circuit.
4.3
Device Suspension
As was mentioned earlier, the electromigration work received the bulk of the effort in
this project and thus little work was done on device suspension. The work that was
done, however, shows that the suspension recipe presented in Sec. 3.4.4 fails to achieve
the necessary undercut of the SiN mechanical layer and that some aspect of the HF
and KOH etching steps causes deformation of the gold electrical layer. SEM images
of the FIB’d device after etching are shown in Fig. 4.9. These images clearly show
that the recipe needs to be rethought. One possibility is ongoing work by J. Campbell
that aims to replace the HF etch for SiN undercut. Instead of undercutting the SiN
to remove the material under the junction, a trench is FIB milled through the gold
and SiN in the area of the junction. This trench is stepped so that it is entirely
through the SiN layer just in the region where the break will form. Once milled, a
second round of electron beam lithography and metal deposition is done to replace
the gold that was milled through. In this way, the mechanical layer underneath the
junctions is removed, leaving the centre beam free to move, without the need for
carefully controlled chemical undercutting. This also has the benefit of not changing
the thickness of the beam, making the fabrication of thin beams easier.
CHAPTER 4. EXPERIMENTS AND RESULTS
(a) Test pattern milled into a bond pad. The gold is
peeling away from the centre of the pattern.
(b) The device from Fig. 3.15 after etching. The gold
leads have moved and the SiN in the junction region
is no thinner
Figure 4.9: SEM images of the milled patterns after HF and
KOH etching.
104
Chapter 5
Conclusions and Future Work
This is the end
My only friend, the end
Of our elaborate plans, the end
— The Doors, “The End”
In hindsight, the stated goal of this project, the fabrication and testing of a
nanomechanical electron shuttle, may have been somewhat ambitious.† This is largely
because the development of a working electromigration system and procedure involved
far more time and effort than had been expected. Single junction electromigration to
the 10–100 kΩ range was successfully achieved, albeit with a mediocre yield. Achieving two junctions with electromigration proved to be more difficult, however these
difficulties resulted in improvements to the electromigration system, such as using
the 1.5 K cryostat for reliable electrical contact and the development of the cut-off
circuit, each of which moved the process closer to success. It is the belief of the
†
Not that there is anything wrong with that.
105
CHAPTER 5. CONCLUSIONS AND FUTURE WORK
106
author that the fast ∼ 150 ns operation of the cut-off circuit will prove to be the key
to successful double junction formation.
Where, then, does this leave the nanomechanical electron shuttle project? Recall
that there were three fabrication areas that had to be developed for the creation
of the shuttles: device definition (lithography and evaporation), junction formation
(electromigration), and device suspension (FIB and etching). As was discussed above,
double junction formation is on the cusp of being achieved. The device definition
recipes and techniques have all been developed and proven, so that area is taken care
of. The device suspension techniques, however, are a different story. Little work
was done in this area but it was enough to show that, although FIB milling for the
definition of the suspended regions was successful, the junction undercut and device
suspension recipes need to be rethought. In short, much work has been done towards
the fabrication of the nanomechanical electron shuttle using electromigration, but
there is still work that needs to be done in order to realize these devices.
5.1
Future Work
Looking beyond the fabrication work discussed above, there are some things that
should be considered in order to move forward with electron shuttling experiments.
5.1.1
Device Suspension
As was discussed in Sec. 4.3, there is some ongoing work in the group to mill trenches
though the junction regions instead of undercutting them; this would is also applicable to these devices and, if successful, this technique should be considered for the
CHAPTER 5. CONCLUSIONS AND FUTURE WORK
107
fabrication of the suspended devices. Also on the topic of device suspension, it might
be possible to use reactive ion etching (RIE) to perform the milling and etching steps
for device suspension. Although the FIB milling seemed to work well for this task
and RIE facilities do not exist at Queen’s, this technique is still worth mentioning in
the event that further problems arise with the current suspension techniques.
5.1.2
Shuttle Fabrication
Recall that the original shuttle specification called for an electrically isolated island
travelling between two electrodes, but that the double-junction parallel electromigration done for this work required that the beam suspending the island be metallized.
