The Atomic Point Contact as a Detector of
Nanomechanical Motion
byNathan Edward Flowers-Jacobs
B.S., California Institute of Technology, 2001

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2010This thesis entitled:
The Atomic Point Contact as a Detector of Nanomechanical Motion
written by Nathan Edward Flowers-Jacobs
has been approved for the Department of Physics

Prof. Konrad W. Lehnert

Prof. Charles T. Rogers

Date

The final copy of this thesis has been examined by the signatories, and we find thatboth the content and the form meet acceptable presentation standards of scholarly
work in the above mentioned discipline.

iii
Flowers-Jacobs, Nathan Edward (Ph.D., Physics)
The Atomic Point Contact as a Detector of Nanomechanical Motion
Thesis directed by Prof. Konrad W. Lehnert

I experimentally demonstrate that an atomic point contact (APC) is a sensitive
detector of nanomechanical motion. With a microwave technique, I increase the measurement speed of APCs by a factor of 500. This measurement is fast enough to detect
the resonant motion of nanomechanical structures at frequencies up to 150 MHz. I√√
measure displacement with a shot-noise limited imprecision of Sx = 0.29 fm/ Hz√√
and simultaneously observe a SF = 61 aN/ Hz backaction force. A quantum limited
detector would operate at the limit imposed by the Heisenberg uncertainty principle,√√
Sx SF ≥ h̄; for this APC detector Sx SF = 168h̄. Because the measurement noise isdominated by the shot noise of tunneling electrons, the non-ideality of the APC detector
is likely due to a backaction force in excess of that required by quantum mechanics. Although I cannot unambiguously determine the origin of this excess backaction force, I am
able to eliminate certain possible origins. For example, the observed linear dependenceof the backaction force SF on APC current is inconsistent with a noisy electrostatic attraction mediated by the mutual capacitance between the APC electrodes. In contrast,a model of the backaction force that invokes a momentum impulse delivered by eachtunneling electron correctly accounts for the observed scaling. However, each electron
would have to deliver a momentum impulse greater than 20 times the Fermi momentum
which seems implausibly large. I also observe the signs of molecular vibrations in the
APC using inelastic electron tunneling spectroscopy. At the bias energy associated withvibrations I measure a resonant increase in the backaction force. This observation suggests that the excess backaction may arise from the interaction of tunneling electrons
and molecular vibrations.

iv

Acknowledgements

I would first like to acknowledge and thank my thesis advisor, Prof. Konrad W.
Lehnert. Konrad was willing to spend an enormous amount of time introducing me tocryogenic/microwave techniques and physics as a profession, working beside me during
long nights and providing a constant source of motivation and intellectual stimulation.
I am tremendously grateful.
I first started working in Konrad’s lab with Dan Schmidt, a post-doc, and ManuelCastellanos-Beltran. I am thankful that Dan was always willing to provide advice when
I knew, basically, nothing. I have always appreciated having Manuel across the room;
having someone who shared a similar perspective to talk physics, and gossip, with wasinvaluable.
Similarly, I am glad that I was able to interact with the members of the optomechanics experiment in the Lehnert Lab: Cindy Regal, John Teufel, Jennifer Harlow,
Tobias Donner. It was a pleasure to have another group of people readily available
with whom I could discuss the details of cryogenic nanomechanical fabrication andmeasurement. I also acknowledge and appreciate conversations with professors Aash
Clerk, Steve Girvin, John Price, and Chuck Rogers.
The support staff at JILA are amazing. I would especially like to thank David
Alchenberger who made it possible to perform nano-fabrication at JILA; the wonderful
JILA instrument shop people who were always willing to provide advice on (among otherthings) how not to kill yourself while dealing with heavy loads and spinning blades , inv
particular Tracy Keep for designing and machining the bending apparatus used in thisexperiment; the electronics shop for advice on microwave (and low frequency) electronics
and for allowing me to almost permanently “borrow” spectrum analyzers and network
analyzers; and finally, the computing team and supply office for quickly dealing with
my frantic queries.
On a more personal note, I also immensely appreciate how my partner Celeste
Yang, my friends in Colorado (including Bryan Eastin, Matt Reese, and Samantha
Bench Reese who likes getting her name in theses), and my family have supported meemotionally and physically. I am in their debt, most recently for their willingness to
put up with me during the thesis writing process.

vi

Contents

Chapter
1 Introduction

1

2 Background: Mechanics Coupled to Mesoscopics

9

2.1

2.2

2.3

Nanomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.1.1

Nanomechanics as a gentle probe of the nanoworld . . . . . . . .

12

2.1.2

Applications of nanomechanical oscillators . . . . . . . . . . . . .

15

2.1.3

Quantum behavior of macroscopic degree of freedom . . . . . . .

17

Mesoscopic Amplifiers Coupled to Nanomechanical Oscillators . . . . . .

18

2.2.1

SET nanomechanical displacement detector . . . . . . . . . . . .

19

2.2.2

SQUID nanomechanical displacement detector . . . . . . . . . .

23

2.2.3

QPC nanomechanical displacement detector . . . . . . . . . . . .

25

2.2.4

Carbon nanotube displacement detector . . . . . . . . . . . . . .

27

The Atomic Point Contact as a Mesoscopic Detector . . . . . . . . . . .

28

2.3.1

APC as a widely used detector of static displacements . . . . . .

28

2.3.2

APC as a quantum-limited displacement detector . . . . . . . . .

35

3 The Theory of an Atomic Point Contact Coupled to a Nanomechanical Oscillator 41
3.1

Electron Tunneling Across an Atomic Point Contact . . . . . . . . . . .

42

3.1.1

Nano-scale ballistic contact . . . . . . . . . . . . . . . . . . . . .

43

3.1.2

Electron shot noise . . . . . . . . . . . . . . . . . . . . . . . . . .

48

vii
3.1.3

Tunneling probability . . . . . . . . . . . . . . . . . . . . . . . .

50

APC as a Quantum-Limited Displacement Detector . . . . . . . . . . .

58

3.2.1

The standard quantum limit . . . . . . . . . . . . . . . . . . . .

59

3.2.2

APC displacement detection. . . . . . . . . . . . . . . . . . . .

62

3.2.3

APC backaction and the quantum limit . . . . . . . . . . . . . .

64

Simple Harmonic Motion of the Nanostructure and APC . . . . . . . . .

67

3.3.1

The modes of a nanomechanical beam . . . . . . . . . . . . . . .

68

3.3.2

Mechanical modes modeled as harmonic oscillators . . . . . . . .

74

3.3.3

The influence of the APC on the nanomechanical modes . . . . .

79

3.3.4

Calibrating nanomechanical motion . . . . . . . . . . . . . . . .

85

4 Experimental realization of an APC embedded in a nanomechanical oscillator

91

3.2

3.3

4.1

4.2

APC Creation Using Electromigration . . . . . . . . . . . . . . . . . . .

92

4.1.1

Introduction to electromigration . . . . . . . . . . . . . . . . . .

92

4.1.2

Design of devices for electromigration . . . . . . . . . . . . . . .

96

4.1.3

Fabrication of devices for electromigration . . . . . . . . . . . . . 100

4.1.4

Electromigration procedure . . . . . . . . . . . . . . . . . . . . . 103

4.1.5

Electromigration failures and frustrations . . . . . . . . . . . . . 109

APC Creation Using a Mechanically Controllable Break Junction . . . . 113
4.2.1

Mechanically controllable break junctions . . . . . . . . . . . . . 113

4.2.2

MCBJ bending apparatus in a 3 He cryostat . . . . . . . . . . . . 118

4.2.3

MCBJ device fabrication . . . . . . . . . . . . . . . . . . . . . . 124

4.2.4

Creating an APC using a MCBJ . . . . . . . . . . . . . . . . . . 129

5 Electrical measurement of APC Resistance
5.1

131

Shot Noise Limit in Idealized APC Displacement Measurement . . . . . 134
5.1.1

Noise in an APC resistance measurement . . . . . . . . . . . . . 134

5.1.2

Noise in an APC displacement measurement . . . . . . . . . . . 139

viii
5.2

5.3

Conventional APC Resistance Measurement . . . . . . . . . . . . . . . . 146
5.2.1

Limited bandwidth of a conventional measurement . . . . . . . . 146

5.2.2

Description of low frequency resistance measurement . . . . . . . 149

Microwave APC Resistance Measurement . . . . . . . . . . . . . . . . . 152
5.3.1

Improved bandwidth using microwave measurement . . . . . . . 152

5.3.2

Network analysis of microwave measurement . . . . . . . . . . . 160

5.4

Measurement of APC resistance at the Mechanical Resonance Frequency 169

5.5

Using Shot Noise to Calibrate the Microwave Measurement . . . . . . . 176

6 Sensitive Position Measurement using Electromigrated APC
6.1

181

Initial Demonstration of the APC Displacement detector . . . . . . . . . 183
6.1.1

Comparison between APC detection and magnetomotive detection 184

6.1.2

APC detector calibration using thermal noise . . . . . . . . . . . 188

6.1.3

The Heisenberg uncertainty principle and the standard quantum
limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6.1.4
6.2

The experimental APC detector compared to the quantum limits 199

Another Similar Electromigrated Device . . . . . . . . . . . . . . . . . . 202
6.2.1

Improved thermal noise calibration . . . . . . . . . . . . . . . . . 205

6.2.2

Measurement backaction . . . . . . . . . . . . . . . . . . . . . . . 211

7 Results: Mechanically Controllable APC coupled to nanomechanical motion

216

7.1

Mechanically Controlling the APC Resistance . . . . . . . . . . . . . . . 217

7.2

Changes in APC Resistance Effect Nanomechanical Modes

7.3

APC Resistance Effects Displacement Sensitivity . . . . . . . . . . . . . 228

7.4

APC Detector Backaction . . . . . . . . . . . . . . . . . . . . . . . . . . 231

7.5. . . . . . . 225

7.4.1

Unintentional feedback caused by the backaction force . . . . . . 232

7.4.2

Correlations between shot noise and nanomechanical motion. . 241

APC Backaction and Tunneling Spectroscopy . . . . . . . . . . . . . . . 246

ix
8 Conclusions and Future Directions

256

8.1

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

8.2

Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Bibliography

263

Appendix
A Fabrication of Nanostructures on Gallium Arsenide Substrates

278

B Fabrication of Nanostructures on Silicon Substrates

281

C Fabrication of Nanostructures on Phosphor Bronze/Polyimide Substrates

285

D Algebra Related to Unintentional Feedback

290Chapter 1

Introduction

The goals of the experiments described in this thesis were to create and evaluate
the atomic point contact (APC) as a detector of nanomechanical motion. The firststep towards these goals was to couple the APC, a mesoscopic device, to the dynamics of mechanical displacement. By making the relative position of the APC atoms adynamical variable, I hoped to extend the field of mesoscopic electronics to mesoscopic
electromechanics. The second step was to increase the speed of APC measurements sothat the APC can be used to sensitively observe nanomechanical motion at frequencies between 10 MHz and 150 MHz. The final step was to measure both the APC’s
displacement sensitivity and the backaction force created by the APC detector. These
measurements were used to evaluate theoretical claims that the APC detector can be aquantum-limited displacement detector.
An APC is formed by locating an atomically sharp metal tip near another conducting mass (figure 1.1), as in a scanning tunneling microscope (STM). When thevacuum gap between the tip and the mass is on the order of a nanometer, electrons
incident on the APC can quantum-mechanically tunnel across the gap between atoms
in the APC. The probability that an electron will tunnel and be transmitted across thegap, instead of reflecting off of the gap, is exponentially dependent on the width of thegap. In most metals, including the gold APCs used in this thesis, the length scale λ that
controls the exponential dependence is approximately λ ≈ 100 pm. A voltage applied

2
across the gap (figure 1.1) creates a difference in chemical potential for electrons in thetwo electrodes. The flow of electrons that results from the chemical potential difference is an electrical current which is extremely sensitive to the size of the separation
between the mass and the point. If the mass is attached to a spring (figure 1.1), thenthe measured changes in the current through the APC is a sensitive measurement of
the displacement of the mass-on-a-spring harmonic oscillator.

x

Atomic Point
Contact (APC)

VAPC
Figure 1.1: A conceptual diagram of an APC displacement detector. A metal mass (redbox) is placed about one nanometer away from an atomically sharp, metal tip (green
pyramid). The atomic point contact is formed between two atoms (orange circles).
A voltage bias VAP C will cause electrons to tunnel across the APC and the tunnelingcurrent is a sensitive measurement of the displacement x of the metal mass. If the metal
mass is attached to a spring (blue), then the APC displacement detector measures thedisplacement of the resulting harmonic oscillator.

There were three main goals behind this investigation of the APC displacement
detector. First, I wanted to expand the field of mesoscopic electronics to mesoscopic
electromechanics by using the APC, a mesoscopic device, to sense the dynamical motionof a nanomechanical structure instead of the static position of atoms. Mesoscopic elec-

3tronic devices are those devices in which the discrete charge, spin, or quantum phase ofelectrons is manifest in the device behavior. The current in mesoscopic devices cannot
be modeled as the flow of a classical charged fluid. The APC is a mesoscopic device
where the discreteness of electrons is manifest because electrons pass across the APC
by tunneling through a barrier. The probabilistic nature of discrete electrons tunnelingacross the barrier creates shot noise; in comparison, a resistance which reduces the flowof a charged fluid would not create shot noise. The devices I study as part of my thesis
will couple the discreteness of charge to a mechanical oscillator.
APCs are widely used as atomic-scale mesoscopic sensors. The atomic resolution
in a STM is provided by the APC formed between the STM’s atomically sharp tip andthe substrate. APCs are used in mechanically controllable break junctions to study how
metals fracture and flow at the atomic scale. They are also used to electrically contact
single molecules. In each of these examples, the APC combines the mesoscopic effects
associated with the discreteness of charge with a sensitivity to the static position of
atoms. This thesis was motivated by the possibility of using this sensitivity to detectthe dynamic motion of nanomechanical structures.
Experimentally, the first step in achieving this goal was creating the APC in situ
as part of a nanomechanical structure (figure 1.2, for example). The frequency of thenanostructure’s fundamental mode is between 10 MHz and 100 MHz. The APC is created in the ultra-high vacuum environment provided by a 4 K cryogenic system using
either electromigration or a mechanically controllable strain. The displacement of thenanostructure strains the APC and changes the width of the APC gap, so the APC displacement detector can sensitively measure the harmonic motion of the nanostructure.
The final step towards sensitively detecting nanomechanical displacement, andthe second goal of this project, was the development of a high bandwidth readout ofan APC. In the past, only about 100 kHz of an APC’s intrinsically large bandwidth isusually realized because the APC, which is necessarily high-impedance RAP C > 20 kΩ, is

4

VAPC

1μm

Atomic Point
Contact (APC)

Figure 1.2: A scanning electron micrograph of a nanostructure prior to APC creation.
The APC will be created at the constriction between the stationary electrode (green) andnanomechanical beam (red); the entire structure (red and green) is made entirely outof gold. A voltage bias VAP C applied between the electrode and the nanomechanicalsupports will cause electrons to tunnel across the APC and is used to measure thedisplacement of the nanomechanical beam.

shunted by the large cable capacitance between the device and the remote measurement
electronics. I overcame this limitation by embedding the high-impedance device in a LC
resonant circuit; this is the same microwave technique that was used to create the radio
frequency single electron transistor (RF-SET). This microwave APC measurement hasa bandwidth that is controlled by the quality factor of the resonant circuit which, inthis case, is greater than 30 MHz [11].
The final goal of this project was to evaluate theoretical claims that the APC
can be a quantum-limited displacement detector. The noise in any displacement detector will result in an imprecise measurement of displacement; the imprecision can be
expressed as a displacement noise spectral density Sx . The detector will also effect themechanical system that is being measured and create a noisy backaction force with aspectral density SF . A continuous linear measurement of position is subject to the√
Heisenberg constraint Sx SF ≥ h̄. When the detector is measuring the displacement ofa harmonic oscillator, this quantum-mechanical constraint also limits the total uncer-

5tainty in the displacement measurement due to the sum of the measurement imprecisionand the random motion created by the backaction force. The point of minimum total
uncertainty is called the standard quantum limit and occurs, on-resonance, at an imprecision SxSQL = h̄/mγω0 and backaction SF SQL = h̄mγω0 where m is the mass ofthe oscillator, γ is the damping, and ω0 is the resonance frequency.
In a basic phenomenological model of an APC displacement detector at the quantum limit, the imprecision Sx is due to the shot noise of tunneling electrons and the
backaction SF is due to the random momentum kick imparted by each tunneling electron. Unlike many other mesoscopic detectors, the large 5 eV energy scale of the APC
barrier allows a large voltage bias (50 mV in this experiment) to be applied acrossthe contact without significantly affecting the mesoscopic properties of the detector.
Because I use both a large voltage bias and the microwave measurement techniquementioned earlier, I can make a displacement measurement whose noise is dominatedby the fundamental shot noise of tunneling electrons instead of the other, less interesting, sources of noise such as the noise of following amplifiers. This is important because
it is a prerequisite for making quantum-limited displacement measurements. Experimentally, I use the APC detector to measure nanomechanical motion with a shot-noise
limited imprecision Sx = 0.08 fm2 /Hz = 20SxSQL .
I also measure the noisy backaction force created by the detection process. Since
Sx > SxSQL , if this APC detector was operating at the quantum limit then SF < SF SQL .
Instead, I measure a backaction force with a spectral density SF = 3750 aN2 /Hz =√1400SF SQL and an imprecision-backaction product Sx SF = 168h̄ which is 168 times
larger than required by quantum mechanics. The total imprecision on-resonance Sx +SF H(ω0 ), where H(ω) is the harmonic oscillator response function, is 730 times thetotal imprecision at the standard quantum limit.
Since the imprecision is shot-noise limited, I conclude that the likely source of
non-ideality in the APC displacement detector is a backaction force in excess of that

6
required by quantum mechanics. If the entire backaction force is due to the momentum
kicks of tunneling electrons, each electron would have to deliver a momentum equal to
23 times the Fermi momentum pF . Since an electron with an energy equal to the heightof the tunnel barrier would only have a momentum equal to about 2pF , the electron
momentum required to account for the observed backaction is implausibly large.
The momentum kicks from tunneling electrons would result in a backaction force
proportional to the current IAP C through the APC, but there are also other possible
sources of detector backaction which depend on the voltage VAP C across the APC.
The capacitance of the APC changes as the width of the gap changes, creating an
2electrostatic backaction force proportional to VAP
C . The presence of trapped charges

near the APC would result in a backaction force proportional to VAP C . Finally, the2power VAP
C /RAP C dissipated near the APC could effectively create a backaction force

by increasing the temperature of the nanostructure.
I have attempted to determine the origin of the backaction force using four different approaches. While all of the results are consistent with a backaction force created bymomentum kicks from tunneling electrons, further work is required to both conclusively
demonstrate the origin of the backaction and understand the physical mechanism.
First, the simplest approach is to measure multiple devices in different configurations. I have measured the random backaction force of three different devices using
two different nanostructure geometries and APC creation techniques. In all three cases,
the random backaction force is consistent with each tunneling electron delivering amomentum between 20pF and 35pF ; the factor of two difference is contained within
the large systematic error caused by my poor knowledge of the shape of the nanomechanical modes. Although the origin of the backaction force remains unknown, thesemeasurement do allow me to determine that the size of this random force SF is linearly
proportional to the current or voltage across the APC. This observation is inconsistent with a random force which arises from the potential fluctuations across the mutual

7
capacitance between the APC electrodes.
Second, I use an unintentional feedback loop created by the APC measurement
to determine that the backaction force is more consistent with a force proportional to
IAP C , such as the momentum kicks of tunneling electrons, than with a force proportional
to VAP C , such as a force created by the presence of trapped charges. The feedback
loop is unintentional because I do not measure displacement and then create a force
proportional to the measured displacement. Instead, the feedback loop is caused by the
motion of the nanostructure changing the current or voltage across the APC and thuschanging the backaction force. The magnitude of the feedback depends upon both theorigin of the backaction force and the electrical circuit surrounding the APC. As an
example, if the APC is voltage biased then the voltage VAP C does not depend on RAP C
and only a backaction force proportional to current IAP C would result in feedback. I
observe a feedback force which is more consistent with a backaction force proportional tocurrent. The observed feedback cannot be used to determine the physical origin of thebackaction force, but if the backaction force is due to the momentum kicks of tunneling
electrons then each electron would have to deliver a momentum kick between pF and
40pF .
Third, I attempt to determine whether the observed backaction force is correlatedwith the shot noise of tunneling electrons. If the backaction force is caused by heating
due to electrical power dissipation, then it would not be correlated with the shot noise.
On the other hand, a backaction force caused by the fluctuating VAP C or IAP C would
be correlated with the shot noise. The correlation measurement is plagued by the noise
added by the first amplifier in the measurement circuit; I am only able to determinethat the observed signal is consistent with an electron momentum < 50pF . An amplifier
with less noise would be required to better determine the correlations between electricalshot noise and the observed backaction force.
Finally, I simultaneously measure the backaction force and differential resistance8
RAP C of the APC as a function of the dc voltage across the APC. This type of tunnelingspectroscopy has been used in other experiments to investigate the vibrational modes ofmolecules trapped in the APC. I sometimes observe features in the differential resistanceas a function of voltage that are similar to the vibrational features observed in otherexperiments. At the same dc APC voltage where I observe these features, I also observe
an large increase (> ×2) in the backaction force. This opens the possibility that thevibration of nearby atoms or molecules could play a role in the large backaction force.
This thesis is organized into four broad sections: background, theory, experimental techniques, and results. In the next chapter (chapter 2), I discuss in more detailthe past work involving nanomechanics and atomic point contacts that is relevant toand helped motivate this thesis. I then explain the necessary theoretical background inchapter 3. The experimental techniques used to create an APC coupled to a nanomechanical structure and perform the fast, shot-noise limited measurement of the APC
are described in chapters 4 and 5, respectively. In chapters 6 and 7 I present in detailthe experimental results of my evaluation of the APC as a displacement detector; theseresults are briefly summarized in this introduction. Finally, I conclude this thesis inchapter 8 and discuss some possible directions for further research.

Chapter 2

Background: Mechanics Coupled to Mesoscopics

This thesis describes an experiment that combines the fields of nanomechanics andmesoscopics, using the mesoscopic physics of tunneling electrons to create a sensor of
nanomechanical motion: the atomic point contact (APC) displacement detector. Bothof these fields spring from the idea of nanotechnology [12,13]: that shrinking the overall
size of systems to the nanometer scale and beyond can lead to new and interesting
physics and applications. Nanomechanics is the the study of mechanical systems on thenanometer (or 100s of nanometers) scale [14–16]. Mesoscopic physics usually involves
studying electrical systems created using physical structures on a similar scale, however
the important feature of mesoscopic systems is that the flow of electrical current canno longer be described by the flow of a charged fluid. Instead, the discreteness of anelectron becomes important [17–20].
In this chapter I am going to summarize the work in nanomechanics and mesoscopics that has motivated and influenced this thesis. Starting with the broad motivation forinterest in nanomechanics, I will describe the important role nanomechanical systems
play in nanoscale probes and other sensors [21–23]. More specifically, the desire to createa gravitational wave detector twenty years ago led to the first investigations into how
mesoscopic physics could be used to sensitively detect nanomechanical motion [24–28].
In the past ten years that interest has shifted towards using nanomechanical systems todemonstrate quantum limited measurements of nanomechanical modes and observing

10
quantum mechanical effects in a macroscopic degree of freedom [3–5, 29–34].
After covering this background on nanomechanics, I will focus on how four mesoscopic systems have been coupled to nanomechanical motion in attempts to make quantum limited measurements. The first technique is to use a single electron transistor(SET) which is capacitively coupled to the nanomechanical device [3, 31, 32]. The effectthat the SET displacement detector has on the system being measured, that is, theSET’s backaction, has been measured [31]. This is a prerequisite for quantum limited
detector, though in the case of the SET experiment the backaction was twenty times
larger than theoretically predicted. A second technique involves magnetically couplinga superconducting quantum interference devices (SQUIDs) to a nanomechanical beam.
As a third example, I will describe how nanomechanical motion has been coupled toquantum point contacts (QPCs) using both piezoelectricity [33] and capacitance [5].
This technique using QPCs is especially interesting because the APC and the QPC
measurements are theoretically similar and have the same tunneling Hamiltonian. I
will then briefly discuss how the nanomechanical motion of carbon nanotubes can bemeasured using the transistor properties of semi-conducting carbon nanotubes [6,35–37].
I also describe previous work using atomic point contacts (APCs). APCs havebeen widely used as detectors of static changes in position, since they provide theatomic resolution for imaging and manipulation in scanning tunneling microscopes(STMs) [7, 38, 39]. STMs have a large mechanical support structure to allow the tip toscan over a large three dimensional area; however, APCs have also been fabricated aspart of a stiffer, more compact nanostructure. The experiments in this thesis build onthese fabrication techniques, where the APC is formed from a constriction in the nanostructure either by passing a large current through the constriction thus creating an APC
using electromigration [40–43] or by stretching the constriction to form a mechanicallycontrollable break junction (MCBJ) [8, 44–48]. In the past these techniques have beenused to measure static properties of the contact and study how metals fracture and flow

11
at the nanoscale (summarized in [8]) as well as the electrical properties of molecules
trapped in the contact [40, 49–57].
Finally, I will discuss theoretical proposals and evaluations of the APC displacement detectors. In the late 1980s and early 1990s the APC displacement detector was
proposed as a possible means to detect the strain created by gravitational waves [58–63].
At the time, the APC displacement detector was evaluated theoretically and predicted
to be a quantum-limited detector [64–66]. Since 2002 there has been renewed theoretical interest in the APC displacement detector in the context of quantum-limited
measurement [67–77].

2.1

Nanomechanics
Research into new detectors of nanomechanical motion, such as the research de-scribed in this thesis, is partially motivated by the wide use of nanomechanical motion
in other sensors. This motivation comes from the possibility that a new detector of
nanomechanical motion will result in better sensors, either because the new detector
has a better absolute sensitivity or because the change in constraints on the sensor due
to the change in detection method results in a better sensor. I am therefore going toemphasize the importance of nanomechanics by describing two ways in which nanomechanical devices are used to probe the nanoscale world, atomic force microscopy (AFM)
and magnetic resonance force microscopy (MRFM), as well as some other applicationsof nanomechanical devices.
The other motivation for research into new detectors of nanomechanical motionis the desire to measure the quantum behavior of a macroscopic degree of freedom.
The initial interest in the quantum limits of position detection revolved around interestin designing gravity wave detectors. However most current gravity wave detectors use
optical interferometry, and mesoscopic experiments have moved away from this focus on
gravity wave detection. Instead, more recent mesoscopic experiments have focused on

12
two goals: detecting nanomechanical motion with an imprecision at the quantum limit
and cooling a nanomechanical mode to the ground state defined by quantum mechanics.
I will end this section (section 2.1) by discussing both of these goals in the context ofnanomechanical motion; in the next section (section 2.2) I will describe a number ofmesoscopic experiments aimed at creating a quantum-limited position detector.

2.1.1

Nanomechanics as a gentle probe of the nanoworld

Two different techniques use the exquisite force sensitivity of nanomechanicaloscillators to create images with atomic-scale resolution. In magnetic resonance force
microscopy (MRFM, figure 2.1), a nanomechanical oscillator is used to sense the force
due to spins rotated by an oscillating microwave field in a static magnetic field with alarge magnetic field gradient [78–80]. The combination of the microwave field at frequency ω0 and static magnetic field with a large gradient Bz (y) results in the resonant
excitation of spins in a thin slice of the sample where Bz (y) = ω0 /γ (γ is the gyromagnetic ratio of the sample). The frequency of the microwave field is usually modulatedslightly around ω0 , causing the spins to Rabi flop and creating a magnetic moment in theẑ direction mz (t) with a sinusoidal time dependence [1, 78, 81–88]. Since this magnetic
moment is in a static magnetic field gradient ∂Bz /∂y, there is a force on the cantilever
due to the precession of the spins
Fy (t) = mz (t)

∂Bz
∂y

(2.1)

The frequency of the force can be matched to the < 10 kHz nanomechanical resonancefrequency, where the nanomechanical system has the best force sensitivity, by matching
the mechanical resonance frequency to the rate at which the spins flip. This rate iscontrolled by the frequency of the modulations to the frequency of the microwave field.
The location of the resonant slice can be moved by changing the location of thesample in the magnetic field. Since this slice can be as thin as a few nanometers, the13Figure 2.1: Basic diagram of Magnetic Resonance Force Microscopy (MRFM), figure
from reference [1].

MRFM resolution is on the nanometer scale [81]. By sensitively detecting the force
on the cantilever as the resonant slice is moved through the sample it is possible, forexample, to determine the location of a single electron spin [78] or image a tobacco
mosaic virus [81].
One future direction of MRFM research is pushing the observation frequency to
the Larmor frequency ω0 which is usually in the MHz to GHz regime. This increasein frequency could be helped by new types of detectors designed to monitor smaller,higher frequency nanomechanical resonators. The optical interferometer technique currently used to detect cantilever motion works best with large, and thus low frequency,
cantilevers.

14

Figure 2.2: Basic diagram of an Atomic Force Microscope (AFM), figure from reference[2]. The surface is imaged by measuring the force between the tip of the cantilever andthe surface as the tip is scanned over the surface.

In atomic force microscopy (AFM), a surface is imaged by measuring the force
between a sharp tip attached to a lever and the surface as the tip is moved over thesurface (figure 2.2). In general, the force is larger when the gap between the tip andthe surface is smaller. This map of force versus tip position can be used to infer thethree-dimensional surface topology [21–23]). If the tip is atomically sharp then it ispossible to image the surface with atomic resolution [21, 89–92].
In the twenty five years since the first AFM experiments, this technique has
progressed from it’s origin as a method to measure forces in an scanning tunnelingmicroscope (STM) to it’s current place as a widely used tool for imaging nanostruc-15tures and measuring force in both biological and material sciences. The initial AFM
measurements where taken using a diamond STM tip which was attached to the free
end of a macroscopic singly-clamped beam, a 0.8 mm long by 0.25 mm wide by 25 µm
thick piece of gold foil. The force on the beam was then detected by measuring thebeam’s 5.8 kHz resonant motion using the STM’s tunneling current between the tip
and the sample [23]. However, in modern AFM measurements a smaller beam withmicron-scale dimensions and an optical readout is usually used and changes in force areusually detected by measuring changes in the beam’s resonance frequency [21]. AFMs
have become important and common laboratory instruments, commercially produced
for general surface characterization and imaging. AFMs are also the key measurement
tool in continuing research into understanding nanoscale forces, macromolecules, andbiological materials [21, 22].

2.1.2

Applications of nanomechanical oscillators

Nanomechanical oscillators are not only used to probe the nanoscale world, butthey also play a role in other sensors both in a research setting and commercially.
Two examples of the role of nanomechanical sensors in current research are in sensitive
detectors of mass [93–96] and in the related area of detecting the presence of molecules
for biological applications [97–101].
In general, mechanical oscillators can be used to detect mass by measuring theeffect on the oscillator’s resonance frequency of placing mass on the oscillator. The
deposition of a small amount of mass causes a measurable change in the resonance frequency of a nanomechanical oscillator [95, 96] because, compared to large oscillators,
nanomechanical oscillators have a low total mass. To be more quantitative, recent experiments with nanomechanical systems have demonstrated a mass sensitivity of about√
10−23 kg/ Hz using a doubly clamped SiC beam which is 2 µm long by 150 nm wide by
100 nm thick and has a resonance frequency of about 150 MHz with a quality factor of16√5000 [94]. Further improvements in sensitivity, to about 10−25 kg/ Hz, have been madeby using a singly-clamped carbon nanotube to make a lighter mechanical resonator [93].
Nanomechanical oscillators are also promising sensors of biological molecules. The
standard approach combines the mass sensitivity described in the previous paragraphwith the application of a molecular coating to the nanomechanical oscillator which
selectively binds the molecule of interest [100,102–104]. The presence and concentrationof the molecule of interest can then be deduced by measuring the changes in the massof the nanomechanical oscillator. In general, it is desirable to operate the mass sensor
in a room temperature and pressure environment (or in a fluid) which has the effectof lowering the mechanical quality factors and complicating the entire measurement;
this has led to some novel approaches such as flowing solvents through, instead ofaround, a micron-scale mechanical oscillator [102]. Sensors based on this techniqueare being developed [99, 101]. There is also research into using the mass sensitivityof a nanomechanical oscillator to effectively create a mass spectrometer, determining
the identity of an absorbed macromolecule based purely on the detected mass of themolecule [97].
Finally, nanomechanical and micromechanical oscillators also have many commercial applications [105–114], including as inertial sensors for automotive and navigation
applications [109, 114] and in analog signal processing [109, 112]. While I am not going
to describe these devices in detail, in general the use of micromechanical oscillators
allows an intertial sensor to be compact, low power, and cheap without losing sensitivity [114]. In the realm of analog signal processing, the mechanical oscillator’s motion
at it’s resonance frequency can be used as an electrical filter with a high quality factor.
In comparison with a filter built from discrete inductors and capacitors, the mechanicaloscillator is both high-Q and compact. The frequency of the mechanical resonance canalso be tuned over a wide range in frequency and dissipates very little power during
operation [112]. This combination makes electrical filters created from micromechanical17
oscillators ideal for many different applications, particularly for wireless communication [109].

2.1.3

Quantum behavior of macroscopic degree of freedom

The initial interest in quantum limited position detection in the 1970s, 1980s,
and 1990s was motivated by a desire to detect gravitational waves. Gravitational waves
effectively create a classical force on a mechanical oscillator. Therefore there was alarge amount of theoretical interest in the limits imposed by quantum mechanics onthe detection of classical forces [24, 34] as well as in general limits on measurement
precision [115, 116].
At the most basic theoretical level, the quantum limit on a measurement is described by the Heisenberg Uncertainty principle. For a measurement of the position ofa particle, the quantum limit is ∆x∆p ≥ h̄/2 where ∆x is the uncertainty in the position measurement and ∆p is the uncertainty in the momentum of particle due to themeasurement [115, 117, 118]. This formula describes the effect of a single measurement,
however from the perspective of force detection a more interesting case is the continuous
measurement of position. In this case, the quantum limit on a continuous measurement
is Sx SF ≥ h̄2 (see chapter 3 for more details), where Sx is the spectral density of thedisplacement imprecision and SF is the spectral density of the random force caused by
the detector, that is, the backaction of the detector [72, 115, 116]. This thesis focuses on
evaluating a specific detector, the atomic point contact (APC) displacement detector,
in the context of this quantum limit.
Accompanying these theoretical discussions of the quantum limit were proposals
describing detectors which could approach these limits. Many different types of detectors
where considered [24,25,34,115], including optical interferometers, Weber-type resonantbars coupled to SQUID amplifiers, and detectors using electron tunneling in atomic
point contacts. The most sensitive gravitational wave detectors in use today employ

18
optical interferometers [119, 120], however Weber-type resonant bar detectors are still
being used [121–126] and in active development [127]. Gravitational wave detectors thatused electron tunneling were never implemented, though the analysis provided in the
proposals, which will be discussed in more detail in section 2.3.2, helped motivate theresearch in this thesis.
Since 2000, interest in mesoscopic detectors at the quantum limit has largely
shifted from a desire to detect gravitational waves towards the dual goal of demonstrating a quantum limited detector and then detecting the quantum behavior of a macroscopic degree of freedom [29, 30]. This has resulted in a change in focus, from detecting
the motion of large masses which are better coupled to gravity waves to detecting the
motion of nanomechanical objects which are easily integrated with mesoscopic detectors
and are both higher frequency and lower mass [3–5,11,31–33]. The move towards higher
frequencies is important in the context of the second goal, because the first step towards
detecting quantum behavior is usually the ability to cool the oscillator to it’s quantum
mechanical ground state Tosc ≈ h̄ω0 /2, where ω0 is the frequency of the mechanicaloscillator. Making lighter oscillators is also important because it implies larger, moreeasily detectable, motion when the oscillator is in the ground state in comparison withmore massive oscillators that have the same resonance frequency.

2.2

Mesoscopic Amplifiers Coupled to Nanomechanical Oscillators
Having discussed some of the motivations for sensing nanomechanical motion, I

now describe four different types of mesoscopic position detectors. The first type of detector capacitively couples a nanomechanical beam to a single electron transistor (SET)and uses the mesoscopic physics of electrons tunneling through multiple Josephson junctions to sensitively detect the change in capacitance due to motion of the nanomechanical beam [3, 31, 32]. The second type of detector also uses Josephson junctions, but inthis case the junctions are used to form a superconducting quantum interference device

19(SQUID) which is inductively coupled to a nanomechanical beam [4]. The third typeof detector couples nanomechanical motion to a quantum point contact (QPC), andnanomechanical motion changes the probability of electrons tunneling through a barrier
in a two-dimensional electron gas [5, 33]. While physically different from the detector
described in this thesis, which involves electrons tunneling through a vacuum gap between atoms, the theoretical descriptions of the APC and QPC detectors are equivalent.
I will finally describe how the motion of carbon nanotubes results in a change in theelectrical properties of the nanotube, thus creating a sensor of motion that uses themesoscopic properties of the nanotube [6, 35–37].

2.2.1

SET nanomechanical displacement detector

The displacement detectors based on single electron transistors (SETs) leverage
the extraordinary charge sensitivity of the radio frequency SET (RF-SET, [128]) to detect the motion of a charged nanomechanical beam [3,31,32]. One type of single electrontransistor is composed of a conducting metal wire interrupted by two barriers of thininsulating material placed in series; the island between the two barriers is capacitively
coupled to a gate electrode [129, 130]. Each insulating barrier is a capacitively-shunted
tunnel junction: the metallic leads separated by an insulating layer forms a capacitor
and the insulating layer is thin enough that electrons can tunnel across the gap. In
comparison to an APC or QPC, the tunnel junctions in an SET have a large number ofchannels through which electrons can tunnel, though each channel has a much smallertransmission probability. A voltage bias is applied across the junction-island-junction
circuit (effectively the drain-source voltage, see figure 2.3b), and current flows when an
electron sequentially tunnels across the first barrier and then the second barrier.
If the drain-source voltage is small compared to the voltage needed to charge all
the capacitances in the SET with one electron, then the mesoscopic physics of Coulomb
blockade makes the current through the junctions very dependent on the SET gate

20junctions

beam

1 μm

Figure 2.3: SET detector of nanomechanical motion, figure from reference [3]voltage. If the voltage on the gate is such that there is an integer number of electrons
on the island, then it is energetically unfavorable for electrons to tunnel on or off of theisland and current does not flow. However, if the voltage on the gate is such that the
charge on the island is equivalent to half of an integer number of electrons then current
will flow and, by appropriately biasing between these two extremes, the SET drainsource current is a sensitive measurement of the charge on the gate [130]. The SET
displacement detector is created by capacitively coupling the nanomechanical beam to
the SET island, that is, the beam is an SET gate. When the beam is held at the correct,
usually large, potential compared to to the SET island, the drain-source current of theSET is modulated by the nanomechanical motion and can be used to sensitively infer
the displacement of the beam.
One problem with SET measurements is that SETs, like the atomic point contacts

21(APCs) described in this thesis, are intrinsically high impedance devices RSET > 26 kΩwhich makes the fast measurements required to detect the 10s of MHz nanomechanicalresonance frequencies more difficult. In general, the stray capacitance to ground of theleads to the SET and the high impedance of the SET limit the typical bandwidth of themeasurement to less than 1 MHz. However, Schoelkopf et al demonstrated a techniquethat increased the bandwidth of the SET to more than 100 MHz by using a microwave
matching circuit to create the radio frequency SET (RF-SET, [128]). Experimentally,
RF-SETs are used to detect nanomechanical motion [3, 31, 32], and I use this same microwave technique to greatly increase the bandwidth of APC measurements (see chapter
5, [11]).
The other problem with SET measurements is that the intrinsically near-quantum
limited performance of an SET amplifier is not realized in practice. In an SET, as in
an atomic point contact, the fundamental source of noise is the shot noise of electrons
tunneling through the insulating gaps in the junctions. Ideally, this noise source should
dominate the noise in the measurement system. However, if the shot noise of tunneling
electrons is small, because the current through the junctions is small, then the system
noise is dominated by the added noise of the measurement amplifiers that follow the
SET and it is not possible to make a quantum limited measurement. In an SET it isdifficult to pass a large current through the junctions because the maximum voltage
across the junctions is limited by the physics of the Coloumb blockade. Therefore in theexperiments that have used an SET to detect nanomechanical motion the displacement
imprecision was dominated by the noise of following amplifiers and the measurement
was not quantum limited [3, 31, 32].
The first experiments using an SET to detect nanomechanical motion were madeby Knobel and Cleland [3] in 2003 and by LaHaye et al in 2004 [32]. Knobel measured
the 117 MHz, Q = 1700, fundamental mode of a doubly-clamped nanomechanical beamthat was 3 µm long by 250 nm wide by 200 nm thick. The beam was made from a22
GaAs heterostructure and topped with a thin layer of aluminum. They calibrated theirexperiment using a magnetomotive measurement, and determined that the SET mea√
surement had an imprecision of 2 fm/ Hz. A good way of determining the significanceof the measurement precision is by comparing the precision to the zero-point motion of
the nanomechanical system which is closely related to the standard quantum limit [29].
In this case, the imprecision is 180 times the zero-point motion of the nanomechanicalbeam. LaHaye measured the 19.7 MHz, Q ≈ 40 × 103 , fundamental mode of a doublyclamped nanomechanical beam that was 8 µm long by 200 nm wide by 100 nm thick. It
was made from a SiN membrane and topped with a 20 nm layer of gold. The imprecision of this SET detector was calibrated using the thermal motion of the beam between
100 mK and 600 mK and they determined that the SET measurement had an impre√
cision of 3.8 fm/ Hz which is 5.4 times the zero-point motion of the nanomechanical√beam when comparing the imprecisions in units of m/ Hz.
More recently, in 2006 Naik et al [31] measured the classical backaction of a SET,
that is, the effect of the SET measurement on the nanomechanical beam. This backaction force is caused by the electrical potential of the SET island which creates anelectrostatic force on the nanomechanical beam through the dependence of the islandbeam capacitance on the position of the beam. Niak measured the backaction of asuperconducting SET near a bias point where there is resonant transport of electrons
through the SET. In this case, the backaction can effect the mechanical resonance frequency and quality factor. They measured a backaction force twenty times larger thantheoretically expected and used this backaction to change the mechanical damping. Thisadditional damping cooled the temperature of a mode of the nanomechanical beam by
about a factor of two.