In order to get back to the isolated island required for proper shuttling, this electrical
connection to the island must somehow be cut after electromigration. The original
thought for this was to deposit aluminium on the beam instead of gold and remove it
with MF-319 after electromigration, however the formed junctions proved too fragile
for wet processing like this. One possibility for cutting this connection would be to
remove a section of the metal with the FIB while being careful not to cut too far into
the stiff mechanical structure of the beam. Considering that the FIB has a reasonably
linear dose-depth profile (Fig. A.3), this would be possible with careful alignment.†
Another possibility is to make contact to the island in some other way, for example
with an AFM tip. The main drawback of the AFM tip is that it is time consuming
to find and contact the island of a sample. Once the yield of the electromigration
process has been improved, this will be less of a problem.
†
Provided the region of the junctions was not imaged with the FIB, as this might damage the
junctions.
CHAPTER 5. CONCLUSIONS AND FUTURE WORK
5.1.3
108
Other Experiments
Even without removing the metal from the beam, there are still interesting experiments that can be done. For example, a system having two point contact displacement
detectors on either side of a conducting, vibrating beam, like that proposed by Doiron et al. [11], can be used to reduce back-action and improve the precision of position
measurements.
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Appendix A
Detailed Fabrication Recipes
A.1
Spin Coating
Spin coating of the wafers was performed under a fume hood in a cleanroom with a
Laurell WS-400A-6NPP-LITE spin coater [82]. This model features a vacuum chuck
to secure wafers and programmable speed, timing, and acceleration control. First,
LOR 7A was spun onto the wafer and then baked. This was followed by spinning and
baking a layer of PMMA to form a bilayer resist structure. The resist application
parameters are given in Table A.1.
Table A.1: Parameters for spin coating a bilayer of PMMA on
LOR 7A
LOR 7A
Spin speed 4000 rpm
Spin time
1 min
Layer thickness
600 nm
Bake temperature
200 ◦ C
Bake time ∼5 min
116
PMMA
4000 rpm
45 sec
120 nm
180 ◦ C
∼40 min
117
APPENDIX A. DETAILED FABRICATION RECIPES
A.2
Electron Beam Lithography
The electron beam lithography was done on a JEOL JSM-6400 scanning electron microscope fitted with an ion pump, LaB6 filament, computer-controlled beam blanker,
and the NPGS lithography control system. Most of the samples were written at an
accelerating voltage of 40 kV and a working distance of 15 mm. Later samples were
written at 30 kV, but this did not seem to have an effect on the final devices. The
pattern run-file parameters are given in Table A.2. Offset correction was performed
on the 100× magnification patterns, i.e. taking the 1000× centre to be (0, 0). The
correction amounts are not given because they change whenever the SEM column is
re-aligned. Finally, it should be noted that a small amount of silver dust was applied
to the corner of the sample prior to insertion into the SEM. This allowed the SEM to
be focused properly at the height of the resist surface. This focus was usually close
to, but not quite equal to, the focus for the gold on carbon focusing standard. Astigmatism correction was only performed on the gold standard; it was assumed that
this would not change between the standard and the sample. As for all small-scale
lithography, proper focusing and astigmatism correction was very important for these
devices. Without good focus, devices like Fig. A.1 are the result.
Table A.2: Parameters for the SEM run-file
Magnification
Current
Dose
Point Spacing
Line Spacing
CL Setting
Small Features
1000×
10 pA
1450 µC/cm2 lead tips & island
900 µC/cm2 small leads
1.91 nm
1.91 nm
14
Medium & Large Features
100×
2000-4000 pA
675-700 µC/cm2
19.11 nm
19.11 nm
7
APPENDIX A. DETAILED FABRICATION RECIPES
118
Figure A.1: Example of poor focus during lithography. The
sharp corners are rounded and the pointed regions truncated.
A.3
Chemical Processing
All of the chemical processing was performed under a fume hood in the QFAB cleanroom. While it is beyond the scope of this thesis to provide detailed safety instructions, it must be noted that many of the chemicals used in these processes can be
dangerous if proper procedures are not followed. The need for formal training in the
use of these chemicals and for the strict adherence to all safety protocols cannot be
overemphasized.
A.3.1
Development
After the resist has been exposed by EBL, it must be developed. The procedure was
as follows:
APPENDIX A. DETAILED FABRICATION RECIPES
119
1. Rinse the sample in deionized water (DI H2 O) to remove the silver dust and
any other loose particles. Dry with compressed nitrogen gas. Do not immerse
in IPA or acetone, as these will attack the PMMA layer.
2. Immerse the sample in a 1:3 solution of MIBK:IPA for 2 minutes. Rinse in IPA
and dry with N2 .
3. Immerse the sample in MF-319 for 10-30 s. This develops the LOR 7A layer.
This layer was originally LOR 7B, but it was found that 7B developed too
quickly to be repeatable. The variability of the development time was thought
to be due to differences in the age of the resist and developer as well at the
amount of time since the resist was baked. For this reason, the sample was
removed from the developer and imaged with an optical microscope at intervals
through the development time. No further development was done once sufficient
undercut was achieved to avoid collapse of the upper PMMA layer.
4. Rinse the sample with DI H2 O, dry with N2 , rinse with IPA and finally dry
again with N2 whenever it is removed from the MF-319.
A.3.2
Liftoff
After metal evaporation, the layers of resist and unwanted metal were lifted off in
order to reveal the device. This was done by immersing the sample in NanoTM
Remover PG [65] at 78 ◦ C. The sample was left in the Remover until the metal surface
appeared to have bubbled. This indicated that the metal was ready to come off and
usually happened in less than 5 minutes, although additional time in the Remover
did not seem to affect liftoff success. At this point, a pipette was used to squirt the
APPENDIX A. DETAILED FABRICATION RECIPES
120
sample with Remover, causing the unwanted layers to come loose from the sample.
Occasionally, especially in cases where there had been PMMA collapse, liftoff was not
completely successful and unwanted metal remained on the device, connecting leads
or bond pads. An example of this is shown in Fig. A.2. In these cases, liftoff could
sometimes be improved by “shaking loose” the extra material with a few seconds in
the sonicator, but this had an equally good chance of also removing the small features
of the device and thus was not often successful.
Figure A.2: Optical microscope image of material remaining
on device after liftoff. In this case, only the lower device was
ruined since the three leads going to the upper device are still
electrically distinct.
A.3.3
Wet Etching
After FIB milling, wet etching was done to undercut the junction regions (HF) and
undercut the mechanical sections of the device (KOH) and thus free it. Most of
APPENDIX A. DETAILED FABRICATION RECIPES
121
the details involved in wet etching are from the procedures for working with these
chemicals and only the timing is specific to this work. First, the sample was immersed
in HF for 12 minutes. Too long an etch would undercut the beam and leads too much,
eventually removing them all together, so care was taken to not etch too long. After
proper rinsing of the sample, it was etched for 2 minutes in KOH to undercut the
beam.
A.4
Evaporation
As with wet etching, most of the electron-beam evaporation procedure does not
change based on project. For the Cr and Au on the devices, the layers were
• 50 Å Cr deposited normal to the surface at ∼2 Å/s (the sticking layer)
• 300 Å Au deposited normal to the surface at ∼3 Å/s
• 300 Å Au at +15.3◦ to the surface (+17 steps of the stepper motor) at ∼3 Å/s
• 300 Å Au at -14.4◦ to the surface (-16 steps of the stepper motor) at ∼3 Å/s
The three Au evaporations were done in rapid succession, i.e. without adjusting the
electon beam between them. The positions of +17 and -16 steps were chosen to give
close to the required throw of 30◦ .
A.5
FIB Milling
The FIB milling was done on a Micrion 2500 [71] with FIB Assist software from
Fibics Inc. [72]. All milling was done using a 75 µm aperture and a beam current of
APPENDIX A. DETAILED FABRICATION RECIPES
122
∼80 pA. Prior to milling, samples were coated with 70 Å of Al to prevent charging of
the sample surface. In order to ensure good grounding of this metal coating, copper
tape was used to connect the surface to the FIB stage in addition to the clip used
to secure the sample. After the usual start-up and beam optimization procedures,
the device was located and the stage was rotated to align the sample with the FIB
writing axes. From here, the procedure was
1. Near the shuttle, zoom in to the smallest field of view (FOV), 1 µm. Continuously image this area for a few seconds and thus mill a 1 µm square into the
substrate.