23
2.2.2

SQUID nanomechanical displacement detector

A recent experiment by Etaki et al [4] in 2008 demonstrated the direct couplingof a nanomechanical beam to a dc superconducting quantum interference device (dc
SQUID). A dc SQUID is a sensitive magnetometer composed of a pair of Josephson
junctions: thin, electrically insulating barriers across which both electrons and Cooper
pairs can tunnel. The junctions are connected in parallel and form a loop with an input
and an output (see figure 2.4). In this experiment, the SQUID is current biased witha current that is greater than the critical current of the SQUID, that is, current biased
so that a voltage has developed across the superconducting loop due to the presenceof the Josephson junctions. Since the loop is superconducting, a screening current willdevelop in the loop which ensures that the flux through the loop is an integer numberof flux quanta, Φ = N h/2e. This screening current changes the current through each
Josephson junction, changing the voltage across the SQUID. Small changes in the flux
through the loop will therefore cause changes in the voltage across the SQUID, thus the
SQUID can be used to create a sensitive magnetometer [131].
Nanomechanical motion is coupled to a SQUID by designing a nanostructure
where the motion of a nanomechanical element changes the magnetic flux through the
SQUID loop. Etaki coupled the nanomechanical beam to the SQUID by suspending
one arm of the SQUID loop, creating a nanomechanical beam whose motion will change
the area of the SQUID loop (figure 2.4). The device was placed in a magnetic field andchanges in the size of the SQUID loop due to nanomechanical motion will change theflux through the loop. Nanomechanical motion is detected by monitoring the voltage
across the SQUID loop. While SQUID amplifiers have a long history of use in attempts
to detect gravitational waves by measuring the vibrations of large metal masses [26,121–
127, 132], in these earlier experiments the mechanical system was macroscopic rather
than nanoscale. Also, mechanical motion was not directly coupled to the SQUID but

24
a

b

Josephson
junctions

J

50 μm

J
R

c

HeightVibration amplitude

Sd
Cryogenic

IB

IF

SQUID

Strip

LF
amp

B

Piezo

HF
amp

Filter–100 dB

FilterFigure 2.4: SQUID detector of nanomechanical motion, figure from reference [4]instead was capacitively or inductively coupled to the SQUID through superconducting
circuits.
Theoretical investigations have suggested that this SQUID displacement detector
can be quantum limited [133, 134], with an imprecision that is set by the shot noise ofelectrons and Cooper pairs tunneling through the Josephson junctions and a backaction
due to the Lorentz force from a noisy current passing through the beam in a magnetic
field. In the experiment described by Etaki et al, the imprecision was limited by con√ventional electronics to Sx = 10 fm/ Hz, which is 50 times the zero-point motion of
the 2 MHz, 6.1 × 10−13 kg, doubly-clamped beam. The measurement was not sensitive
enough to detect the SQUID backaction.

25
2.2.3

QPC nanomechanical displacement detector

In a quantum point contact (QPC) displacement detector, nanomechanical motion changes the transmission through a weak link in a two-dimensional electron gas.
In general, a quantum point contact is formed in a conductor by creating a constriction
with a width that is comparable to the Fermi wavelength [135]. This constriction effectively creates an electron waveguide with a small number of conducting channels; eachfully conducting channel has a conductance Gq = 2e2 /h (the factor of two comes fromspin degeneracy) because of the small number of modes, or channels, in the waveguidecompared to the large number in the surrounding conductor.
Experimentally, a quantum point contact is usually formed by restricting theregion through which electrons flow in a 2-Dimensional Electron Gas (2DEG) [135].
The 2DEG is formed at the interface between two different layers of material, often theinterface of a GaAs-AlGaAs heterostructure created on top of a GaAs substrate using
molecular beam epitaxy. The flow of electrons in the 2DEG is constricted forming a
QPC by applying a potential between gates on top of the heterostructure and the 2DEG
(figure 2.5c, for example) which locally depletes the electron gas. The conductance of
the QPC is controlled by controlling the voltage on the gates.
The conductance of the QPC is a step-like function of the gate voltage. The
size of the constriction is a continuous function of the gate voltage, but the number ofconducting channels is an integer quantity. As electrons are allowed to flow through a
new channel, the conductance increases in steps of Gq = 2e2 /h. This step is smeared out
by thermal energy allowing access to otherwise closed modes and by electrons tunnelingthrough partially closed modes. The abrupt change in conductance as a function ofvoltage is used to sensitively detect changes in the gate voltage and create, for example,
charge detectors to investigate quantum dephasing [136] or detectors of nanomechanicalmotion [5, 33].

26

a
x
y

b

z

2 μm

Cantilever

Laser
10 μm
QPC

c

10 μm

1 μm

Figure 2.5: QPC detector of nanomechanical motion, figure from reference [5]The theoretical descriptions of the QPC displacement detector and the atomic
point contact (APC) displacement detector are very similar. In both detectors, nanomechanical motion is detected by measuring changes in the transmission of electrons
through a barrier. Using this simple description, both detectors are described by thesame Hamiltonian (see [75] and section 2.3.2).
Cleland [33] used the peizoelectric effect in GaAs to couple nanomechanical motion to the conductance of a QPC. The 2DEG was fabricated as part of the layered
GaAs-AlGaAs-GaAs nanomechanical structure: a beam that was 680 nm thick by 4 µm
wide by 13 µm long. The QPC gates were fabricated on top of the nanostructure. Thisfabrication procedure significantly degraded the quality of the 2DEG which hurt thedisplacement sensitivity. Out-of-plane nanomechanical motion effectively lengthens thebeam, stressing the GaAs and creating out-of-plane peizoelectric fields. These fields also
deplete the 2DEG, creating a change in the QPC conductance. The nanomechanical27
motion is detected by measuring the small changes in QPC conductance. In this experiment, the QPC displacement detector was able to detect nanomechanical motion with√
an imprecision of 3 pm/ Hz which was limited by the voltage noise of the amplifier
used to measure the QPC conductance.
More recently, Poggio [5] used a QPC to detect the motion of a micron-scalecantilever by capacitively coupling the cantilever to the QPC (figure 2.5). In this case,
the QPC and the cantilever were fabricated separately which simplified the fabricationof the QPC and improved the quality of the 2DEG. The cantilever was 350 µm long by
3 µm wide by 1 µm thick and was made out of single-crystal silicon with a gold tip anda platinum coating. After placing both the QPC and the cryostat in a 4 K cryostat, theend of the cantilever was positioned 70 nm above the QPC. Because the tip was close to
the QPC, the tip was capacitively coupled to the QPC and acted as another, moveable
QPC gate. The x-y position of the tip was also optimized for maximum coupling between
the movement of the cantilever and the QPC conductance. The minimum imprecision√of this QPC displacement detector was 1 pm/ Hz and was limited by the noise causedby charges fluctuating near the QPC.

2.2.4

Carbon nanotube displacement detector

Figure 2.6: Carbon nanotube displacement detector, figure from reference [6]28
The nanomechanical motion of carbon nanotubes has been measured using thecarbon nanotube as both the sensing element and the mechanical oscillator [6, 35–37].
The resistance of the nanotube is effected by the charge induced on the nanotube. Thiseffect is used to detect mechanical motion by voltage biasing a nearby gate electrode
that is capacitively coupled to the nanotube (figure 2.6) and measuring changes innanotube resistance. Mechanical motion changes the nanotube-gate capacitance; theresulting change in the charge induced on the nanotube creates a detectable shift in theresistance of the nanotube. The nanomechanical motion of nanotubes and larger silicon
nanopillars has also been detected using the field emission of electrons from the tip ofthe nanotube or pillar [137–139].

2.3

The Atomic Point Contact as a Mesoscopic Detector
In this section, I briefly review the use of atomic point contacts (APCs) as meso-

scopic detectors and the theoretical descriptions of APC displacement detectors. The
APC has been widely used to detect the position of atoms in scanning tunneling microscopes (STMs) and to investigate both mechanical and electrical properties on theatomic scale. In this thesis, I combine the displacement sensitivity of the APC with afast measurement technique to detect nanomechanical displacement and evaluate theperformance of the APC as a detector of displacement. Theoretical descriptions of thistype of APC detector indicate that it is possible for the APC to be used to detect motion
at the limits imposed by quantum mechanics.

2.3.1

APC as a widely used detector of static displacements

Broadly speaking, the use of APCs as detectors of static displacement can bedivided into two categories. The first category contains STMs where the APC is createdby bringing an atomically sharp tip close to another surface. The position of the tip canbe scanned over the surface and controlled in three-dimensions, creating a versatile tool

29
for investigating mesoscopic phenomenon. Second, the APC can be created in situ froma constriction composed of more than 106 atoms either using electromigration or bystraining the constriction; the constriction is often part of a suspended nanostructure.
This second type of APC trades the ability to scan the tip for simplicity and stability.
Usually the mechanical circuit is entirely on-chip resulting in a much stiffer support
structure than can be obtained in a STM. Experiments that investigate how metals
fracture and flow on the atomic scale and the electrical properties of single atoms andmolecules can be carried out using either type of detector, but benefit from the improved
stability of the second category of APCs which are supported by a nanostructure.
APCs provide the atomic resolution in STMs [7, 22, 38, 39]. As implied by thename “scanning tunneling microscope”, an STM scans a tunneling probe, i.e., APC,
over a surface and uses the changes in the probability that an electron tunnels to create
a topographic image of the surface (figure 2.7). The probability that an electron willtunnel is exponentially sensitive to the size x of the gap and so the resistance of the APC
is RAP C = R0 exp [2x/λ]. The length scale λ that controls the probability of electrons
tunneling is determined by the height of the energy barrier through which the electrons
tunnel. The energy barrier is usually set by the work function φ of the material; inmost metals, φ is on the order of 5 eV resulting in a length scale λ ≈ 0.1 nm. The
exponential dependence of the APC resistance on a sub-atomic length scale allows an
STM to image the surface of a pure material with atomic resolution. Experimentally,
the gap x is determined by applying a voltage VAP C across the APC and measuring thetunneling current IAP C = VAP C /RAP C as the tip of the APC is swept over a surface(figure 2.7).
The high-impedance nature of the APC limits the speed, or bandwidth, of theposition measurement. STMs are usually operated with APC gaps > 1 nm and RAP C >1 MΩ. There is also capacitance in parallel with the APC resistance (figure 2.7); thisstray capacitance Cs is usually dominated by the electrical wiring connecting the APC to

30

VAPC

Cs
RAPC

electrons tunnel
across APC (IAPC)

Figure 2.7: Basic diagram of Scanning Tunneling Microscope (STM), figure adaptedfrom reference [7]. The topography of the surface is determined by scanning the tip(triangular arrangement of atoms) over the surface and determining the relative change
in the probability of electrons tunneling from the tip. The tip is voltage biased VAP C ,and measured changes in the tunneling current IAP C = VAP C /RAP C are used to infer
the probability of electron tunneling.

external amplifiers. The APC resistance RAP C > 1 MΩ and stray capacitance Cs > 1 fF
form a low-pass RC filter with a resonance frequency less than 100 kHz [11, 140]. Any
signal at frequencies above 100 kHz will be attenuated by this unintentional RC filter,
limiting the bandwidth of the resistance measurement.
There have been a number of attempts to overcome this bandwidth limitation.
Standard STMs have been improved to the point where the bandwidth of the electricalmeasurement is on the order of 500 kHz [141, 142]; in these experiments, the focus isoften on improving the speed with which the APC can be scanned over a surface. There
have also been attempts to directly measure radio-frequency spin precession [143, 144]
using an STM tip. The sample is placed in a static magnetic field and the STM tip is31
located near a spin center. The spin precesses at the Larmor frequency (500-800 MHz inthese experiments). The changing magnetic moment couples to the tunneling electrons,causing modulations at the Larmor frequency in the probability of electrons tunneling.
This signal frequency is significantly higher than the frequency of the unintentional
low pass filter, so the signal should be significantly attenuated. These experiments did
not focus on improving the bandwidth of the STM, but instead simply increasing thetunneling current until the 500-800 MHz signal could be detected despite the attenuation[144].
Ekinci [140], following the work described in this thesis [11], recently used a
microwave technique to increase the STM bandwidth and create the RF-STM. This
technique involves creating an LC matching circuit with a bandwidth > 10 MHz andwill be described in detail in chapter 5. It was first used in mesoscopic physics bySchoelkopf [128] to increase the bandwidth of another high-impedance device, the SET,and create the RF-SET. Using this technique, Ekinci demonstrated an RF-STM witha bandwidth of 10 MHz [140]. Ekinci also measured the motion of a 65 µm by 70 µm
by 40 nm silicon nitride membrane coated with 20 nm of gold. The best imprecision√obtained in this measurement was 15 fm/ Hz and was limited by amplifier noise, notthe smaller fundamental noise due to the shot noise of tunneling electrons.
STMs are not only used to create topographic images, but also to probe the mechanical and electrical properties of small numbers of atoms and molecules. In fact,
the initial motivation behind the STM was to create a local spectroscopic probe [7].
Electrical properties of samples are determined by measuring the dependance of thetunneling current IAP C on the voltage VAP C across the APC. In the simple tunneling
picture, the tunneling current is proportional to voltage RAP C = VAP C /IAP C wheneVAP C  φ. However, changes in the local density of states due, for example, todeviations in the tunneling barrier from a rectangular barrier or the presence of vibrational energy levels will result in a voltage-dependent tunneling current and resistance32
2
d2 RAP C /dVAP
C 6= 0 [145]. This spectroscopic technique has been widely used to inves-tigate the electrical properties of everything from high temperature superconductors tographene [146]. I will discuss the voltage dependence of the tunneling current in chapter
3 and the effect of vibrational levels in chapters 3 and 7.
STMs are used to probe mechanical characteristics on the atomic scale in two
different ways. First, the STM can be combined with an atomic force microscope (AFM)
to measure the force required to change the size of the APC gap [147, 148]. This canbe used to understand how metals fracture and flow on the atomic scale. Second, theinteraction between the STM tip and the surface atoms can be used to manipulate theposition of atoms and understand how atoms slide over surfaces [149, 150].
The second, different, way in which APCs are often created is by fabricating athin, narrow section of material and removing atoms in situ until electron transport
across the junction is due to tunneling between two atoms: an APC. The thin, narrowsection of material is usually a constriction in a larger nanostructure created on-chipusing the tools of nanolithography (for example, figure 2.8). The micron-scale support
structure is both simpler and stiffer than the cm-scale structure used to support andscan the APC in an STM.
Two different techniques are commonly used to remove atoms from the constriction and create an APC. First, a large current can be passed through the constriction.
In a process called electromigration, atoms will move from regions of high current density to low current density eventually forming an APC. Second, the constriction canbe strained by increasing the distance between the supports (figure 2.8); under strain,
constriction will flow, reducing the cross-section and eventually fracturing forming an
APC. This type of junction is called a mechanically controllable break junction (MCBJ).
Unlike an electromigrated APC, the resistance of a MCBJ can be controlled and cycled
between completely closed, with > 103 atoms separated by < 0.1 nm, and open withthe resistance controlled by the > 100 nm separation between two atoms.

Suspended Constriction

Supports

33

1μm

Figure 2.8: Example of the nanostructure used in a mechanically controllable breakjunction (MCBJ), figure adapted from reference [8]. In this isometric scanning electron
micrograph, the constriction is suspended above the substrate over a length of 1 µm by
two supporting electrodes.

The experiments described in this thesis have used both electromigrated APCsand APCs created by applying mechanical strain. The creation of these APCs is discussed in detail in chapter 4. A more detailed review of the creation and properties(which are briefly discussed below) of such atomic-sized conductors is also given inreference [8].
MCBJs have been used to study the fracture and flow of metals at the atomic
scale [8]. In one technique, the resistance of the MCBJ is cycled between closed andopen and hysteretic patterns in the resistance as a function of strain are observed due
to atom rearrangement. This hysteretic behavior has, for example, been used to study
how chains of gold atoms form and break under strain [151]. Repeatable hysteretic
loops in gold APCs have also been used to determine the adhesion between the two
atoms that form a gold APC. The data is explained by a simple elastic model of a gold

34
dimer attached to elastic electrodes; the electrodes are modeled as springs with a spring
constant between 5 N/m and 32 N/m [9].
A similar technique has also been used to investigate how the electrical propertiesof the junction change as the junction fractures. Instead of analyzing just a single cycle
between closed and open, thousands of cycles are averaged in a histogram. It is hopedthat using many cycles averages over mechanical effects due to the position of atoms
and reveals the preferred values of the electronic system [8]. In gold, these histograms
generally show a clear peak at the quantum of resistance, RQ = h/2e2 = 12.9 kΩ andsmaller, less pronounced peaks at RQ /2 and RQ /3. This technique has also been used
to investigate the change in the electrical properties of the APC created by the presenceof other atoms or molecules in the junction [8, 54].
APCs created using both electromigration and mechanical strain have been used
to electrically contact and study single molecules [8]. These experiments usually use goldor platinum contacts which are inert and easy to clean. The molecules are incorporated
into the APC using two different techniques. First, after the APC has been createda gas of molecules can be added to the APC environment [49, 50, 54]. One (or more)
molecule condenses on the APC, changing the electrical and mechanical properties of
the APC. Second, molecules can be placed on the surface of the nanostructure prior to
APC creation, usually by placing the nanostructure in a dilute solution that includes
the molecule of interest. The APC is then created using either electromigration (forexample, [40, 52, 56]) or mechanical strain (for example, [51, 53, 57, 152]). In a fraction(usually > 10%) of the APCs there is a molecule trapped in the middle of the APC. The
presence of the molecule is confirmed based on the mechanical and electrical propertiesof the APC.
As with the STM, the electrical properties of samples are determined by measuring the dependance of the tunneling current IAP C on the voltage VAP C across the APC.
This has been used to study molecular conductance and APC-molecule coupling [54,57]

35
as well as, more specifically, the Kondo effect [51, 56] and Coloumb blockade physics inmolecules (which also requires a nearby capacitive gate) [40, 52, 53, 56]. This spectroscopic measurement has also been used to study the conformational changes [49] andvibration [10,40,50,52,54] of molecules attached to the APC. A static change in the orientation of the molecule in the junction shifts the resistance of the APC. The coupling
between electron tunneling and molecular vibration causes a change in the resistance of
the APC at a voltage VAP C where the energy of the electron eVAP C is large enough toexcite the molecular vibration h̄ωm . This is discussed in more detail in chapters 3 and
7 and references [8, 10]. The same procedure has also been used to study the vibrationof chains of gold atoms created using mechanical strain [152].

2.3.2

APC as a quantum-limited displacement detector

In this thesis, I combine the displacement sensitivity of the APC with a fastmeasurement scheme and implement an APC displacement detector. The APC displacement detector has two main advantages over optical displacement detectors andother mesoscopic displacement detectors: it is inherently a nanoscale detector, and, unlike many other mesoscopic detectors, the measurement imprecision can be limited by
the the shot noise of tunneling electrons which is the fundamental source of noise in an
APC. This idea is not new; the use of an APC to detect the vibration of micron andnano-scale mechanical structures was proposed in the late 1980s and early 1990s in thecontext of gravitational wave detection [58–63]. At the time, theoretical evaluations of
the APC detector based on an analysis of electron tunneling through a barrier indicatedthat an APC detector could be quantum limited [64–66]. Starting in 2002 there has
been renewed theoretical interest in the APC displacement detector in the context ofquantum-limited measurement [67–77].
In 2007 I demonstrated an APC measurement with a 30 MHz bandwidth [11]using the same microwave technique that was used by Schoelkopf in creating the RF-

36
SET [128]. This increased bandwidth adds a dynamic component to a previously static
measurement of APC resistance. Instead of just measuring the average resistance of the
APC, I can measure the fast modulation in resistance caused by nanomechanical motion.
In comparison, previous measurements of displacement due to molecular vibrations
were only able to measure the indirect effect of molecular vibration energy levels on
the average resistance. I not only detect nanomechanical motion, but the noise in themeasurement is also dominated by the shot noise of tunneling electrons (see chapter 5).
This is important, because it is the fundamental source of noise in the APC detector.
In detecting nanomechanical motion, the APC detector has a number of advantages over both optical detectors and other mesoscopic detectors. First, unlike the otherdetectors it is inherently a detector of atomic-scale motion. The optical detectors are
not sensitive to the motion of objects that are much smaller than the wavelength ofthe light used in the detector. Most other mesoscopic detectors either rely on capacitive coupling between the nanostructure and the detector or measure the total strain
created by nanomechanical motion. In both cases, the coupling becomes smaller as thedimensions of the nanostructure are decreased. On the other hand, the APC detector
measures the change in the relative position of two atoms. The coupling between thatatomic gap and motion of the surrounding nanostructure is not inherently decreased byshrinking the size of the nanostructure.
Second, as mentioned above, the noise in the APC displacement measurement
can be dominated by the shot noise of tunneling electrons. This is important because inan APC detector and many other mesoscopic detectors the fundamental source of noise
is shot noise. Additional sources of noise such as external amplifiers can also contributeto the total measurement noise, however these sources of noise are not a fundamental
part of the measurement. Achieving a quantum limited measurement requires that theadditional sources of noise are negligible compared to the fundamental noise, that is,
the shot noise. Two approaches are used to minimize the additional noise in comparison

37
to the shot noise. First, the additional noise is made as small as possible using carefully
designed electrical circuits and low-noise cryogenic amplifiers (see chapter 5). Second,
the shot-noise is made as large as possible. This is somewhat counter-intuitive, since
it increases the total noise in the measurement. However, the size of the signal due tomechanical motion (in displacement units) is linearly proportional to the current while
the shot noise is proportional to the square root of the current. Increasing the shotnoise therefore results in a more precise measurement of displacement and decreases thedisplacement noise added by the additional, non-fundamental noise sources.
In this context, the APC has an advantage over other mesoscopic detectors because it can be used to create larger shot noise. The maximum voltage that can beapplied to the APC is determined by the height of the tunneling barrier. In most metals, the height of the barrier is about 5 eV; the effective barrier in most other mesoscopic
devices is much smaller. This comparatively large energy scale implies that a large voltage can be applied to the APC resulting in a large tunneling current and large shotnoise. In other words, the APC can have much larger gain than other mesoscopic detectors. Given the magnitude of additional noise added by commercial low-noise amplifiers,
APC displacement measurements can be limited by shot noise while displacement measurements made using other mesoscopic detectors, such as SETs, have been limited by
the additional noise of amplifiers. It is relatively easy to use an optical detector to makedisplacement measurements which are limited by photon shot noise.
One disadvantage of the APC detector is that atomic-scale contamination andmesoscopic effects will affect the displacement measurement. In STMs at resistances> 10 MΩ the tunneling barrier appears consistent with a simple rectangular barrier.
However, at low resistances and in the presence of contaminant atoms or molecules themesoscopic physics becomes more complicated. This physics will effect the displacement
measurement and is generally expected to reduce the measurement precision (that is, increase the measurement imprecision). Atomic rearrangements also alter the mechanical38
resonance frequency and dissipation rate. When the APC is unstable, the mechanicalresonance is unstable; it is then not possible to determine the properties of the APC
displacement detector.
The idea of using an APC to directly detect mechanical motion near the quantum
limit is not new; in 1988 Niksch and Binnig proposed that an APC could be used tocreate a broadband detector of gravitational waves [63]. This proposal was expanded
upon by Bocko, Stephenson, and Koch over the next three years [58, 59, 61, 62]. Theyevaluated the detector from an experimental perspective, including a generic amplifier,and estimated that the displacement detector could operate at the limit imposed by the
Heisenberg uncertainty principle Sx SF ≥ h̄2 .
The spectral density SF of the backaction force noise is estimated using a simple model where each tunneling electron delivers a momentum kick to the mechanicalstructure. The average momentum that each electron delivers is assumed to be about
equal to the Fermi momentum pF . Because there is shot noise with spectral density
SI = 2eIAP C in the current IAP C through the APC, the average momentum delivered byeach electron creates a noisy backaction force with spectral density SF corr = 2p2F IAP C /e.
This force is correlated with the electron shot noise, so it is theoretically possible to apply a force to the nanostructure proportional to the shot noise and cancel the effect of
SF corr . Therefore this component of the backaction force is not fundamental and is notincluded in the comparison of the APC to the Heisenberg limit.
The variance in the size of the momentum kick that is delivered by each electron also causes a backaction force. This backaction force is not correlated with theelectron shot noise and enforces the Heisenberg limit. Bocko, Stephenson, and Kochassume that the act of tunneling effectively “measures” the location of the electron withan uncertainty ∆x ≈ λ/2 where λ is the length scale that controls the probability ofelectron tunneling. Using the uncertainty relationship ∆x∆p ≥ h̄/2, the uncertainty ineach electron’s momentum is at least ∆p = h̄/λ. A current IAP C through the APC will

39
therefore create a noisy backaction force with a spectral density SF = (h̄/λ)2 (IAP C /2e).
The estimated spectral density of the displacement noise due to the shot noise of tunneling electrons is Sx = λ2 /2eIAP C . This simple estimate of the backaction force anddisplacement imprecision of an APC detector predicts that the product of the backaction
and imprecision is equal to the limit imposed by quantum mechanics Sx SF = h̄2 .
More careful theoretical evaluations of the APC displacement detector in theearly 1990s also predicted that the detector could be used to make a quantum limited
measurement of position [64–66]. They model the APC as an arbitrary one-dimensionaltunnel barrier. A fraction of the electrons incident on the barrier tunnel through thebarrier, creating a current of tunneling electrons. The average fraction of electrons
that tunnel can be used to measure the width of the barrier; this is a familiar problem
used in basic quantum mechanics courses. The uncertainty in the fraction of tunneling
electrons, that is, the shot noise, determines the imprecision ∆x in the measurementof the width of the barrier. They also calculate the momentum transported across thebarrier, and determine the uncertainty in the momentum ∆p which is not correlatedwith the imprecision. For a rectangular barrier, they determine that a quantum limited
uncertainty product ∆x∆p = h̄/2 can be achieved.
The APC displacement detector attracted more theoretical interest after 2002 inthe context of quantum-limited measurements [67–77]. In comparison to the early 1990s,
these theoreticians take a different approach to calculating the effect of the measurement on the mechanical oscillator. In general, they calculate the effect of mechanicaldisplacement on the tunneling current and the backaction of the tunneling current on
the mechanical oscillator by describing the detector using a tunnel Hamiltonian witha small linear coupling to displacement. For example, reference [75] uses the detector
Hamiltonian
Hdet = τ0+ eiν τ 0 x̂ X 
2πΛ

k,k0



Y † c†R,k cL,k0 + H. C.  − eV (t)m̂

(2.2)

40
where m̂ denotes the number of electrons that have tunneled, the operator Y † increasesm̂ by one, the operators cR,k and cL,k destroy an electron state on the right and left
side of the APC respectively, τ0 is the displacement-independent tunneling amplitude,and Λ is the conduction-electron density of states. Displacement x̂ is linearly coupled
to the tunneling amplitude with a proportionality constant τ 0 and relative phase ν.
It is assumed that the displacement will only create small changes in the tunneling
amplitude τ0  τ 0 x. The state of the oscillator x and current m is described using
a reduced density matrix. The time evolution of the reduced density matrix is used
to determine the full counting statistics of the tunneling electrons m and the effect of
displacement x. It is generally found that the detector will create a backaction force. In
qualitative agreement with earlier theoretical work, part of the force is correlated withthe shot noise of tunneling electrons and part is uncorrelated. Unlike early work, thedetector is also predicted to modify the intrinsic dissipation of the mechanical system
γtot = γintrinsic + γdet . The fluctuating force created by the detector can be describedby an effective temperature Tef f = eVAP C /2 and damping constant γdet which depends
on the strength of the position measurement.
In this thesis, I will evaluate the APC displacement detector in the context ofthe phenomenological model described by Bocko, Stephenson, and Koch [58, 59, 61, 62](see chapter 3 for details). The precision of the measurement and the strength ofthe coupling between displacement and tunneling are not large enough to be able todistinguish between this phenomenological model and the more detailed expectationsof the other more complicated models. In addition, even though the theoretical models
make a distinction between the backaction forces that are correlated and uncorrelatedwith the shot noise, in this experiment it is not possible to separate the two types offorces. Finally, it is difficult to extract detailed information about the backaction force
because it is obscured by the larger 4 K thermal force on the nanostructure.

Chapter 3

The Theory of an Atomic Point Contact Coupled to a
Nanomechanical Oscillator

In this chapter, I start by considering the physics of an electron incident on an
atomic point contact (APC). Even if there is no scattering, that is, no tunnel barrier,the junction is still resistive because there are only a limited number of channels forelectrons to flow through the junction. In an atomic point contact composed of two
atoms with no tunnel barrier, there is one spin-degenerate channel and the resistance
RAP C = h/2e2 . When electrons scatter with a transmission probability D, RAP C =h/2e2 D and there will be shot noise in the tunneling current. In the limit where D  1
and the temperature is zero, the noise is Poissonian and results in a voltage noise with
spectral density SV = 2eVAP C RAP C . The probability D that an incident electron willtunnel through a vacuum gap is calculated using the Wentzel-Kramers-Brillouin (WKB)
method and is approximately equal to D = exp [−2x/λ] where x is the width of thetunneling gap and λ is the length scale that controls the probability of electron tunneling(in gold, λ ≈ 90 pm).
I also compare the APC displacement detector, as described by a phenomenological model, to a quantum-limited detector. The imprecision-backaction product of acontinuous, linear quantum-limited detector with uncorrelated backaction SF and imprecision Sx is equal to the minimum allowed by the Heisenberg uncertainty constraint
Sx SF ≤ h̄2 . Using a phenomenological model of the APC detector where the impreci-

42sion is caused by the shot noise of tunneling electrons and the backaction is caused by
the momentum kicks of tunneling electrons, I estimate that the APC detector can bequantum-limited.
Finally, I describe the flexural motion of a nanomechanical beam and the effect ofan APC on the mechanical structure. The vibration of the structure is described using
a set of orthogonal modes; each mode has an eigenfrequency and can be thought of asa simple harmonic oscillator. The thermal motion of a single eigenmode at a known
temperature can be used to calibrate a displacement measurement [11].

3.1Electron Tunneling Across an Atomic Point Contact
In this thesis, I generally assume that a small change δx in the size of the APC

gap causes an exponential change in the resistance RAP C of the APC. In this section I
justify this assumption that RAP C = R0 exp(2δx/λ). I start by calculating the conductance of a junction where electrons flow through a single channel without scattering,that is, without a gap and with a transmission probability D = 1. This is also calledballistic transport and occurs, for example, in a chain of gold atoms. The resistance of aballistic junction, including spin degeneracy, is equal to half the quantum of resistanceR = h/2e2 . The quantum of resistance sets the scale of the tunneling resistance. Whenelectrons scatter off of, for example, a potential barrier with a transmission probability
D then the resistance of the junction is RAP C = h/2e2 D. I also describe how the probabilistic nature of electron tunneling causes shot noise in the current passing throughthe tunnel barrier. Finally, I calculate the probability D(x) that an electron will tunnel
through a barrier of width x concentrating on the experimentally relevant case of arectangular barrier and a small applied voltage bias.

43
3.1.1

Nano-scale ballistic contact

While the conductance of macroscopic systems G = σ(A/L) simply scales withthe cross-sectional area A and length L of the conductor, this scaling breaks down when
considering nano-scale junctions. At small scales the quantization in the directions
perpendicular to L, that is, the transverse directions, becomes important. In this thesis, quantization becomes important when the transverse dimensions are on the atomic
scale, though, depending on the material, quantization can effect devices with transverse dimensions up to hundreds of nanometers. The quantization of the transverse
momentum limits the number of available incoming and outgoing modes in the junction
and the resistance of the junction is due to the transition between the small number ofmodes in the junction and the large number of modes in the surrounding contacts. This
description follows the derivations in references [17, 18, 20, 153].
I will consider the conductance of a ballistic junction between two contacts (figure 3.1). To simplify the discussion below, I assume that the junction is a thin, narrowrectangular conductor. An external potential is applied between the two contacts causing electrons to flow through the junction. Since there is no scattering, the current
through the junction and the resistance of the junction are determined by the numberof accessible momentum states in the direction perpendicular to the electron flow. In
other words, the resistance depends on the number of transverse modes or channels.
The different transverse modes, which are similar to the modes traveling downa waveguide, are well-defined with distinct energy levels. These are the modes of ajunction which is a conductor (figure 3.1) with length L in the x̂ direction, width W
in the ŷ direction, and a height Z in the ẑ direction where Z  W so the junction
is effectively two dimensional. There is also a confining potential U (y). Under these
conditions, the motion of the electrons (in the absence of a magnetic field) is describedcontact #1

44contact #2
junction(ballistic)yx

Figure 3.1: A thin, narrow ballistic junction (grey) is attached to large contacts (black).
Even though electrons do not scatter in the junction, the transition from large contacts
to a small junction creates resistance.

by
"

#
p2y
p2x
Es +
+
+ U (y) Ψ(x, y) = EΨ(x, y)
2m 2m

(3.1)

Expressing Ψ in the form of plane waves traveling through the junction and assumingthat U (y) is a parabolic confining potential U (y) = mωy2 y 2 /2, which is a good approximation for electron waveguides,1
Ψ(x, y) = √ exp [ikx x] φ(y)
L
"
#
2
2
py
(h̄kx )1Es +++ mωy2 y 2 φ(y) = Eφ(y)
2m
2m 2

(3.2)
(3.3)

The electron eigenenergies are(h̄kx )2
1
E(kx , ny ) = Es ++ ny +h̄ωy ,
2m2

ny = 0, 1, 2, . . .

(3.4)

The use of a parabolic potential is convenient, but not necessary; the important featureof this equation is the discrete energy levels due to confinement in the ŷ direction.

45
For a given electron energy Ee and momentum kx , the number of available modes
is determined by the width of the conductor. For a wide conductor, such as the region
contacting the junction, the confining potential is weak and the frequency ωy is small(compared to a narrow conductor). The eigenergies E(kx , ny ) of the wide conductor
are more tightly spaced, so the number of available modes with an eigenergy less thanEe is much larger than for a narrow conductor. In a narrow conductor, such as thejunction, the electrons are tightly confined by a potential with a larger frequency ωy .
The resulting eigenergies are more widely spaced, resulting in fewer available modes.
When the narrow junction is connected to wide contacts (figure 3.1), there is a transition
between the large number of transverse modes in the contacts to the small number oftransverse modes in the junction.
The change in the number of transverse modes results in an effective resistancethat is inherent to any mesoscopic system and is not due to quantum interference or
uncertainty. In calculating the resistance, I start with a ballistic conductor, as describedabove, which is connected to two reflectionless contacts (figure 3.1). Since electrons do
not reflect of off the contacts, all of the k+ modes which contribute to the current in
the positive x̂ direction are due to excitations in contact #1 (left, figure 3.1) and allof the k− modes which contribute to the current in the negative x̂ direction are due toexcitations in contact #2 (right, figure 3.1). There will not be any mixing caused byinternal reflection off of the contact/conductor interfaces. In other words, the occupationof the k+ modes is purely a result of the chemical potential µ1 of contact #1 and theoccupation of the k− modes is purely a result of the chemical potential µ2 of contact#2.
The current through the junction due to an applied bias VAP C is found by summing the electron transport due to all of the occupied modes. The probability that amode with energy E in contact #1 and contact #2 is occupied is given by the Fermi
function f1 (E) and f2 (E) respectively. The total current from contact #1 to contact

46#2 is the sum of the current −evk /L due to each available mode weighted by the Fermi
function
X evk x
f2 (Ekx ,ny ) − f1 (Ekx ,ny )
L

(3.5)


∂Ekx ,ny 
f2 (Ekx ,ny ) − f1 (Ekx ,ny ) Dkx dkx
∂kx

(3.6)

I=

kx ,ny

eX
I=
h̄ n

Z

y

where h̄vk = ∂Ekx ,ny /∂kx and Dkx is the one-dimensional density of states. A voltage
V applied to contact #1 will change the chemical potential µ1 = µ − eV of contact #1
in comparison to µ2 = µ for contact #2. At low temperatures the Fermi functions are
fm (E) = θ(µm − E). Since only available modes with an eigenergy E(kx = 0, ny ) lessthan the electron energy contribute to the current, the total current isI=

e X
2πh̄ nyZ ∞dE [θ(µ − E) − θ(µ − eV − E))]

(3.7)

e2 X
e2
V
θ(µ − E(kx = 0, ny )) = N V
h
h
n

(3.8)

E(kx =0,ny )I≈ywhere N is the number of occupied modes. The conductance of the junction is equal to
the quantum of conductance e2 /h per mode. In an atomic point contact, the two statesof the electron spin are degenerate and the total resistance is
RAP C =h 1
2e2 N 0(3.9)

where N 0 is the number of available spin-degenerate channels. For a more detailed
discussion of the resistance of atomic point contacts see reference [8].
Electron scattering due, for example, to the presence of a vacuum gap will increase the resistance of a junction. Instead of assuming that the junction is a ballistic
conductor, I include the possibility that an electron will scatter. The junction is effectively composed of three parts (figure 3.2): ballistic lead #1 connects contact #1 tothe second part, a set of scatterers, which are then connected by ballistic lead #2 tocontact #2. Because of the scatterers, an electron in mode n will scatter and transmit
with a probability Dn or reflect with probability 1 − Dn .

contact #1{

junction contact #2

47

scattering#1#2
ballistic leadsyx

Figure 3.2: A junction is composed of scatterers (blue) and thin, narrow ballistic leads(grey) which are attached to large contacts. The resistance of the junction is increasedby the presence of scatterers in comparison with a ballistic junction.

The total current through the junction is equal to the sum of the incident andreflected current as well as the transmitted current. The magnitude of the current
incident on the scatterers from ballistic lead #1 due to a voltage V is equal to thecurrent in a ballistic conductor (equation 3.7). For each mode, a fraction 1 − Dn of theincident electrons are reflected and a fraction Dn are transmitted into ballistic lead #2
so the total current I flowing through the junction is
!
!
N0
N0
N0
X
X
X
2e2
2e2
2e2
I=
V −
(1 − Dn )
V =
Dn
V
h
h
h
n=1

n=1

(3.10)

n=1

The resistance of this junction is
0

R=

N
X
n=1

2e2
Dn
h

!−1
(3.11)and the expected resistance of a ballistic junction is recovered by setting Dn = 1 (all of
the electrons are transmitted).

48
3.1.2

Electron shot noise

The discreteness of electrons and the probabilistic nature of electron tunnelingcreates shot noise in the tunneling current. The shot noise of tunneling electrons is
the fundamental source of noise in mesoscopic measurements that use tunnel junctions,such as the APC displacement detector. The shot noise in mesoscopic junctions has
been reviewed in detail in references [8, 19, 154, 155]. The current noise spectral density
SI (ω) is equal to the Fourier transform of the correlation function
Z ∞SI = 2ˆˆdteiωt h∆I(t)∆I(0)i(3.12)−∞

ˆ = I(t)ˆ − hI(t)iˆ
where ∆I(t)and the factor of two is included because I use a single-sided
spectral density which is only defined for positive frequencies.
I will briefly sketch the calculation of shot noise described in detail in references [8]ˆ is defined in terms of electron annihilation andand [19]. The current operator I(t)creation operators. The operators (α =1 or 2, figure 3.2) âαn (E) and â†αn (E) annihilate
and create incoming electrons with energy E in channel n of contact #α; similarly,b̂αn (E) and b̂†αn (E) annihilate and create outgoing electrons. The occupation of theincoming modes is determined byhâ†αn (E)âβm (E 0 )i = δαβ δmn δ(E − E 0 )fα (E)(3.13)

where fα (E) is the Fermi function in contact α. The outgoing modes are related tothe incoming modes by the scattering matrix which also determines the transmission
probabilities Dn and reflection probabilities 1 − Dn . The current operator in contact
#α is
N0 Z

2e2 X
Iˆα (t) =hhi0dEdE 0 ei(E−E )t/h̄ â†αn (E)âαn (E 0 ) − b̂†αn (E)b̂αn (E 0 )(3.14)

n=1

and the expectation value hIˆα (t)i is, as expected, equal to the average current calculated
earlier using a different approach (equations 3.6, 3.7, 3.10).

49
Calculating the shot noise (equation 3.12) involves finding the expectation value of
products of two creation and two annihilation operators. After performing the necessary
algebra, the shot noise in the two contacts is
N0 Z

2e2 X
SI = 2hdE Dn2 (f1 (1 − f1 ) + f2 (1 − f2 )) +n=1

Dn (1 − Dn )(f1 (1 − f2 ) + f2 (1 − f1 ))(3.15)

where the transmission probability Dn and the Fermi distributions f1 and f2 are allenergy E dependent. In this equation, the terms on the top line correspond to thethermal noise and disappear at zero temperature while the terms on the bottom line
correspond to the shot noise contribution. The derivation of this equation also assumes
that the measurement frequency ω and energy h̄ω are much smaller than the otherfrequencies and energies in the problem.
Equation 3.15 is simplified by assuming that the applied voltage is significantly
smaller than the Fermi energy. In this case, only electrons which have an energy close
to the Fermi energy contribute to the net current, so the transmission probability isequal to the probability at the Fermi energy and is approximately constant. Using thisassumption, the shot noise spectral density is


N0 
2e2 X
eV
2
SI = 2
2kB Tcryo Dn + Dn (1 − Dn )eV cothh
2kB Tcryo(3.16)

n=1

where Tcryo is the temperature of the junction. At low temperatures Tcryo → 0
N0

2e2 X
[Dn (1 − Dn )]SI = 2eVh(3.17)

n=1

The expected result for a Poisson process is recovered when there is only a small probability of electron tunneling Dn  1
0

SI = 2e

N
X
n=1

Dn

2e2
V = 2eIh(3.18)The expected magnitude of the shot noise in this thesis is determined from equation 3.16. The shot noise is the fundamental source of noise in the measurement. The

50
measured shot noise is caused by electrons tunneling across a gold atomic point contact (APC). The APCs used to detect nanomechanical motion in this thesis are usuallycomposed of a single spin-degenerate channel N 0 = 1 so RAP C = h/(2e2 D) and



2e2
eV
2
SI = 2
2kB Tcryo D + D(1 − D)eV cothh
2kB Tcryo(3.19)In order to maximize the measured signal I usually apply a voltage V  kB Tcryo /ewhere Tcryo ≤ 10 K. At these large voltages,
SI = 22e2
D(1 − D)eVh

(3.20)

Finally, when analyzing the noise in a circuit it is often more convenient to express theshot noise contribution as a voltage noise source in series with the junction. The voltage
noise has a spectral density2
SV = SI RAP
C = 2eV RAP C (1 − D)

3.1.3

(3.21)

Tunneling probability

In a gold atomic point contact, the connection between two atoms can be thoughtof as a single, spin-degenerate conducting channel. A gap between the two atoms will
scatter the incident electrons and, at the experimentally relevant voltages, electrons
will either reflect off of the barrier created by the vacuum gap or tunnel through thegap. In this section I calculate the the transmission probability D as a function of the
gap width x. As will be shown below, the transmission probability D is exponentiallysensitive to the size of the gap with a length scale, in gold, of about 0.1 nm. Because
this length scale is smaller than the atomic radius, the conductance of an atomic point
contact is usually dominated by the least resistive channels: either a sum of ballisticchannels where Dn = 1 or, if there are no ballistic channels, a single channel m with
Dm = D  Dn , n 6= m. The atomic-scale roughness implies that the channel between
the two atoms with the smallest separation will dominate the conductance.

Energy (E or V)
incident
particle

reflection

51

transmission

Position (x)Figure 3.3: A particle (blue) incident from the left will scatter off of a potential barrier(black) and will either be reflected or tunnel through the barrier and be transmitted.

I calculate the transmission probability by modeling the vacuum gap as a onedimensional barrier V (x) and using the Wentzel-Kramers-Brillouin (WKB) method to
calculate the approximate probability D that an incident particle (an electron) withenergy E and mass m will tunnel through the barrier (figure 3.3). The WKB method
is described in most quantum mechanics textbooks, for example references [156, 157].
The goal of the WKB method is to determine the particle’s wave function ψ(x)from the Schrodinger equationh̄2 d2 ψ+ V (x)ψ = Eψ2m dx2(3.22)The WKB method uses a trial wave function ψ(x) = exp[u(x)] and then finds a solution
for u(x). For an arbitrary barrier V (x) there is not an analytic solution for u(x).
However, if V (x) = V0 is constant for −l < x < l then over that range u(x) is constantand the wave function isψ(x) = A exp[ikx] + B exp[−ikx](3.23)

52
where k =q
2m[E − V0 ]/h̄2 if E > V (x) or elseψ(x) = C exp[κx] + D exp[−κx]where κ =(3.24)q
2m[V0 − E]/h̄2 = ik if E < V (x).

The WKB method extends this solution to potential barriers V (x) that change
slowly, that is, where |k 0 (x)|  |k 2 (x)|. Substituting ψ(x) = exp[u(x)] into the Schrodinger
equation andiwhere k(x) =

d2 u
−
dx2



du
dx

2

+ k(x)2 = 0(3.25)q
2m[E − V (x)]/h̄2 . An approximate solution for u(x) is obtained whend2 u/dx2 is small, as is the case for a free particle, using an iterative approach. The
initial approximation
Z x
u0 (x) = ±k(x)dx + c(3.26)

assumes that d2 u/dx2 = 0. More accurate approximations are found iteratively bysolving


dun
dx

2
=i

d2 un−1+ k(x)2dx2(3.27)

The approximate WKB wave functions ψ(x) = exp[u1 (x)] are found by solving theabove equation for n = 1
 Z x

 Z x

B
ψ(x) ≈ p
exp +ik(x)dx + p
exp −ik(x)dxk(x)k(x)
A(3.28)

for E > V (x) and
 Z x

 Z x

D
ψ(x) ≈ pexp +κ(x)dx + p
exp −κ(x)dxκ(x)κ(x)
C(3.29)

for E < V (x).
This approximation is only valid in the region where |k 0 (x)|  |k 2 (x)| which
implies that these approximate WKB wave functions are inaccurate in the region where53
E is close to V (x). It is necessary to use a different approximations to connect thesolutions for E < V (x) to the solutions for E > V (x), or, in terms of equations 3.28
and 3.29, express C and D in terms of A and B. One method, which is appropriate
for the potential barriers considered in this thesis, is to use a linear approximation forthe barrier near the point where E = V (x). That is, if E = V (a), then near x = a
the potential is approximated by V (x) − E = gx − ga. Using this approximation, thesolutions to the Schrodinger equation are Airy functions Ai [· · · ] and Bi [· · · ]ψ(x) ≈ aAi

2mg
h̄2




(x − a) + bBi



2mg
h̄2




(x − a)(3.30)The Airy functions provide the desired connection between the approximate WKB
waveforms for E > V (x) and E < V (x). The asymptotic expressions for the Airyfunctions can be expressed in terms of the approximate WKB wave functions by appropriately choosing A and B for E > V (x) (equation 3.28) and C and D for E < V (x)(equation 3.29). Explicitly, for a turning point at x = a where E > V (x) when x < aand E < V (x) when x > a (figure 3.4a)
 Z x

 Z x

B
ψ(x)|xa ≈ p
exp i
k(x)dx + p
exp −ik(x)dx
k(x)
k(x)
 Z x

Z x

A − iB−iA + Bexp −κ(x)dx + p
exp
κ(x)dx
ψ(x)|xa ≈ p
2 −iκ(x)−iκ(x)
A(3.31)(3.32)

and for a turning point at x = b where E < V (x) when x < b and E > V (x) when x > b(figure 3.4b)
 Z x

Z x

D
exp −
κ(x)dx + p
exp
κ(x)dx
ψ(x)|xb ≈ p
κ(x)
κ(x)
 Z x

 Z x

C − iD−iC + D
2
2
ψ(x)|xb ≈ p
exp ik(x)dx + p
exp −ik(x)dx−ik(x)−ik(x)
C(3.33)(3.34)

I apply the WKB approximation to a simple high potential barrier V (x) (figure3.4c) where V (x) < E for x < a or x > b and V (x) > E for a < x < b. The solution to(a)Energyincident
particlex(b)Energyincident
particle

x=a

x

x=b

Energy (E or V)(c)incident
particle

54

reflectionxx=atransmission

x=b

Figure 3.4: (a & b) Diagrams of the WKB turning points. (a) In the vicinity of x = a,the energy of the particle (blue) is greater than the potential of the barrier (black) whenx < a. (b) In the vicinity of x = b, the energy of the particle (blue) is greater thanthe potential of the barrier (black) when x > b. (c) Particles (blue) with energy E are
incident from the left on a potential V (x) where V (x) > E when a < x < b.

the Schrodinger equation with particles incident from the left (from −∞) is
 Z x

 Z x

B
ψ(x)|x<a = p
exp +ik(x)dx + p
exp −ik(x)dx
k(x)
k(x)
a
a
 Z x

Z x

C
D
ψ(x)|a<x<b = p
exp −κ(x)dx + p
exp
κ(x)dx
κ(x)
κ(x)
a
a
 Z x

F
ψ(x)|x>b = p
exp +ik(x)dxk(x)bA(3.35)(3.36)(3.37)

55
Using the connection formulas above, I solve for the transmission probability D = |F/A|2
 Z b

−2
 Z b
1
D = exp −2
κ(x)dx 1 + exp −2
κ(x)dx(3.38)4aa
A model for the barrier of caused by a vacuum gap of length L = b − a in an
atomic point contact (APC) is shown in figure 3.5. When there is no applied voltage,
the barrier is rectangular and, at zero temperature, the electron states with energies
less than the Fermi energy EF will be filled on both sides of the barrier. The height ofthe barrier is equal to the Fermi energy plus the work function of the material EF + φ.
Since there are no states which are filled on one side of the barrier and empty on theother side of the barrier, there will be no tunneling. When a bias voltage VAP C isapplied to the contact on the left then the height of the barrier on the left will increase
to EF + φ + eVAP C and electron states with an energy up to EF + eVAP C will be filled.

incident
particle

reflection

barrier

Energy (E or V)

Work
Functiontransmission

Fermi
EnergyApplied Voltage

x

x=a

x=b

L=b-a

Figure 3.5: Electron tunneling across an APC. When the tunneling probability D  1,the height of the potential barrier (grey) above the Fermi energy (purple) is approximately equal to the work function of the material (green). Incident electrons (blue) cantunnel across the potential barrier from filled states (purple) to empty states (white).
An applied voltage (red) raises the potential on the left side of the barrier.

56
The barrier height decreases linearly to EF + φ on the right side of the barrier, andthe electron states on the right side will still only be filled up to the Fermi energy EF .
Electrons with an energy between EF and EF + eVAP C can therefore tunnel from filledstates on the left hand side of the barrier to empty states on the right hand side of thebarrier.
The voltage dependence of the tunneling barrier and available electron energies
implies that the APC resistance RAP C = h/(2e2 hDi) is also dependent on VAP C . In
figure 3.6a I plot RAP C as a function of VAP C for APCs with various gap sizes L. I
assume typical values for the work function φ = 5.1 eV and Fermi energy EF = 5.5 eV
in gold. As seen in figure 3.6a, a voltage |VAP C | > 1 V is required to significantly modify
the tunneling resistance.
Over the experimentally relevant voltages |VAP C | < 50 mV φ/e ≈ EF /e thebarrier can be modeled as a rectangular barrier with height EF + φ and the electronenergy is approximately equal to the Fermi energy EF . Experimentally, the transmission
probability is approximately−21
D = exp [−2L/λ] 1 + exp [−2L/λ]
4(3.39)q
where λ = h̄2 /(2mφ) = 1/κ is the length scale that controls the probability of electron
tunneling (figure 3.6a, black lines). In gold, λ ≈ 86 pm. Experimentally, the resistanceof the APC is approximately
RAP C =

2

h
1
exp
[2L/λ]
1+exp[−2L/λ]2e2
4(3.40)

I plot the resistance RAP C of the APC versus L/λ in figure 3.6b. When L/λ > 1 theresistance of the APC is exponentially dependent on the width of the gap L
RAP C =
(figure 3.6b, black lines).

hexp [2L/λ]
2e2(3.41)

57(a) 5 R10APC6

2  106L  0.05 nm
L  0.1 nm
L  0.15 nm
L  0.2 nm
L  0.25 nm

1  106
5  105
2  1051  105
0.01

RAPC 5  100.05 0.10

6

1  106
5  105

0.50 1.00

Λ2RAPC RAPC L
1.

(b)

0.95
0.9

(c)

0.85

1  105
5  1041  104
0.0

Voltage V

0.8

0.5

1.0

1.5

2.0

L/λ (λ=86 pm)

2.5

3.0

0.0 pm
0.5
LΛ Λ86
0.75

1.0

1.5

2.0

L/λ (λ=86 pm)

2.5

3.0

LΛ Λ86 pm

Figure 3.6: (a) APC resistance RAP C as a function of bias voltage V across the APC
for various barrier widths L, assuming an ideal gold junction. At voltages V > φ/10 =500 mV, the effect of the applied voltage on the barrier is noticeable; φ = 5.1 eV is
the work function of gold. (b) APC resistance RAP C as a function of the width of
the junction L. The resistance is calculated assuming a bias voltage V  φ and using
the WKB method (red) and assuming an exponential dependence (black). The WKB
result deviates from an exponential dependence when the width is small L < λ. (c) The
effective length scale (1/2RAP C )(∂RAP C /∂L) as a function of the width of the junction
L. When L > 0.8λ, the difference between the effective length scale calculated using
the WKB method (red) and calculated assuming an exponential dependence (black) is
less than 10%.

58
The length scale λ also determines the sensitivity of the resistance RAP C to smallchanges δL in the the width of the gap. From equation 3.40, a small change in thewidth L → L + δL creates a change in resistance RAP C → RAP C + δR
λ
δR = δL
2RAP C

4 exp [2L/λ] − 1
4 exp [2L/λ] + 1

(3.42)The quantity λ (4 exp [2L/λ] − 1) / (4 exp [2L/λ] + 1) is an effective length scale that
controls the probability of electron tunneling. In figure 3.6c I plot this length scale (redline) as a function of L/λ. The difference between this length scale and λ (black line)
is less than 10% when L/λ > 0.8. In this thesis I generally assume that
δR = δL

λ
2RAP C(3.43)and that λ can depend weakly on the resistance of the APC as in figure 3.6c (red line).
Deviations of the barrier potential from a rectangular barrier will also cause λ to changewith resistance; these deviations are expected to have a larger effect when the size ofthe gap L/λ < 1.

3.2

APC as a Quantum-Limited Displacement Detector
The atomic point contact (APC) displacement detector uses the exponential de-pendence of the APC resistance RAP C on the width of the gap between atoms in the
APC to sensitively detect nanomechanical motion. This detector has been theoretically
predicted to be a quantum-limited detector [65–67, 69, 71, 72, 75, 77]. Near the quantum limit, the displacement imprecision is due to the shot noise of tunneling electrons
and the detector backaction is caused by the momentum kicks due to tunneling electrons [58–62]. I use this phenomenological model of the APC backaction to compare
the APC detector to a quantum-limited detector.

59
3.2.1

The standard quantum limit

The Heisenberg uncertainty relationship places a limit on the minimum imprecisionbackaction product of a displacement detector. A quantum-limited displacement detector operates at this minimum. The uncertainty relationship in the simple case of asingle strong measurement of the position of a particle is calculated in most quantum mechanics textbooks [156–158]. The measurement of position has an uncertainty∆x2 = h(x̂ − hx̂i)2 i = hx̂2 i − hx̂i2 . This measurement of position necessarily effects
the state of the particle; this backaction effects the momentum of the particle with anuncertainty ∆p2 = h(p̂ − hp̂i)2 i = hp̂2 i − hp̂i2 . For any two operators Â and B̂
∆A2 ∆B 2 ≥DÂ − hÂiE 2 11
B̂ − hB̂i= |h[A, B]i|2 + |h{A, B}i|244(3.44)

Since x̂ and p̂ do not commute [x̂, p̂] = ih̄, the product of the uncertainty in the position
measurement ∆x and the uncertainty in the detector backaction ∆p has a lower bound∆x∆p ≥h̄2(3.45)The APC displacement measurements performed in this thesis are not strong,single-shot position measurements but instead are weak, continuous position measurements. A qualitative understanding of the Heisenberg constraint on a weak continuous
measurement can be found by considering a sequence of weak measurements with imprecision ∆x and backaction ∆p [115]. A detector performs a sequence of N displacement
measurements separated by a time τ . Each measurement has an imprecision ∆x andbackaction ∆p. I assume that there is no correlation between ∆x and ∆p or between themeasurements at different times. Averaging the sequence of measurements taken over aptime t = N τ , the average displacement uncertainty ∆xavg (t) = ∆x τ /t decreases andp
the average backaction ∆pavg (t) = ∆p t/τ increases as in a random walk.
This sequence of measurements becomes a weak continuous measurement in thelimit that the time between measurements τ → 0 and each measurement’s uncertainty

60∆x → ∞. Since each measurement in the sequence is uncorrelated, the imprecision
spectral density Sx = 2(∆x)2 τ of the continuous measurement is frequency independent.
The factor of two is present because I use single-sided spectral densities which are only
defined for positive frequencies. So that the imprecision spectral density is independent√of τ , the measurement uncertainty is increased as ∆x ∝ 1/ τ . In this case the averageppdisplacement uncertainty ∆xavg (t) =Sx /t = ∆x τ /t is also independent of themeasurement spacing. Similarly, in the limit of a continuous measurement the detector’s
backaction ∆p → 0 and creates a white backaction force with a spectral density SF =√2(∆p)2 /τ . I set the backaction ∆ ∝ τ so that the spectral density and the averagep√backaction ∆pavg (t) = SF t = ∆p t/τ are independent of the measurement spacing
τ.
Using this description of a weak continuous measurement with uncorrelated backaction SF and imprecision Sx I can find the Heisenberg constraint on the imprecisionbackaction product. The imprecision-backaction product is 2(∆p)2
Sx SF = 2(∆x) τ= (2∆x∆p)2τ

2(3.46)

From the Heisenberg relationship on a single measurement above, ∆x∆p ≥ h̄/2, theconstraint on a weak continuous measurement is
Sx SF ≥ h̄2(3.47)

This formula applies when the imprecision Sx and backaction SF are uncorrelated andthe spectral densities are single-sided. Weak continuous measurements with correlations
are considered in detail in references [29, 72, 75, 115].
The total imprecision in the measurement of the position of a harmonic oscillator,
such as a mass on a spring, is limited by quantum mechanics. The spectral density ofthe total displacement noise Sxtot is composed of two types of noise. First, there are theapparent fluctuations in the position of the harmonic oscillator due to the measurement

61
noise Sx which I assume has a white spectral density (as above). Second, the backaction
force SF creates real fluctuations in the position of the harmonic oscillator SxBA . A
force F at frequency ω on a mass m with spring constant mω02 and damping γ will resultin displacement x = H(ω)F where H(ω) is the harmonic oscillator’s response function(see section 3.3)
H(ω) =1
 2
m ω0 − ω 2 + iγω

(3.48)The backaction force SF creates random motion with a spectral density
SxBA = |H(ω)|2 SF =

m2

SF
 2

(ω0 − ω 2 )2 + γ 2 ω 2(3.49)

These two sources of noise, one due to apparent motion and and one due to actual
motion, result in a total displacement measurement noise with spectral density
Sxtot = Sx + SxBA = Sx +

m2

SF
 2

(ω0 − ω 2 )2 + γ 2 ω 2(3.50)

Using the Heisenberg constraint
h̄2
 2

Sx m2 (ω0 − ω 2 )2 + γ 2 ω 2(3.51)2h̄p
2
m (ω0 − ω 2 )2 + γ 2 ω 2(3.52)

Sxtot ≥ Sx +which has a minimum
Sxtot ≥

when the contribution to the noise from the apparent motion and actual motion are
equal
h̄
p
2
m (ω0 − ω 2 )2 + γ 2 ω 2q
SF = SF SQL = h̄m (ω02 − ω 2 )2 + γ 2 ω 2
Sx = SxSQL =(3.53)(3.54)

This minimum is called the standard quantum limit (SQL). Experimental displacement

62
detectors are usually compared to the SQL on-resonance ω = ω0 where
2h̄
mγω0
h̄
SxSQL =
mγω0
Sxtot ≥SF SQL = h̄mγω0(3.55)(3.56)(3.57)The quantum limits discussed above apply to a continuous, linear position measurement. It is possible to evade these limits by, for example, only measuring a single
quadrature of a harmonic oscillator by modulating the strength of the measurement at
the harmonic oscillator resonance frequency. This type of measurement is called a backaction evading measurement and is discussed in more detail in references [29, 115]. The
displacement measurements in this thesis are continuous, linear position measurements.

3.2.2

APC displacement detection

The physics of an atomic point contact (APC) can be used to create a sensitive detector of the motion of a nanomechanical system. I use the same approach as
was proposed in reference [62] and measure the change in APC resistance caused bynanomechanical motion. Conceptually, the APC resistance change is measured by placing the APC in series with a dc voltage source and a high impedance voltage amplifier(figure 3.7). Changes in APC resistance create a voltage signal at the amplifier. In thissubsection I assume that the amplifier noise is negligible compared to the shot noise. Experimentally (chapter 6 and 7) this is usually true at large bias voltages VAP C > 10 mV;
the amplifier noise and actual measurement circuit is discussed in detail in chapter 5.
A change in APC resistance will change the voltage measured by the amplifier.
From the circuit in figure 3.7, the voltage measured by the amplifier is
Vamp = Vb

Ramp
Ramp + RAP C(3.58)

Small changes in the APC resistance RAP C → RAP C + δR(t) creates a small change in

63

VAPC

SV

RAPC

Vb

Ramp

Vamp

Figure 3.7: Circuit diagram used in a basic APC displacement measurement. An applied
voltage Vb creates a voltage VAP C across the APC resistance RAP C and shot noise in
the tunneling current creates a voltage noise source SV = 2eRAP C VAP C . The APC isin series with a voltage amplifier with input impedance Ramp . The applied voltage Vb
creates a voltage Vamp at the input of the amplifier. Changes in the APC resistancealso changes the measured voltage Vamp creating a signal δV . The voltage fluctuations
SV limit the precision of the measurement.

the voltage at the amplifier Vamp → Vamp + δV (t)δV (t) = −δR(t)Vb Ramp(Ramp + RAP C )2

(3.59)As discussed earlier, changes in the width of the APC tunneling barrier δx(t) willchange the APC resistance (equation 3.43) δR = δx(t)λ/2RAP C where λ is the lengthscale that controls the probability of electron tunneling. The APC is fabricated as partof a nanomechanical structure. Resonant motion of the structure can be modeled as amass on a spring system (see section 3.3) with mass mef f , resonance frequency ω0 , anddisplacement y(t). Nanomechanical displacement will change the width of the gap
δx(t) = χy(t)(3.60)

64
and create a voltage signal
δV (t) = −y(t)χ 2Vb RAP C Ramp
λ (Ramp + RAP C )2

(3.61)The shot noise of tunneling electrons will create voltage noise at the amplifier.
The shot noise is modeled by a voltage noise source in series with the APC (figure3.7) with spectral density (from equation 3.21) when the tunneling probability is small
RAP C  h/2e2
SV = 2eRAP C |VAP C |(3.62)

where VAP C is the voltage across the APC. This voltage noise creates uncertainty inthe displacement measurement. The displacement imprecision spectral density is
Sx = SVy 2e
λ 2 RAP C (Ramp + RAP C )2=2
δV
2|VAP C | χ
Ramp(3.63)The minimum imprecision occurs when Ramp  RAP C . I also assume a simple masson-a-spring structure where χ = 1, so the APC displacement detector can be used tomeasure impression with a sensitivity
Sx =e|λ|2 RAP C
2 |VAP C |(3.64)

Using a gold atomic point contact (λ ≈ 86 pm) and an experimentally realizable current
VAP C /RAP C = 0.1 µA, it should be possible to achieve a displacement imprecision√√Sx ≈ 80 am/ Hz.

3.2.3

APC backaction and the quantum limit

I follow Bocko, Stephenson, and Koch [58–62] and approximate the APC detector
backaction by estimating the momentum kick delivered by each tunneling electron. The
experimentally applied APC voltages VAP C ≤ 50 mV are much smaller than both the
Fermi energy in gold EF = 5.5 eV and the height of the barrier φ = 5.1 eV above theFermi energy. Therefore the tunneling electrons have an energy approximately equal

65
to the Fermi energy and are expected to apply an average momentum kick pe to thenanostructure equal to the Fermi momentum pF ≈ 10−24 N·sec. The shot noise in thetunneling current creates a random backaction force with a spectral density
p2 |VAP C |SF corr = 2 e
e RAP C

(3.65)

This backaction force is correlated with the shot noise. Technically, the force due to theaverage momentum of each tunneling electron pe can be canceled. The shot noise canbe detected electronically, and an external force can be applied to the nanostructurewith equal magnitude but in the opposite direction.
The uncertainty ∆pe in the magnitude of the momentum of each tunneling electron also creates a random backaction force. This backaction force is not correlatedwith the shot noise of tunneling electrons and enforces the quantum limits describedearlier. Each tunneling electron imparts a random momentum kick ∆pe which createsa random force with spectral density
SF ∆ = 2(∆pe )2 |VAP C |e
RAP C(3.66)

Following Bocko [62], I use the Heisenberg uncertainty principle to estimate theuncertainty ∆pe in the electron momentum. During electron tunneling, the positionof the electron is localized with an uncertainty ∆xe that is about equal to the lengthscale λ/2 (equation 3.41) that controls the probability of electron tunneling. This effective measurement of the electron’s position results in an uncertainty in the electron’s
momentum
∆pe ≈

h̄
h̄
≈
2∆xe
λ

(3.67)

Using this estimate of ∆pe , there is a backaction force
SF ∆ = 2h̄2 VAP C
eλ2 RAP C

which is not correlated with the shot noise or the imprecision Sx .

(3.68)

66
I calculate the imprecision-backaction product from the imprecision Sx (equation 3.64) and the uncorrelated backaction force SF ∆ (equation 3.68). Based on theseestimations, the APC displacement detector can achieve the Heisenberg limit
Sx SF ∆ ≈ h̄2(3.69)

More detailed and accurate calculations of the backaction and imprecision also indicatethat the APC displacement detector can be quantum-limited [64–77].
In the experiments described in this thesis I do not attempt to cancel the correlated backaction force SF corr , so the total backaction has a spectral density
2
SF ba = SF corr + SF ∆ =eh̄2 VAP C
2
pe + 2λ
RAP C(3.70)In gold the work function and Fermi energy are approximately equal, so pe ≈ h̄/λ.
Ignoring correlations, the best imprecision-backaction product is expected to be
Sx SF ∆ = 2h̄2(3.71)

There are also other ways in which measuring displacement with the APC detector
can create a backaction force. The fluctuating voltage across the APC, due to shot noise,
will create a backaction forceSF C = 2eVAP C RAP C

∂C
VAP C
∂x

2
(3.72)

because the capacitance C of the APC depends on the size of the APC gap x. The fluctuating voltage can also launch surface acoustic waves onto the surface of the substrate,shaking the nanostructure’s supports and creating a backaction
SF V = 2eVAP C RAP C σ 2(3.73)

where the proportionality constant σ depends on the substrate material and the coupling
between the nanomechanical motion and the motion of the supports. The presence oftrapped charge near the nanostructure will also create a backaction force with the same67
spectral density (equation 3.73), but in this case σ will depend on the number, location,and sign of the trapped charges. Finally, passing a current IAP C through the APC
2will result in power IAP
C RAP C being dissipated in the contacts. If this dissipatedpower heats the nanostructure by a temperature ∆T then the measurement creates anadditional backaction force
SF T = 4mef f γkB ∆T(3.74)

where γ is the nanomechanical damping and mef f is the effective mass. The presence of
any additional backaction force will keep the APC displacement detector from operatingwith a Heisenberg limited imprecision-backaction product.

3.3

Simple Harmonic Motion of the Nanostructure and APC
The atomic point contact (APC) is fabricated as part of a nanostructure. Mechan-ical motion of the nanostructure can be described by adding together a set of normal
modes; each mode can be modeled as a simple harmonic oscillator. I calculate the modeshapes and frequencies of a long beam with a small cross-section. Each mode shape is a
~ (x, y, z) which describes the normalized displacement, in three dimensions, at
vector U
every point in the structure. The physical displacement of the nanostructure is equal to~ (x, y, z) times a magnitude u(t). The magnitude u(t) is the coordinate of a simple
the U
harmonic oscillator with frequency ω0 , an effective mass mef f , and damping constantγ; these three quantities can be different for each mode. I normalize the mode shapes
so that the effective mass is equal to the mass of the nanostructure.
The nanostructures used in this thesis would be, in the absence of the APC,
either doubly or singly clamped beams. However, the addition of an APC adds a more
complicated element. I estimate the effect of the APC on the structure’s vibrationalmodes using a finite element model (FEM).
I calibrate the measured voltage signal in units of displacement using the known

68
spectral density Sx of the nanostructure’s Brownian motion. Determining the measurement sensitivity to displacement at the APC requires knowing the mode shape; themode shape is difficult to determine because of the presence of the APC. However, thecomparison between the APC detector’s characteristics and the standard quantum limit
is independent of the mode shape.

3.3.1

The modes of a nanomechanical beam

In general, the displacement of a mechanical system can be described by defining~ (x, y, z) which gives the displacement of a small volume dV of a mechanicala vector U
system from it’s equilibrium position at (x, y, z). The elastic nature of the mechanicalstructure implies that each displaced volume dV will feel a restoring force. The normal
mode shapes and frequencies are the solutions to the equations (and boundary conditions) which describe the relationship between displacement and force. The solutions
cannot be found analytically for most cases, but there are analytic solutions for small
displacements of simple geometries.
One of these simple cases (covered in numerous text books such as [14, 159, 160])
is that of a thin rectangular beam, that is, a beam whose width and height are much~smaller than it’s length. Any mechanical system’s motion X(x,y, z, t) can be describedby a sum of normal modes weighted by a time-dependent coordinate un (t)~X(x,y, z, t) =NX

~ (x, y, z)
un (t)U

(3.75)

n=1

where N → ∞ for a continuous system with an infinite number of degrees of freedom. In
the simple case of a beam, the modes split into flexural modes (motion perpendicular tothe beam’s long direction), rotational modes (rotational motion around the axis definedby the beam’s long direction), and compressional modes (motion parallel to the beam’s
length). The flexural modes are much easier to excite, therefore it is possible to describe
the experimentally observed displacement by ignoring the rotational and compressional

69
modes.
This simplifies the analysis, since small volumes of the beam are only going to be
displaced in a direction perpendicular to the beam’s length (figure 3.8a). Therefore, I
can define a small volume by slicing the beam into small sections with length dx so thateach element has a volume dV = Adx, where A = wh is the area of the beam, w is thewidth in the ŷ direction, and h is the height in the ẑ direction. Each of these elements
can be described by it’s position along the length of the beam and the displacement of

z


h

x
y

w

dx

M(x)

F(x)

M(x+dx)F(x+dx)dx
Figure 3.8: (a) Diagram of a thin, narrow nanomechanical beam with length l, heighth, and width w where w, h  l. (b) I calculate the mode shapes in the ŷ direction bysumming the forces on a small volume element hwdx due to the neighboring elements.
For small displacements, rotational inertia and shear forces can be ignored and therelevant forces are the elastic force F (x) in the ŷ direction and the torque M (x) aroundthe ẑ axis.

70~~
each element is X(x,y, z, t) = X(z,t).
Using rectangular coordinates, the flexural normal modes can be further divided
into modes which have displacement in the ŷ direction and modes which have displacement in the ẑ direction. I am going to analyze motion in the ŷ direction (an identicalanalysis can be applied to motion in the ẑ direction). Therefore, the displacement of~each element in this analysis is X(x,y, z, t) = ŷX(z, t).
I can now balance the forces (all applied along the ŷ direction) and torques (allapplied around the ẑ axis, that is, NOT rotation around the long axis of the beam) on asingle element (figure 3.8). The neighboring element on the left will exert a force −F (x)and the neighboring element on the right will exert a force F (x + dx); the resulting
acceleration of the element under consideration is∂ 2 X(z, t)
∂t2
∂F
∂2X
= ρA 2∂x∂t−F (x) + F (x + dx) = ρAdx

(3.76)(3.77)

Similarly, I sum the torques exerted on the left side of the element (at position x). theneighboring elements will directly exert a torque −M (x) and M (x + dx). The force
F (x + dx) on the right side of the element (at position x + dx) also exerts a torqueF (x + dx)dx. In the case of flexural vibrations there should be no net torque and sothese torques must balance:−M (x) + M (x + dx) + F (x + dx)dx = 0(3.78)∂M∂x(3.79)

F =−I assume that the beam is made out of a single material with a constant Young’s
modulus Y and has a moment of inertia about the ŷ axis I, therefore
M = EI∂2X∂x2(3.80)and so the partial differential equation relating the beam’s displacement to the beam’s

71
acceleration is
∂2
− 2
∂x



∂2X
∂2X
EI
=
ρA
∂x2
∂t2
∂4X
∂2X
−EI
= ρA 24∂x∂t(3.81)(3.82)

This partial differential equation is similar to that of a mass on a spring, but hasa fourth order partial derivative with respect to displacement instead of a second order.
The solutions to this equation have the formX(x, t) = [A cos(ωn t) + B sin(ωn t)][an cos(kn x) + bn sin(kn x) + cn cosh(kn x) + dn sinh(kn x)](3.83)

This solution is usually split into a normalized mode shape
Un (x) = an cos(kn x) + bn sin(kn x) + cn cosh(kn x) + dn sinh(kn x)(3.84)

and a coordinate
un (t) = A cos(ωn t) + B sin(ωn t)(3.85)

whose overall magnitude depends on the normalization of the mode shape. Using thissolution in the partial differential equation above, the eigenfrequency ωn is related to
the wave vector knsωn = ±

EI 2
k
ρA n

(3.86)

I now have a general solution to the mode shape, but the boundary conditions
and normalization must also be specified in order to determine the constants above (an ,bn , cn , dn , and kn ). I am going to look at two different sets of boundary conditions:a doubly clamped beam and a singly clamped beam. In this section I normalize themagnitude of the mode shapes so that the average of the mode shape squared is equal
to one (see section 3.3.2).

72

Mode Shape
1.5
1.0
0.5

n1
n2
n3

0.0

0.2

0.4

0.6

0.8

1.0

xl

0.5
1.0
1.5

Figure 3.9: The mode shapes U (x) of the fundamental (n = 1), second, and third modeof a doubly-clamped nanomechanical beam are plotted as a function of the position x
along the beam. Since the beam is clamped at both ends, both the displacement andslope of the beam is zero at both ends.

A doubly clamped beam is constrained so that both ends of the beam do not
move and are not sloped (that is, U (0, t) = U (l, t) = 0 and ∂U (x, t)/∂x|x={0,l} = 0).
Therefore an = −cn , bn = −dn ,bn = −

cos(kn l) − cosh(kn l)ansin(kn l) − sinh(kn l)(3.87)

andcosh(kn l) cos(kn l) = 1(3.88)

This transcendental equation does not have an analytic solution, but it can be solved numerically and the first few solutions are kn l = {0, 4.73004, 7.8532, 10.9956, 14.1372, . . .}.
Therefore, the resonance frequencies of the first few ŷ flexural modes of a doubly clamped
narrow, thin beam aresω=EI{22.3733, 61.6728, 120.903, 199.859, . . .}
ρAl4

(3.89)

73

Mode Shape
2

1

n1
n2
n3

0

0.2

0.4

0.6

0.8

1.0

xl

1

2

Figure 3.10: The mode shapes U (x) of the fundamental (n = 1), second, and third modeof a single-clamped nanomechanical beam are plotted as a function of the position x
along the beam. The displacement and slope of the beam is zero at x = 0 because thebeam is clamped, but the beam’s displacement and slope at x = l is unconstrained.

and the mode shapes are given in figure 3.9.
The solution for a singly clamped beam proceeds along a similar path, but only
one end is clamped (U (0, t) = 0 and ∂U (x, t)/∂x|x=0 = 0). The other end is freeso U (l, t) and ∂U (x, t)/∂x|x=l are unconstrained. However, there is also no force or
torque on the free end so ∂ 2 U (x, t)/∂x2 |x=l = 0 and ∂ 3 U (x, t)/∂x3 |x=l = 0. Therefore
an = −cn , bn = −dn ,bn =cos(kn l) + cosh(kn l)ansin(kn l) + sinh(kn l)cosh(kn l) cos(kn l) = −1

(3.90)(3.91)

This transcendental equation must also be solved numerically, and the first few solutions
are kn l = {1.8751, 4.69409, 7.85476, 10.9955, 14.1372, . . .}. Therefore, the resonance frequencies of the first few ŷ flexural modes of a singly clamped narrow, thin beam ares
EIω={3.51602, 22.0345, 61.6972, 120.902, 199.86, . . .}(3.92)
ρAl4

74
and the mode shapes are given in figure 3.10.

3.3.2

Mechanical modes modeled as harmonic oscillators

Detectors of nanomechanical displacement such as the APC detector are usuallyused to measure resonant mechanical motion. The simplest type of mechanical resonanceis a single-degree-of-freedom system such as a mass on a spring in one dimension. The
use of the word ”single” to describe the system can be confusing because there are
actually two degrees of freedom: the position and the velocity. Using the example ofa mass on a spring, the spring provides a restoring force F = −kx proportional to thedisplacement x of the mass m. In the absence of any driving force, the free undamped
motion x(t) is found by solving the differential equation derived from Newton’s laws,mẍ = −kx. The solution has the form x(t) = aeiω0 t + be−iω0 t . The displacement occurs
p
at the system’s natural resonance frequency ω0 = k/m and with a magnitude that
depends on the initial position x(0) and velocity ẋ(0)x(t) = ẋ(0)  iω0 t
x(0)  iω0 te+ e−iω0 t +e− e−iω0 t
22i(3.93)

I can include a generic viscous damping force −γ ẋ and calculate the damped displacement x(t) in response to a force F (t) = F0 eiωt . The displacement is also found by solving
a similar differential equation derived from Newton’s laws, mẍ = −kx − γ ẋ + F (t). The
solution has the form x(t) = H(ω)F0 eiωt where H(ω) is the mechanical response function
H(ω) =1
 2
m ω0 − ω 2 + iγω

(3.94)The largest response occurs when the force is applied at the mechanical resonancefrequency ω = ω0 .
As was described earlier for the simple case of the flexural vibrations of a beam,~
the displacement X(x,y, z, t) of a volume dV of the mechanical structure located at

75~ n (x, y, z) eachthe point (x, y, z) can be decomposed into a sum of orthogonal modes U
weighted by a time-dependent coordinate un (t)~X(x,y, z, t) =NX

~ n (x, y, z)
un (t)U(3.95)

n=1

where N is the number mechanical degrees of freedom and N → ∞ for a continuous
mechanical structure. I describe the normalization of each mode shape using a coefficientαnZ~m · U~ n dV = δmn αn V
U

(3.96)

where the integral is over the mechanical structure and V =R

dV is the volume of themechanical structure. The units and magnitude of αn determine the units and magni~ n (x, y, z). Two common choices of normalization are αn = 1
tude of the mode shapes U
and αn = L2 where L is the significant length scale of the mechanical structure. In thefirst case αn = 1, the coordinate un (t) is equal to the root-mean-squared displacementof the mechanical structure along the normal mode and the mode shapes are unit-less. I
will use this normalization αn = 1 in the rest of this thesis. In the second case αn = L2 ,the mode shapes have units of displacement and the coordinate un (t) is unit-less.
The response of the coordinate un (t) to an applied force per unit volume F~ (x, y, z)eiωt
is effectively that of a harmonic oscillator. When the displacement is small comparedto the dimensions of the mechanical structure, the differential equation that is used toP~~calculate the response X(x,y, z, t) = N
n=1 un (t)Un (x, y, z) to the applied force usuallysimplifies to
!
−ρω

2

X

~n
un (t)U!
= F~ eiωt(3.97)~ n = F~ eiωtρ(ω02n − ω 2 )un (t)U

(3.98)

+ρ

n

X

~n
ω02n un (t)Un

Xn

~ n are orthogonal, this response can be separated into N independent
Since the modes U
equations, each equation describing the response of the time-dependent coordinate un (t),ZZ
X
~m · U~ n dV = eiωt U~ m · F~ dV(3.99)
ρ(ω02n − ω 2 )un (t) Un76
Including a mode-dependent damping γn and assuming that γn  ω0n , the response of
the coordinate un (t) is approximatelym(ω02n − ω 2 + iγn ω)un (t) = eiωt fn(3.100)where m is the mass of the mechanical structure (I assume the density ρ is constant)R~ n · F~ dV is the mean force driving mode n. This is equivalent to a SHO
and fn = U
with frequency ω0n , mass mef fn , displacement uef fn (t), and an applied force fef fn
mef fn (ω02n − ω 2 + iγn ω)uef fn (t) = eiωt fef fn(3.101)

Because only the displacement uef fn (t) and the coordinate un (t) are time-dependent,
they are related by a constant χn uef fn (t) = un (t). The values of the effective mass,force, and proportionality constant χn are related to the total mass m and mean force
fn by
mef fnfef fn=mχnfn(3.102)In this thesis and in many other experiments, the measured mechanical motion
is dominated by a single mode n and the motion is described by a single effective SHO.
Mechanical motion is measured at frequencies ω near the eigenfrequency ω0n of the moden. The difference between this eigenfrequency and all other eigenfrequencies is usuallylarge compared to both the measurement bandwidth and the width of the mechanicalresonance γ, so |ω02k − ω 2 |  ωγn for all modes k 6= n. The measured displacement
is also generally due to a force that effectively drives the mode n of interest; that is,R~ k · F~ dV andthe force per unit volume F~ (x, y, z)eiωt generates effective forces fk = U
the force fn on the mode n of interest is not significantly smaller than the force fk onother modes. With these restrictions, the contribution of the modes k 6= n to the total~
motion X(x,y, z, t) is going to be attenuated by a large factor |ω02k − ω 2 | compared tothe contribution from mode n. The total motion (from equations 3.95 and 3.100) is

77
approximately~X(x,y, z, t) ≈fn
~ (x, y, z)eiωt
U 2m ω0n − ω 2 + iγn ω(3.103)

I place additional constraints on the parameters of the effective SHO (equation
3.101) by requiring that the temperature of the effective SHO and the mechanical structure is the same. The effective SHO has two degrees of freedom, position and velocity,so the average total energy of the SHO in thermal equilibrium at temperature T is
hEef f i = kB T , as described by the equipartition theorem. Thermalization occurs because of a random force on the SHO with a one-sided spectral density SFT ef f . I calculate
the magnitude of this force spectral density from the average kinetic energy
11
kB T = mef f ω02 hu2ef f i
2
2(3.104)

which is in response to SFT ef f
kB T
1
= mef f ω02
2
2

Z ∞

dω
2π

(3.105)

SFT ef f ≈ 4mef f γkB T(3.106)

SFT ef f |H(ω)|20

where H(ω) is the SHO response function (equation 3.94) and I assume that γ  ω0and h̄ω0  kB T .
Similarly, the thermalization of a continuous mechanical structure at temperature
T occurs because of a random force per unit volume F~ T (x, y, z, t). The random force
at different points in the structure is uncorrelated. The random force V F~ T (x, y, z, t)
can be decomposed into a sum of mode shapes weighted by uncorrelated random forces
fnT (t)1
F~ T (x, y, z, t) =V
where fnT =RX~ n (x, y, z)
fnT (t)U(3.107)n~ n · F~ T dV . The coordinate un (t) is equal to the root-mean-squaredUmotion of the structure and the average kinetic energy of the mode is1
1
mω02n hu2n i = kB T
2
2

(3.108)

78
As for the single SHO above, the spectral density SFTn of fnT derived from the averagekinetic energy is
SFTn ≈ 4mγn kB T

(3.109)The requirement that the temperature of the effective SHO and the continuous
structure are the same places an additional constraint on the parameters of the SHO
in equation 3.102. This constraint can be derived in two different ways. First, theequipartition theorem implies that the thermalized average kinetic energy of the effective
SHO and the relevant mode of the continuous structure are the same (equations 3.104
and 3.108). The mean-squared displacement of the effective SHO and the mean-squared
displacement of the mechanical mode are related by the proportionality constant χ2n =h(un )2 i/hu2ef fn i. This leads to the constraintm=

mef fnχ2n(3.110)This same constraint can be found by examining the thermal force on the effective
SHO and on a mode of the continuous structure. From equations 3.106 and 3.109,
the constant temperature T implies that the ratio of the effective SHO force fef fn toR~ n · F~ dV is
fn = Uv
u T
ru SF f f n
fef fn
mef f
e
t
=
=
(3.111)TfnmSFn
Applying this constraint to equation 3.102 shows that this constraint and equation 3.110
are equivalent.
Since there are only two constraints on the three parameters in the effective
SHO, there is a single free parameter. A common approach is to treat χn , that is,
the proportionality constant χn uef fn = un , as the free parameter. Given χn , the mass
mef fn of the effective SHO is
mef fn = χ2n m(3.112)

79
and the force fef fn on the effective SHO isZ
fef fn = χn fn = χn

~ n · F~ dV
U(3.113)The parameter χn is often set equal to one or chosen so that the measured displace~ 0 , y0 , z0 )| =ment at a specific point (x0 , y0 , z0 ) in the structure due to mode n, |X(x~ n (x0 , y0 , z0 )|, is equal to the displacement of the effective SHO
un (t)|Uχn =1~ n (x0 , y0 , z0 )||U(3.114)

If the displacement is measured at a spot on the structure where the mode’s displacement
is larger than the root-mean-squared displacement, then χn < 1, the effective mass
mef fn will be smaller than the total mass m, and the effective force fef fn will also
smaller than the force fn .