2. Zoom out to the writing FOV, 10 µm, and focus on the 1 µm square. Stop
imaging when good focus is achieved.
3. Zoom out, if necessary, centre on the writing area, and return to the writing
FOV. Take single images of the sample when necessary for navigation; the
sample should not be imaged once the image has been zoomed in to writing
FOV. The centring tool is useful for this step.
4. Wait at least 15 minutes for the stage to settle. Do not move the stage during
this time. This helps to minimize drift while writing. If the stage is moved,
restart the 15 minute wait.
5. While waiting, take the time to set up the mills. For the work in this thesis,
bitmap images (.xbm) were created to define the milling regions. The images
were created in the GIMP [83] by overlaying on SEM images of the fabricated
devices. The mills were done to a dose of 0.4 nC/µm2 for a mill depth of ∼70 nm.
APPENDIX A. DETAILED FABRICATION RECIPES
123
This is more than enough to get through the 50 nm of nitride and thin Al layer.
See Fig. A.3 for the dose-depth profile of the FIB.
6. After waiting, take a single image of the writing region, align the mill pattern
to the image, and perform the mill. Repeat as necessary for additional devices.
Figure A.3: Dose-depth profile measured on the FIB. The Al
layer on this sample was 30 nm thick.
Appendix B
Electromigration Per-Sample
Results
The per-device results for single and double junction electromigration are given in the
following tables. For the single junction devices, the B3.x devices were fabricated by
J. Campbell; the author fabricated the cxx devices. For the double junction results,
the devices broken at 78 K were broken in the probe station; the others were done in
the 1.5 K cryostat. Devices which failed prior to reaching the trial temperature are
not listed.
The devices that made it to electromigration testing represented the end of a
long process of developing fabrication recipes and sample handling techniques. Over
the length of the project, 75 single junction and 212 double junction devices were
patterned with the SEM. By the end of the project, ∼ 90% of devices patterned were
suitable for electromigration testing, but these numbers show that this was the result
of extensive testing and refinement of the process.
124
APPENDIX B. ELECTROMIGRATION PER-SAMPLE RESULTS
Table B.1: Table of the single junction devices that were tested
Device
Temp (K)
B3.5 P3 D1
300
B3.5 P4 D2
300
B3.5 P4 D1
300
B3.7 P1 D3
78
78
c8 p1a
c8 p3a
78
78
c8 p4b
c7b p4aR
78
c8b p1aL
78
c8b p1aR
78
c14b p1bR
300
Initial R (Ω) Final R (Ω)
54.4
4×107
42
2×108
58
9.8×104
33.7
2.0×104
26.7
6×105
27.7
4.2×103
36.3
3×108
63.9
4×109
42.3
3.1×104
56.3
1.0×104
25.7
4.4×105
Table B.2: Table of the double junction devices that were tested
Device Temp (K)
c6b p2a
78
c6b p2b
78
c7b p1a
78
78
c7b p1b
c7b p2b
78
c8b p1b
78
c8b p2b
78
c8b p4
78
c12b p4b
77
c14b p3a
300
c14b p2a
4.2
c15b p2a
4.2
c15b p3a
4.2
c3c p3a
78
c3c p2a
4.2
c3c p4b
4.2
c1c p1b
4.2
c1c p2a
4.2
Initial R (Ω)
33.7
43.4
27.5
38.2
49.2
30.6
27.8
21.7
140
96.1
224
114
83
20.5
15
13
134
43.5
Final R (Ω)
4×109
5.4×103
1.5×104
7×107
7.8×105
5×103
1.3×107
2.1×105
4.5×109
5.3×105
(open)
(open)
2.1×107
1.7×103
(open)
(open)
1×103
7.3×105
125
Appendix C
1.5 K Cryostat and Circuit Board
The 1.5 K cryostat is shown in Fig. C.1. It was designed by Cory Dean, a summer
student with the group several years ago, and built by Cory Dean & Gary Contant.