3.3.3

The influence of the APC on the nanomechanical modes

As discussed above, for very simple mechanical structures it is possible to find ananalytic solution for the low frequency modes of the structure. However, the presence ofan APC makes finding an analytic solution difficult. The APC introduces a mechanicallyweak spot in the system, which I often model as a spring, and removes some of the usefulsymmetries. In addition, a physically accurate model would require a difficult transition
from bulk gold to the atomic-scale properties near the APC.
While the presence of the APC creates an intractable analytic problem, I can still
attempt to numerically model the structure’s bulk mode-shapes and frequencies. Thesenumerical approaches fit under the broad heading of finite element modeling (FEM)
and approach the problem by splitting the mechanical system into small elements withanalytically understood behavior. These small elements are linked together, creating amatrix which describes the effect of each element on all of the other elements. In thesimple of case, this matrix is inverted in order to determine the mechanical system’s

80
normal mode shapes and resonance frequencies. This approach has been used in creating numerous different professional software packages; I have been using the ANSYS
software package.
Finite element modeling programs are quite often used to model the resonancefrequencies and mode shapes of nanomechanical systems, where the model of the mechanical system is designed using scanning electron microscope (SEM) pictures of thedevice and knowledge of the materials used in fabrication. In this case, there is also
the question of how to model the APC. The FEM software is not designed for use withindividual atomic/molecular bonds, and, in any case, I do not know the atomic configuration. However, it is possible to continue with the idea of a finite element model andattempt to model the effect of the APC on the mechanical system by replacing the APC
with other elements. I can then modify the parameters of the APC element in order toreproduce the observed mode frequencies using the FEM.
I should mention that, while this is a good way to gain a qualitative understanding of the mechanical modes (especially if it is possible to accurately reproduce theobserved mode frequencies), there are problems with progressing towards quantitative
predictions. Specifically, this technique will attempt to compensate for any differences
between the mechanical model and the physical system by changing the parameters of
the APC element. Also, one of the most interesting parts of these models is the strain
at the APC (that is, the quantity which is directly measured by the APC displacement
measurement), however this is also the quantity which is represented by the least physical part of the model: the finite element that is chosen to represent the physical effectof the APC.
There are two different ways in which I have attempted to include the effect of
the APC in a FEM. In both cases, the first step is to create the usual FEM of thenanomechanical system with a gap in place of the APC. In the next step, the couplingcreated by the APC is modeled by connecting the two sides of the gap using either an81ideal spring or a small piece of material with the same properties as the rest of thenanomechanical system. Finally, I modify either the spring constant (in the case of theideal spring) or the dimensions of the material (in the case of the small piece of material)
until the frequencies of the FEM modes approximately match the observed mechanicalresonance frequencies.
I measure the motion of two types of nanomechanical structures in this thesis.
The first type of structure is a nanomechanical beam which is touched in the middle byan APC (figure 3.11a). The APC is created using electromigration (see chapter 4) andthe nanomechanical displacement measurements are discussed in chapter 6. In this case,
I model the effect of the APC on the mechanical mode shapes by replacing the APC
with a spring. A spring constant of 180 N/m results in FEM modes with resonancefrequencies which approximately match the observed resonance frequencies.
The spring constant of APCs has also been measured in other experiments. The
repeated breaking and reformation of the connection between two gold atoms (“training
behavior,” see chapter 7) has been used to infer APC spring constants ranging from 5
to 32 N/m [9]. However, unlike the APC modeled above, these junctions were formed
using mechanical strain and carefully trained so that the effect of the leads connectedto the atoms on the spring constant is minimal. The static force needed to displace
an APC connected to a tuning fork (as in an AFM) has also been used to determinethat the spring constant of contacts composed of a few (< 30) atoms is between 30130 N/m [8, 161]. These junctions are also unlike the APC modeled above because the
APCs are low resistance RAP C ≤ 2 kΩ, thus composed of multiple atoms. Despite thedifferences between the contacts and the simplicity of the FEM, the spring constant I
infer from the FEM model above is within a factor of 5 of the spring constants observedin other experiments.
The presence of the APC significantly effects the shape of the nanostructure’s
modes (figure 3.11b). The nanostructure is partially clamped by the APC and the

82

a)

APC

1μm
b)

18.4 MHz39.2 MHz

z
y
APC

APC

43.1 MHz54.2 MHz

x

APC

APC

57.2 MHz

APC

Figure 3.11: (a) Scanning electron micrograph of a gold nanostructure (yellow) containing an APC, prior to the in situe creation of the APC. The APC will be created at theconstriction between the top electrode and the suspended nanomechanical beam. (b)A simple finite element model is used to estimate the mode shapes associated with thenanostructure in (a). The APC is modeled as a spring which is connected to the bottomof the beam, because the constriction is thinner than the beam, and close to the middleof the beam. The presence of the spring causes the mode shapes to be a combination of
the in-plane, out-of-plane, and torsional modes expected from a simple doubly-clamped
beam. The sensitivity of the APC displacement detector to a mode can be estimatedfrom the normalized displacement of the point on the nanostructure connected to thespring.

displacement at the APC, that is, the quantity measured by the APC detector, depends
on the mode. For the 18 MHz mode, which is similar to the 2nd mode of a doublyclamped beam, the APC is close to a node of the system. The APC detector should be83
relatively insensitive to 18 MHz displacement. On the other hand, the 43 MHz mode
has an anti-node near the APC so the APC detector should be more sensitive to 43 MHz
displacement. This aspect of the FEM is qualitatively consistent with the experimentallyobserved sensitivity. The presence of the APC also causes the nanostructure’s modes tohave in-plane, out-of-plane, and torsional components; a good example of this mixing
is the 54 MHz mode.
The second type of structure is a nanomechanical beam which is interrupted in
the middle by an APC (figure 3.12a). The APC is created by mechanically straining thebeam (see chapter 4) and the nanomechanical displacement measurements are discussed(a)

100 nm(b)

65 MHz
APC

APC

APC

115 MHz
APC

Figure 3.12: (a) Scanning electron micrograph of a gold nanostructure (white/grey)
containing an APC, prior to the in situ creation of the APC. The top SEM is a topdown view of the beam and the bottom SEM is an angled view. The APC will be createdat the constriction in the middle of the suspended nanomechanical beam. (b) A finite
element model is used to estimate the mode shapes associated with the nanostructure in(a). The APC is modeled as a thin element connecting the two singly-clamped beams.
The presence of the connection causes the mode shapes to be similar to the mode shapesof a doubly-clamped beam. The sensitivity of the APC displacement detector to a modecan be roughly estimated from the strain at the constriction.

84
in chapter 7. I model the effect of the APC on the mechanical mode shapes of thisnanostructure by replacing the APC with a thin, narrow constriction with the samemechanical properties as the rest of the structure. I vary the height, width, and lengthof the constriction until the FEM mode frequencies approximately match the observed
resonance frequencies.
This APC model is less intuitive because bulk material is used instead of a spring,but I used the more realistic FEM to improve the design of the nanostructure. This FEM
model was used create a nanostructure with a larger coupling between the fundamental
mode of the nanostructure and APC displacement, compared to the type of structure
discussed earlier. The more realistic modeling of the structure’s boundary conditions
was also used avoid interference between flexural modes in the structure’s supports andthe APC measurement of the fundamental mode of the nanostructure. Low quality
factor modes were observed below 500 MHz due to the flexing of the structure’s supports. The frequency of these modes made it difficult to measure the fundamental modewith the APC displacement measurement. The FEM model was used to design stiffer
supports which pushed the frequency of the low quality factor modes above 500 MHz
and successfully removing their effect on the APC measurement.
The observed linear coupling between resistance RAP C and nanomechanical displacement can also be understood in the context of the FEM. In the first type of structure (figure 3.11), it is easy to see that small nanomechanical motion will create a linearchange in the APC gap. However, the linear coupling observed in the second type ofstructure (figure 3.12) is less intuitive. If the measured change in RAP C was causedby the total change in the length of the beam, then the resistance RAP C would be,
for small displacements, proportional to the absolute value of the displacement and thedominate voltage signal would be at a frequency equal to twice the frequency of thedisplacement. However, I observe that a driving force creates a voltage signal at thefrequency of the driving force. This linear coupling is due to the resistance’s depen-

85

#1

115 MHz#2

APC/constriction
Figure 3.13: A view from the top of the FEM solution (color) to the shape of the 2ndmode of the nanostructure in figure 3.12; the grey shadow shows the relaxed shape ofthe nanostructure. The APC is located somewhere in the nano-scale constriction. If
the APC is located near point #1 then the APC gap is currently expanded and will
be compressed a half cycle later, leading to a linear coupling between nanomechanicaldisplacement and the width of the APC gap. It is only if the APC is located near point#2 in the middle of the constriction that the width of the APC gap will depend on theabsolute value of the nanomechanical displacement.

dence on the atomic-scale strain. Figure 3.13 contains a close-up view of a section of
the nanostructure near the constriction used to model the APC. The length of the topof the constriction is increased, while the length of the bottom of the constriction isdecreased. Since the APC resistance depends on this local strain, the APC resistancewill be linearly coupled to displacement unless the APC is located in the center of theconstriction.

3.3.4

Calibrating nanomechanical motion

I use the known thermal force SFT = 4mef f γkB T , discussed in subsection 3.3.2, tocalibrate the APC displacement detector. The measured nanomechanical displacement
and backaction force need to be calibrated on two similar scales: in real units represent-

86
ing the displacement of the nanostructure and the force on the nanostructure, and incomparison to the quantum limits.
Earlier in this chapter I calculated the expected sensitivity of an APC displacement detector composed of gold atoms based on the work function of bulk gold. However, the atomic-scale system is complicated and, especially at low resistances RAP C <100 kΩ, the actual sensitivity may deviate from the expected sensitivity. The actual sensitivity is determined by measuring the signal in response to a known force. In general,
it is difficult to calculate the exact magnitude of an applied force and the complicatedmode shapes caused by the presence of the APC increase this difficulty. However, thespectral density of the thermal force SFT = 4mef f γkB T is known so the spectral density of the measured voltage signal SVT (ω) (figure 3.14) at a temperature T is used tocalibrate the transduction G between displacement and voltage
G2 =

SxT
1
1 4γkB T
= T
2
T
2
2
2 2
m
SV
SV
ef f (ω0 − ω ) + γ ω(3.115)In chapters 6 and 7, I discuss a more complicated version of this calibration which I use
to analyze the experimental data.
The APC detector’s imprecision Sx and backaction SFBA are determined from thewhite noise background in the voltage signal SVimp and the additional voltage signal
above the thermal noise SVadd , respectively (figure 3.14), as well as the transduction
factor G. On-resonance where ω = ω0 , the transduction is G = 4kB T /SVT mef f γω02 . The
measurement imprecision on-resonance in displacement units Sx is
Sx = G2 SVimp =SVimp 4kB T
SVT mef f γω02

(3.116)The measurement backaction SFBA is inferred from the additional on-resonance signal
SVadd
 S add
SFBA = G2 SVadd m2ef f ω02 γ 2 = VT 4γkB T mef fSV(3.117)

87

SV

10

add

SV

on-resonance

8

SVT

6

on-resonace
thermal motion

4

2

imp

0
49.8

49.9

SV

50.0

50.1

50.2

Freq MHz

Figure 3.14: An example of a voltage spectral density SV used to measure the APC
imprecision and backaction. The shot noise of tunneling electrons results in a whitebackground SVimp (black). The Brownian motion of the nanostructure has a Lorentzianline shape (blue, imprecision plus Brownian motion) and is used to calibrate the voltage
measurement in units of displacement. The backaction force causes additional random
motion of the nanostructure and an additional Lorentzian signal SVadd (red, imprecision
plus backaction plus Brownian motion). Both the shot noise (black arrow) and backaction (on-resonance, red arrow) contribute to the total uncertainty in the displacement
measurement of, for example, the Brownian motion of the nanostructure (on-resonance,
blue arrow).

The values of the displacement imprecision and backaction force in displacement
units depend on the effective mass of the SHO. As discussed in subsection 3.3.2, whenmeasuring the root-mean-square displacement of the nanostructure the effective massof the SHO is equal to the total mass of the nanostructure. In this thesis, I use the total
mass of the nanostructure in the calibration and quote the displacement imprecision

88
in terms of the root-mean-square displacement of the structure. I could attempt touse the mode shapes from the FEM models to find the effective mass associated withdisplacement at the APC. I do not attempt to use this calibration because it would add
a large source of systematic error to the calculation; on the other hand, the total massof the nanostructure is known with an error of about about 20%.
The choice of an effective mass is also less important because it is irrelevant
when comparing the APC displacement detector, calibrated using the thermal force,to the quantum limits.

Both the backaction force at the standard quantum limit

SF SQL = h̄mef f γω0 (equation 3.55) and the thermal force have the same mass dependence. Using the thermal calibration above, the ratio of the measured backaction
force to the backaction force at the standard quantum limit does not depend on theeffective mass
SFBA
S add 4kB T= VT
SF SQLSV h̄ω0(3.118)Similarly, the ratio of the measured imprecision and the imprecision at the standard
quantum limit does not depend on the effective mass
Sx
SxSQL

=

SVimp 4kB T
SVT h̄ω0(3.119)Finally, the product of the imprecision and the backaction, which is equal to h̄2 if thedetector is quantum limited, is also independent of the effective mass
SVimp SVadd 4kB T 2BA
Sx SF = T
≥ h̄2
T
ω
SV SV
0

(3.120)The mode shape is important when estimating the magnitude of a specific force
required to produce the observed backaction force. While this is true for any typeof force, I am going to specifically consider the phenomenological description of thebackaction force FI due to tunneling electrons. I assume that the force is applied near
the APC at the point (xAP C , yAP C , zAP C ) and in a direction parallel to the changing

89
APC gap. The force fn on mode n due to the backaction force FI is thenZ
fn =~ n · F~ dV ≈ |FI ||U~ n (xAP C , yAP C , zAP C )|U

(3.121)~ n (xAP C , yAP C , zAP C )| relates the change in the
The proportionality constant χn = 1/|Usize of APC gap xn to the displacement of the mode un = xn χn and can be used todetermine an effective mass, as discussed earlier in subsection 3.3.2.
Based on the FEM simulations described in the previous section, I estimate thatχ ≈ 1/2 for the modes and nanostructures discussed in chapters 6 and 7. However, theFEM models are very rough approximations and assume that the atomic-scale strain is
the same as the 10 nm scale strain. Such an estimate is probably accurate to within anorder of magnitude, but making a more precise estimate based on mechanical models isbeyond the scope of this thesis.
I also independently estimate that χ ≈ 1 by comparing the expected sensitivityof the APC detector to the measured sensitivity. At large resistances RAP C > 500 kΩ(chapter 7) the measured sensitivity, which is parameterized by a length scale λ, isconsistent with the theoretical predictions to within a factor of two. This data implies
that the root-mean-squared motion of the mode is equal to the change in the size of the
APC gap and χ ≈ 1 to within a factor of two.
These two estimates are consistent and in chapters 6 and 7 I assume that χ = 1
unless noted otherwise. Specifically, this assumption effects my estimates of the momentum impulse delivered by each tunneling electron. I generally estimate each electron’s
momentum kick by assuming that the entire backaction force is due to momentum kicks
from tunneling electrons and is described by the phenomenological model discussed
above. If χ is actually > 1, then the motion at the APC is smaller than the rootmean-squared motion of the structure and I will have underestimated the magnitude ofthe electron momentum needed to explain the observed backaction force. On the other
hand, if χ is actually < 1 then the APC is closer to an anti-node of the mode and the90estimated electron momentum in chapters 6 and 7 is too large. The uncertainty in χ isa major source of systematic error in the estimates of the momentum kick delivered byeach tunneling electron.

Chapter 4

Experimental realization of an APC embedded in a nanomechanicaloscillator

In this chapter I describe the two different methods that I use to create an atomic
point contact (APC) which is coupled to a nanomechanical oscillator. Both of thesetechniques are commonly used to create APCs (as described in chapter 2), but theprevious work concentrated on the static properties of the APC and did not measure
the dynamics of nanomechanical motion.
The first method of APC creation is called electromigration and involves passing
a large current through a constriction. Atoms in the high current-density region causedby the constriction migrate into the low current-density region until finally the flow ofcurrent is due to the connection between only two atoms: an atomic point contact.
The main advantage of this technique is the simplicity. The fabrication procedure
involves relatively few steps (compared to many nanomechanical devices) and minimal
infrastructure is required to pass a current through a constriction. However, it is very
difficult to control the resistance of the APC while electromigrating which leads toa number of disadvantages: the procedure is unreliable with more than 50% of theelectromigrated devices have an unmeasurably high resistance, the resistance of the
APC after electromigration is largely random, and it is usually not possible to reduce
the resistance of the APC using electromigration.
The second method of APC creation is to physically stretch a constriction until it92
flows and fractures to form an APC. This type of device is commonly called a mechanically controllable break junction (MCBJ), and the resistance of a MCBJ can be tuned
from less than 100 Ω to more than 10 GΩ by changing the strain (that is, the amountthat the constriction is stretched). In this method, the constriction is fabricated on a
flexible substrate, and the strain is applied by bending the substrate. The infrastructure required to controllably bend the substrate makes the second method both more
complicated and more controlled than electromigration.

4.1

APC Creation Using Electromigration

4.1.1

Introduction to electromigration

The interest in electromigration, the motion of atoms in a conductor due to theapplication of an electric field, has gone through three distinct phases. In the firstphase prior to the 1960s, most research in electromigration involved basic studies ofbulk materials, concentrating on mass transport in metals and the effect of charge
carriers on conductors [162]. By the late 1960s electromigration had become a problem
for the integrated circuit industry because the metal lines in integrated circuits were,by then, small enough that electromigration could create gaps in the lines over time.
In this second phase, the industrial problem led to a great deal of both experimental
and theoretical research with the goal of understanding electromigration in thin films
and preventing the failure of integrated circuits [162–164]. This type of applied researchcontinues today (for example, [165]), but a third phase began in the late 1990s as
electromigration became a tool for creating electrodes with a small (sub-5 nm) gap [41].
This tool is used in ongoing research into the electrical and mechanical characteristics
of atomic point contacts (for example, [166]) and single molecules (starting with [40]).
In the context of this thesis, the third phase of research led to the most convenient theoretical description of electromigration. The basic idea of an electromi-

93
gration force from earlier work [167] is combined with thermal effects [168, 169]. In
theoretical explanations of electromigration, the force on an impurity is divided into
two parts [162, 167, 168, 170]. The first part, called the “direct force” Fd , is due to the~ acting directly on an impurity atomelectric field E~
F~d = Zd eE(4.1)

where Zd is the effective electric charge of the atom. The second part, called the“electron wind force” Fw , is due to the net momentum transfered to the atom byscattered electrons~
F~w = Zw eE(4.2)

where Zw = K/ρ, ρ is the total resistivity, and K is a phenomenological constantrelated to the mobility of the impurity. Therefore the atom behaves as if it had a charge
Z ∗ = Zd + Zw though in general |Zd |  |Zw |, [168–170].
From reference [169], the flow of atoms in a constriction can be described using
a theory based on the basic description of the force on an atom Fd and an evaluationof the flow of electrons, atoms, and energy using the thermodynamics of irreversible
processes. This description assumes electromigration occurs at a constriction which canbe treated in the continuum limit. If the constriction is composed of many atoms, theconstriction has a resistance RAP C (t) = ρL/A(t) and electromigration does not change
the length L or electrical resistivity ρ but does have the effect of decreasing the averagecross-sectional area A(t) over time t. (I should also point out that I am going to use
the symbol RAP C to refer to the constriction resistance even when the constriction
resistance RAP C < 100 Ω and there is no APC, that is, no electron tunneling).
With these assumptions, the expected flux of atoms Jm due to the flow of a large
electrical current with density j through a thin, narrow gold wire that is heated by the94flow of current from a temperature T0 to a temperature T isαEa
Jm = (|j| − jmin ) exp −
T
kB T

(4.3)

In this model, the cross-sectional area A(t) of the constriction, and thus RAP C , changes
slowly or not at all (dRAP C /dt < 100 µΩ/sec) unless the atom flux Jm is greaterc which is experimentally determined. In this equation, thethan some critical value Jm

activation energy of gold diffusion on a surface is Ea = 0.12 eV. The other constants, αand jmin , are experimentally determined though it is expected that α ∝ Z ∗ ρ and jmin ∝1/(Z ∗ ρ). If the inelastic scattering length is smaller than the size of the constriction,then it is possible to express the temperature of the constriction as a function of thecurrent density T = T0 + βj 2 . This assumption is valid for the relatively large voltage
biases (> 15 mV) and bath temperatures (T0 ≈ 4 K) used here. Then the equation for
the flow of atoms is


α
Ea
Jm =
(j − jmin ) exp −(T0 + βj 2 )kB (T0 + βj 2 )(4.4)

where β is also experimentally determined. In the experiment described by referencec /α ≈ 1.5 × 104 A/K cm2 ,[169], β ≈ 1.7 × 10−15 K cm4 /A2 , jmin ≈ 108 A/cm2 , and Jm

though these constants are dependent upon the device geometry and material.
Using equation 4.4, it is possible to gain a better understanding of how the resistance of the constriction RAP C changes during electromigration and understand why
controlling electromigration requires controlling the voltage across the constriction. Experimentally, the process of electromigration starts by slowly increasing the current
IAP C that flows through a constriction with a resistance RAP C < 100 Ω. This increasing current has the effect of increasing both the current density j and the temperature
T = T0 + βj 2 in the constriction. Eventually the combination of a rising temperatureand increasing force due to the increasing current density causes the flux of atoms Jm
c where the constriction cross-section begins to shrink.
to pass a critical value Jm

95
As the average cross sectional area A(t) of the constriction shrinks, the APC
resistance RAP C (t) = ρL/A(t)) also begins to change and controlling the rate of change
requires controlling the voltage across the constriction. The simplest way to control thevoltage is to voltage bias the constriction (i.e., impose a voltage) so that the voltage
VAP C across the constriction is independent of the resistance RAP C (t). In this case,the current density j = IAP C /A(t) = VAP C /ρL will be constant. The atom flux Jm
and the rate of gap formation (dRAP C /dt) will also be constant and under control. On
the other hand, if the constriction is current biased so that the current IAP C passing
through the constriction, instead of the voltage VAP C , is imposed then the current
density j(t) = IAP C /A(t) = IAP C RAP C (t)/ρL and the constriction temperature T
increases as the cross-sectional area A(t) of the constriction shrinks. This creates a
feedback loop which causes the rate of atom flux away from the constriction Jm andthus the rate at which the area A(t) of the constriction shrinks to increase almost
exponentially until the assumptions in the model are violated (usually either the material
melts or the way in which electrons pass through the constriction changes from thediffusive regime to the tunneling regime). Since it is very difficult to stop the exponential
decrease in area A(t) and increase in RAP C by effectively duplicate a voltage bias andquickly lowering the current bias IAP C , voltage biasing the constriction is an important
step towards controlling the process of electromigration.
To proceed with this description of how the cross-sectional area of the constriction shrinks during electromigration, the area of the constriction eventually reaches theatomic scale where the discrete nature of atoms becomes relevant [8, 169, 171, 172]. At
that point, the change in resistance RAP C is no longer continuous but proceeds in jumps
as the configuration of atoms changes. This rearrangement is uncontrollable. Once theconduction through the constriction is dominated by only a few atoms (that is, theresistance RAP C ≥ 1 kΩ) then the change in conductance is usually equal to the conductance quanta Gq = 2e2 /h = 1/(12.9 kΩ) (see chapter 3 and reference [8] for more

96
details on conductance quanta).
As these rearrangements continue under the influence of a large current density,
the conductance will eventually be dominated by electrons tunneling between two atoms
thus forming an atomic point contact (APC). The resistance RAP C ≥ 12.9 kΩ is anexponential function of the gap size (with a length scale of about 1 Å, see chapter3) therefore atomic rearrangements have a large effect on RAP C making it impossible
to stop electromigration at a predetermined value. It is possible to continue passing acurrent with a large density through the APC and hope that a favorable reconfiguration
occurs, however in the tunneling regime the reconfigurations almost always result in ahigher APC resistance.

4.1.2

Design of devices for electromigration

The desire to control the location and speed of electromigration places a number of
constraints on the design of the nanomechanical structure. First, and most importantly,
the current density at the desired location of the APC must be as large as possible incomparison to the current density everywhere else. There are two other similar ways ofstating this condition: in comparison with the rest of the device, there should be a thin,
narrow constriction at the desired location of the APC; and the resistance of the circuit
should be dominated by the resistance at the constriction. Otherwise, the location of
the APC is poorly defined and it is not possible to voltage bias the constriction.
Second, the constriction (that is, the desired location of the APC) should belocated near the point which is most likely to melt when current is flowing through thenanostructure. As described above, the process of electromigration is thermally assisted
and a large local temperature can result in gap formation even if there is a larger current
density at other, cooler, locations (see equation 4.4). In structures which are fabricatedon a substrate and thus cooled through the substrate, the large resistance of the desiredsite of the APC ensures that this constraint is naturally satisfied. However, with a97suspended nanomechanical structure it is possible to create hot spots because regions
close to the supports are cooler than those which are remote (in the middle of thesuspended structure), and the structure can either electromigrate or melt at these hot
spots instead of at the desired location.
Third and finally, there should not be a large stress on the suspended structure

(a)
1 μm
(b)
1 μm
(c) beam(d) beam
electrode

100 nmelectrode

APC
constriction

APC
constriction

20 nm
thick
120 nm thick 120 nm thickFigure 4.1: Scanning electron micrographs of devices prior to electromigration; the APC
will be formed in the thin, narrow constriction between the red nanomechanical beamand green electrode. (a) top down view of device. (b) isometric view of different device.
(c) close up view of constriction. (d) schematic representation of an ideal constriction.
A triple-angle evaporation is used to make the constriction thin in comparison to thebeam and the electrode.

98(particularly at the location of the APC). If the stress is too large, the mechanicalstructure will eventually not be able to withstand the stress as atoms are removed fromthe desired location of the APC and the constriction will fracture resulting in a large(on the atomic scale) gap. This is a particular problem when a suspended structure iscooled to cryogenic temperatures because the relative thermal contraction of different
materials can result in a large stress.
The nanomechanical structure that I use to satisfy these three constraints is shownin figure 4.1. The main structure is composed of a doubly-clamped beam (red, figure4.1) that is about 10 µm long by 250 nm wide by 120 nm thick and is contacted in
the middle by another electrode (green, figure 4.1). The beam and part of the otherelectrode are suspended above the gallium arsenide (GaAs) substrate (figure 4.1b) andthe suspended structure is made entirely out of gold. The APC will be formed using
electromigration at the constriction between the center electrode and the beam (labeled”APC constriction” in figure 4.1c, d).
The constriction in figure 4.1c is 20 nm thick by 75 nm wide and therefore satisfies the first constraint required for electromigration (described in detail above); theconstriction is thinner and narrower than the rest of the electrode and the beam. In
devices that have been successfully electromigrated, the resistance RAP C of the constriction is greater than the resistance R0 +[Rbeam /4] of the rest of the structure (figure 4.2).
Experimentally, for electromigration to succeed the constriction must have a resistance
RAP C ≥ 30 Ω while the resistance of the remaining structure is R0 + [Rbeam /4] ≈ 15 Ω.
The smallest achievable constriction size and therefore the largest achievable constriction resistance (RAP C ≤ 60 Ω) is limited by the resolution of the fabrication technique.
In order to satisfy the second constraint and avoid melting the nanomechanicalbeam instead of electromigrating the constriction, it is necessary to take care when
designing the beam’s dimensions. When power is dissipated in the beam in figure 4.1a,b
the beam is cooled through the three connections to the substrate: the two clamped

99

VB
R0
electrode

I0

APC
constriction

RAPC
beam

I0/2
I0/2
Rbeam/2
Figure 4.2: A simple electrical diagram of the DC circuit used during electromigration.
The resistance R0 represents the series resistance between the constriction RAP C andthe voltage bias Vb , including the cryostat wiring and on-chip electrode. The current issplit at the beam by grounding both sides of the beam; since the constriction touches
the middle of the beam, the resistance of and current through the beam are split into
two equal parts. Typical values are R0 ≈ 5 Ω, Rbeam ≈ 40 Ω, and 30 Ω < RAP C < 60 Ωends (in red) and the central electrode (in green). The beam will first melt at the
points that are furthest from these cold reservoirs, so in this geometry there are two
hot spots which are halfway between the central electrode and the ends of the beam.
Because these hot spots are cooled through the body of the beam, there is a compromise
between designing a more compliant beam, which is longer and narrower, and designinga beam which will not melt during electromigration. The beam design in figure 4.1 canhandle a current of about 5 µA; the resulting 10 µA of current through the constriction
is sufficient to electromigrate constrictions similar to that in figure 4.1c. The required
current is very sensitive to the size of the APC constriction.

100
The nanomechanical structure in figure 4.1 also satisfies the third and final condition that the strain at the constriction is small by limiting the distance between the
beam support and the central electrode support. Most of the strain in the system isdue to the different coefficients of thermal contraction of the metal nanostructure andthe semiconductor substrate. If a relaxed nanostructure (that is, not under strain) atroom temperature is cooled to 4 K, then the gold nanostructure will shrink by about∆l/l = 0.003 while the GaAs substrate will only shrink by about ∆l/l = 0.001 resulting
in a gold nanostructure with a stress consistent with a strain of ∆l/l = 0.002. In otherwords, at 4 K the relaxed size of the nanostructure is 0.2% smaller than the actual size.
The distance between the central electrode support and the beam is only 500 nm. The
maximum separation of the APC gap due to the strain is therefore only 1 nm which isthe same order of magnitude as the expected size of the APC gap. Some of the tension
is also released during electromigration since the large temperature at the constriction
effectively anneals the constriction. Experimentally, the total strain is small enough toallow successful electromigration.

4.1.3

Fabrication of devices for electromigration

The device in figure 4.1 is fabricated using the standard tools of nanolithography. A detailed description of the recipe is given in appendix A, however the basic
steps are described in chronological order in this subsection (and in figure 4.3). Thisfabrication process is simpler than most nanomechanical fabrication recipes because thenanomechanical structure is made purely out of metal instead of a semiconductor/metal
bilayer. Although I will describe the fabrication process used with a GaAs substrate,
later devices were made using a silicon substrate because silicon has a smaller peizoelectric response than GaAs. The main difference in fabrication is just the use of different
techniques to etch the GaAs and the silicon.
When using a GaAs substrate, the process starts with a GaAs wafer which

101
is cleaved into chips that are about 5 mm on a side. An organic polymer bilayer(PMGI/PMMA) is spun on the chip (figure 4.3a). This layer is exposed in the desired
two dimensional pattern using the electron beam from a scanning electron micoscope(SEM). The exposed organic polymer is developed and removed. There is now a three
dimensional pattern in the organic polymer determined by a combination of the pattern
written by the electron beam and the chemistry of the organic polymer (figure 4.3b).
Gold is evaporated onto the entire chip at three different angles in order to createa thick beam and thick electrodes as well as a thin region at the desired APC location(figure 4.3c). The organic polymer and all of the gold that is not touching the substrate
is removed using a chemical wash (figure 4.3d). Finally, the nanomechanical structure issuspended using a citric acid/hydrogen peroxide wet etch (figure 4.3e). An SEM pictureof the final device is shown in figure 4.3f. As described in the previous subsection (andfigure 4.1), the APC will be created at the constriction where the triangular central
electrode touches the doubly-clamped nanomechanical beam.
One major difference between the nanomechanical structure used in this experiment and most other nanomechanical devices is the use of an all-metal (gold) structure;
the semiconductor material is only used as a sacrificial layer (that is, the semiconductor
is removed by etching). Most nanomechanical devices are made of a single-crystal semiconductor material in order to maximize the nanomechanical quality factor. However,in this experiment the quality factor is limited by the presence of the APC. The useof an amorphous metal structure instead of a single-crystal is not expected to significantly reduce the quality factor and does make the fabrication simpler by avoiding theneed to pattern and etch the single-crystal structural layer. (Reference [14, 15] or any
textbook on nanomechanics will contain a more detailed description of the fabrication
steps involved in creating a nanomechanical structure out of single-crystal semiconductor material).
After the initial measurements of devices on top of GaAs substrates, I realized

(a)

102

(d)

(b)
(e)
(f)
(c)

1 μm

100 nmFigure 4.3: Fabrication steps on a GaAs substrate; for more detail, see A. (a) spin alayer of PMGI (light purple) and then PMMA (dark purple) resist on the GaAs surface(orange), shown in cross-section. (b) after exposing a two-dimensional pattern using
the electron beam of an SEM, the PMGI/PMMA bilayer is developed removing theexposed resist. The top view shows a cross-section at the APC constriction; the crosssection is taken at the black line in the top-down view. (c) Gold is evaporated at three
angles, shown by the green arrows. When the gap in the resist is narrow, then theangled evaporations do not deposit gold on the GaAs surface, but when the gap is large
then a thick layer of gold is deposited. (d) The resist is removed by rinsing in acetone,
leaving behind the gold which was deposited directly on the GaAs substrate. (e) The
GaAs is etched using an isotropic wet etch of citric acid/hydrogen peroxide, suspending
the naomechanical beam and undercutting the larger electrodes. (f) Scanning electron
micrographs of the finished structure; in the lower figure, the area where the gold
contacts the GaAs is shown in orange. The grey area of the gold is suspended.

103
that the peizoelectric property of GaAs was complicating my measurements of forces applied to the nanomechanical structure. I therefore switched to using silicon substrates.
An electrostatic force can be applied to the nanomechanical structure by voltage biasinga gate electrode that is capacitively coupled to the structure. However, the peizoelectric
nature of GaAs implies that this voltage also launches surface acoustic waves onto thechip. These surface acoustic waves shake the supports of the nanomechanical structures,
creating a force on the nanomechanical structure. The etched surface of the chip is acomplicated boundary condition and therefore the magnitude of the force on the structure has a complicated frequency dependence due to the constructive and destructive
interference of surface acoustic waves rebounding from the boundaries. When using
GaAs substrates, the magnitude of the surface acoustic waves and the frequency dependence interferes with measurements of the electrostatic force. Therefore later devices
were made on a silicon substrate where the peizoelectric response is small enough tobe ignored. This change required small modifications to the fabrication procedure (therecipe is in appendix B) including a different recipe to etch silicon.

4.1.4

Electromigration procedure

Once a device with a constriction attached to a nanomechanical beam has been
fabricated, the device is placed in the ultrahigh vacuum environment of a 4 K cryogenic
system and the constriction is electromigrated to form an atomic point contact (APC)by passing a current through the constriction. The magnitude of the current must be
carefully controlled to create an atomic point contact with a small gap instead of alarger gap with a vanishing probablity of electron tunneling. This control takes two
forms: the constriction should be voltage biased (as explained in section 4.1.1) and a
feedback loop is needed to slow the rate of resistance change to a more controllable paceby lowering the voltage. Even with feedback, the electromigration procedure becomes
uncontrollable at resistances greater than 1 kΩ where only a few atoms are involved and104
it is not possible to control the final resistance.
In order to simultaneously measure the resistance of the constriction RAP C andapply a stiff voltage bias, I use the circuit in figure 4.4. In this figure, the area in the

Rcr

Ra

Va

Rsh

Vcr

VB

Thèveninequivalent

RAPC
Rb

Vth

Rth

RAPC

Figure 4.4: DC electromigration circuit. The black portion of the circuit diagram is
at room temperature, and the resistors Ra and Rsh are usually chosen to be small andequal in comparison to the sum of the resistance in the cryostat, shown in blue. Rcr
is the resistance of the cryostat wiring and on-chip electrodes, RAP C is the resistanceof the constriction, and Rb is the resistance between the constriction and ground dueto the nanomechanical beam. The area in the green dashed-box is a one-port network
biasing the constriction RAP C and, according to Thévenin’s theorem, is equivalent to a
voltage source Vth in series with a resistor Rth .

105
blue-dashed box represents the resistance in the cryostat which can be divided into threeparts: the resistance of the wires in the cryostat and on-chip leads Rcr , the resistanceof the beam Rb (Rb = Rbeam /4, see figure 4.2), and the resistance of the constriction
RAP C . This circuit diagram also includes two resistors whose values I can change during
electromigration: a resistor Ra with a known resistance which is used to measure thecurrent flowing through the circuit and a shunt resistor Rsh .
Successful electromigration requires a clear hierarchy in the values of the resistors
in figure 4.4, Ra ≈ Rsh  (RAP C + Rcr + Rb ) and RAP C  Rcr + Rb . This relationshipwill be described in more detail below, but, as a brief outline, the measured resistanceof the APC is more precise if Ra ≈ Rsh , voltage biasing the restriction requires Rsh 
RAP C , and RAP C  Rcr + Rb in order to both voltage bias the constriction and satisfy
the conditions for electromigration described in section 4.1.2. In this experiment, Rb ≈10 Ω, Rcr ≈ 5 Ω, and the initial resistance of the constriction before electromigration is60 Ω ≥ RAP C ≥ 30 Ω, thus the third and last condition is satisfied.
I will describe the DC resistance measurement in more detail in chapter 5, however
the basic technique is to measure the voltage Va across the known resistor Ra and thevoltage drop across the cryostat Vcr (see figure 4.4). These quantities are a direct
measurement of the resistance Rtot , the parallel combination of the resistance in thecryostat RAP C + Rcr + Rb and the shunt resistor Rsh ,VcrRa
Rsh (RAP C + Rcr + Rb )= Rtot =Va
Rsh + RAP C + Rcr + Rb

(4.5)

and since the resistances Rsh and Ra are known it is easy to calculate the resistance inthe cryostat RAP C + Rcr + Rb
RAP C + Rcr + Rb =Rtot Rsh
Vcr Ra Rsh=
Rsh − Rtot
Va Rsh − Vcr Ra(4.6)

In order to most accurately measure Rtot the two voltages Vcr and Va should beapproximately equal, and since (Rcr + Rb + RAP C )  Rsh the resistance Rtot ≈ Rsh .