The cryostat closes with a brass can that fits over the leads and 1 K pot and seals on
a taper. This taper seals with a small amount of cerrolow low-temperature solder.
Getting the can to seal can be difficult. Some pointers:
• Do not overheat the taper. Many of the plumbing connections to the taper
are done with indium solder. Its 200◦ C melting point is higher than cerrolow’s
50◦ C, but not by much. Overheating could cause these joints to soften, loosen,
and ultimately leak. As a rule of thumb, if the taper is too hot to touch, it is
too hot.
• For best results, make sure the taper and the inside of the can are clean and
smooth. Some water and a soft abrasive pad can be used for cleaning, if necessary.
• Both the taper and inner can surfaces should be tinned thoroughly with cerrolow
126
APPENDIX C. 1.5 K CRYOSTAT AND CIRCUIT BOARD
127
before attempting to seal.
• Heating the can more than the taper seems to allow the solder to remain hot
enough long enough to flow around and seal well, without overheating the indium solder.
• Finally, if the seal leaks and this leak can be localized, locally heating the
cerrolow in the area of the leak with a soldering iron can close the leak without
needing to remove the can and start over. It also avoids heating with the heat
gun, which applies heat to such a large area that it may cause other leaks to
develop elsewhere.
The circuit board that attaches to the DC leads at the bottom of the cryostat is
shown in Fig. C.2. Once the design for the board had been made, it was sent to
ExpressPCB [84] for fabrication. For mechanical stability, the board mounts to a
copper piece on the end of the cryostat with a 4-40 bolt.
APPENDIX C. 1.5 K CRYOSTAT AND CIRCUIT BOARD
Figure C.1: 1.5 K cryostat
128
129
APPENDIX C. 1.5 K CRYOSTAT AND CIRCUIT BOARD
Letters�refer�to�the�pins�on�the�19-pin�connector.
Resistances�are�for�the�DC�leads�at�room�temperature.
Pins�B�&�C�and�D�&�E�are�connected�to�either�ends�of�a
carbon�resistor�for�4-wire�temperature�measurement.
0.8''
1.2''
0.2''
4-40�clearance
(0.1285''�φ)
0.4''
0.025''�φ
U�->�166�Ω
0.06''
N/C
0.18''
T�->�166�Ω
F�->�165�Ω
S�->�174�Ω
G�->�165�Ω
R�->�175�Ω
H�->�170�Ω
J�->�168�Ω
0.8''
A�->�162�Ω
0.4''
P�->�167�Ω
V�->�164�Ω
K�->�172�Ω
N�->�162�Ω
L�->�167�Ω
M�->�164�Ω
Figure C.2: Cryostat circuit board
Appendix D
The Breaking Cut-off Circuit
The detailed circuit diagram for the cut-off circuit is shown in Fig. D.1. The OP27
operational amplifier is connected as a transimpedance amplifier with gain of 5090
(given by the value of the feedback resistor). This takes the current coming from the
device and turns it into a voltage according to Vi = −Ri I, with Ri = 5.09 kΩ. The
pair of 411 op amps are simply buffers on the output of the transimpedance amplifier.
The voltage signal from the OP27 goes to one of the inputs of the LM311 comparator.
The output of this device switches from low to VCC when the voltage on the inverting
input is higher than the voltage on the non-inverting input.† When the comparator
triggers (more on this in a moment), the high signal causes the MC14027 flip-flop
to be set high. The output of the flip-flip will remain high until the reset switch is
used, ensuring that the output is locked in the cut off state once the comparator has
triggered. The flip-flop output‡ is used to drive (through the TC4425 chip, a dual
†
Note that the voltages here are considered as they lie on the number line, not as the magnitude
of the voltages. That is, the signs matter.
‡
Actually, the compliment of the flip-flop output is used. This is due to its behaviour on Reset
when Set is still high.
130
APPENDIX D. THE BREAKING CUT-OFF CIRCUIT
131
MOSFET driver chip with both inverting and non-inverting circuits) two MOSFETs.