106
Therefore, the resistor Ra should be chosen so that Ra ≈ Rsh as suggested in theprevious paragraph.
In section 4.1.1, I described how a voltage bias limits the current density inthe constriction. The same behavior that limits the current density also limits thepower dissipated in the constriction. Because the electromigration is thermally assisted,
limiting the dissipated power helps slow the rate of electromigration and allows the final
resistance to be somewhat controlled.
The simplest way to properly choose the value of Ra and Rsh so that the powerat the constriction PAP C decreases during electromigration is to employ a Thévenin
equivalent circuit to describe the circuit which biases the constriction RAP C (figure4.4). As discussed earlier, I am going to set Rsh = Ra . Thévenin’s theorem states thata one-port network of voltage sources, current sources, and resistors can be describedusing a single voltage source Vth in series with a single resistor Rth [173]. For the circuitin figure 4.4, the Thévenin equivalent is
Rsh
Ra + R + sh
Rsh (Rb + Rcr ) + Ra (Rb + Rcr + Rsh )Rth =Ra + Rsh
Rsh
Rth Ra =R =+ Rb + Rcrsh2
Vth = Vb

(4.7)
(4.8)
(4.9)

and the power dissipated in the constriction PAP C is
PAP C = VthRAP C
RAP C + Rth

2

1
RAP C

(4.10)The power PAP C is plotted in figure 4.5a versus the constriction resistance RAP C . Electromigration will be controllable when the power dissipated in the constriction PAP C
decreases as the constriction resistance RAP C increases, that is, when RAP C > Rth given
a constant Vth and Rth .
Achieving the goal of RAP C > Rth at all constriction resistances requires carefulfabrication of the initial constriction so that the constriction resistance RAP C prior

107

!a"

2 !1
PAPC #!Rth #Vth
"

0.200
0.150
0.100
0.070
0.050

0.030
0.020
0.015
0.010
0.01

0.1

1

10

100

RAPC #RthRth HWL
5000

HbL
1000
500

100
50

10
1

10

100

1000

104

Rsh =Ra HWL

Figure 4.5: Analysis of Thévenin equivalent circuit. (a) power dissipated in the constriction PAP C versus the constriction resistance RAP C , where both values are normalizedbased on the Thévenin equivalent circuit. For constant bias conditions, the power at
the constriction will decrease as the constriction resistance increases if RAP C > Rth . (b)The Thévenin resistance Rth as a function of the shunt resistor Rsh , where Ra = Rsh
and Rb + Rcr = 15 Ω. The device fabrication should ensure that before electromigration
Rb + Rcr < RAP C and then the shunt resistor Rsh can be chosen so that Rth < RAP C .

108
to electromigration is larger than the typical impedance in the cryostat Rb + Rcr =15 Ω. The Thévenin resistance Rth is plotted in figure 4.5b as a function of Rsh , where
Ra = Rsh and Rb + Rcr = 15 Ω. At large constriction resistances RAP C > 100 Ω, it
is possible to choose a shunt resistor Rsh > 2RAP C so that RAP C > Rth as desired.
At small constriction resistances Rth is dominated by Rcr + Rb . To ensure that thepower at the constriction decrease with increasing RAP C it is necessary that the initialfabrication of the constriction results in a constriction resistance RAP C > Rcr + Rb .
While having Rth > RAP C is sufficient to ensure that the power at the APC does notincrease during electromigration (figure 4.5a), electromigration is more often successfulwhen Rth > 2RAP C and the power at the APC decreases significantly with increasing
RAP C .
Although in some nanoscale structures an APC (with RAP C < 1 MΩ) can becreated just by ramping a stiff voltage bias [41, 169, 174], I have found that it is also
necessary to implement a feedback loop (as in references [171,172,175–177]) that lowers
the applied voltage bias in order to slow the rate of resistance change. Because there
are multiple detailed descriptions of this type of feedback control in the literature, I am
only going to describe the basic procedure.
Electromigration is initiated by slowly increasing the voltage bias Vb while continuously monitoring the resistance of the APC. Increasing the voltage bias heats thesystem which initially causes the measured resistance to change in a reversible way because the resistivity of the constriction material is temperature dependent. Eventually
the measured resistance will begin to change irreversibly while the voltage bias is held
constant; this usually occurs when a few microamps of current flow through the APC. At
this point, a feedback loop modifies the voltage bias to keep the rate of fractional resistance change dRAP C /RAP C dt constant. This procedure works well up to a constriction
resistance RAP C ≈ 1 kΩ.
When RAP C ≈ 1 kΩ, the constriction resistance begins to change in discrete,109
uncontrolled steps (see section 4.1.1, references [8, 169, 171, 172]). This is not a surprise;the constriction has gone from the continuum limit where the resistance is due to many
atoms to the atomic scale where there are only a couple of atoms contributing to theresistance. The rearrangment of these atoms then leads to discrete changes in theresistance. In a successfully electromigrated junction, the resistance jumps from thekΩ regime to a resistance less than 1 MΩ, but in an unsuccessful electromigration theresistance jumps to completely open (> 10 GΩ). An example of the resistance evolution
during a successful electromigration procedure is given in figure 4.6.

Figure 4.6: Resistance as a function of time during a successful electromigration. Thechange in resistance at about 5 min is successfully controlled using a feedback loop, after
which the contact is composed of less than 100 atoms. After this point, the resistancechanges in steps due to atomic rearrangement until an uncontrolled adjustment results
in an atomic point contact with 23 kΩ < RAP C < 100 kΩ.

4.1.5

Electromigration failures and frustrations

Electromigration has multiple failure modes, the most spectacular of which iswhen the nanomechanical beam melts (figure 4.7). As mentioned earlier, this usually110
occurs when the resistance of the APC is not large enough compared to the resistanceof the beam. The points of the nanomechanical beam farthest from the cold reservoir
provided by the supports (labeled A and B in figure 4.7) will melt if a large enoughcurrent is passed through the beam. This occurs when the constriction at the desired
APC location (labeled C in figure 4.7) is not weak enough and the beam melts (or gets
hot enough to electromigrate) before the APC location begins to undergo electromigration. This problem has a simple solution: better fabrication resulting in a smaller
constriction at the desired APC location.

1 μm
A

C

B

Figure 4.7: Scanning electron micrograph of device after failed electromigration. In thiscase, the current through the beam was enough to melt the beam at points A and B
before the current was large enough to electromigrate the constriction at point C.

Another failure mode is when electromigration begins at the appropriate location,
but as the electromigration progresses the resistance jumps from a resistance less than
1 kΩ to completely open (> 10 GΩ). In some of these cases there is a gap at the desired
location of the APC which is noticeable in the SEM (figure 4.8), but in general it isdifficult to determine the reason for electromigration failure. One possible explanation

111
is that the third constraint in section 4.1.2 is not satisfied. The nanomechanical beam
could separate from the APC electrode in order to relieve stress which could result in alarge gap. Another possible explanation is that the temperature near the constriction
becomes larger than the melting point of gold, resulting in a large scale motion of atoms
and a large gap. This would usually happen either because the APC location is not
sufficiently in the limit of a voltage bias resulting in increasing power dissipation as the
APC resistance increases or because the power dissipated at the constriction melted thethin, suspended gold at a lower current density than would have been needed to startelectromigration.

100 nmpre-electromigration

100 nmpost-electromigration

Figure 4.8: Unsuccessful electromigration. (a) Scanning electron micrograph of the constriction prior to electromigration. (b) Scanning electron micrograph of the constriction
after electromigration. While the initial device appears promising, the gap created byelectromigration is greater than 20 nm and therefore the device is useless.

At the end of the previous subsection I mentioned the final way in which electromigration fails: electromigration proceeds in a promising fashion up to about 1 kΩ,
but then jumps to completely open. One possible explanations for this failure is thatthe final adjustment of the position of the final couple of atoms could randomly results
in a very large gap. A second explanation is that, as described in the previous para-

112
graph and section 4.1.2, there could still be a small residual stress at the APC when theconstriction cools down (due to the lower power dissipated at higher resistances) which
results in a large final gap. In general, I do not have a method for imaging the final
atom configuration and therefore it is not possible to confirm any of these hypothesis.
Unfortunately this failure of electromigration is irreversible; once there is a large gap
resulting in a > 10 GΩ APC resistance, the device is useless.
A final frustration with electromigration is the possibility of contaminants becoming integrated into the atomic point contact. This contamination will change thebarrier through which electrons tunnel when crossing the APC and can also effect the
APC’s mechanical properties [8, 178–181]. While the contaminants can have a positiveeffect on the mechanical stability of the APC, the changes in the barrier often result in adecrease in the barrier height [178,179,182] and thus decrease the sensitivity of the APC
to nanomechanical motion (see chapter 3). Since the actual electromigration occurs in
the ultrahigh vacuum provided by a vacuum space surrounded by 4 K liquid helium,
there are two main sources of contaminants: residual contamination from fabricationand contamination by the helium exchange gas (used to cool the cryogenic system to4 K) which has condensed on the entire surface of the chip. While it is hoped that thetemperature during electromigration will cause contaminates to evaporate, a better wayof dealing with possible contaminates would be to use the STM tip-cleaning procedureof pressing together the two sides of the APC and then retracting multiple times toremove contaminants and form a clean APC.
This list of frustrations and failures leads to the main problem with electromigration: it is not possible to reliably control the size of the gap. The initial atomic
point contact creation requires careful device design and fabrication; even then it isunreliable. After the initial creation it is not possible to control the APC resistance. At
best, an uncontrollable change to an almost always larger resistance can be achieved byapplying a large voltage across the atomic point contact. Unfortunately, this procedure

113
often results in a completely open APC. It is also impossible to recreate the APC. In
scanning tunneling microscopes a common method of cleaning the tip is to press the tip
into the substrate, creating a low resistance contact, and then pulling away from thesurface resulting in a tip composed of a new set of atoms. This control of the atomic
separation is not possible when the APC is created using electromigration. Therefore
the technique of electromigration allows the study of a single atomic point contact witha unique resistance and configuration coupled to a unique nanomechanical system (assuming that electromigration is successful). Being able to change the resistance andconfiguration of the APC, as well as to measure the motion a single nanomechanicalsystem with multiple different APCs, would facilitate the study of the effect of the APC
on the nanomechanical system and make it possible to optimize the APC displacement
measurement.

4.2

APC Creation Using a Mechanically Controllable Break Junction

4.2.1

Mechanically controllable break junctions

In fact, there is a widely used technique for creating an atomic point contact
with an adjustable gap size: the mechanically controllable break junction (MCBJ) [8,44, 47, 48, 183, 184]. MCBJs have been used to study both how a metal constrictionstransitions from containing many atoms, where the discreteness of atoms can be ignored,
to containing only a small number of atoms and the electrical and mechanical propertiesof these junctions (reference [8] summarizes these studies). MCBJs composed of two
atoms have also been used to electrically contact single molecules (for example, [51, 53,
57]).
A constriction in a section of material goes through three distinct phases understrain. First, the constriction stretches until it fractures creating a MCBJ; usually a

114
5% strain is sufficient to fracture a metal constriction. Second, this MCBJ stretches
until eventually the narrowest part of the MCBJ is composed of only a small number ofatoms. Finally, under additional strain the atoms in the MCBJ will rearrange creating
an atomic point contact (APC) where the electrical contact through the constriction isdue to tunneling between two atoms. In this way, a MCBJ can be used to create an
APC and the width of the tunneling gap can be controlled by changing the strain on
the constriction.
Using a two-stage design, it is possible to both apply a large enough strain to
fracture a metal constriction and minimize the effect of mechanical vibrations or noiseon the MCBJ. In the first stage, a strain is applied to the top surface of a chip by pushingon the back surface of the chip with a plunger (figure 4.9a). Mechanical noise couples tothe MCBJ by vibrating the plunger, but the displacement at the MCBJ is less than 10−3
times the motion of the plunger. Thus the MCBJ is partially protected from mechanicalnoise. In the second stage, the strain from the first stage is concentrated at a constrictionsuspended above the top surface of the chip (figure 4.9b). This concentration has two
effects: it determines the location of the MCBJ (if the strain was constant, then themetal would fracture at an unknown location) and more importantly makes it possible
to fracture metal using a chip that is compatible with nano-lithography.
Given this basic description of a two-stage MCBJ apparatus, it is possible to
make a more quantitative description of the strain applied by the first stage based on
the typical bending structure in figure 4.9a. A chip with some thickness t is supportedon two sides (brown supports in figure 4.9a) and then the chip is bent using a moveable
plunger (in green in figure 4.9). The movement of the plunger displaces the middle ofthe chip by an amount u. If the two supports are separated by a distance L, then thestrain δD/D (that is, a length D is stretched to a length D + δD) on the top surface of

115
the chip (the surface not touched by the plunger) is
δD
3ut
= 2
D
L

(4.11)

assuming the ideal case of homogenous strain [8, 46, 183].

(a)

fixed supports

t

u

L

moveable plunger

(b)

D
area

A

d

area

a

Figure 4.9: Basic setup used to create a mechanically controllable break junction. (a)The chip, in blue with a thickness t, rests on two fixed supports (brown) which areseparated by L. A moveable plunger displaces the center of the chip by an amount u,
straining the top surface of the chip. (b) A device is fabricated on the top surface of thechip which concentrates the strain at a constriction. The constriction has a length d andcross-sectional area a. It is suspended above the top surface of the chip by a structurewith cross-sectional area A; the suspended structure has a total length D along thestrained direction.

The concentration of strain due to the second stage of a MCBJ apparatus andthus the total strain at the constriction can be calculated from the typical structure in
figure 4.9b. A constriction of length d and cross-sectional area a is suspended above
the substrate by a larger structure of total length D and cross-sectional area A. When116
the substrate is strained, then the supports will separate by an amount δl, as describedby equation 4.11, however the strain will not be constant throughout the suspended
structure. Instead, the mechanically weak constriction will experience a strain that islarger by a factor  [44].
The simplest way to calculate  is to treat the system as two springs in series,
where the constriction is the softer spring with relaxed length d and spring constant k =Ea/d (E is the elastic constant of the material) and the constriction’s support structureis the stiffer spring with a relaxed length D − d and spring constant K = EA/(D − d).
Stretching the two springs by a total amount δD will stretch the constriction spring by
an amount δdδd = δDK
k+K(4.12)and therefore the strain in the constriction is
δd
δD
3ut
D
=
= 2
d
d + (D − d)(a/A)
L d + (D − d)(a/A)(4.13)and the strain concentration  = (δd/d)/(δD/D) is therefore=D
d + (D − d)(a/A)(4.14)The design of the MCBJ bending structure and on-chip device is determined byequation 4.13 and the main requirement of applying a strain δd/d > 0.05 plus thesecondary desire to isolate the MCBJ from mechanical noise and vibrations. In general,the maximum amount of deflection u = umax that can be applied is limited either
for semiconductor substrates by the substrate shattering or for metallic substrates bybending the substrate past the point where the strain is homogenous. While this limit
on metallic substrates is actually a limit on the applicability of equation 4.11 and 4.13,
it is preferable to stay close to this limit because otherwise the strain becomes very
dependent upon the way in which the chip is supported and the location of the device
on the chip. This dependence makes δd/d both difficult to calculate and a nonlinear

117
function of u. Mechanical noise will couple to the MCBJ by vibrating the plunger andmaking u a noisy variable. In order to minimize the effect of noisy plunger vibrations
on the MCBJ strain, the dependence of δd/d on u should be minimized by making
α = (3t/L2 )D/(d + (D − D)(a/A)) as small as possible. This is a secondary goal, but it
implies that α should be designed so that the constriction will fracture at the maximum
plunger displacement umax α ≈ 0.05 which minimizes the effect of noise on the MCBJ.
In most MCBJ setups α < 10−3 [8, 44].
As the constriction is transformed into an APC using strain, equation 4.13 becomes irrelevant (and inaccurate) and instead the change in the size of the APC gap
due to strain becomes important. Equation 4.13 relies on the assumption that the constriction is a continuous cylinder of material; however, this model becomes incorrect as
the constriction stretches and fractures under strain. In fact, once the MCBJ is composed of only a small number of atoms then calculating the strain and the effect on theatomic configuration becomes a very complicated problem [8, 185–187]. However, once
the junction in the MCBJ is an APC composed of only two atoms with a gap w then the
problem becomes much simpler. The two atoms are in a potential well with an effective
spring constant that is much weaker than all of the other effective spring constants inthe problem, so any additional strain due to the motion of the plunger δu will result ina change in the size of the gap δw with
δw = δD =

3tD
δu
L2

(4.15)

This equation for δw can be used in two ways: first to calculate the effect of
mechanical noise, and second for calibration purposes. First, as described earlier, mostmechanical vibrations effectively couple to the MCBJ through the plunger u. Therefore,
when the APC has formed, the mechanical noise will be attenuated by a factor α0 =3tD/L2 . Second, equation 4.15 can be used to compare the expected change in resistancedue to the change in the tunneling gap RAP C (δw) = R0 exp [2δw/λ] (see chapter 3) with118
the experimentally observed resistance change, either to calibrate α if λ (which is related
to the height of the barrier, see chapter 3) is known or to calibrate λ if α is known. WhileI can measure the dimensions of the bending apparatus, chip, and on-chip structure inorder to calculate α and I am interested in measuring λ, the chip is often bent past the
point of where the substrate is homogeneously strained and therefore I have not beenable to use equation 4.15 to accurately measure λ.

4.2.2

MCBJ bending apparatus in a 3 He cryostat

Having given a general description of a MCBJ, I now describe the MCBJ bending
apparatus I use to implement the conceptual diagram in figure 4.9a and strain the topof the chip. The main constraint on my experimental realization of figure 4.9a is mydesire to create an APC using a MCBJ in the ultralow pressure cryogenic environment
provided by a 3 He cryostat. The APC’s stability and purity both benefit from this low
pressure environment. Low temperatures are also needed to detect small forces on thenanomechanical system. The bending apparatus must therefore both be compatible withthe physical structure of the cryostat and also must not disrupt the carefully designed
thermal isolation in the cryostat.
The need for the bending apparatus to be physically compatible with the cryostat creates three main design constraints. First, and simplest, the chip must fit in
the cryostat and therefore the maximum separation L of the supports (figure 4.9a) isdetermined by the size of the cryostat. I successfully strain a constriction using chips
with supports that are about L = 5 mm apart and a cryostat with a diameter greaterthan 5 cm, so the chip fits easily into the cryostat. Second, there should not be any
risk of bending or twisting the cryostat while bending the chip. This constraint will be
discussed in more detail below, but basically implies that the bending structure should
be largely self-contained and not exert a force on any of the physically delicate partsof the cryostat. Finally, the entire bending apparatus must be carefully designed to fit

119
into the cryostat. Tracy Keep from the JILA shop designed the the MCBJ bending
apparatus and machined the parts. He also created the nice 3D CAD drawings used inthis thesis.
In order to understand the thermal constraints on the MCBJ bending apparatus
it is first necessary to understand the basic operation of the 3 He cryostat, shown in figure
4.10. Starting with the coldest part of the cryostat, the base temperature stage (bluestage in figure 4.10) is cooled to about 300 mK by pumping on liquid 3 He. However
the cooling power of this process is not very large, so the base stage must be thermallyisolated from the other temperatures in the system.
The initial step in this thermal isolation is placing the base stage in a vacuum
space surrounded by liquid helium (see figure 4.10) which helps thermally isolate thebase stage in three different ways. First, it cools the walls of the vacuum can so thatany remaining gas in the vacuum space freezes on the 4 K walls creating an ultra-low
pressure environment and removing any thermal connection through gas. Second, thecold walls and the use of baffles in the line-of-sight port minimizes the amount of thermalradiation. Finally, the liquid helium bath provides a 4 K heat sink (red stage in figure4.10) which can be used to cool everything in the vacuum space to 4 K.
The final step is therefore to isolate the base stage from the 4 K liquid helium bath.
This is accomplished by attaching the base stage to 4 K using thin-walled stainless steel(black connections in figure 4.10) which has a low thermal conductivity. The total heatload on the base stage is also decreased by interrupting the heat load at an intermediatestage (green stage in figure 4.10) which is cooled to 1 K by pumping on liquid helium.
In order to avoid creating a large heat load on the base stage, the thermal designof the MCBJ bending apparatus should be similar to the thermal design of the entire
3 He cryostat.

Any radiation from 300 K room temperature should be blocked, andthe thermal load from room temperature should be small and should be heat sunk to
the 4 K liquid helium bath. As in the basic 3 He cryostat, low thermal conductivity

120line-ofsight port

300 K

bafflesLiquidHelium
4K
1K
300 mK
vacuum canvacuum space

Figure 4.10: 3 He cryostat; the actual cryostat is on the left and a diagram is on theright. When the cryostat is in use, the brown vacuum can is surrounded by liquid helium(pink in the diagram) and the inside of the vacuum can is under vacuum (the whitearea in the diagram). The top of the cryostat (orange) is at room temperature, and the
4 K stage (red) is directly cooled by liquid helium and protected from 300 K radiation
through the line of sight port by a set of baffles. Thin-walled stainless steel (black) isused to connect the 4 K stage to the 1 K stage (green) and then the 1 K stage to the
300 mK stage (blue).

121
structures should then be used to connect the 4 K part of the MCBJ bending apparatus
first to a heat sink at 1 K and then from 1 K stage to the base temperature stage thus
minimizing the heat load on the base temperature stage.
The MCBJ bending apparatus used in this experiment, which satisfies both thethermal and mechanical constraints described above, is shown in figure 4.11. The basic
idea of the bending apparatus is to use a straight rod from the top of the cryostat
to turn a screw. The rotation of the screw then coarsely determines the position of aplunger which is used to bend a chip. Fine positioning of the plunger is accomplishedusing a piezo crystal (see figure 4.12b,c for details).
In order to reduce the thermal conductivity of the straight rod, it is split into
three different pieces. The piece between 300 K and 4 K is made out of a low thermal
conductivity ceramic rod and is passed through baffles which reduce thermal radiation(figure 4.11). At 4 K, the ceramic rod turns a screw with 100 threads/inch which is
attached to the second piece of the straight rod, a thin-wall stainless steel tube (figure
4.12a). The screw passes through a brass piece (figure 4.11 and 4.12a, shown in red)which is anchored to 4 K, thus blocking any remaining thermal radiation and heatsinking the top of the stainless steel tube to 4 K. The stainless steel rod turns the
top half of a fork-coupling and the bottom half of the fork coupling is attached to ascrew (also 100 threads/inch) which moves the plunger (figure 4.12b, the screw is red).
Thermal isolation is provided both by the thin-wall stainless steel tube and by the fork
coupling. The fork coupling can be disengaged, thus breaking the thermal connection
to 4 K, when the position of the plunger does not need to be coarsely adjusted.
While this long, multi-stage rod does provide thermal isolation, it also resultsin two types of imperfections in the conversion of rotation at the top of the cryostat
into plunger motion at the bottom of the cryostat. The first type of imperfection isbacklash, that is, in order to reverse the direction of plunger motion it is first necessary
to rotate the knob at the top of the cryostat by about 110◦ in the reverse direction.

122

300 K

4K

baffles

1K

300 mK

Figure 4.11: 3 He cryostat with MCBJ bending apparatus. The diagram on the far
right shows the entire apparatus, starting with a vacuum feedthrough at 300 K whichconnects to a rod that passes through the baffles in the line-of-sight port. The middle
diagram and left picture show a close-up of the outside of the MCBJ bending apparatus.
Thin stainless steel tubing, in grey, is used to rigidly connect the 4 K brass piece, 1 K
copper piece (upper green piece), and the sample holder at 300 mK (blue). The final
connection between the stainless steel tubing and the sample holder is made using fourthin fiberglass tubes. The 4 K brass piece, 1 K copper piece, and 300 mK sample holder
are heat sunk to the appropriate 3 He cryostat stages using flexible oxygen-free copper
straps, though the 4 K brass piece is also rigidly connected to the 3 He cryostat.

(a)

123

(b) fork coupling
screw

4K

plunger
1K

(c)

piezo
plunger

300 mK

chip

Figure 4.12: Details of MCBJ bending apparatus. (a) cut-away diagram of apparatus
from 4 K to base temperature. The heat sunk pieces (4 K in red, 1 K in green, and
300 mK in blue) are connected by thin-wall stainless steel tubing (light grey), while athin stainless steel rod (dark grey) passes through the center of the support structure
and turns the fork coupling. (b) cutaway view of fork coupling (pink and blue) andsample holder. The fork coupling is used to turn a screw (red) and thus coarsely control
the plunger (green) position. (c) cut-away view of chip holder (orange) and plunger(green). The piezo crystal is used to finely control the plunger position.

124
Most of this hysteresis is due to the fork coupling, but some is also due to the torsional
spring constant of the other pieces of the multi-stage rod. Since this type of backlash isrepeatable, it can be accounted for during operation and does not affect the resolution ofplunger motion. However, the static friction in the screws and the torsional complianceof the long multi-stage rod causes the rotation to occur through a stick-slip process
which does limit the resolution of plunger motion. In this stick-slip process, rotation
at the top of the cryostat does not immediately cause the screw at the bottom of thecryostat to rotate, but instead causes force to build in the system (i.e, the system
sticks). When the force is large enough to overcome the static friction, then the screwat the bottom of the cryostat finally rotates (i.e, slips) and this rotation has a minimum
step-size. In this system, that minimum step size is about 0.5◦ or, in terms of plunger
motion, δu = 0.1 mm.
The MCBJ bending structure must be supported in space so that no torque istransfered by the rod to the fragile base stage of the cryostat. The sample is supported
by a structure with low thermal conductivity that is rigidly connected to the stiff 4 Kstage in the cryostat (red, figure 4.10). At the 4 K stage, a thin-wall stainless steel tube
is soldered to the brass piece discussed in the previous paragraph (figure 4.11 and 4.12a).
The other end of this stainless steel tube is soldered to a copper block (figure 4.12a,green) which is heat sunk to 1 K . Another thin-wall stainless steel tube is soldered tothe other side of the copper block. Finally, the sample holder and the threads for the
final screw that is used to adjust the plunger position are attached to this stainless steel
tube using two thin fiberglass tubes. The sample holder is then thermally connected tothe base stage (blue, figure 4.10) using a flexible, gold-plated, oxygen-free copper plate.

4.2.3

MCBJ device fabrication

I now describe the design and fabrication of the on-chip nanomechanical structure
I use to concentrate strain on a constriction (the experimental realization of figure 4.9b)

125thus creating a MCBJ coupled to nanomechanical motion. This design can be split into
two pieces. First, designing a nanostructure where the strain δd/d at a constriction(see equation 4.13) is enough to fracture the constriction δd/d > 0.05, creating a MCBJ
before the plunger displacement u reaches it’s maximum displacement umax . Second,
designing a nanostructure so that nanomechanical motion will create a large strain at
the MCBJ. Coupling nanomechanical motion and strain at the MCBJ is an important
step towards creating a sensitive detector of nanomechanical motion (see chapter 3).
Achieving the first goal of fracturing a constriction to create a MCBJ is dependent
upon both the chip material and the nanomechanical structure on the chip’s surface. I
follow the work of van Ruitenbeek, et. al. [44] and use a chip made out of phosphor
bronze with an insulating layer of polyimide spun on the top surface. More specifically,
my chips are 5 mm squares of 0.35 mm thick polished phosphor bronze shim stock.
Experimentally, these chips can be bent by an amount umax ≈ 0.2 mm before the strain
is no longer homogenous and plastic deformation begins. It is possible to apply a greater
strain u > umax without catastrophic consequences because the chip is made of metal;
however, as described earlier, the amount of strain varies greatly over the surface of thechip and is no longer described by equation 4.13.
Given the characteristics of the phosphor bronze chip from the previous paragraphand the MCBJ bending apparatus described in the previous subsection, the maximum
strain (δD/D)max that can be applied to the top surface of the chip is given by equation
4.13. The maximum strain occurs when the plunger has displaced the center of the chipby an amount u = umax = 0.2 mm, therefore


δD
D


=
max

3umax t3(0.2 mm)(0.35 mm)== 0.0084
L2(5 mm)2(4.16)and so the maximum strain (δD/D)max = 0.8% on the surface of the chip is less thenthe 5% strain usually needed to fracture metal.
To fracture a metal trace and create a MCBJ it is necessary to concentrate the126strain on the constriction using a suspended nanostructure as described in subsection
4.2.1 and in figure 4.9b. The actual nanomechanical structure that I use in this experiment is a doubly-clamped nanomechanical beam with a constriction in the middle ofthe beam (figure 4.13). The structure is made entirely out of gold, the beam is 1 µm
long by 100 nm thick by 150 nm wide, and the constriction located in the middle of thebeam is about 50 nm long by 20 nm thick by 100 nm wide. These dimensions are used
to calculate the factor  by which the strain is concentrated at the constriction using
equation 4.14
=

(a)

D
1 µm
=
= 5.7
d + (D − d)(a/A)
50 nm + (950 nm)(7.5)(4.17)

100 nmconstriction(b)
constriction
Figure 4.13: Scanning electron micrographs of nanomechanical structure with constriction prior to stretching. In general, the scanning electron beam caused electrical charge
to be trapped in the insulating polyimide and made it difficult to image devices. (a)top-down view, the gold which is supported by polyimide is colored orange and thegold structure in white is suspended. The contrast difference between the constriction
and the supporting structure is due to the difference in the thickness of the gold. (b)isometric view of the same device, where it is possible to more directly observe the thinconstriction.

127
The constriction in this nanostructure should fracture when the chip is bent by anamount close to, or a little more than, umax .
While the dimensions of the beam and constriction are constrained by the needto fracture the constriction, the location of the constriction can be chosen to maximize
the strain created at the MCBJ by nanomechanical motion. I choose to concentrate on
the fundamental, lowest frequency, mode of the nanomechanical system which is usuallymore sensitive to forces than the higher frequency modes. In a simple doubly-clamped
beam without a constriction, the strain caused by motion of the fundamental mode islargest at the midpoint of the beam. While the presence of a MCBJ changes the modeshape (see the finite element analysis in chapter 3) and complicates the analysis of thestrain, placing the constriction in the middle of the beam is still a good choice.
The fabrication recipe used to create this nanomechanical structure is very similar
to the fabrication recipe used to create the nanomechanical structure on GaAs for electromigration, described in subsection 4.1.3 and figure 4.3. The differences in the recipe
are due to the chip material; instead of a semiconductor, the MCBJ nanostructure isfabricated on top of an insulating sacrificial layer which is spun on top of the polishedphosphor bronze substrate. This sacrificial layer is composed of a 15 µm thick layer of
PI-2611 polymide from HD MicroSystems. The polyimide also creates a smooth top surface for nano-fabrication and electrically isolates the nanostructure from the phosphor
bronze which is electrically grounded to the cryostat. After the metal nanostructure has
been created on the surface of the polyimide using the same basic process as for GaAs
electromigration devices, the nanostructure is suspended by etching the polyimide using
a simple SF6 reactive ion etch (for a detailed recipe, see appendix C).
Using a polyimide sacrificial layer does have one negative effect on the fabrication procedure; imaging the devices in the scanning electron microscope (SEM) after
processing is difficult and can destroy the device. When imaging the structure on topof the insulating polyimide with a SEM, some of the imaging electrons do not flow to128ground through the insulating polyimide but instead collect on the nanostructure andin the polyimide. The build-up of charge results in poor image quality and can pull
the suspended conducting nanostructure to the surface of the chip, breaking the nanostructure at the weak constriction. This problem is reduced by connecting the metalnanostructure to ground during imaging.
In comparison to the work of van Ruitenbeek et. al. [44], I make fast measurements of the MCBJ resistance and therefore need to minimize the capacitance between
the nanomechanical device and ground (chapter 5 explains why it is important to minimize the capacitance in parallel with the APC). This total capacitance is decreased
by minimizing the capacitance between the grounded phosphor bond substrate and thebond pads which are used to electrically contact the MCBJ. The Leiden group of vanRuitenbeek uses a 3 µm thick layer of polyimide to insulate the bondpads and nanomechanical structure from the phosphor bronze substrate. They also electrically connect
to the device by using tweezers to press a wire into a macroscopic pellet of indium thatwas placed on a large, 1 mm by 1 mm bond pad [44]. The combination of large bondpads and a thin insulating layer results in a large stray capacitance Cs = 3 pF, which
limits the resistance measurement bandwidth (see chapter 5).
In contrast, I use a 15 µm thick layer of polyimide and wirebond to smaller 100 µm
by 100 µm bondpads, resulting in a stray capacitance Cs = 10 fF which is negligible
compared to other sources of stray capacitance. Pressed indium bonds were used by
van Ruitenbeek instead of wirebonds because he found the wirebonding process to bevery delicate and unreliable [44]. I also found this to be true, however wirebonding was
possible by carefully adjusting the wirebonder settings and, more importantly, by usinga very thick titanium/gold layer for the bondpads. It is still only possible to make one or
two wirebonding attempts before the bondpads separated from the polyimide surface.

129
4.2.4

Creating an APC using a MCBJ

Using the nanomechanical structure in figure 4.13 and the MCBJ bending apparatus in figure 4.12, I fracture a metal constriction and create a MCBJ. After applying
strain and fracturing the metal constriction, it is possible to remove the strain, forcing the two sides of the fracture together and healing the fracture. An example of thehysteretic loop formed by this breaking and then healing cycle is shown in figure 4.14,
where the plunger position is manipulated using the piezo (figure 4.12) in the ultra-highvacuum provided by the 4 K cryogenic environment.

Figure 4.14: MCBJ plunger resistance versus plunger position, where the plunger’s
position is finely controlled using the piezo crystal. The rearrangement of atoms aroundthe MCBJ causes a hysteretic loop as the strain is applied (red) and then removed(blue).

Looking in more detail at the hysterisis loop, the constriction is initially composedof many atoms and then stretches under strain to a single atom contact, that is, an
APC. The constriction stretches under applied strain, the number of atoms at thefracture point decreases, and the resistance increases in a roughly continuous manner.
Eventually the discreteness of atoms becomes important and larger jumps are observed
as the rearrangement of single atoms causes large fractional changes in the resistance.
As this rearrangement continues due to increasing strain, gold atoms have a tendency

130
to form a chain composed of single atom links [8,151,186,187], which causes the plateau
at about 13 kΩ in figure 4.14. The large jump from 13 kΩ then occurs when the chain
breaks under increasing strain. At this large resistance (R  13 kΩ) the MCBJ is an
APC, since electrons flow through the MCBJ by tunneling through the vacuum gap
between two atoms. While it is possible to apply additional strain thus increasing the
APC gap and increasing the resistance of the APC, in figure 4.14 I begin to remove
the strain after breaking the chain of gold atoms. Since the atoms that had been partof the chain have been reincorporated into the sides of the MCBJ [151], the resistancedecreases in a hysteric manner as strain is removed until the fracture in the constriction
has effectively been healed and the resistance is approximately equal to the original
constriction resistance. In general, it is possible to sweep out a hysteretic loop from< 100 Ω to greater than 10 GΩ and back to < 100 Ω. This type of hysteretic loop will
be discussed in more detail in chapter 7.

Chapter 5

Electrical measurement of APC ResistanceIn this chapter I describe the electrical measurement of the atomic point contact(APC) resistance. The main goal of this measurement is to enable the detection of
nanomechanical motion with an imprecision that is limited by the fundamental sourceof noise in the measurement, the shot noise of tunneling electrons (see chapter 3). Thisis important, because it is a prerequisite for making a quantum limited measurementof position. I accomplish this goal by using the same microwave matching techniquethat was used to create the radio-frequency single electron transistor (RF SET, [128])
to make a fast, large bandwidth, measurement of small changes in the APC resistance.
I start by explaining, in general terms, how the precise measurement of a small
resistance change is limited by amplifier noise and shot noise. By comparing the effectof the amplifier noise and shot noise, I calculate the voltage VAP C across the APC thatis required for the shot noise contribution to the measurement noise to be greater thanor equal to the contribution from the amplifier noise. Because the APC is only stablemax ≈ 50 mV, realistic measurement circuits are, at best, only shot noise
at VAP C < VAP
Climited over a finite bandwidth determined by the frequency dependent noise added by
the measurement circuit.
The noise and bandwidth of the resistance measurement can be used to calculate
the expected imprecision in the measurement of nanomechanical motion. I consider
three similar types of resistance measurements. First, I can make a baseband mea-

132
surement: I apply a dc voltage across the APC and changes in the resistance RAP C atfrequency ω creates a voltage signal at frequency ω. Ideally, the noise in a baseband
measurement would be dominated by shot noise at the mechanical resonance frequency
over a bandwidth that is about an order of magnitude greater than the < 1 MHz widthof the mechanical resonance. Second, I can make a double sideband measurement byapplying a microwave voltage at frequency ωb across the APC instead of a dc voltage.
Changes in the resistance RAP C at frequency ω creates a voltage signal at frequencies
ωb ± ω, assuming ωb  ω. To make a shot-noise limited measurement of both sidebands, the noise in the double sideband measurement would have to be dominated byshot noise at the microwave voltage frequency ωb over a bandwidth slightly greater than
twice the mechanical resonance frequency. Finally, I consider a single sideband measurement which uses the same microwave voltage as the double sideband measurement
but only measures the voltage signal of a single sideband at ωb + ω or ωb − ω. This
technique has the advantage that it only requires a shot-noise limited bandwidth thatis about an order of magnitude greater than the < 1 MHz width of the mechanicalresonance at ωb + ω or ωb − ω and can be used to measure nanomechanical motion
at an arbitrary frequency, but has the disadvantage that the displacement imprecision
spectral density is twice that of the double sideband measurement. The imprecision of
the double sideband measurement is about the same as the imprecision of the baseband
measurement.
I then analyze whether two specific amplifier configurations have a large enoughbandwidth to detect nanomechanical motion. The first amplifier configuration uses a
low-noise high-impedance amplifier. The cable capacitance shorts the amplifier input athigh frequencies; thus the measurement noise is only limited by shot noise at frequencies
less than 100 kHz. This 100 kHz bandwidth is less than or about equal to the widthof a nanomechanical resonance, and therefore is not sufficient for sensitively detecting
nanomechanical motion. However, this configuration is useful for measuring the DC

133
resistance of the APC and performing IV spectroscopy.
The second configuration uses a 500 MHz resonant circuit and a low-noise 50 Ω
amplifier, as in the RF SET [128], in order to make a shot-noise limited measurement
over a large bandwidth. I also analyze this circuit from a microwave measurement
perspective, where the change in APC resistance results in a change in the fraction of
the incident microwave voltage bias that is reflected from or transmitted through theresonant circuit. By using a single sideband measurement and appropriately choosing
the frequency of either a reflected or transmitted microwave voltage bias, resistanceschanges at frequencies between dc and greater than 300 MHz can be observed withshot-noise limited precision.
I also show that it is possible to add to the measurement circuit an additional
electrical resonance at the < 200 MHz nanomechanical resonance frequency and a corresponding low frequency amplifier. This resonant circuit is used to measure the shotnoise at the nanomechanical resonance frequency, which is important for studying thebackaction of the APC detector (see chapter 3). However, the relatively large noise of
amplifiers available at the nanomechanical resonance frequencies limited the utility ofthis technique.
The shot noise of the atomic point contact is used to calibrate the microwave
measurement circuit. The dependence of the shot noise due to electrons tunnelingacross the APC on the dc voltage across the APC and device temperature, as describedin chapter 3, is used to calibrate the noise temperature and the gain of the microwave
electrical measurement. In addition, the shot noise due to a known dc voltage canbe compared to the shot noise due to an uncalibrated microwave voltage in order tocalibrate the magnitude of the microwave voltage bias across the APC.

134

5.1

Shot Noise Limit in Idealized APC Displacement Measurement

5.1.1

Noise in an APC resistance measurement

I now describe the basic measurement of small changes in the APC resistance andthen calculate the effect of shot noise and amplifier noise on the measurement. The basic
measurement circuit is composed of an APC with resistance RAP C and amplifier with
input impedance Ra ; a voltage bias Vb is applied to this series combination (figure 5.1).
The voltage at the input of the amplifier Va and the voltage across the APC resistor
VAP C are therefore
RAP C
RAP C + Ra
Ra
Va = Vb
RAP C + Ra

VAP C = Vb

(5.1)
(5.2)

Small changes δR in the resistance of the APC can be detected by measuring

VAPC
RAPC
Vb

Ra

Va

Figure 5.1: Diagram of a simple measurement circuit. VAP C is the voltage across RAP C
due to the voltage bias Vb . Va = Vb − VAP C is the voltage across the amplifier input
impedance Ra which will be amplified and measured in order to detect changes in theresistance RAP C of the APC.

135
small changes in the voltage across the amplifier δVa . Expanding Va above to first orderin δR implies that the signal at the amplifier due to a small change in APC resistanceisδVa = −Vb

Ra
δR
(RAP C + Ra )2(5.3)

The total measurement noise, that is, the effective noise at the input of the amplifier, is due to both the shot noise of tunneling electrons and the noise of the amplifier.
The shot noise is modeled by a voltage noise source in series with the APC (figure5.2) which has a voltage noise spectral density SV sn . For relevant APC resistances
RAP C > h/2e2 = 12.9 kΩ the conduction of electrons through the APC is usually dominated by a single (spin degenerate) conducting channel (see chapter 3) with electron
tunneling probability D yielding a resistance RAP C = h/2e2 D. In this case, and for low

VAPC

Noise SVtot
Signal Va

RAPC SVsn
Vb

SVamp

Ra

Figure 5.2: Diagram of a simple measurement circuit including the major sources ofnoise. SV sn is a noise source due to the shot noise of tunneling electrons. SV amp is theamplifier noise, which includes the noise of the entire amplifier chain referred back to
the input of the amplifier.

136
temperatures kB T  eVAP C ,SV sn = 2e|VAP C |RAP C ζΞ = 2eζΞ|Vb |2
RAPC
RAP C + Ra

(5.4)

where Ξ = 1 − D and Ξ → 1 as RAP C → ∞. The constant ζ is unit-less and accounts
for the difference between the shot noise created by a dc voltage VAP C and the average
shot noise created by a peak-to-peak voltage VAP C at microwave frequencies. For a dc
bias Vbζ = ζdc = 1(5.5)

and for an ac bias Vb cos(ωb t)ωb
ζ = ζac =2π

Z 2π/ωb|cos(ωb t)| dt =
0

2
π

(5.6)

These equations for the shot noise are only valid at the experimentally relevant frequencies ω  kB Tcryo /h̄ and ω  eVAP C /h̄.
The other major source of noise is the voltage amplifier. The noise added by
the amplifier can be modeled by a voltage noise source with spectral density SV amp
in parallel with the input impedance of the amplifier (figure 5.2); the magnitude andfrequency dependence of SV amp is a characteristic of the amplifier. The total noise at
the input of the amplifier has a voltage spectral density SV tot

SV tot = SV sn
SV tot = 2eζΞ|VAP C |RAP CRaRAP C + Ra

2

Ra
RAP C + Ra

2

+ SV amp(5.7)+ SV amp(5.8)

The final measurement of nanomechanical position, derived from the signal δVa ,is only quantum limited if the noise in the measurement of SV tot is dominated by thefundamental shot noise due to electrons tunneling through the vacuum gap in the APC
instead of the amplifier noise. From equation 5.8, the shot noise contribution dominates

137
the total noise when2Ra2eζΞ|VAP C |RAP C≥ SV amp
RAP C + Ra
SV ampRAP C 2
|VAP C | ≥1+2eζΞRAP C
Ra


(5.9)
(5.10)The voltage VAP C across the APC required for the shot noise to dominate the measurement noise is minimized by making the amplifier impedance Ra large as possible andthe APC resistance RAP C equal to the amplifier impedance. This second condition is
an impedance matching condition. It ensures that the shot noise power is efficiently
coupled into the amplifier.
This analysis of a simple measurement circuit can also be applied to more realistic measurement circuits (figure 5.3). The entire circuit connected to the APC is aneffective amplifier with a complex, frequency dependent input impedance Ra = Za (ω)
and frequency dependent amplifier noise SV amp (ω). This measurement circuit contains
a voltage amplifier with an input impedance Ra0 and noise SV0 amp . The input impedanceZa (ω) is just the impedance of the entire circuit connected to the APC resistance RAP C(figure 5.3). If there is a voltage Va across the effective amplifier input Za (ω), then it
is possible to calculate the voltage Va0 across the amplifier in the complex measurement
circuit with input impedance Ra0 . The effective amplifier noise is then
SV amp (ω) = SV0 amp



Va
Va0

2
(5.11)

In other words, SV amp (ω) is given by the voltage noise of the amplifier SV0 amp referredto the input of the effective amplifier.
max = 50 mV, equation 5.10 can be
Since an APC is only stable when VAP C < VAP
Cused to find the maximum bandwidth of a the amplifier. Rewriting equation 5.10 using
the frequency dependent amplifier noise and input impedance,
SV amp (ω)
RAP C 2
|VAP C | ≥1+2eζΞRAP C
Za (ω)(5.12)

138

VAPC

measurement voltage
circuitamplifier Ra

RAPC SVsn
Vb

SVamp

effective
amplifier

VAPC
RAPC SVsn
Vb

Va

SVamp
Za

Va

Figure 5.3: For the purpose of determining the noise in a measurement of small resistancechanges, a complicated measurement circuit (top, green box) can be treated as a simple
effective measurement circuit (bottom, green). A voltage amplifier (red, top) combinedwith a complex measurement circuit (blue, top) is effectively an amplifier with a different
amplifier noise SV amp (ω) and a complex input impedance Za (ω) equal to the impedanceof the entire complex measurement circuit and amplifier (top, green box).

139
When this inequality is satisfied, the measurement is in the desired regime where the shotnoise dominates the measurement noise. Given a specific measurement configuration
with RAP C , SV amp (ω), and Za (ω), the frequencies at which this inequality is satisfied
define a bandwidth and that bandwidth is going to be as large as possible when VAP C =max . It is also possible that this inequality is never satisfied for a voltage V
VAP
AP C ≤Cmax , in which case the measurement noise is always dominated by the noise of the
VAPCamplifier.

5.1.2

Noise in an APC displacement measurement

Using the relationship between resistance and displacement from chapter 3 andthe signal analysis in the previous subsection, it is possible to calculate the expectedmagnitude and frequency of the voltage signal due to nanomechanical motion and thedisplacement imprecision due to noise in the voltage measurement. I take the voltage measurement to be dominated by shot noise over a bandwidth γmeas centered ata frequency ωmeas . There are three main ways in which I use this voltage measurement to detect nanomechanical motion at frequencies near a nanomechanical resonancefrequency ωm . These three measurement techniques are summarized in table 5.1 andfigure 5.4. The first technique is a baseband measurement, where a DC voltage bias
VAP C is applied across the APC and nanomechanical motion at a frequency ω creates a voltage signal at ω (see figure 5.4a). Using this technique, it is desirable for thenanomechanical resonance to be centered in the measurement band, ωmeas = ωm . Whenωmeas = ωm , the required measurement bandwidth γmeas is controlled by the width γmof the nanomechanical resonance (usually γmeas ≥ 10γm ).
The second type of measurement uses a double sideband technique, where a microwave voltage VAP C cos(ωb t) is applied across the APC and nanomechanical motion
creates a voltage signal composed of equal sidebands at frequencies ωb ± ωm (figure5.4b). Measuring both sidebands requires a large measurement bandwidth γmeas ≈ 2ωm

140
measurement type
Basebanddesired measurement bandwidth
10γm , usually 2 MHz

measurement
frequency
ωmDouble-Sideband

2ωm , desire > 100 MHz

ωbSingle-Sideband√
larger (× 2) imprecision

10γm , usually 2 MHz

ωb ± ωm

Table 5.1: Describes the required shot-noise limited bandwidth and measurement frequency for different measurement types. γm is the width of the nanomechanical resonance, ωm is the nanomechanical resonance frequency, and ωb is the frequency of theapplied voltage bias. Since ωb can be easily varied, the measurement frequency of the
double and single sideband measurements can be tuned.

a)

V

1.0

DC BiasVoltage

BasebandMeasurement
width ~10 γm
frequency ωm

0.8
0.6

Voltage
Signal

0.4
0.2

b)

0

20

40

V

1.0
0.8

ωm

60

80

Voltage
Signal0.4

Single Sideband
Measurement
width ~10 γm
frequency ωb+ωm

0.2
0.0
400

450

ωb-ωm

500

ωb

Freq MHz

Double Sideband
Measurement
width ~2 ωmfrequency ωb

Microwave
Bias Voltage

0.6

100

550

ωb+ωm

600

Freq MHz

Figure 5.4: (a) baseband measurement (b) double and single sideband measurements

141(assuming ωm  γm ) and the bandwidth should be centered at the frequency of the
microwave voltage bias ωmeas = ωb . Since ωb is easily tuned in situ, the frequency ofthe sidebands can be matched to the frequency of a measurement bandwidth. This
is in contrast to the baseband measurement, where the frequency of the measurement
bandwidth ωmeas , which cannot be tuned in situ, must be matched to the frequency ofnanomechanical motion ωm . While it is possible to make a double sideband measurement using a smaller bandwidth when ωmeas ≈ ωb < ωm , I am going to analyze andconcentrate on the experimentally relevant case where ωmeas ≈ ωb  ωm .
The third method is a single sideband measurement which is very similar to the
double sideband measurement. The same microwave voltage VAP C cos(ωb t) is applied
across the APC and nanomechanical motion creates voltage sidebands at frequencies
ωb ± ωm . However, in the single sideband measurement, as the name suggests, onlya single sideband is measured (figure 5.4b). This technique has the advantage that,in comparison to the double sideband measurement, the needed bandwidth is muchsmaller γmeas ≈ 10γm which is the same as the baseband measurement. The frequencyof the measured sideband (either ωb + ωm or ωb + ωm ) can also be tuned to the center
frequency of the measurement bandwidth ωmeas = ωb ±ωm by changing the frequency ofthe measurement bias ωb . This method can therefore be used to detect high frequency
nanomechanical motion, limited just by the technical problem of applying a measurement bias at ωb = ωmeas ± ωm . The disadvantage of of the single sideband measurement
is that, since half of the voltage signal is ignored, the displacement imprecision spectral
density is twice that of the double sideband measurement.
Having summarized the three different types of measurements (table 5.1), I calculate and compare the voltage signal, noise, and displacement imprecision of the different
measurements. The voltage signal due to nanomechanical displacement can be found using the results from chapter 3. A small change δx in the position of the nanomechanical142
structure will result in a small change in resistance δRδR =2RAP Cχδx
λ(5.13)

where λ is the length scale that controls the probability of electrons tunneling throughthe APC and χ relates changes in nanomechanical position to changes in the width of
the APC gap (see chapter 3). From equation 5.3, the signal at the amplifier is thereforeδVa = −

χVb 2RAP C Ra
χVAP C2Ra
δx = −δx
2
λ (RAP C + Ra )λ RAP C + Ra

(5.14)

Both δx and VAP C can have a time dependence. Harmonic motion of the nanomechanicaloscillator at frequency ω is described as the sum of two quadrature components, δx (t) =x1 cos(ωt) + x2 sin(ωt).
The baseband measurement uses a dc bias VAP C (t) = Vdc which does not have atime dependence. The signal at the amplifier using a baseband measurement is therefore
δVabb (t) = V1bb cos(ωt) + V2bb sin(ωt)(5.15)

where for j = {1, 2}
Vjbb = −

χVdc
2Ra
xj
λ RAP C + Ra

(5.16)

That is, the nanomechanical motion at frequency ω is going to create a voltage signalat ω.
The double sideband measurement uses a microwave bias VAP C (t) = Vac cos(ωb t)
and the nanomechanical motion at frequency ω creates a voltage signal at the amplifier
at frequencies ωb ± ωδVads (t) = V1ds cos(ωt + ωb t) + V2ds sin(ωt + ωb t) +V1ds cos(ωt − ωb t) + V2ds sin(ωt − ωb t)(5.17)

where for j = {1, 2}
Vjds = −

χVac
2Ra
xj
2λ RAP C + Ra

(5.18)

143
The single side band measurement uses the same microwave bias as the double
sideband measurement, but ignores one of the sidebands in equation 5.17. In this case
I choose to analyze the measurement of the upper sideband at frequency ωb + ω; theanalysis of the measurement of the lower sideband is identical. Having chosen the upper
sideband, in the single sideband measurement nanomechanical motion at a frequency ωcreates a signal at the amplifier at frequency ωb + ωδVass (t) = V1ss cos(ωt + ωb t) + V2ss sin(ωt + ωb t)(5.19)

where for j = {1, 2}
Vjss = −

χVac
2Ra
xj
2λ RAP C + Ra

(5.20)

Even if the mechanical oscillator is stationary, the measured voltage is will fluctuate. This voltage noise will cause a purely apparent fluctuation in the inferred positionof the beam, described by a spectral density Sx . As described in subsection 5.1.1, therandom voltage noise in the measurement has a spectral density SV tot due to the shotnoise of tunneling electrons and amplifier noise. The noise in a small band ∆f  γm /2π
around a voltage signal at frequency ωV creates uncertainty in the measurement of thevoltage signal’s quadrature amplitudes V1 and V2 with variance (for j = {1, 2})σV2j = SV tot (Vb , ωV )∆f2

(5.21)

where SV tot is a function of both the voltage bias Vb and frequency of the voltage signal
ωV . The noise in different quadratures and at different frequencies is uncorrelated (see,for example, reference [188]).
For each measurement type, this variance can be used to calculate the variance ofthe displacement measurement and thus the displacement imprecision spectral density.
In the baseband measurement, displacement xj is related to the voltage Vjbb at frequency
ω by equation 5.16, therefore(σxbbj )2 =λ RAP C + Ra 2∆f
λ RAP C + Ra 2 2
σVj =SV tot (Vdc , ω)
Vdc
2Ra
Vdc
2Ra
2

(5.22)

144
and the imprecision spectral density Sxbb of the baseband measurement is
Sxbb (ω) =λ RAP C + Ra 2
SV tot (Vdc , ω)Vdc
2Ra

(5.23)

Similarly, in the single sideband measurement the displacement xj at frequency
ω is related to the voltage Vjss at frequency ωb + ω by equation 5.20, therefore(σxssj )2 =2λ RAP C + Ra 2∆f
2λ RAP C + Ra 2 2
σ Vj =SV tot (Vac , ωb + ω)
Vac
2Ra
Vac
2Ra
2

(5.24)and the imprecision spectral density Sxss of the measurement of nanomechanical motion
at frequency ω using the single sideband technique is
Sxss (ω) = 4

λ RAP C + Ra 2
SV tot (Vac , ωb + ω)Vac
2Ra

(5.25)The double sideband measurement is equivalent to simultaneously making two singlesideband measurements at frequencies ωb + ω and ωb − ω with uncorrelated noise. The
variance of this displacement measurement is(σxdsj )2 =2λ RAP C + Ra 2 [SV tot (Vac , ωb − ω) + SV tot (Vac , ωb + ω)] ∆f
Vac
2Ra
4
2

(5.26)and the imprecision spectral density Sxds of the double sideband measurement is
Sxds (ω) =λ RAP C + Ra 2
[SV tot (Vac , ωb − ω) + SV tot (Vac , ωb + ω)]Vac
2Ra(5.27)

While the displacement imprecisions derived above apply to an arbitrary voltage
noise spectral density SV tot , I most interested in the case where the amplifier noise isnegligible (SV amp → 0) and SV tot is dominated by the fundamental source of noise in
the problem, the shot noise of tunneling electrons. In the baseband measurement with
VAP C (t) = Vdc , the shot noise creates a fluctuating voltage with a spectral density (seesubsection 5.1.1)
SVbbsn (Vdc , ω) = 2eΞζdc |Vdc |RAP CRaRAP C + Ra2
(5.28)

145
therefore the shot-noise limited displacement imprecision of the baseband measurementiseΞζdc λ2 RAP C
2|Vdc |

Sxbb (ω) =(5.29)In the sideband measurements with VAP C (t) = Vac cos(ωb t), the shot noise creates afluctuating voltage with a spectral density (see subsection 5.1.1 and chapter 3)
SVsssn (Vac , ω) = SVdssn (Vac , ω) = 2eΞζac |Vac |RAP CRaRAP C + Ra2
(5.30)

Using this voltage noise spectral density, the displacement imprecision spectral densityof the double sideband measurement is
Sxds (ω) =eΞζac λ2 RAP C|Vac |(5.31)and the displacement imprecision spectral density of the single sideband measurement
isSxss (ω) =2eΞζac λ2 RAP C|Vac |(5.32)

which is twice the imprecision of the double sideband measurement Sxss = 2Sxds .
The comparison of the baseband imprecision Sxbb and double sideband imprecision
Sxds is complicated by the difference between a dc voltage Vdc and microwave voltage
Vac cos(ωb t). When comparing noise, the signal and noise from dc voltage Vdc is often
equated with a microwave voltage that dissipates the same amount of power, that is,√
Vac = 2Vdc . This is most applicable if the voltage noise SV is independent of bias
voltage; in this case (not calculated above), Sxbb and Sxds would be equal when Vac =√2Vdc . Shot noise SV sn does depend on the bias voltage; in the shot noise dominatedlimit Sxbb and Sxds are equal when Vac = (ζdc /ζac )Vdc = (π/2)Vdc . Therefore if the bias
magnitude is limited by a maximum power that can be dissipated in the APC then thesmallest imprecision can be achieved using a double sideband measurement. However,if the bias magnitude is limited by a maximum voltage that can be sustained across the

146
APC then the smallest imprecision can be achieved using the baseband measurement.
Experimentally, the maximum voltage or power that can be handled by the APC is notstable enough for the difference in the best imprecision of these two techniques to be
relevant.

5.2

Conventional APC Resistance Measurement
I now show that a simple low frequency measurement with a high impedance

amplifier has an unacceptably small bandwidth of about 100 kHz starting at 0 Hz. This
low frequency measurement can still be used to measure the average and differential
resistance RAP C of the APC even though the measurement is not fast enough to resolve
nanomechanical motion.

5.2.1

Limited bandwidth of a conventional measurement

The simplest way to implement the basic measurement circuit in figure 5.1 is touse a high-impedance amplifier, however the stray capacitance in the physical circuit
must also be included (figure 5.5). As desired, The impedance of the amplifier itself
Ra0 is much larger than the APC resistance, however making the effective amplifier
impedance Za large over a large range in frequency is actually rather difficult becauseof the capacitance shunting the amplifier input. Any capacitance between the physicalline connecting the APC to the amplifier and the circuit’s ground (Cs and Cc in figure5.5) will shunt the amplifier input. In a cryogenic system this capacitance can be large
because the cable between the amplifier and the device results in a capacitance toground of about 100 pF per meter of cable. Figure 5.5 contains a more realistic circuit
diagram including the shunting capacitance to ground on and near the chip Cs and thecapacitance to ground of the cable Cc as well as the internal impedance of the voltage
amplifier Ra0 . The effect of the cable inductance can be ignored because that impedanceis much smaller than the internal impedance Ra0 of the amplifier. From the circuit in147
figure 5.5, the effective amplifier impedance Za as a function of frequency is
Za (ω) =Ra0
1 + iω(Cs + Cc )Ra0

(5.33)and taking the ideal case where Ra0 → ∞ thenZa (ω) =1iω(Cs + Cc )VAPC

Noise SVtot
Signal Va

RAPC SVsn
VB

Cs(5.34)

Ra

SVamp

Cc

Figure 5.5: Diagram of the measurement circuit using a high-impedance amplifier, including the capacitance to ground Cs from connections near and on the chip and thecapacitance to ground Cc due to the cable between the chip and the amplifier at the
top of the cryostat .

With this frequency-dependent amplifier input impedance Za (ω), equation 5.12
can be used to calculate the bandwidth over which the noise in this measurement circuit
is dominated by shot noise given actual values for the noise SV amp = SV0 amp of the highimpedance amplifier, the total capacitance Cs + Cc , and the APC resistance RAP C .
From equation 5.12, the measurement noise is dominated by shot noise when
SV amp|1 + iω(Cs + Cc )RAP C |22eΞζdc RAP Cs1
RAP C |VAP C |ω≤2eΞζdc−1
RAP C (Cs + Cc )SV amp|VAP C | ≥(5.35)(5.36)

148
Commercially available high-impedance amplifiers typically have voltage noise spectral
densities of about SV amp = 10−17 V2 /Hz, and using conservative assumptions the total
capacitance in the system is Cs + Cc = 200 pF. Using these values, for voltages suchthat |VAP C |ΞRAP C > 31 VΩ there will be a bandwidth where the measurement noise
is limited by shot noise. This bandwidth starts at ω = 0 and extends to a maximum
frequency given by the equality in equation 5.36. This maximum frequency, which isalso the measurement bandwidth, is plotted in figure 5.6 as a function of APC resistancemax = 50 mV and V
RAP C for different APC voltages ranging between VAP C = VAP
AP C =
C

0.4 mV.
The bandwidth of this low frequency measurement with a high-impedance amplifier is not large enough to implement a shot-noise limited measurement of nanome-

Maximum Frequency HkHzL
200

VAPC = 50 mV
VAPC = 10 mV
VAPC = 2 mV
VAPC = 0.4 mV

100

50

20

10

5

1 ´ 104

5 ´ 104

1 ´ 105

5 ´ 105

1 ´ 106

5 ´ 106

RAPC HWL

1 ´ 107

Figure 5.6: In this figure the maximum frequency at which the measurement noise of alow frequency measurement with a high impedance amplifier is dominated by shot noiseis plotted as a function of APC resistance RAP C for various different APC voltages
VAP C . I use reasonable values for SV amp = 10−17 V2 /Hz and Cs + Cc = 200 pF. The
max = 50 mV. The bandwidth of this measurement ismaximum stable APC voltage VAP
Cbetween 0 Hz and the maximum frequency plotted in this figure.

149
chanical motion. All of the measurement schemes in table 5.1 require a bandwidth thatis greater than 1 MHz, however the largest bandwidth achieved with this low frequency
measurement is less than 150 kHz at an APC resistance RAP C = 25 kΩ and the largest
max = 50 mV (see figure 5.6); at higher frequencies the cablestable APC voltage VAP
Ccapacitance short circuits the voltage amplifier. A microwave technique must be used
to detect nanomechanical motion as described in section 5.3.

5.2.2

Description of low frequency resistance measurement

A low frequency measurement using a high impedance amplifier may not be able
to sensitively detect nanomechanical motion, but it can still measure four important
quantities: the dc voltage VAP C across the APC, the dc current IAP C flowing throughavg
the APC, the average resistance of the APC RAP
C = VAP C /IAP C , and the differential

resistance of the APC RAP C = ∂VAP C /∂IAP C . This measurement uses a circuit (figure5.7) that has some small differences from the circuit described in the previous section.
The biggest difference is the use of a second high impedance amplifier. One amplifiermeasures the voltage drop VAP C across the APC due to a voltage bias VLF ; the otheramplifier measures the current IAP C = VR /Rb flowing through the APC by measuring
the voltage drop VR across a known resistor Rb . By simultaneously measuring VAP C andIAP C as a function of bias VLF , I determine the APC current IAP C , average resistanceavg
RAP
C , and differential resistance RAP C as a function of the voltage VAP C across the

APC. In mesoscopic systems, these electrical characteristics are often summarized byplotting the APC current IAP C versus APC voltage VAP C ; this plot is generally called
an I-V curve.
avg
For an ideal resistor the average resistance RAP
C and differential resistance RAP C
avg
are independent of the voltage bias and RAP
C = RAP C , however the APC is not an

ideal resistor. In particular, the shape of the tunnel barrier under large voltage bias,
as described in chapter 3, results in a decrease in resistance at large bias voltages and

150

VR
Rb
VLF

RAPC

C

VAPC

Figure 5.7: Circuit diagram of the low frequency measurement circuit with two highimpedance amplifiers that are used to measure the voltage VAP C across the APC RAP C
and the voltage VR across a known resistor Rb . C is the capacitance to ground in thecryostat, including the cable to the device.

therefore a nonlinear I-V curve. Other mesoscopic effects also create nonlinearities in
I-V curve of an APC. The use of an I-V curve and the differential resistance RAP C asa function of VAP C (or IAP C ) to understand the characteristics of the APC is called IV
or tunneling spectroscopy (see chapter 2, 3, and 7 for more details).
avg
In order to measure the average resistance RAP
C and differential resistance RAP C ,I apply a small square wave voltage bias VLF with a mean value V0 and a peak-to-peakvoltage V1 at a frequency between 5 Hz and 100 Hz. I usually choose the frequency
so that the the shunting capacitance (C, figure 5.7), which smoothes the square wave
transition, can be ignored. I use the amplifiers (figure 5.7) to measure the averageTTvoltages at the top of the square wave hVAP
C i and hVR i as well as the average voltages at
B i and hV B i (see figure 5.8). Using these measured
the bottom of the square wave hVAP
C
R

151

VLF

Varb

V

0

2.5

V

1

T
V
APC

} }
} }

VAPC

3.0

2.0
1.5

VR

T
R

V

B
APC

V

1.0
0.5

B
R

V
1.0

0.5

0.5

1.0

Time 2ΠΩ0.5

Figure 5.8: In this figure I plot the voltages VLF (red), VAP C (blue), and VR (green)used in the DC measurement as a function of time, using as an example RAP C = 2.3Rb .
The low frequency bias voltage VLF is a square wave at frequency ω with an offset V0
and peak-to-peak V1 . Two amplifiers are used to measure both the resulting voltage
VAP C across the APC and the resulting voltage VR across the known bias resistor Rb .
The APC resistance is calculated from the average measured voltages at the top of theTT
BB
square wave hVAP
C i and hVR i and the bottom of the square wave hVAP C i and hVR i.

voltages, the average resistance is approximately
avg
RAP
C =

B i
hV T i + hVAP
VAP C
C
Rb
≈ AP CIAP C
hVRT i + hVRB i(5.37)and the differential resistance is approximately
RAP C =B i
hV T i − hVAP
∂VAP CC≈ AP C
Rb
T
B∂IAP ChVR i − hVR i(5.38)

Ignoring noise in the problem, the approximations become equalities when the peakto-peak magnitude of the square wave goes to zero, V1 → 0. IV spectroscopy is acavg
complished by varying the mean value V0 of the square wave and measuring RAP
C and

RAP C . V1 determines the resolution of the measurement; that is, the spectroscopy canresolve structure in the IV curve on the scale of V1 . The addition of a microwave voltage
bias across the APC will also limit the resolution of the spectroscopic measurement.

152

5.3

Microwave APC Resistance Measurement
Using a 50 Ω amplifier embedded in a 500 MHz resonant circuit (as in the RF-

SET [128]), I can make a measurement of nanomechanical motion whose measurement
noise is dominated by the shot noise of tunneling electrons over a bandwidth greaterthan 30 MHz. Using a double sideband measurement this enables shot-noise limited detection of nanomechanical motion at frequencies up to 15 MHz. Using a single sideband
measurement, nanomechanical motion at frequencies between 0 Hz and > 300 MHz
can be detected with an imprecision that, while still shot noise limited, is twice theimprecision of the double sideband measurement (see section 5.1.2 and table 5.1). This
technique employs a microwave voltage bias which is applied using either a transmission or reflection circuit; in a microwave circuit analysis, changes in APC resistance aredetected by measuring changes in the resonant circuit’s reflection or transmission coefficients. The signal to shot-noise ratio of these two measurements is identical, but there
are advantages to using a reflection circuit when making a double sideband measurement (table 5.1) and advantages to using a transmission circuit when making a single
sideband measurement (table 5.1).

5.3.1

Improved bandwidth using microwave measurement

I use a microwave circuit composed of a low-noise 50 Ω amplifier embedded in aresonant circuit (figure 5.9, top) in order to make a large bandwidth measurement ofchanges in the APC resistance. Using a low impedance amplifier avoids the problem
created by the cable capacitance to ground; this problem was demonstrated in the previous section. The Ra0 = 50 Ω amplifier input impedance is matched to the impedance ofthe cable, therefore the cable capacitance is effectively canceled by the cable inductanceand the cable can be ignored in the circuit analysis. The cable will just shift the phaseof the voltage signal by an amount determined by the length of the cable.

153

VAPC

measurement voltage
circuitamplifier Ra

RAPC SVsn

Cs

Vb

SVamp
VAPC

Va

effective
amplifier

SVamp

RAPC SVsn
Vb

L

Za

Va

Figure 5.9: (top) Circuit diagram of the microwave measurement circuit. A bias voltage
Vb creates a voltage VAP C across the APC, also creating shot noise with a spectral
density SV sn . The microwave measurement circuit includes a stray capacitance Cs toground and an inductor L. The bias also creates a voltage Va0 across an amplifier which
has amplifier noise SV0 amp and input impedance Ra0 = 50 Ω. This input impedance ismatched to the characteristic impedance of the cable connected the amplifier and theinductor, thus the cable only contributes an overall phase shift which will be ignored.
(bottom) As described in subsection 5.1.1, the measurement circuit and amplifier (top,green box) can be described as an effective amplifier (bottom, green circuit) with acomplex frequency dependent impedance Za , amplifier noise SV amp , and amplified (thatis, measured) voltage Va .

154
Simply using a Ra0 = 50 Ω amplifier will not result in a shot-noise limited measurement noise. Low-noise cryogenic amplifiers at 500 MHz can have a voltage noise spectral
density as low as SV amp ≈ 3×10−21 V2 /Hz. However, this voltage noise and a Za = 50 Ω
amplifier input impedance implies that an APC bias voltage |VAP C | > 200 mV would
be required to make a measurement whose noise is dominated by shot noise (equation
5.12). Since the APC is generally unstable for an APC bias voltage |VAP C | > 50 mV, it
is necessary to modify the measurement circuit so that the effective impedance of theamplifier is larger than 50 Ω.
I therefore use a microwave resonant circuit (figure 5.9, top) to create an effective
amplifier with a large input impedance near the electrical resonance frequency. The
electrical resonance is created by adding an inductor L to the stray capacitance Cs toground present on and around the chip (figure 5.9, top). This type of circuit is also
called a matching circuit, since it is used to match the impedances of the high impedance
APC and the low impedance amplifier. It’s use in mesoscopic physics was pioneered by
Schoelkopf et al [128] to create the radio-frequency single electron transistor (RF-SET).
As described in section 5.1.1 and shown in figure 5.9, the entire measurement circuit iseffectively an amplifier with a frequency-dependent input impedanceZa =Ra0 + iω(L − ω 2 L2 Cs − Cs Ra02 )(1 − ω 2 LCs )2 + (ωCs Ra0 )2(5.39)

and an frequency-dependent effective amplifier noise (from equation 5.11)
SV amp (ω) = SV0 amp

Va 2
Ra02 + ω 2 L2
0
=
S
V amp
Va0
Ra02

(5.40)

This effective amplifier description of the measurement circuit can be used to
calculate the measurement bandwidth γmeas . The measurement noise will be dominatedby shot noise when equation 5.12
SV amp (ω)
RAP C 2
|VAP C | ≥1+2eΞζac RAP C
Za (ω)

155
is satisfied where SV amp is the effective amplifier noise given by equation 5.40 and Za
is the impedance of the measurement circuit given by equation 5.39. The frequencies ω
at which this inequality is satisfied defines a bandwidth γmeas centered at ωmeas within
which the measurement noise is dominated by shot noise.
In order to maximize the measurement bandwidth γmeas , it is simpler to determine
the optimal electrical resonance quality factor Q and frequency ω0 instead of optimizing
the inductance L and capacitance Cs . In this change of variables, the quality factorp
Q = L/(Ra02 Cs ) and the undamped (that is, Ra0 → 0) resonance frequency is ω0 =√1/ LCs . Using these variables and solving equation 5.12,h ω pi
p0α+β+ α−β
2
i
h p
p
α+β− α−βγmeas = < ω0

ωmeas = <(5.41)(5.42)

where
α=1−Q2 Ra021−
2
2Q2 2RAP
C(5.43)

and
1
β =
4Q2
2



Q2 Ra0
+1
RAP C

2 


Q2 Ra021
2Ra0
2eζac ΞRa02 |VAP C |
+
−
−
4
+
2
Q2 RAP CRAP C SV0 amp
RAPC(5.44)

When β is imaginary then there is no frequency at which the measurement noise isdominated by shot noise and γmeas = 0. While this equation for the measurement
bandwidth γmeas is a complicated function of the quality factor Q, the optimal quality
factor Qopt that maximizes γmeas is simples
Qopt =RAP C
Ra0(5.45)

Qualitatively, this optimal quality factor Qopt can be understood in terms of the
impedance Za of the measurement circuit. On resonance (ω = ω0 ) the measurement
circuit’s impedance is
Za (ω0 ) = Ra Q(Q − i)(5.46)

156
Since Qopt  1, when Q = Qopt then the measurement circuit will have an impedance
that is matched to the APC impedance Za (ω0 ) = RAP C . Since the maximum shot noise
power is coupled into the measurement circuit when the impedance of the APC and themeasurement circuit are matched, it is not surprising that the optimal quality factor
enforces this condition.
For a given RAP C and VAP C , the measurement bandwidth γmeas is not only
dependent upon the quality factor of the electrical resonance but also on the electricalresonance frequency ω0 and the amplifier noise SV0 amp . Since γmeas ∝ ω0 , theoreticallythe bandwidth can be made infinite by taking electrical resonance frequency ω0 → ∞.
Since a large Q is still optimal, this is equivalent to taking Cs → 0.
The desire for a small capacitance Cs can also be understood in terms of the
Bode-Fano limit. The Bode-Fano limit is a limit on the bandwidth over which it ispossible to make a good match to a complex impedance [189]. In this case the complex
impedance is the resistance RAP C of the APC shunted by the capacitance Cs , and thelimit on the bandwidth is proportional to 1/(RAP C Cs ). Therefore taking Cs → 0 is
equivalent to taking the Bode-Fano limit on the bandwidth to infinity.
Experimentally, maximizing γmeas by minimizing Cs involves carefully designing
the physical connections between the APC and the inductor, including the on-chipelectrodes, wirebond, and inductor solder-pad. The bond-pads are designed to have asmall area and a large separation from ground (see chapter 4). I also use a short wire
between the bondpad and the inductor and use a small surface-mount inductor. In
addition, the stray capacitance to ground is further decreased by flipping the inductor
and directly wire-bonding to the inductor’s soldering pad (see figure 5.10). However,even after using these techniques to limit the stray capacitance there is still Cs ≈ 200 fF.
The desire to use a surface mount inductor in this experiment instead of an onchip resonant circuit limits the electrical resonance frequency ω0 to frequencies less thanabout 500 MHz. Using surface mount inductors simplifies device fabrication, however

157

wire

inductor

bondpads
groundplane

Figure 5.10: Pictures of chip and circuit board used to minimize stray capacitance on
the right, higher magnification picture of chip and inductor on the left. The capacitance
between the APC and the inductor is minimized by using small bond-pads and directly
wirebonding to the solder pad of a surface-mount inductor which is turned over andglued to the circuit board.

it is generally difficult to find inductors that, at frequencies above about 500 MHz, havea large inductance and a large quality factor. I use Coilcraft surface mount inductors
from the 0805CS series. These inductors have a large internal quality factor at 4 K(> 50 at 500 MHz) due to the use of high purity copper and therefore the loss in the
inductor has little effect on the quality factor Q of the electrical resonance; most of
the loss in the circuit is due to the amplifier impedance Ra0 as desired. In general,
inductors are shunted by a geometrical capacitance which limits the frequency range
over which the inductor has a large inductance; that is, an impedance which increases
with frequency. This frequency, where the contribution to the inductor’s admittancefrom the inductance and geometrical capacitance is equal in magnitude but opposite
in sign, is called the self resonance frequency. The Coilcraft 0805CS inductors have asmall shunting capacitance and a large self resonance frequency of about 500 MHz. This
limits the desired electrical resonance frequency to frequencies less than about 500 MHz.

158
While the Bode-Fano limit is important, it does not directly limit the bandwidth γmeas
where the noise in the measurement is dominated by shot noise because γmeas can always
be increased by making the amplifier noise SV0 amp smaller.
Since the use of a surface mount inductor limits the measurement frequency toless than about 500 MHz, the measurement bandwidth γmeas is maximized by usinga Ra0 = 50 Ω amplifier with a measurement bandwidth centered at 500 MHz. I use a
low-noise Berkshire U-500-2 amplifier which has a noise temperature less than 2 K overa bandwidth of about 100 MHz centered around 500 MHz. This is a cryogenic amplifier
and to achieve the quoted noise temperature it is placed in the 4 K LHe bath. (Thetransistors in the amplifier are encapsulated, hence they can withstand the frost that
deposits when the cryostat is extracted from LHe bath). Specifying an amplifier noise
temperature Tamp and input resistance Ra0 is equivalent to specifying the voltage noise
spectral density
SV0 amp = 4kB Tamp Ra0 = 5 × 10−21 V2 /Hz(5.47)

This choice of ω0 = 2π × 500 MHz and the minimum stray capacitance Cs ≈ 200 fF
implies a desired inductance of about L ≈ 500 nF. In general there are small variationsin the stray capacitance depending on the details of the configuration of the device, chip,and inductor; I select an inductor value that yields an electrical resonance frequency
close to 500 MHz.
The stray capacitance Cs = 200 fF, inductance L = 500 nH, and amplifier noise
SV0 amp = 5 × 10−21 V2 /Hz used in this experiment determine the measurement bandwidth γmeas as a function of the APC resistance RAP C and bias voltage VAP C . In figure5.11 I plot the bandwidth as a function of the APC resistance RAP C at four differentmax ≈APC bias voltages VAP C . For the largest stable APC bias voltage VAP C = VAP
C

50 mV, the bandwidth of the measurement is limited by the bandwidth of the cryogenic
amplifier at low APC resistances (RAP C < 200 kΩ) to approximately 100 MHz. The

159
Bandwidth HMHzL200

VAPC = 50 mV
VAPC = 10 mV
VAPC = 2.5 mV
VAPC = 1.8 mV

100

50

20

10

5

2

1
1 ´ 104

5 ´ 104

1 ´ 105

5 ´ 105

1 ´ 106

5 ´ 106

RAPC HWL

1 ´ 107

Figure 5.11: The bandwidth of the microwave measurement is plotted as a functionof the APC resistance at four different APC voltages, using typical circuit parameters
Cs ≈ 200 fF and L ≈ 500 nF. This bandwidth is centered at around 500 MHz. While thisplot assumes a constant amplifier noise temperature Tamp = 2 K, the actual amplifier
is in fact as good as Tamp ≈ 1 K at 530 MHz but becomes Tamp > 2 K below 450 MHz
or above 580 MHz. Therefore, the actual bandwidth is limited to < 100 MHz.

bandwidth is also greater than about 2 MHz both using the largest stable APC bias
voltage at RAP C < 5 MΩ and using a small APC bias voltage VAP C ≈ 2 mV at an APC
resistance RAP C ≈ 100 kΩ.
I use this large bandwidth to measure nanomechanical motion. With 100 MHz
of bandwidth, it is possible to implement the double sideband measurement describedin table 5.1 and make a shot-noise limited measurement of nanomechanical motion at
frequencies ≤ 50 MHz. Using a single sideband measurement (table 5.1), it is possible
to make a shot-noise limited measurement of nanomechanical motion at a wide rangeof nanomechanical frequencies (< 300 MHz) using a small bandwidth (≈ 2 MHz) measurement. Because the microwave measurement has a shot-noise limited bandwidth> 2 MHz at a wide range of APC voltage biases and APC resistances, the single side-160
band technique can be be used to make a sensitive measurement of nanomechanicalmotion at a wide range of nanomechanical resonance frequencies, APC voltage biases,
and APC resistances. However, the single sideband measurement has the disadvantagethat it’s shot-noise limited displacement imprecision spectral density is twice as large asthat of the double sideband measurement because the single sideband technique ignores
half of the voltage signal created by nanomechanical motion (subsection 5.1.2).

5.3.2

Network analysis of microwave measurement

In this subsection I describe the APC and resonant circuit as a two-port networkin order to decide whether the microwave voltage bias should be applied in transmission,as was analyzed in the previous section, or in reflection. As I will show, the type ofvoltage bias does not effect the signal due to small changes in the APC resistance or themeasurement noise when they are parameterized as functions of the voltage VAP C acrossthe APC. Therefore the measurement bandwidth calculated in the previous subsection
and the displacement imprecision will be the same for the two types of voltage biases.
However, there is also a tone at the frequency of the voltage bias, which can be used toinfer the average resistance of the APC. This signal may be so large that it saturates
the low-noise amplifier. By carefully choosing the frequency of the microwave bias andwhether it is applied in transmission or reflection, this saturation can often be avoided.
From a microwave engineering perspective, the APC and resonant circuit is a
two-port network, a type of network which is analyzed in detail in many microwave
textbooks (for example, [189]). The basic circuit diagram is shown in figure 5.12, where
the circuit elements form a simple T-network and there is a voltage wave incident on
V1+ and emerging from V1− port 1 and similarly voltage waves V2+ and V2− incident
on and emerging from port 2. These voltages are related using a scattering matrix, or161Port 1

V1+

Port 2

RAPC

V1

Cs

L

V2

V2+

Figure 5.12: Circuit diagram of the microwave measurement circuit as a two-port network. The RAP C , Cs , and L are contained within a network (blue box) with two-ports,
labeled “Port 1” and “Port 2.” These ports are connected to the outside world bycables with a characteristic impedance Z0 = 50 Ω. Voltage waves can travel along thesecables: a wave V1+ (V1− ) travels towards (away from) port 1 and similarly a wave V2+(V2− ) travels towards (away from) port 2.