In the low-input state, the MOSFET circuit connects the PID to the device under
test (DUT). Once the comparator has triggered, however, the states of the transistors
flip and the device is grounded, removing the applied voltage and quickly stopping
electromigration. The LED is lit to indicate that the input-to-device path is active;
it is extinguished when the circuit triggers.
The remaining issue concerns the second input to the comparator. This is where
the target resistance is set. At any given time, the current through the device depends
on the resistance of that device and the voltage across it (the voltage being Set by
the PID); this current is transformed to a voltage by the OP27 circuit. At this
set voltage, a device having the target resistance will also have some current and
this current can also be represented by a voltage. In fact, since the gain of the
transimpedance amplifier is known, this target current can be represented by a voltage
that is a fixed fraction of the Set voltage. A voltage divider can thus be used to set
the target resistance. The current through a device of resistance Rstop is given by
Istop = Vset /Rstop . Is can be transformed to a voltage using
Vstop = −Ri Istop
= −Vset
Ri
,
Rstop
which is simply a voltage divider. It was decided to take the stopping voltage across
a 1 kΩ resistor, making the target resistance set by the resistor connected in series,
as given by
1000
Ri
=
.
Rset + 1000
Rstop
APPENDIX D. THE BREAKING CUT-OFF CIRCUIT
132
Working this through gives
Rset = 1000
Rstop
−1 .
Ri
(D.1)
For a target resistance of 10 kΩ, this gives Rset = 965 Ω ' 1 kΩ.† From this point, the
third 411 amplifier is simply connected as a unity gain inverting amplifier. This is to
change the sign of the target voltage to give it the same sign as the output from the
transimpedance amplifier.
It should also be noted that there is a bypass switch on the PID In and To
DUT connections. This switch allows those two connections to simply be shorted
together without involving the transistors, effectively bypassing the cut-off circuit.
This functionality is useful for performing a resistance measurement on the device
without needing to change the target resistance to prevent unwanted triggering of the
circuit.
Finally, a few words need to be said about the 1 MΩ pull-down resistor on the
gate of the transistor that grounds To DUT. Testing has shown that this resistor is
required for the transistor to open properly after being closed. However, it is not
clear why this is the case. After discussion with R. Knobel and D. Stopps, the best
theory is that there may be a problem with the TC4425 MOSFET driver. Any future
changes to this circuit should keep this in mind, and it might be worth testing with
a new TC4425 chip.
†
With Rset = 1 kΩ, the target resistance becomes 10.9 kΩ. Factoring in the 200 kΩ input
impedance of the difference amplifier, this corresponds to a device resistance of 11.5 kΩ.
Figure D.1:
circuit.
From
DUT
K
4
R
MC14027
16
5
Q
Q
2
1
2
4
0.1�µF
-Vs
4
6
0.1�µF
3
6
0.1�µF
TC4425
+Vcc
7
G
5
S
D
-Vs
G
0.1�µF
3
J
S
3
LF411
7
Vi
Output
2
3
S
D
6
6
6
8
2
+Vs
6
6
4
R
5
8
0.1�µF
1
Bypass
Switch
7
Vcc
1�kΩ
External�Box
LM311
0.1�µF
0.1�µF
+Vcc
3.3�kΩ
0.1�µF
R
Rset
In
To
DUT
PID
In
Vset
7 8
-Vs
4 0.1�µF
OP27
7
+Vs
0.1�µF
0.1�µF
-Vs
4
LF411
0.1�µF
+Vs
3
2
1
3
2
+Vs
4
7
3
LF411
7
5.09�kΩ
1�kΩ
2
Vcc
*�All�power�lines�have�0.1�µF�to�ground.
*�Flip-flop�and�MOSFET�driver�single-sided�power,
op�amps�and�comparator�are�bipolar.�Power�is�±12�V.
*�OP27�and�LM311�have�trimmers�attached�to
appropriate�terminals.�Consult�data�sheets�for�details.
APPENDIX D. THE BREAKING CUT-OFF CIRCUIT
133
-Vcc
1�MΩ
Detailed circuit diagram of the breaking cutoff
0.1�µF