S-matrix,− V1 









S12   V1+ 

 S11
=


V2−
S21 S22V2+(5.48)

The approaching and departing waves are carried on a cable with a characteristic
impedance Z0 = 50 Ω, and I make the simplifying assumption that these cables are
terminated in a 50 Ω load, in the form of an amplifier input impedance, signal generator source impedance, or other circuit element that can be treated as a 50 Ω load
termination. Therefore, there will not be any further reflections of V1− or V2− .
In this picture, changes in the resistance of the APC are detected by measuringchanges in the S-matrix coefficients. Section 5.1.1 and the previous subsection analyzed
the relationship between the voltage bias V1+ and the voltage at the amplifier V2− using
an analysis of the voltage and current through different circuit elements. However, thisis equivalent to measuring changes in the transmission coefficient S21 due to changes in
the APC resistance. Similarly, if an incident voltage wave V2+ is used then the same

162
amplifier can be used to measure changes in the reflection coefficient S22 due to changes
in the APC resistance. In fact, this defines the two different types of biases used in
the experiment: V1+ is a microwave voltage bias applied in transmission and V2+ is a
microwave voltage bias applied in reflection (figure 5.13).
Calculating the scattering matrix coefficients S21 and S22 for the network in figure5.12 can be done rather simply using the conventions and information in a microwave
engineering textbook (for example, [189]). The voltages and currents at the two ports

Z0

RAPC

Cs

directional
coupler

L
Z0

2V1+

-

V2

Z0

2V2+

Figure 5.13: Circuit diagram describing the microwave transmission measurement. In
reflection, a microwave voltage bias 2V1+ with a source impedance Z0 = 50 Ω is applied
to the right side of the two-port network (blue) and the transmitted voltage V2− ismeasured with an amplifier that has an input impedance Z0 = 50 Ω. In transmission,a microwave voltage bias 2V2+ with a source impedance Z0 = 50 Ω is applied to the left
side of the two-port network (blue) and, as in reflection, the transmitted voltage V2− ismeasured with an amplifier that has an input impedance Z0 = 50 Ω. The directional
coupler at port 2 separates incoming (green, V2+ ) and outgoing (red, V2− ) waves.

163
are related by an ABCD matrix  V1   1 + iωCs RAP C

=
I1
iωCs

RAP C + iωL + ω 2 Cs LRAP C
1 − ω 2 Cs L  V2 I2(5.49)

Using a general transformation which converts this ABCD matrix into a scatteringmatrix, the reflection coefficient S21 and transmission coefficient S22 are
2Z0
2
RAP C (1 − ω Cs L) + iω(L + Cs Z0 RAP C )RAP C (1 − ω 2 Cs L) + iω(L − Cs Z0 RAP C )S22 =RAP C (1 − ω 2 Cs L) + iω(L + Cs Z0 RAP C )S21 =(5.50)(5.51)

where Z0 = 50 Ω  RAP C is the impedance of the cables, amplifier, and voltage source.
The signal at the amplifier created by small changes in the APC resistance canbe calculated from the scattering coefficients:−2Z0 (1 − ω 2 Cs L + iωCs Z0 )∂S21=
∂RAP C[RAP C (1 − ω 2 Cs L) − iω(L + Cs Z0 RAP C )]2
2Z0
∂S22
=
∂RAP C[RAP C (1 − ω 2 Cs L) − iω(L + Cs Z0 RAP C )]2(5.52)(5.53)−
when a voltage bias is applied in transmission due to
The signal at the amplifier δV2T

a small change in APC resistance δR is therefore transmission
−
=
δV2T




∂S21
−2Z0 (1 − ω 2 Cs L + iωCs Z0 )δRδR V1+ =V1+ (5.54)∂RAP C[RAP C (1 − ω 2 Cs L) − iω(L + Cs Z0 RAP C )]2−
and the signal at the amplifier δV2R
when the voltage bias is applied in reflection is
−
δV2R
=




∂S22
2Z0 δRδR V2+ =V2+ (5.55)∂RAP C[RAP C (1 − ω 2 Cs L) − iω(L + Cs Z0 RAP C )]2−−
While these signals δV2R
and δV2T
are of different magnitude for the same mag-nitude incident voltage (V1+ and V2+ respectively), the resulting displacement measurements have the same imprecision and bandwidth. The voltage signals will be examinedin more detail below, with the conclusion that they are the same magnitude for thesame voltage VAP C across the APC. The voltage VAP C across the APC is the relevant
quantity because the physics of the APC itself constrains VAP C and the noise at the164amplifier SV tot is due to shot noise which is controlled by VAP C and amplifier noise
which is independent of bias. Therefore the bandwidth of the measurement, describedin the previous subsection, and the displacement imprecision are only dependent upon
the voltage VAP C across the APC and not dependent on whether VAP C is due to a
voltage bias applied in transmission V1+ or reflection V2+ .
−−To demonstrate that the voltage signals δV2R
and δV2T
are identical functions of

the voltage VAP C across the APC, I start by calculating the voltage VAP C across the
APC due to a bias 2V1+ or 2V2+ and then substitute VAP C for V1+ or V2+ in equations
T
5.54 and 5.55. From the circuit in figure 5.13, the voltage VAP
C due to a voltage bias

applied in transmission V1+ is+T
VAP
C = 2V1

RAP C (1 − ω 2 Cs L + iωCs Z0 )RAP C − ω 2 LCs RAP C + iω(L + Cs RAP C Z0 )(5.56)

R
assuming that Z0  RAP C . Similarly, from the circuit in figure 5.13, the voltage VAPC

due to a voltage bias applied in reflection V2+ is+R
VAP
C = 2V2RAP C
RAP C − ω 2 LCs RAP C + iω(L + Cs RAP C Z0 )(5.57)

assuming that Z0  RAP C .
These relationships between the voltage VAP C across the APC and the applied
voltage biases can be substituted into equations 5.54 and 5.55 to demonstrate that thevoltage signal due to a voltage bias in transmission or reflection is the same function of
VAP C . When the voltage bias is applied in transmission then signal at the amplifier asa function of the voltage across the APC is
−
δV2T
=−

T
VAP
2Z0 eiθδRC
RAP C RAP C (1 − ω 2 Cs L) − iω(L + Cs Z0 RAP C ) 2(5.58)and similarly when the voltage bias is applied in reflection then signal at the amplifier
as a function of the voltage across the APC is
−
δV2R
=

R
VAP
δR
2Z0 eiθC
RAP C RAP C (1 − ω 2 Cs L) − iω(L + Cs Z0 RAP C ) 2(5.59)

165
where in both cases
θ = 2 tan−1ω(L + Cs RAP C Z0 )RAP C (1 − ω 2 Cs L)(5.60)

Comparing equation 5.58 and 5.59, the magnitude of the signal due to small changes
δR in the resistance of the APC is the same function of the voltage VAP C across the
APC for a voltage bias applied in transmission and reflection.
Although the bandwidth and sensitivity of the displacement measurement are
independent of whether the voltage bias is applied in reflection or transmission, thetotal voltage at the input of the amplifier depends strongly on the type of bias. To
avoid saturating the amplifier, it is important to choose the type of bias that minimizes
the voltage at the amplifier for a given voltage VAP C across the APC, that is, for a given
measurement sensitivity.
Qualitatively, the appropriate type of bias can be understood by examining thecharacteristics of the resonant circuit. As was discussed in the previous section, theresonant circuit is used to create an effective amplifier with a large input impedance.
This input impedance is large over approximately the bandwidth of the electrical resonance ωe /Q ≈ 20 MHz and the maximum input impedance is at the electrical resonancefrequency where Za ≈ 50 kΩ (using the circuit values above). Then if the circuit is biased in reflection, there will be little reflection when the effective amplifier impedance
and the APC impedance are similar and no reflectance if they are the same. On theother hand, if the circuit is biased in transmission then the ratio of the voltage acrossthe APC to the voltage across the amplifier is going to be proportional to the input
impedance of the effective amplifier. Based on this qualitative understanding, it is best
to bias in reflectance when the APC resistance is not much bigger than the effective amplifier impedance. This occurs approximately when the frequency of the bias is within
the bandwidth of the electrical resonance up to some maximum APC resistance. Beyond this maximum APC resistance or at bias frequencies outside the bandwidth of the166electrical resonance, it is better to bias in transmission.
More quantitatively, the appropriate type of voltage bias, V1+ in transmission orV2+ in reflection, is the one that has the smaller ratio of the voltage at the amplifier V2−
to the voltage VAP C across the APC. When a voltage V1+ is applied in transmission andusing the previous results for the scattering matrix−
V2T= S21 V1+ =

2Z0
V+
2
RAP C (1 − ω Cs L) + iω(L + Cs Z0 RAP C ) 1(5.61)

When a voltage V2+ is applied in reflection, then−
= S22 V2+ =V2R

RAP C (1 − ω 2 Cs L) + iω(L − Cs Z0 RAP C ) +V
RAP C (1 − ω 2 Cs L) + iω(L + Cs Z0 RAP C ) 2(5.62)

Therefore, when the voltage bias is applied in transmission the ratio of VAP C (equation−is5.56) to the voltage at the amplifier V2T
−
V2T
Z0
=
T
2
R
(1
−
ω
Cs L + iωCs Z0 )
VAP C
AP C

(5.63)

and when the voltage bias is applied in reflection then the ratio of VAP C (equation 5.57)−
isto the voltage at the amplifier V2R−
V2R
RAP C (1 − ω 2 Cs L) + iω(L − Cs Z0 RAP C )=R
2RAP C
VAP
C(5.64)

−
−
R
T
/VAP
These two ratios, V2T
C and V2R /VAP C , are plotted in figure 5.14 versus the fre-quency ω of the bias voltage for the circuit values Cs = 200 fF, L = 500 nH, and
Z0 = 50 Ω near the center of the measurement bandwidth at 500 MHz.
Examining figure 5.14, the voltage at the input of the amplifier is minimized byapplying the voltage bias in transmission when the APC resistance is large RAP C >150kΩ or when the frequency of the bias is off resonance. However, when the APC
resistance is small RAP C < 150kΩ and the frequency of the bias is close to the electricalresonance frequency then the amplifier can be protected by applying the voltage bias inreflection. Figure 5.14 also highlights the importance of protecting the amplifier. For
example, if a large enough voltage was applied in reflectance to create VAP C = 50 mV167
V2- VAPC
0.100

0.050

0.020

0.010

RAPC = 20 kW
RAPC = 50 kW
RAPC = 150 kW
0.002
RAPC = 500 kW
0.005

0.001
470

480

490

500

510

520

530

Freq HMHzL

Figure 5.14: The ratio V2− /VAP C of the voltage at the amplifier to the voltage acrossthe APC versus the frequency of the applied bias for various APC resistances; thebias voltage is applied in reflection (solid lines) or transmission (dashed lines). This
calculation uses typical circuit values Cs = 200 fF, L = 500 nH, and Z0 = 50 Ωwhich implies that the measurement bandwidth, calculated in the previous section,is centered approximately around 500 MHz. The low-noise cryogenic amplifier canbe overwhelmed by large voltages, such as the worst case scenario on this graph ofmax ≈ 4 mV. Therefore the type of bias should be chosen to minimize
V2− = (0.08) × VAP
CV2− /VAP C based on the APC resistance and the frequency of the voltage bias.

at a frequency far from the electrical resonance, then the amplifier would experience a
voltage V2− > 1 mV at the input. This voltage is sufficient to saturate the low-noise
cryogenic amplifier. Finally, while I am not going to discuss the procedure in detail, it
is also possible to create a large VAP C and a negligible voltage V2− at the amplifier bysimultaneously applying voltages in transmission V1+ and reflection V2+ with the correct
phase and magnitude.
I have now described the dc measurement of APC resistance (subsection 5.2.2)
and the fast microwave measurement of changes in the APC resistance (subsection
5.3.1) which can be accomplished using a voltage bias applied in reflection or trans-

168

Low Freq
Measurement

VLF
Rb

Rapc

bias teeCs

L

directional
coupler
Z0

Z0

Z0

VR
VT

High Freq
Measurement

Figure 5.15: Diagram of measurement circuit, including both the low frequency measurement and high frequency measurement. The low frequency and microwave portionsof the circuit are separated using a bias tee; the inductor is open at high frequencies,
while the capacitor is open at low frequencies. Similarly, both sides of the beam (red)are shorted at low frequencies using an inductor, but the RAP C (between the red beamand green electrode) can be biased at high frequencies in transmission using VT . It isalso possible to bias the circuit at high frequencies in reflection using VR . For moredetails on the low frequency measurement see figure 5.7 and subsection 5.2.2, as well as
figure 4.2 for details around the APC. The high frequency transmission and reflection
measurements are described in more detail in figure 5.13.

169
mission (subsection 5.3.2). In the actual experiment, all of these measurements can be
performed simultaneously. While the details of the measurement circuit have evolved
over time, figure 5.15 contains a representative circuit diagram showing the combined
measurements.

5.4

Measurement of APC resistance at the Mechanical ResonanceFrequency
While the 500 MHz voltage measurement described in the previous section canbe used as a shot-noise limited displacement measurement, I would also like to makea shot-noise limited measurement of voltage signals around the nanomechanical resonance frequency. Such a voltage measurement can be used with a dc voltage to detectnanomechanical motion (baseband measurement, subsection 5.1.2) and, as describedin chapter 3, can be used to detect correlations between nanomechanical motion andshot-noise due to tunneling electrons at the nanomechanical resonance frequency. This
correlation is important because may be created by the momentum kicks of tunneling
electrons, the fundamental source of measurement backaction in the APC displacement
detector (chapter 3).
There are two obstacles to making the noise in a voltage measurement aroundthe nanomechanical resonance frequency dominated by shot-noise. First, it requires
a knowledge of the mechanical resonance frequency prior to the creation of the APC
at cryogenic temperatures. Second, it requires a low-noise amplifier at the mechanicalresonance frequency. In this thesis, the relatively large noise of the available amplifier
and the design of the measurement circuit prevent a shot-noise limited measurement
and make measuring correlations difficult.
The first obstacle, correctly choosing the electrical resonance frequency, can be
avoided using two strategies. First, the mechanical resonance frequencies can be predicted by using a combination of finite element modeling and knowledge of the mechan-

170
ical resonance frequencies of similar devices. The second more effective and complicatedstrategy is to use a mechanically controllable break junction (MCBJ, see chapter 4) tomeasure the mechanical resonance frequency before choosing the electrical resonancefrequency. Using a MCBJ, a constriction can be cooled to cryogenic temperatures,
stretched into an APC, used to measure nanomechanical motion, returned to almost
it’s original state, and then returned to room temperature. At room temperature, theelectrical resonance frequency can be adjusted to match the measured frequency ofnanomechanical motion. Then when the device is again cooled to cryogenic temperatures and a new APC is created using the MCBJ, the resulting mechanical resonancefrequency will match the electrical resonance frequency. This procedure can be repeatedmultiple times, though there is always the worry that contaminates are incorporated
into the constriction and APC.
In this project I was unable to overcome the second obstacle, of finding a lownoise amplifier at the mechanical resonance frequency, and the ability to measure correlations was severely limited by the relatively large noise of the amplifier. In general, thenanomechanical resonances which were most strongly coupled to the APC had resonant
frequencies below 100 MHz. Therefore a low-noise cryogenic HEMT (high electron mobility transistor) amplifier, like the amplifier used in the previous section at 500 MHz,cannot be used. HEMT amplifiers have 1/f noise which is small at 500 MHz but becomes unacceptably large at frequencies less than 100 MHz. Instead, I used a room
temperature amplifier with a 70 K noise temperature, implying a voltage noise with aspectral density of 2 × 10−19 V2 /Hz. This amplifier noise is about 30 times worse than
the 500 MHz cryogenic amplifier and makes it difficult to measure both shot noise at themechanical resonance frequency and correlations between the shot noise and mechanicaldisplacement.
I use the circuit in figure 5.16 to simultaneously measure voltage fluctuations
at a 70 MHz nanomechanical resonance using a relatively noisy amplifier and voltage

171

Low Freq (70 MHz)
Measurement

VR2
Z0

Z0

Low Pass Filter
L2

RAPC
Z0

VT

Cs

LPF
L1 C1

High Pass Filter

High Freq (500 MHz)
Measurement

HPF

Z0

Z0

VR1

Figure 5.16: Circuit diagram including both a low frequency (70MHz) measurement(blue box) and a high frequency (500 MHz) measurement (purple box). The two measurements are electrically separated using filters (LPF and HPF) and an additional
capacitance C1 : the purple box is an open circuit around 70 MHz and the blue box isan open circuit around 500 MHz.

172
fluctuations at 500 MHz using a low-noise cryogenic amplifier. The two signals are
separated by including a low pass filter on the 70 MHz measurement line and a highpass filter on the 500 MHz measurement line. The low pass filter turns the 70 MHz
measurement circuit (blue box, figure 5.16) into an open circuit at frequencies above
400 MHz, and therefore the 70 MHz measurement circuit can be ignored when analyzing
the 500 MHz measurement (purple box, 5.16). Similarly, the high pass filter (andcapacitor C1 ) allows the 500 MHz measurement circuit to be ignored when analyzing
the 70 MHz measurement.
The values of the circuit elements used in the measurement circuit (figure 5.16)
were chosen based on the desired electrical resonance frequencies and wide (approximately 5 MHz) electrical resonance at 70 MHz. The wide electrical resonance is required because the frequency of the nanomechanical resonance often varies by up to
10% and, in this case, is achieved by adding an additional capacitance to ground sothat Cs = 2.5 pF. As described in the previous section, this large capacitance will havethe undesired effect of decreasing the measurement bandwidth; a better solution, which
could be used in future experiments, would use a more complicated measurement circuit with multiple electrical resonances. The desired 500 MHz resonance frequency ofthe high frequency electrical measurement circuit and the additional series capacitance
C1 = 500 nF determines the inductance L1 = 150 nH; the series capacitance C1 is chosen to ensure that the high frequency measurement circuit is effectively an open circuit
at 70 MHz.
Using these circuit values, I analyze the 70 MHz and 500 MHz measurement
circuits separately to determine the minimum voltage VAP C across the APC that isrequired for the noise in each measurement to be dominated by shot noise. Fromequation 5.12, the noise in 70 MHz measurement using a dc voltage VAP C across the

173
APC will be dominated by shot noise when
|VAP C | ≥SV amp2
RAP C 21+2eΞζdc RAP C
Za2 (ω)(5.65)The noise of the effective amplifier SV amp2 created by the 70 MHz measurement circuitis
SV amp2 = SV0 amp2

Z02 + ω 2 L22
Z02(5.66)and Za2 (ω) is the impedance of the 70 MHz measurement circuit
Za2 (ω) =Z0 + iω(L2 − ω 2 L22 Cs − Cs Z02 )(1 − ω 2 L2 Cs )2 + (ωCs Z0 )2(5.67)

where SV0 amp2 = 2 × 10−19 V2 /Hz is the noise and Z0 = 50 Ω is the input impedanceof the 70 MHz amplifier. Similarly, the noise in the 500 MHz measurement using a
microwave voltage VAP C across the APC will be shot noise limited when
SV amp1
RAP C 21+2eΞζac RAP C
Za1 (ω)2
2(ωC1 Z0 ) + 1 − ω 2 C1 L1
0
SV amp1 = SV amp1
(ωC1 Z0 )2
1 − ω 2 L1 C1 + iωC1 Z0
Za1 (ω) =iω (C1 + Cs − ω 2 C1 Cs L1 + iωC1 Cs Z0 )|VAP C | ≥(5.68)(5.69)(5.70)

where SV0 amp1 = 5 × 10−21 V2 /Hz is the noise and Z0 = 50 Ω is the input impedance ofthe low-noise cryogenic 500 MHz amplifier.
In the previous section I used this type of inequality to calculate a measurement
bandwidth, however in this section I will calculate the frequency-dependent voltage
SN (ω) across the APC at which the amplifier noise and shot noise contribute equally
VAPC
SN is the minimum APC voltage whereto the measurement noise. In other words, VAP
Ceither equation 5.65 or equation 5.68 is satisfied. Using the values for the circuit comSN as aponent describe above and various values of APC resistance RAP C , I plot VAP
Cfunction of frequency ω for the 70 MHz and 500 MHz measurements (figure 5.17a,b).
SN as a function of frequency ω for the original 500 MHz
For comparison, I also plot VAP
C

174
SN

VAPC HmVL
1000

aL

700
500
300
200

RAPC = 20 kW
RAPC = 50 kW
100
RAPC = 150 kW
RAPC = 500 kW
150

50
64

66

68

70

72

74

Freq HMHzL76

SN

VAPC HmVL
1000
700

bL

500
300
200

RAPC = 20 kW
RAPC = 50 kW
100
RAPC = 150 kW
RAPC = 500 kW
150

50
400

450

500

550

600

500

550

600

Freq HMHzLSN
VAPC HmVL

100

cL

50

20

RAPC = 20 kW
RAPC = 50 kW
5
RAPC = 150 kW
RAPC = 500 kW
2
RAPC = 1.5 MW

10

1
400

450

Freq HMHzL

SN is plotted plotted versus frequency for a range of APC
Figure 5.17: The voltage VAPC
SN is applied across the APC, the contribution of shotresistances. When a voltage VAP
Cnoise and amplifier noise to the total measurement noise is equal; at APC voltages above
SN for thethe plotted line, the measurement noise is dominated by shot noise. Since VAP
Ccombined 70 MHz (a) and 500 MHz (b) measurement circuits (figure 5.16) is never lessmax = 50 mV, there is no frequency at whichthan the maximum stable APC voltage VAPCthe measurement noise is dominated by shot noise. On the other hand, the magnitude
SN for the 500 MHz measurement described in the previous section (c) indicatesof VAP
C
that the measurement noise is dominated by shot noise over a wide bandwidth and at
various APC resistances and voltages, as expected.

175
measurement analyzed in section 5.3 (figure 5.17c). In this figure, the measurement
bandwidth over which the measurement noise is dominated by shot noise can be deSN . Since the APC
termined by observing the frequency range over which VAP C ≥ VAPC
max = 50 mV, the maximum measurement bandwidth is theis unstable for VAP C > VAPCSN ≤ V max = 50 mV.
frequency range over which VAPCAP C

As expected, it is possible to make a shot-noise limited measurement using theoriginal 500 MHz measurement circuit (section 5.3, figure 5.17c), but there is no frequency range over which the measurement noise in the new 70 MHz measurement (figure
5.17a) or new 500 MHz measurement (figure 5.17b) is dominated by shot noise. The
new 500 MHz measurement is less sensitive because of the larger stray capacitance,
as expected. The large stray capacitance also decreases the sensitivity of the 70 MHz
measurement, but the main problem is the large noise of the 70 MHz amplifier. The
70 MHz amplifier’s voltage noise spectral density SV0 amp used in figure 5.17a is about
AP C ∝ S 0
30 times worse than that of the 500 MHz cryogenic amplifier. Since VSN
V amp ,

if I could use a 70 MHz amplifier that had the same noise as the 500 MHz cryogenic
SN in figure 5.17a would be scaled down by a factor of 30 and theamplifier then VAP
C

70 MHz measurement would have a shot noise bandwidth > 1 MHz over a wide rangeof APC resistances.
In conclusion, while I can attempt to measure the correlations between shot noiseat the nanomechanical resonance frequency and nanomechanical displacement using
the measurement circuit in figure 5.16, the measurement noise will not be dominated byshot noise and therefore the results are disappointing (see chapter 7). This measurement
could be improved in the future by using an amplifier that contributes less noise and amore complicated measurement circuit.

176

5.5

Using Shot Noise to Calibrate the Microwave Measurement
In this experiment, the shot noise of the atomic point contact is used to cali-brate the microwave measurement circuit. There are three steps in this process. First,
the shot noise is used to determine the frequency and APC resistance dependent gainG(ω, RAP C ) and noise SV amp (ω, RAP C ) of the measurement circuit. Second, a modelfor the measurement circuit, including the value of circuit components and a frequencydependent amplifier gain, is determined by fitting the calculated gain of the model
circuit to the measured gain G(ω, RAP C ). Finally, the magnitude of the microwave
voltage across the APC created by a microwave voltage bias is calibrated by comparing
the shot noise created by the microwave bias to the shot noise created by a known dc
voltage bias.
The measurement circuit’s gain G(ω, RAP C ) and noise SV amp (ω, RAP C ) are determined using the voltage dependence of the shot noise due to tunneling electrons. As

measured SV (μV2/Hz)

5
4
3
2
1

0
−15

−10

−5

0
VAPC (mV)

5

10

15

Figure 5.18: I plot the 500 MHz shot noise versus the dc voltage bias applied to the
APC (red dots) and a fit of the data to equation 5.72 (black line).

177
shown in chapter 3, the shot noise of electrons tunneling across the APC creates voltage
fluctuations with a spectral density



2e2
eVAP C2
2
SV sn (VAP C ) = 2
2kB T D + D(1 − D)eVAP C coth
RAP
C
h
2kB T

(5.71)

when the APC is at temperature T with a dc voltage VAP C across the APC and electrons
tunnel through a single spin-degenerate channel with probability D, that is, the APC
resistance RAP C = h/(2e2 D) > 13 kΩ. The total measured noise SV also includes
the noise of the amplifiers in the measurement circuit and is scaled by the gain of themeasurement circuit
SV (VAP C ) = G [SV amp + SV sn (VAP C )](5.72)

Since the shot-noise SV sn (VAP C ) has a distinct V-shape as a function of VAP C , the gain
and noise of the measurement circuit at a frequency ω can be determined by fitting
equation 5.72 to the measured noise SV at ω (figure 5.18).
An accurate circuit model for the measurement can be determined using thefrequency and APC resistance dependent gain G(ω, RAP C ). The approximate measurement circuit (figure 5.19) is known, however the exact values of the capacitances, circuitcomponents, and amplifier gain need to be determined. The shot noise SV sn will be
amplified by this measurement circuit with a gain
|G| = Ga (ω)

Ra0
Za (ω)
0iωL + Ra RAP C + Za (ω)(5.73)

where Ga (ω) is the frequency dependent amplifier gain, Ra0 = 50 Ω is the amplifier
impedance, Za (ω) is the impedance of the measurement circuit
Za (ω) =iωL + Ra0
1 − ω 2 Cs L + iωCs Ra0(5.74)and iωL is the impedance of the inductor including an inductance Li and shunting
capacitance Ci
iωL =

iωLi
1 − ω 2 Ci Li(5.75)

178

inductor L

Ci

RAPC SVsn

Li

VAPC

Vb

voltage
amplifier Ra

Cs

Figure 5.19: Approximate measurement circuit whose values Li , Ci , Cs , and amplifier
gain Ga are calibrated using shot noise.

Imperfections in this circuit model are included by making the amplifier gain Ga (ω, RAP C )a function of APC resistance as well as frequency ω. This gain Ga (ω, RAP C ) andthe value of Li , Ci , and Cs are found by fitting equation 5.73 to the measured gainG(ω, RAP C ); the difference between the measured gain and the gain of the model circuit is less than 3% over a 15 MHz bandwidth and almost two decades in APC resistance(figure 5.20).
The shot noise of tunneling electrons can also be used to calibrate the voltageac
VAP
C across the APC created by a microwave voltage bias. In general, this quantity isdifficult to calibrate because the attenuation of the microwave bias lines in the cryostat
at cryogenic temperatures cannot be measured easily. As described in section 5.1.1, theshot noise SVdcsn (equation 5.71) due to a dc voltage VAP C across the atomic point contact
is related to the average shot noise SVacsn due to a microwave voltage VAP C cos(ωac t)

Measurement Gain

179
2

10

1

10

Fractional Residuals

490

495
Frequency (MHz)

500

505

0.02
0
−0.02
490

495

500

Frequency (MHz)505

3863 kΩ1202 kΩ
740 kΩ
480 kΩ
299 kΩ
183 kΩ
112 kΩ
87 kΩ

Figure 5.20: Calibrated measurement gain as a function of frequency at different APC
resistances.

across the atomic point contact bySVacsn (VAP C ) =2 dc
S(VAP C )π V sn(5.76)

when eVAP C  kB T and the measurement frequency ω  eVAP C /h̄ and ω  kB T /h̄.
Examining the case of a microwave voltage bias VT cos(ωac t) applied in transmission,
the microwave voltage VAP C cos(ωac t) across the APC due to this microwave bias is
VAP C = AVT

RAP C
RAP C + Za (ωac )(5.77)

where A is the attenuation of the microwave lines and Za (ωac ) is the impedance of themeasurement circuit at the frequency of the microwave bias ωac . Since the impedance180
Za (ω) of the measurement circuit is known, the attenuation A can be found by comparing the shot noise due to a microwave bias VT to the shot noise due to a known dc
bias VAP C (equation 5.76, figure 5.21). A similar procedure can be used to calibrate a
microwave voltage applied in reflection.

measured SV (μV2/Hz)

3.5
3
2.5
2
1.5
1
0.5
0
0

50
100
microwave bias Vb (mV)

150

Figure 5.21: The 500 MHz shot noise is plotted versus the magnitude at the top of
the cryostat of a 437 MHz microwave voltage bias Vb . Using this shot noise data, I
determine that the microwave bias creates a voltage across the APC VAP C = 0.10Vb
which is consistent with the known attenuation in the cryostat.

Chapter 6

Sensitive Position Measurement using Electromigrated APC

In this chapter, I demonstrate a measurement of nanomechanical motion using an
atomic point contact (APC) with shot-noise limited imprecision and an unexpectedly
large backaction. I start by discussing the first measurements I made using an electromigrated APC. The resonance modes detected using this APC are compared to those
detected using magnetomotive detection, confirming that the measured changes in APC
resistance are due to nanomechanical motion. The sensitivity of the APC to nanomechanical motion is calibrated using the random thermal motion of the nanomechanicalbeam. The APC detector’s backaction is also determined by measuring the thermal
motion over a range of device temperatures. The properties of the APC displacement
detectors are compared to the quantum limits. While the imprecision is limited by shotnoise which is the fundamental source of noise in the APC detector, the APC backaction
is much larger than required by quantum mechanics.
I have also observed this unexpectedly large backaction in another electromigrated
devices. Using a similar calibration based on thermal motion, I observe a large randombackaction force that is proportional to shot noise and is not dependent upon whether
the shot noise is created by a dc voltage or microwave voltage across the APC.
I finally compare these experimental APC detectors to the phenomenologicalmodel described in chapter 3 in which the backaction force is caused by momentum
kicks from tunneling electrons. While the backaction force spectral density is propor-

182

a)

b)

c)

Figure 6.1: Scanning electron micrographs of nanomechanical structure prior to electromigration (a,b) and after electromigration (c). (a) large scale structure, with the gold
nanostructure in grey/white and the GaAs substrate in black. The nanomechanicalstructure is suspended, and the white outline seen on the gold (grey) electrodes showsthe extent of the undercut created by the etch. (b) prior to electromigration there is athin, narrow gold contact (black) created using a triple angle evaporation (chapter 5).
(c) after electromigration there is a clear gap between the triangular electrode and thenanomechanical beam. This APC was destroyed when a SMA connector broke at the
top of the cryostat. The resulting spark and changes as the system warmed to room
temperature caused a > 10 nm gap.

183
tional to the shot noise spectral density as in the model, the APC backaction is muchlarger than expected; each electron would have to deliver a momentum equal to about
thirty times the Fermi momentum.

6.1

Initial Demonstration of the APC Displacement detector
I first demonstrated the use of an APC as a detector of nanomechanical motion

using the nanomechanical structure in figure 6.1 and an electromigrated APC. The mechanical system is composed of a doubly clamped 5.6 µm long by 220 nm wide by 100 nm
thick nanomechanical beam next to an atomically sharp point. The total mass of thenanostructure is about 2.3 × 10−15 kg. The fabrication of the device and APC creation
using electromigration was described in chapter 4; in this case, the electromigration at
cryogenic temperatures resulted in an APC with a resistance of 33 kΩ.
Nanomechanical motion is measured using a single sideband measurement, with

Figure 6.2: The average resistance of the APC, measured using a dc measurement(right axis, red) and a microwave reflectance measurement (left axis, blue) is plotted
as a function of a Lorentz force (x-axis) applied by running a dc current through thenanomechanical beam. The resistance of the APC is varied from 3 kΩ to 380 kΩ anddepends on the history of the applied force (black arrows indicate increasing force fromlow resistances and decreasing force from high resistances).

184
the bias voltage applied in reflection (see chapter 3 for details). All of the nanomechanical measurements were taken at the initial electromigrated resistance of 33 kΩ before
the resistance of the APC was adjusted using a Lorentz force.
Unlike most electromigrated devices, the resistance of this electromigrated junction could be varied between 3 kΩ and 380 kΩ using a Lorentz force. A dc current
is passed through the nanomechanical beam (figure 6.1) in a 9 T magnetic field whichis perpendicular to the substrate. As the magnitude and orientation of the Lorentz
force is varied betwen ±50 nN, the size of the APC gap and thus the resistance of the
APC changes in a hysteretic loop (figure 6.2). This type of hysteretic loop is commonly
seen in mechanically controllable break junctions (MCBJ, see chapter 4), where a mechanically applied strain is used to vary the APC resistance instead of a Lorentz force,
however I am not usually able to control the resistance of electromigrated APCs. In
all other electromigrated devices, the current passed through the nanomechanical beam
melted the beam before the Lorentz force was large enough to appreciably change theresistance of the APC.

6.1.1

Comparison between APC detection and magnetomotive detection

In order to confirm that the APC detector is observing nanomechanical motion
and not a different mesoscopic effect, the resonant modes observed with the APC detector can be compared to the modes observed using magnetomotive detection. Magnetomotive detection involves measuring changes in magnetic flux created by nanomechanical motion (figure 6.3, references [14, 190]). The nanomechanical structure is fabricatedas part of a loop, and the loop is placed in a constant magnetic field. Nanomechanicalmotion changes the area of the loop which also changes the amount of flux enclosed by
the loop; this creates a voltage signal which is equal to the rate of change in magnetic
flux, Vsig = dΦ/dt.
This magnetomotive technique is generally used to detect nanomechanical dis-

185

Figure 6.3: Representative circuit diagram used with magnetomotive drive and detection. The network analyzer is used to pass an ac current I~ at a frequency near a~nanomechanical resonance through the nanomechanical beam (red) in a magnetic B~~~
field. The beam reacts to the Lorentz force per unit length F /l = B × I, and the
motion creates an electromotive force Vsig = dΦ/dt. This voltage signal mixes with theapplied voltage and is detected using the network analyzer. The network analyzer iseffectively measuring the transmission through a parallel RLC circuit, with a frequency
and damping determined by the mechanical resonance.

placement due to a magnetomotive driving force (figure 6.3, references [14, 190]). This
type of force is created by passing an ac current I~ = I~ac cos(ωac t) through the nanome~ creating a Lorentz force per
chanical structure in the presence of a magnetic field B,~ × I.
~ When using the combination of magnetomotive drive andunit length F~ /l = B
detection, as in figure 6.3 and below, the nanomechanical structure has an electricalresponse that is equivalent to a parallel RLC circuit at frequencies near a mechanicalresonance; the electrical RLC resonator has the same resonance frequency and width as
the nanomechanical resonance [190].
The magnetomotive technique is not sensitive to the mesoscopic details of electron186transport in the APC. Both sides of the APC are at the same potential as ground, so
there is no current flowing through the APC or voltage across the APC and the electricalcurrent flowing through the beam is not affected by the presence of the APC (figure
6.3).
The shape of the nanomechanical modes, which is influenced by the mechanicalproperties of the APC (see chapter 3), affects the sensitivity of the magnetomotive
and the APC measurements in different ways. The sensitivity of the magnetomotive
measurement to a specific mechanical mode depends on the change in the magnetic flux
enclosed by the measurement loop per unit motion. For example, if resonant motion
does not change the amount of magnetic flux enclosed by the measurement loop thenthe resonant motion will not be detected by the magnetomotive measurement.
On the other hand, the APC measurement is coupled to those resonant modes
which result in a strain at the APC; if the APC is located at a node of a mechanical resonance mode, then the APC measurement of displacement will not detect that
resonant mode. Since the coupling of the APC and magnetomotive measurements tonanomechanical motion is very different, only one type of measurement can detect somenanomechanical modes.
The magnitude of the magnetomotive driving force also depends upon the modeshape. As discussed above, the magnetomotive drive is a Lorentz force per unit length~ × I.
~ The Lorentz force will couple strongly to a nanomechanical mode whenF~ /l = B
the mode shape has a large average displacement in the same direction as the applied
force (see [14, 190] for details). This implies that the modes which couple weakly to themagnetomotive detection will also couple weakly to the magnetomotive drive.
The APC and magnetomotive measurements detect similar resonance modes, confirming that the APC measurement is detecting nanomechanical motion. In figure 6.4
I plot the response to a magnetomotive drive at frequencies near the resonances observed at frequencies below 60 MHz; the response is detected using the magnetomotiveMagnetomotive detection

187

20 kHzMicrowave APC Detection

Figure 6.4: Measure response amplitude (blue) and phase (red) to a magnetomotive
drive as a function of drive frequency for the observed resonance modes; the widthof the black bar in each graph is 20 kHz. In the magnetomotive detection (top), themechanical response is mixed with a background response resulting in a complicatedline-shape. Using the APC detector (bottom), the background response is small enoughthat I generally observe the expected Lorentzian line-shape and 180◦ phase shift. The
18.4 Mhz mode is the only mode that is not detected by both the magnetomotive
detection technique and the APC detector.

technique (figure 6.4, top) and APC technique (figure 6.4, bottom). Five resonances areobserved using the APC technique at frequencies between 18.4 MHz and 57.2 MHz. The
same four higher frequencies modes are also observed using magnetomotive detection,
confirming that the resonances are due to nanomechanical motion.
The lowest frequency resonance detected using the APC measurement is not detected by the magnetomotive measurement; this can be explained by examining themode shapes calculated using finite element modeling (figure 6.5, see chapter 3 for details). The magnetic field is applied in the vertical ẑ direction so the magnetomotive
measurement does not couple to nanomechanical motion in the x̂ or ẑ directions. Ex-

188
amining the mode shapes, the four higher frequency modes include motion in the ŷ
direction that couple to the magnetomotive measurement through a change in magnetic
flux. On the other hand, the lowest frequency mode contains motion that is almost entirely in the x̂ direction and is also reasonably symmetric. Such motion does not change
the magnetic flux through the magnetomotive measurement loop, so it is not surprisingthat the magnetomotive measurement is unable to detect this lowest frequency mode.

18.4 MHzz39.2 MHz

y
x

APC

APC

APC

43.1 MHz54.2 MHz

APC

57.2 MHz

APC

Figure 6.5: Finite element simulations of the five lowest frequency nanomechanicalmodes. The finite element model (FEM) was designed so that the FEM frequencies
match the measured mode frequencies (figure 6.4). In the finite element model, the
APC is represent by a spring attached at the green dot (labeled, in this orientation
at the bottom left side of the middle of the beam). The APC detector is sensitive tomotion that changes the length of the APC spring, that is, motion at the green dot. The
magnetomotive detection technique is sensitive to average motion in the ŷ direction.

6.1.2

APC detector calibration using thermal noise

I use the Brownian motion of the nanomechanical beam to calibrate the APC
detector and determine the measurement backaction and imprecision. Nanomechani-

189
cal displacement is detected by applying a microwave voltage to the APC; changes in
the APC resistance will then result in a voltage signal at the input of the microwave
amplifier (see chapter 5). The frequency and magnitude of the nanomechanical motion
are determined from this voltage signal. Relating the frequency of the voltage signal
ωs to the frequency of nanomechanical motion ωx is simple; the frequency of the voltage signal is detuned from the microwave voltage frequency ωb by the frequency of thenanomechanical motion, that is, |ωx | = |ωb − ωs |.
Determining the magnitude of nanomechanical displacement from the magnitudeof the voltage signal is more difficult and requires a known displacement. As described inchapter 5, the calibration factor dx0 /dVs that relates a small voltage signal Vs to a smallchange in the APC gap x0 is determined by the magnitude of the voltage bias VAP C
across the APC, the resistance RAP C of the APC, the impedance Za of the measurement
circuit, and the length scale λ that controls the probability of electrons tunneling acrossthe APCdx0
1
λ RAP C + Za
=−
dVs
VAP C
2Za
Gamp Λ

(6.1)

where Gamp is the gain of the amplifier chain used in the measurement and Λ = 1/2
for a single sideband measurement (see chapter 5). Each mode of the nanomechanicalstructure is effectively a simple harmonic oscillator (SHO) and the motion x of thatSHO results in a change in the APC gap x0 = χx, where the coupling χ is different foreach mode and depends upon both the mode shape and the details of the APC (seechapter 3). Using this coupling χ, the voltage signal Vs is related to the motion x of ananomechanical mode bydx
1 dx0
λ RAP C + Za
1
=
=−
dVs
χ dVs
χVAP C
2Za
Gamp Λ

(6.2)

While it is possible to measure the resistance RAP C and both VAP C and Za using theshot noise calibration described in section 5.5, the length scale λ and coupling χ cannot

190
be directly measured. Instead, I measure the voltage signal due to a known displacement
to calibrate the measurement and determine dx/dVs .
The known displacement is provided by the Brownian motion of the beam. Because each nanomechanical mode is effectively a SHO coupled to a heat bath at temperature Tbeam , each mode experiences a random force with a spectral density SFT =4mγkB Tbeam where m is the effective mass of the SHO and γ is the SHO damping
constant. This random force creates random displacement with a spectral density
SxT =

SFT
1
m2 (ω02 − ω 2 )2 + γ 2 ω 2(6.3)

The integrated mean squared displacement due to the random force is(∆xT )2 =

Z ∞
0

SxT

dω
=
2π

Z ∞
0

SFT
dω
1
kB Tbeam
=
2
2
2
2
2
2
m (ω0 − ω ) + γ ω 2π
mω02

(6.4)

This displacement is consistent with the equipartition theorem; since each mode is effectively a SHO with two degrees of freedom, the average potential energy is mω02 (∆xT )2 /2 =kB Tbeam /2.
I use the known Brownian motion to determine the calibration factor dx/dVs
which is used to infer nanomechanical displacement from the measured voltage signal.
Near a nanomechanical resonance, the amplifier noise and shot noise will contribute a
white background with a spectral density SVn to the measured voltage spectral density
SVm (figure 6.6). This white background of apparent motion is simply the imprecisionin the measurement. The Brownian motion of the nanomechanical structure creates asignal which is added to the noise whose voltage spectral density SVs has a Lorentzianline shape

SVs =

dx
dVs

−2

SFT

m2 (ω02 − ω 2 )2 + γ 2 ω 2


(6.5)

The integrated strength of the Lorentzian signal is2

Z ∞(∆Vs ) =
0

dω
SVs
=
2π



dx
dVs

−2

SFT
4m2 γω02

(6.6)

191

VAPC=26 mV

Tcryo =10 K

0.5

40
Sx (fm2/Hz)

0.6

T

cryo

=5 K

0.4

SVs (μV2/Hz)

50

30

0.3

Shot
Noise
0.2
Amplifier
Noise
0.1

20
10
0
43.06

Imprecision
43.08
43.1
43.12
Mechanical Frequency (MHz)

43.14

0

Figure 6.6: The spectral density of the fluctuations in the voltage signal (right axis) areplotted as a function of frequency near one of the nanostructure’s mechanical resonances
when the cryostat is at 10 K (red) and 5 K (blue). The Lorentzian peaks in the noiseare due to the Brownian motion of the beam. Using the equipartition theorem andthe integrated strength of the peaks, the y-axis can be calibrated in displacement units(left axis). The Lorentzian peaks are on top of a white background; this apparent
motion is the measurement imprecision (Sx = 5.3 fm2 /Hz for VAP C = 26 mV) due to a
combination of shot noise (green) and amplifier noise (grey).

In the simple case where there is no measurement backaction, the temperature Tbeam of
the nanostructure will be equal to temperature Tcryo of the cryostat and the calibration
factor can be determined from Tcryo and the integrated mean squared (∆Vs )2 voltage
signaldx=
dVsskB Tcryomω02 (∆Vs )2(6.7)

However, I need to include the measurement backaction to accurately describe theexperimental system which complicates the calibration. The effect of the measurementon the nanomechanical system can be divided into two parts. First, the measurement

192
can effect the nanostructure by increasing the temperature of the bulk material. The
microwave current passing through the APC dissipates power in the nanostructure.
This power can raise the temperature of the nanostructure above the temperature of
the cryostat by an unknown amount. Experimentally this effect spoils the calibration
based on Brownian motion, but it is only a problem at low cryostat temperatures < 4 K,
depending upon the dissipated power. This problem is avoided, as discussed in moredetail below, by ignoring data taken at cryostat temperatures where the dissipatedpower has begun to effect the temperature of the nanostructure.
Second, the measurement will necessarily create a random force on the nanostructure which depends on the precision of the measurement (chapter 3). This typeof backaction enforces the Heisenberg Uncertainty Principle, and, unlike the first effect, is interesting in the context of quantum limited displacement measurements. I
use a phenomenological model for the measurement backaction (chapter 3) where themeasurement backaction is a temperature independent, APC voltage bias dependent,
random force with a white spectral density SFBA . This backaction can be parameterized
in temperature units, TBA = SFBA /(4mγkB ), and the mean squared Brownian motion
will be the same as if the nanostructure was at a temperature Tbeam = Tcryo + TBA .
To determine both SFBA and the calibration factor dx/dVs for a given bias VAP C ,it is necessary to collect data at multiple cryostat temperatures Tcryo . Experimentally,
I determine the calibration factor and backaction force of the nanomechanical measurement at four different microwave bias voltages VAP C , or gains, between 4 mV and 39 mV
by measuring nanomechanical motion at cryostat temperatures between 300 mK and
10 K. At each temperature and bias voltage, I measure the fluctuating voltage signal
near the mechanical resonance (figure 6.7) and determine the mechanical resonance frequency ω0 , damping constant γ, and integrated mean squared voltage signal (∆Vs )2 .
In figure 6.7 I plot (∆Vs )2 as a function of cryostat temperature Tcryo for four different APC bias voltages. For a given APC bias voltage, when the temperature of the

0.4

4

0.2

4

6
Tcryo (K)

8

10

0.8

(Δ Vs)2 (mV2/103)(Δ xT)2 (pm2)

0.2

2

4

6
Tcryo (K)

8

10

1
0.5

0
−1 0

0.4

0.4

0
−1 0

0

0.6

0.6

1.5

0.2

0.8

0.8 VAPC=6 mV

2

0.4

2
2

2.5

0.6

2

4

6
Tcryo (K)810

VAPC=3 mV

0

0.15

0.6

(Δ xT)2 (pm2)

0
−1 0

3

0.1

0.4

0.2

0.2

0

0
−1 0

s

(Δ xT)2 (pm2)6

VAPC=12 mV(Δ xT)2 (pm2)

8

0.6

0.8

(Δ Vs)2 (mV2/103)

10(Δ V )2 (mV2/103)

VAPC=26 mV(Δ Vs)2 (mV2/103)

0.8

193

3.5

0.05

2

4

6
Tcryo (K)

8

10

0

Figure 6.7: Integrated strength (∆Vs )2 (data points) as a function of cryostat temperature Tcryo at four different microwave bias voltages. Each plot includes a fit line (solidline) to a phenomenological model which assumes a temperature-independent backaction force SFBA . The data clearly deviates from this model at low Tcryo and high VAP C ,so only the solid data points are included in the fit. This line is not simply proportional to Tcryo , as expected from the equipartition theorem, because the width of theresonance γ changes by 30% between 10 K and 250 mK. Based on this fit the y-axis
can be calibrated in displacement units (left axis) and I include a line (dashed blackline) which shows the expected integrated strength based on the equipartition theorem
in the absence of backaction.

nanostructure is equal to the temperature of the cryostat then2(∆Vs ) =



dx
dVs

−2

kB
mω02



SFBA
Tcryo +
4mγkB

(6.8)

and the integrated strength (∆Vs )2 can be fit using linear fit coefficients a and b(∆Vs )2 = a

kB Tcryo
b
+
2
2
mω0
4m ω02 γ

(6.9)

194
The fit variables are used to determine the calibration factor dx/dVs and the backaction
force SFBA
dx
=
dVs

r

1
a
b
SFBA =
a(6.10)(6.11)The observed backaction force SFBA at each of the four different APC voltage biases isgiven in table 6.1.

1

0.08

0.8
0.6

0.04

0.4

0.02

0.2
0
0

T

TBA (K)(Δ x )2 (pm2)

0.06

2

4

6
Tcryo (K)

8

10

0

Figure 6.8: Backaction temperature TBA = SFBA /(4mγkB ) (left axis) plotted versus
cryostat temperature Tcryo at four different microwave voltage biases: 26 mV (red),
12 mV (magenta), 6 mV (green), and 3 mV (blue). The error bars for the 3 mV data
(blue) are not shown, but are about twice the size of green error bars. The backaction
temperature is proportional to the difference (right axis) between the expected integrated strength due to the equipartition theorem (dashed black lines in figure 6.7) andthe measured integrated strength (data points and solid fit line in figure 6.7). The fit
lines (solid lines) are dependent on temperature because the fit model assumes a constant backaction force SFBA and the width of the resonance γ changes by 30% between
10 K and 250 mK. The hollow data points are excluded from the fit because local heating
causes the temperature of the nanostructure to deviate from the cryostat temperature.

195
In performing the fit to equation 6.9, I ignore the low temperature data points
where the temperature of the nanostructure deviates from the temperature of the cryostat. As discussed above, the local temperature of the nanostructure is increased by
the power dissipated in the APC. The effect of deviation of the local temperature fromthe cryostat temperature is observed in figure 6.7, where the measured values (dots)of the integrated strength deviate from the expected value based on the equipartition
theorem and the cryostat temperature at large bias voltages and low temperatures. Thisdeviation is even more apparent in figure 6.8, where the backaction temperature TBA(∆Vs )2 mω02b
TBA =− Tcryo =akB
4mγkB a(6.12)is plotted as a function of the cryostat temperature. In this figure the solid data points
are used in the fit to equation 6.9 (solid lines), while hollow data points are excluded
from the fit because the temperature of the nanostructure deviates from the cryostat
temperature. As expected, the temperature at which the dissipated power begins tohave a measurable effect decreases as the microwave power decreases.
Having calibrated the detector and determined the backaction force, determining
the detector imprecision Sx in displacement units is simple. I have already measured
the detector imprecision in voltage units; this imprecision is the white background SVnobserved when measuring the magnitude of voltage fluctuations near a nanomechanicalresonances (figure 6.6) and is composed of contributions from the shot noise and theamplifier noise. The voltage fluctuations in figure 6.6 are calibrated in displacement
units using dx/dVs , and the displacement imprecision Sx is simply the magnitude of thewhite background noise in displacement units
dx 2
Sx = SVndVs(6.13)The observed imprecision Sx at each APC voltage bias is shown in table 6.1.
I use the shot-noise contribution to the displacement imprecision Sxsn to determine the effective length scale λ/χ which sets the scale of the APC resistance’s expo-

196
APC voltage bias
2.6 mV
5.5 mV
12 mV
26 mV

Displacement Imprecision Sx
126.8 ± 5 fm2 /Hz
36.1 ± 1 fm2 /Hz
11.29 ± 0.5 fm2 /Hz
5.33 ± 0.2 fm2 /Hz

Backaction Force SF
2800 ± 2000 aN2 /Hz
2800 ± 2000 aN2 /Hz
3900 ± 3000 aN2 /Hz
6100 ± 3000 aN2 /Hz

Table 6.1: The observed imprecision and backaction of a displacement detector createdusing an electromigrated APC with resistance RAP C =33 kΩ and various APC bias
voltages VAP C . This detector is used to measure the 43 MHz mode of the nanostructure
shown in figure 6.1.

nential dependence on nanomechanical displacement x, RAP C = R0 exp [χx/λ]. For thesame microwave bias and RAP C , a smaller length scale will result in a displacement measurement with a smaller imprecision. The length scale is calculated from the shot-noise
contribution Sxsn to the displacement; from chapter 5,"2 # 
Zadx 2
Sxsn = SVsn
RAP C + Za
dVs eζΞRAP C λ 2
Sxsn =
2|VAP C |Λ2 χ(6.14)(6.15)For the 33 kΩ device and 43 MHz mode, ζ = 2/π because the shot noise is due to a
microwave bias and Λ = 1/2 because the measurement is single sideband. Solving forthe effective length scale,ss2
λ2|VAP C |Λπ|VAP C |= Sxsn= Sxsnχ
eRAP C ζΞ
4eRAP C Ξ

(6.16)

and for the 33 kΩ APC detector and 43 MHz mode the effective length scale λ/χ = 3 nm.
The effective length scale λ/χ = 3 nm is larger than in a standard STM because
the APC resistance is relatively small, RAP C = 33 kΩ, and χ, the coupling between
motion at the APC gap and nanomechanical displacement, is less than 1. At large
tunneling resistances, RAP C > 100 MΩ, the length scale in a standard STM is aboutλ/χ = 0.1 nm. In a STM, χ = 1 because the APC support structure is much stiffer thanthe effective spring constant of the APC so λ = 0.1 nm. This length scale is consistent197
with electrons tunneling through a barrier equal to the work function of bulk gold√
φ = 5 eV (λ = h̄/ 2me φ, see chapter 3). However, as the resistance of the tunnelingcontact in an STM is decreased, the height of the tunneling barrier also decreases. This
break-down in the apparent height of the tunnel barrier results in a larger length scale
λ and contributes to the large observed λ/χ = 3 nm in the RAP C = 33 kΩ APC.
The coupling constant χ = x0 /x that relates displacement of the nanomechanicalmodal coordinate x to changes in the APC gap x0 also contributes to the large observed
λ/χ = 3 nm. Based on a finite element simulation of the 43 MHz mode shape (seechapter 3) and assuming that the strain caused by nanomechanical motion on the atomic
scale at the APC gap is equal to the macroscopic strain near the APC, I estimate
χ ≈ 0.5 and therefore λ ≈ 1.5 nm. This APC displacement detector is about an orderof magnitude less sensitive to changes in the APC gap as is possible with a standard
STM, assuming both measurement imprecisions are dominated by the shot noise oftunneling electrons.

6.1.3

The Heisenberg uncertainty principle and the standard quantum
limit

The Heisenberg uncertainty principle limits both the product of the imprecision
Sx and backaction SFBA and, for a harmonic oscillator, the minimum total imprecisionin the measurement Sxtot . I consider the simple case of a continuous linear amplifierof position with uncorrelated imprecision and backaction. In this case, the Heisenberg
constraint is
Sx SFBA ≥ h̄2(6.17)

where I continue to use a one-sided spectral density; if a two-sided spectral density isused, then Sx SFBA ≥ h̄2 /4.
When measuring the position of a harmonic oscillator, the Heisenberg uncertainty

198
principle also limits the total displacement imprecision Sxtot ; the minimum total imprecision is called the standard quantum limit. The random backaction force with spectral
density SFBA results in random displacement with a spectral density at frequency ω
SxBA =

SFBA1
m2 ω 2 − ω 2 2 + γ 2 ω 2
0(6.18)The total displacement imprecision Sxtot is due to the sum of the imprecision Sx andbackaction SxBA
Sxtot = Sx + SxBA = Sx +

SFBA
1

2
m ω2 − ω2 2 + γ 2ω20(6.19)To be clear, while both Sx and SxBA cause displacement noise, Sx is due to voltage
noise added by the measurement process and is not due to actual beam motion. The
backaction contribution SxBA is due to actual beam motion, and creates noise in themeasurement that is very similar to the noise created by Brownian motion. Using theHeisenberg constraint 6.17, the total displacement noise is
Sxtot (ω) ≥ Sx +h̄21m2 Sx ω 2 − ω 2 2 + γ 2 ω 2
0(6.20)

and at the mechanical resonance ω = ω0
Sxtot (ω0 ) ≥ Sx +1h̄2
Sx m2 γ 2 ω02(6.21)At the mechanical resonance ω = ω0 the minimum total imprecision, also called thestandard quantum limit (SQL)
Sxtot (ω0 ) ≥

2h̄
mγω0

(6.22)

occurs when the contributions to the total noise from the imprecision and backaction
are equalh̄mγω0(6.23)

SFBA = SFBASQL = mγh̄ω0(6.24)

Sx = SxSQL =199
When comparing Sx and SFBA to each other and the standard quantum limit, it isconvenient to express both quantities in terms of energy quanta. The backaction force
SFBA increases the energy of the beam by an amount, in quanta,
NF =

SFBA
4mγh̄ω0(6.25)

Similarly, at the mechanical resonance ω0 the noise in the measurement results in an
apparent increase in energyNx =Sx mγω04h̄(6.26)

1
16(6.27)In quanta, the Heisenberg constraint is
Nx NF ≥and the total imprecision
Ntot = NF + Nx ≥1
2

(6.28)

is minimized at the standard quantum limit where
Nx = NF = NSQL =

6.1.4

1
4

(6.29)The experimental APC detector compared to the quantum limits

In this subsection I compare the experimental results from the APC detector
described at the beginning of this section to the the limits imposed by the Heisenberg
uncertainty principle, as described in the previous subsection. In figure 6.9 I plot themeasured imprecision Nx and backaction NF as function of APC bias voltage. At thelargest APC voltage bias, the imprecision is limited by shot noise, the fundamental√
source of noise in an APC detector, and the imprecision Nx = 45/2 is 45 times the√standard quantum limit. At the same voltage bias, the backaction force NF = 38/2
is 38 times the standard quantum limit.

70

100

150 200

300 400

600

900

Amplifier Noise4

10
N (quanta)

200

I APC (nA)

Imprecision
3

10

Shot Noise

2

Backaction

10

2

5

10
VAPC (mV)

20

30

Figure 6.9: Measurement imprecision quanta Nx (red) and backaction quanta (black)as a function of APC voltage VAP C (top axis) or current IAP C (bottom axis). Boththe amplifier noise (blue line) and shot noise (green line) contribute to the imprecision(red line), which is dominated by shot noise at the largest APC bias voltage. The total
imprecision Nx + NF is approximately minimized at the largest APC bias voltage whereNx ≈ NF . The uncertainty (error bars) in the imprecision and backaction is dominatedby the uncertainty in the local temperature of the nanostructure.

APC voltage biasq
Sx SFBA

imprecision Nx

backaction NF

total Ntot

2.6 mV
5.5 mV
12 mV
26 mV
detector at SQL

6000h̄
3000h̄
2000h̄
1700h̄
h̄

54000/4
15000/4
4400/42050/4
1/4

600/4
600/4
900/4
1500/41/427000/2
8000/22700/2
1800/2
1/2

Table 6.2: The observed imprecision-backaction product, imprecision quanta, backaction quanta, and total imprecision quanta of a displacement detector created using
an electromigrated APC with resistance RAP C =33 kΩ and various APC bias voltages
VAP C . This detector is used to measure the 43 MHz mode of the nanostructure shownin figure 6.1. The observed values can be compared to the values of a quantum-limited
detector operated at the standard quantum limit (last line of the table).

201
The imprecision-backaction product at the highest APC bias voltage√

Nx NF =

1700/4 is 1700 times the minimum required by the Heisenberg uncertainty principle√
Nx NF ≥ 1/4. Unlike an ideal detector, this product also becomes larger as the bias
voltage is decreased (table 6.2) because the amplifier noise becomes significant (blueline, figure 6.9).
Since the noise in the APC detector at the highest APC bias voltage is limited byshot noise, the large imprecision-backaction product is due to a large backaction force.
The APC detector is theoretically predicted to be an ideal, Heisenberg limited detecq
tor Nxf und NFf und = 1/4 (see chapter 3). Since the Heisenberg uncertainty principle
does not individually constrain Nx or NF , determining whether Nx , NF , or both are
non-ideal requires understanding the source of measurement imprecision or backaction.
In the APC detector, the fundamental source of noise is the shot noise of tunneling
electrons. At the highest APC voltage, the shot noise accounts for 70% of the measuredq√
Nx , so Nx = 1.2 Nxf und . Since the observed imprecision is close to the theoretical expectation, the observed backaction force is significantly larger than required byq√quantum mechanics, NF = 1400 Nxf und . In the next section I will focus on thislarge backaction force, examining possible sources of the force in the context of thephenomenological model described in chapter 3.
Finally, I have operated the detector near the APC bias voltage which mini√√
mizes the total displacement imprecision Ntot = Nx + NF . This minimum occurs
when Nx = NF , so the total displacement uncertainty at the highest APC voltage bias√√
Ntot = 42/ 2 is close to the minimum achievable Ntot . The total imprecision at thehighest APC voltage bias is 42 times the standard quantum limit, and this Ntot is near
the minimum achievable Ntot and yet still significantly above the standard quantum
limit because the backaction force is significantly larger than that required by quantum
mechanics.

202

6.2

Another Similar Electromigrated Device
I have observed a similar large backaction force in another electromigrated APC.

This APC had a resistance of 16 kΩ, and was created as part of the nanomechanicalstructure in figure 6.10, composed of a doubly-clamped nanomechanical beam (6.3 µm
long by 200 nm wide by 90 nm thick) next to the APC. Nanomechanical motion is detected using a single sideband measurement with the bias voltage applied in transmission(see chapter 3 for details).
This device was initially electromigrated to a resistance of 71 kΩ and later jumped
to 16 kΩ. Using a dc Lorentz force, as described in the previous section, it was possible
to hysteretically modify the resistance between 75 kΩ and 90 kΩ using a force ±30 nN.
Later, the device resistance irreversibly jumped to 16 kΩ while applying a driving force
using a voltage across the APC. The measurements described in this section where taken
while the APC resistance was 16 kΩ. The device was finally destroyed while attempting
to modify the resistance.
I determine the frequencies of the nanomechanical modes by measuring the mechanical displacement in response to an electrostatic driving force. The electrostaticforce FC is created by applying a voltage Vg to the electrostatic gate (figure 6.10 and
6.11). The nanomechanical structure experiences a force
FC =1 dCg 2V
2 dx g(6.30)

where x is the displacement of one of the nanostructure’s modal coordinates and Cg isthe capacitance between the nanostructure and the electrostatic gate. I usually apply
both a dc voltage and an ac voltage at frequency ωg , Vg = Vgdc + Vgac cos(ωg t), to thegate.
Using this type of voltage bias, Vg = Vgdc + Vgac cos(ωg t), simplifies the displacement measurement in two different ways. First, it results in a force FC and displacement

203

0

o

100nm

40

o

1μm
100nm

1μm

Figure 6.10: Scanning electron micrograph (SEM) of nanomechanical structure prior
to electromigration, with the narrow gold structure in white/grey suspended over the
GaAs substrate in black. The APC will be created at the thin narrow constriction
between the nanomechanical beam and the triangular electrode shown in the middle
SEMs. The constriction is created using a triple-angle evaporation (chapter 5). The
two upper left SEMs are taken from directly above the nanostructure. The two lower
right SEMS are taken at a 40◦ angle and show that nanomechanical beam is suspended
above the substrate.

204

Vg = Vdc+ Vac
F=Cg( ( C x ) V + )V
dc

g

ac

x+ , x-

8V

Vdc =electrostaticgate

2xC = x + - x -0V

2xV = x + + x -

8 Vdc

8 Vdc

2 Vdc

2 Vdc

Figure 6.11: A gate electrode (top left, scanning electron micrograph) is voltage biased
to create both an electrostatic force and a piezoelectric force. The electrostatic force isdue to the change dCg /dx in the capacitance Cg between the gate and the nanomechanical structure caused by nanomechanical motion x. Since the GaAs substrate is piezoelectric, a sinusoidal voltage applied to the gate electrode will launch surface acoustic
waves onto the surface of the chip, shaking the nanomechanical supports and creating a
detectable frequency dependent force F = σVac . The complicated, frequency dependent
response to a Vac + Vdc voltage applied to the gate electrode (upper right) can be split
into an electrostatic response (lower left) and a piezoelectric response (lower right) byseparately measuring the total response with both a positive dc voltage, x+ , and negative dc voltage, x− and either subtracting (electrostatic response xC = (x− − x− )/2) oradding (piezoelectric response xV = (x+ + x− )/2) the two responses.

205
at the frequency ωg of the ac voltage
"
#
2

V
dCg
1 dCg
gac
2
FC =
Vgdc
+
1 + cos(2ωg t)2
Vgdc Vgac cos(ωg t) +
dx
2 dx
2

(6.31)

where the displacement due to the dc force and the force at 2ωg are usually negligible.
This force at ωg is an advantage, because a commercial network analyzer can be used
to apply the ac voltage at ωg and measure the magnitude and phase of the resulting
mechanical displacement at ωg . Measuring the magnitude and phase of the response at
2ωg due to an ac signal at ωg is more difficult. Second, it is possible to separate theelectrostatic force from any force that is proportional to voltage FV = σVg (such as apeizoelectric force) by measuring the mechanical response x± (ω) to both a positive andnegative dc voltage ±Vgdc . Changing the sign of the dc voltage will change the phaseof the electrostatic force FC by 180◦ but will not affect the phase of FV . Therefore,
the electrostatic response to +Vgdc is xC = (x+ − x− )/2 and the response to FV is
xV = (x+ + x− )/2 (see figure 6.11).
Using this electrostatic drive, I observed mechanical resonances at 10.46 MHz,
16.29 MHz, 33.82 MHz, 40.69 MHz, 40.93 MHz, 53.31 MHz, and 55.15 MHz. These
resonance frequencies are lower frequency than those observed in the previous section.
This frequency shift is expected, since the nanomechanical beam in this nanostructureis 20% longer. The 34 MHz mode was most strongly coupled to the APC measurement.
In the rest of this section I will concentrate exclusively on this 34 MHz mode.

6.2.1

Improved thermal noise calibration

In order to determine the 16 kΩ APC detector’s displacement sensitivity andbackaction I use a calibration based on the thermal noise. The basic calibration technique is described and applied to the 33 kΩ APC detector in section 6.1 above. The
calibration of the 16 kΩ detector is slightly more complicated. I make two modifications
to the basic technique which improve the calibration by removing systematic errors.

206
First, I measure the displacement in response to an applied electrostatic force as wellas the spectral density of the Brownian motion. The response measurement is used tocompensate for changes in the measurement gain Gamp . Second, I use the temperaturedependence of the mechanical resonance frequency to determine the effective temperature of the nanostructure. Using this effective temperature instead of the cryostat
temperature compensates for the local heating caused by the power dissipated in thenanomechanical structure due to the measurement voltage bias.
I use the measured response to an electrostatic force to compensate for drifts inthe measurement gain Gamp . As described in the previous subsection, the electrostaticgate is used to impose a force
FC =1 dCg 2V
2 dx g(6.32)The measured voltage signal VC due to the displacement xC in response to the force FC
is

VC =

dx
dVs

−1


xC =

dx
dVs

−1
m

FC

2
ω − ω02 + iωγ(6.33)The voltage response VC is going to depend on temperature because the width γ andresonance frequency ω0 depend on temperature. On the other hand, I can define aquantity

A=

dx
dVs

−1

FC
=
m


−

λ

RAP C + Za
1
χVAP C
2Za
Gamp Λ

−1FCm(6.34)

which is frequency independent and only contains variables which should be constantas a function of temperature and time.
Unfortunately, the gain of the amplifier chain Gamp does drift during the >12 hours
needed to complete a thermal calibration. This drift affects both A and the integrated
strength of the measured voltage signal due to Brownian motion (∆Vs )2 , which results
in a systematic error in the backaction force. However, if (∆Vs )2 and A are measured

207
at approximately the same time then the fraction(∆Vs )2
=
A2



m
FC

2

SFT
4m2 γω02(6.35)

is independent of the amplifier gain Gamp . Substituting (∆Vs )2 /A2 for (∆Vs )2 in thebasic thermal calibration (subsection 6.1.2) will results in a more accurate calibrationand measurement of the backaction force SFBA .
Another source of systematic error in the thermal calibration is the difference
between the temperature of the cryostat and the nanostructure. The microwave voltage VAP C used to measure nanomechanical motion also dissipates power PAP C =2
VAPC /RAP C in the nanostructure. As discussed in the previous section, this dissipatedpower can heat nanostructure so that the local temperature of the nanostructure ishigher than the temperature of the cryostat thermometer. In the 33 kΩ APC discussed
in the previous section, this effect was obvious at PAP C > 2 nW and Tcryo < 4 K; I
dealt with this problem by ignoring temperature and power data where there were clear
deviations from the expected linear relationship between integrated strength (∆Vs )2and temperature Tcryo .
Instead of discarding data points, with the 16 kΩ device it is possible to use thetemperature dependence of the mechanical resonance frequency as a local thermometer.
The temperature dependence is determined by measuring the resonance frequency as
a function of temperature using an electrostatic driving force and a small microwave
voltage bias, PAP C = 0.2 nW (figure 6.12, thick blue line). At this small power, thelocal temperature Tlocal of the nanostructure and the temperature Tcryo of the cryostat
are approximately equal. When a large microwave power is dissipated at the APC,
the local temperature Tlocal of the nanostructure can be determined by measuring thefrequency of the nanostructure (figure 6.12 and 6.13).
With these improvements, I use the Brownian motion of the nanostructure to calibrate the displacement measurement and determine both the measurement imprecision

208

Resonance Freq (MHz)33.85

0.2 nW
6 nW
16 nW
45 nW
94 nW
133 nW

33.84533.84
33.83533.83
33.825
33.82
2

4

6
Tcryo (K)8

10Figure 6.12: The nanomechanical resonance frequency is plotted as a function of cryostat
temperature Tcryo with a variety of different powers dissipated in the APC due toa combination of microwave and dc voltages across the APC. The low power (blue)
resonance frequency has a clear dependence on cryostat temperature Tcryo . Using thiscalibration curve, the resonance frequency is a local thermometer and can be used todetermine the effect of the power dissipated at the APC on the temperature of thenanostructure.

and the backaction force. In figure 6.14 I plot the integrated mean squared displacement(∆xT )2 as a function of temperature, measured with various microwave voltage biases
and dc voltage biases across the APC. The measured imprecision and backaction are
given in table 6.3.
The effective length scale λ/χ of the 16 kΩ displacement detector (table 6.3) isabout an order of magnitude larger than in the 33 kΩ detector discussed in the previous
section. The best displacement imprecision obtained with the 16 kΩ device is worse
than with the 33 kΩ device because of the larger effective length scale. The tunnel
barrier is expected to become smaller at low resistances (chapter 3) and even the modelof the APC as a high rectangular tunnel barrier becomes suspect at low resistances, so

209

2.5

133 nW
94 nW
45 nW
16 nW
6 nW
1 nW

Tlocal−Tcryo (K)

2
1.5
1
0.5
0
−0.5
2

4

6
Tcryo (K)8

10Figure 6.13: The difference between the local temperature Tlocal of the nanostructure,derived from the mechanical resonance frequency, and the cryostat temperature Tcryo
is plotted as a function of cryostat temperature for different powers dissipated in the
APC. The temperature shift is larger for large powers and low cryostat temperatures,as expected, and has an uncertainty less than 200 mK.

it is not surprising that the λ would be larger for a 16 kΩ junction. Based on a finite
element simulation, I also expect the 34 MHz mode χ to be smaller than the χ of the
43 MHz mode. Qualitatively, this is because the mode shapes of the 34 MHz modeand the 43 MHz mode are similar, but the 34 MHz mode’s nanomechanical beam isboth longer and has a smaller cross-section than the 43 MHz mode’s beam. Since theeffective spring constant of the nanomechanical beam is smaller, less force is applied to
the APC by nanomechanical motion and χ is smaller.
The effective length scale λ/χ of the 16 kΩ device also depends on the APC
voltage (table 6.3). While the APC voltage is expected to effect λ by effecting theshape of the barrier (see chapter 3), in the simple case of a rectangular 5 eV barrier(with no applied voltage) the effect on λ is only significant when |VAP C | is within an

210

VAPC=6.9 mVRF, 25 mVDC
VAPC=6.9 mVRF, 0 mVDC

1

T

(Δ x )2 (pm2)

1.5

0.5

0
−2

1

2

4

Tlocal (K)

6

8

10

8

10

VAPC=55 mVRF, 25 mVDC
VAPC=14 mVRF, 25 mVDC
VAPC=14 mVRF, 0 mVDC

T

(Δ x )2 (pm2)

1.5

0

0.5

0
−2

0

2

4

Tlocal (K)6

Figure 6.14: The integrated strength in displacement units (∆xT )2 is plotted versus
the local temperature of the nanostructure Tlocal for various different microwave anddc voltages across the APC. The local temperature Tlocal is derived from the observed
mechanical resonance frequency. In these plots, the y-axis has already been calibrated
in displacement units using the integrated strength of the measured voltage signal andthe magnitude of the capacitive response, as described in the main text. Each data set
is fit to a model that assuming a constant backaction force SFBA (lines).

211
order of magnitude of 5 V. In this case, the effective length scale becomes significantly
larger when VAP C < −10 mV; the dependence on the sign of VAP C and the small
magnitude of the voltage scale implies that this APC should not be modeled as a simple
rectangular 5 eV tunnel barrier. The dependence of this 16 kΩ APC’s characteristics
on APC voltage was not investigated in detail, but a similar dependence is discussed
in chapter 7 in the context of a different nanomechanical system and a mechanicallycontrollable break junction (MCBJ).
APC voltage bias
VRF =7 mV, VDC =0 mV
VRF =7 mV, VDC =25 mV
VRF =14 mV, VDC =0 mV
VRF =14 mV, VDC =-13 mV
VRF =14 mV, VDC =25 mV
VRF =55 mV, VDC =25 mV

Imprecision Sx
621 ± 20 fm2 /Hz
1370 ± 60 fm2 /Hz
278 ± 10 fm2 /Hz1940 ± 100 fm2 /Hz
407 ± 20 fm2 /Hz
247.8 ± 6 fm2 /Hz

Backaction SF
1400 ± 500 aN2 /Hz
6400 ± 1000 aN2 /Hz
1260 ± 600 aN2 /Hz
5200 ± 1000 aN2 /Hz
6100 ± 1000 aN2 /Hz
9350 ± 800 aN2 /Hz

λ/χ
20 nm
20 nm
20 nm
50 nm
20 nm
40 nm

Table 6.3: The observed imprecision, backaction, and length scale λ/χ of a displacement
detector created using an electromigrated APC with resistance RAP C =16 kΩ and various
dc and microwave APC bias voltages VAP C . This detector is used to measure the 34 MHz
mode of the nanostructure shown in figure 6.10. The backaction force SF is plotted asa function of the shot noise spectral density SI in figure 6.15 (red).

6.2.2

Measurement backaction

I fit the observed backaction force to the phenomenological model of the APC
detector described in chapter 3. In this model, the backaction of the APC detector isdue to momentum kicks from tunneling electrons. The force is
F =

IAP C
pe
e

(6.36)

equal to the momentum delivered by each electron pe times the number of electrons
that tunnel per unit time IAP C /e. The random backaction force due to the momentumof tunneling electrons is proportional the random number of electrons that tunnel perunit time and has a spectral density proportional to the shot noise spectral density
p2
SFBA = SIsn 2ee(6.37)

212
The magnitude of the observed backaction force SFBA is linearly proportional tothe magnitude of the shot noise current SIsn = 2eζΞVAP C /RAP C . In figure 6.15 I
plot the backaction force SFBA versus the shot noise SIsn observed using the 16 kΩ
APC described in the previous subsection (red) and the 33 kΩ APC describe in theprevious section (blue). I fit each data set to a line through the origin, that is, fit to
the phenomenological model described in the previous paragraph and equation 6.37. A
linear fit to the shot noise current SIsn is better than a fit to higher powers of SIsn (forexample, the black line is a fit to (SIsn )3 ). It is possible that the 33 kΩ APC data has asystematic deviation from a line through the origin, though there are not enough data
points to make a clear determination.
One possible explanation for such a deviation is a poor knowledge of the local
temperature. In the analysis of the 33 kΩ APC data I simply ignored thermal data
points where the local temperature appeared to deviate from the cryostat temperature.
This technique will not compensate for a difference between the local temperature andthe cryostat temperature that is approximately constant between 5 K and 10 K. An
offset ∆T between the local temperature and the cryostat temperature will cause themeasured backaction force to be incorrect by an amount 4mγkB ∆T .
While the linear dependence of the backaction force SFBA on shot noise SIsn isconsistent with a backaction force due to momentum kicks from tunneling electrons,each electron would have to deliver a momentum equal to about 30 times the Fermi
momentum (figure 6.16) which is implausibly large. In the simple phenomenologicalmodel of backaction described in chapter 3, electrons tunnel from filled states near the
top of the Fermi sea on one side of the barrier to unfilled states on the other sideof the barrier. Since the applied voltage bias is small compared to the Fermi energy,
the momentum of each tunneling electron is expected be on the order of the Fermi
momentum pF (for gold, pF = 1.2 × 10−24 N sec).
There are other possible sources of backaction. While I have attempted to com-

213

12000

2

SF (aN /Hz)

10000
8000
6000
4000
2000
0
0

0.05

0.1
SI (pA2/Hz)

0.15

Figure 6.15: The backaction force SFBA observed when using the 33 kΩ APC detector(blue data) and when using the 16 kΩ APC detector (red data) is plotted versus theshot noise in the current SI . The shot noise is due to just a microwave voltage (squares),a microwave voltage plus a dc voltage Vdc = −12.5 mV (circles), or a microwave voltage
plus a dc voltage Vdc = 25 mV. Based on the phenomenological model where the backaction force is due to momentum kicks from tunneling electrons, each data set is fit toa line through the origin (red/blue lines). The red data set is also fit to (SI )3 (blackline), the dependence expected from backaction due to an electrostatic force. This poor
fit is one indication that the backaction force is not electrostatic.

pensate for differences between the local temperature and the cryostat temperature, it
is still possible that heating due to power dissipated at the APC is responsible for theobserved backaction force. As explained earlier, this explanation is unlikely because it
is not consistent with the observed dependence on dc and microwave voltages. I also
expect that there is an electrostatic backaction force due to dCAP C /dx, however thisbackaction force SFBA would be proportional to (VAP C )3 (black line in figure 6.15) which
does not fit the data. I can also determine dCAP C /dx < 10 nF/m directly by measuring
the calibrated mechanical response to a voltage applied across the APC. The resulting

214

70
60

pe/pF

50
40
30
20
10
0

0.05

0.1
SI (pA2/Hz)

0.15

Figure 6.16: The electron momentum pe needed to explain the observed backactionis plotted in units of the Fermi momentum (for gold, pF = 1.2 × 10−24 N sec) as afunction of the shot noise current SI for the 33 kΩ APC detector (blue data) and forthe 16 kΩ APC detector (red data). Fitting the observed backaction forces (figure 6.15)
to the phenomenological model of a backaction force caused by the momentum kicks
from tunneling electrons, the 33 kΩ APC detector backaction would require that eachelectron impart a momentum pe = pF × (32 ± 3) (red line) and the 16 kΩ APC detector
backaction would require pe = pF × (34 ± 1) (blue line).

backaction force
SFBA =

dCAP Cdx

2

2
SIsn RAPC(VAP Crf )2+ (VAP Cdc )22!
(6.38)

is too small to account for the observed backaction.
It is also possible that the observed backaction force is due to the peizo-electric
nature of the substrate or trapped charges near the APC. This type of force is proportional to the voltage across the APC, and thus the spectral density of the backaction
force SFBA would proportional to the shot noise voltage across the APC
2SFBA = σ 2 SV sn = σ 2 SIsn RAP
C(6.39)

215
where σ is the proportionality constant.
Using the data from the two electromigrated APCs (figure 6.15) it is difficult
to determine whether the backaction force is due to a force proportional to current,such as momentum kicks from tunneling electrons, or a force proportional to voltage
because the spectral density of both backaction forces is proportional to the shot noise.
If the proportionality constant is not dependent on the APC resistance, then the typeof backaction force can be determined using the resistance dependence of the backaction
force.
Assuming the proportionality constant is not dependent on the APC resistance,
the backaction force observed in these electromigrated junctions is due to a force proportional to current because this proportionality constant pe /e is the same in the two
devices while the voltage force proportionality constant σ changes by a factor of two.
However, this assumption is suspect because these backaction forces are measured using
different nanomechanical structures and different APCs with different characteristics.
In chapter 7 I will discuss other ways of determining whether the backaction force is
proportional to current or voltage.

Chapter 7

Results: Mechanically Controllable APC coupled to nanomechanicalmotionIn this chapter, I demonstrate that a mechanically controllable break junction canbe used to create an atomic point contact (APC) and measure nanomechanical motion. I
start by describing how the resistance of the junction can be reversibly changed between
40 Ω and open (> 10 GΩ). I use this controllable resistance to measure nanomechanicalmotion at different resistances. The nanomechanical resonance frequency and damping
depend on resistance and generally decrease with increasing resistance.
I also measure the length scale λ that controls the probability of electron’s tunneling at different junction resistances and, in general, observe an increase in λ at resistance
RAP C < 500 kΩ. The best displacement measurement compared to the standard quanp√√
tum limit Sx = 4.4 SxSQL and the Heisenberg constraint Sx SF = 168h̄ is obtained
at a resistance RAP C = 60 kΩ.
Finally, I use the MCBJ to investigate the backaction force created by voltages
or currents through the junction. In general, the magnitude of the backaction force
is similar to that observed in electromigrated devices (chapter 6). The presence of
unintentional feedback indicates that there exists a backaction force proportional tocurrent. I attempt to measure correlations between the shot noise and the backaction,
but the noise in the measurement is large enough that the data is consistent both withthe absence of correlations and with correlations due to the expected backaction force. I

217
also observe an unexpected increase in the backaction at certain dc voltage biases which
could be related to changes in the differential resistance RAP C at the same voltage
biases.

7.1

Mechanically Controlling the APC Resistance
I use a mechanically controllable break junction (MCBJ) to create an atomic point

contact (APC) and control the resistance of the APC. The resistance of the junction
is changed by straining the suspended nanostructure (figure 7.1) using a mechanicalplunger (chapter 4). By moving the plunger with a screw for coarse adjustments and apiezo crystal for fine control, the resistance RAP C of the junction can be increased from40 Ω to open (> 10 GΩ) and then decreased back to about 40 Ω.

100 nmconstriction

65 MHz115 MHz

constriction
Figure 7.1: Scanning electron micrograph of nanostructure (left) composed of a gold
nanomechanical beam (light gray) suspended above a polyimide layer (dark grey/black)on top of a phosphor bronze substrate. The mechanically controllable junction is createdat the constriction by stretching the suspended beam (chapter 4). A finite element
simulation (right) is used to determine the mode shape of the two lowest frequency
nanomechanical modes (see chapter 3).

218
The hysteretic loops in resistance RAP C that occur when the strain is increased
and decreased can be divided into three rough categories. First, the resistance RAP C
can jump from about 100 Ω to greater than 10 GΩ under small increases in strain andthen, as the strain is decreased, jump from greater than 10 GΩ to less than 100 Ω(figure 7.2). Second, in some cases the resistance RAP C could be adjusted almost
continuously between about 10 kΩ and greater than 10 MΩ with only small (< 10%)jumps in resistance (figure 7.3). In other MCBJ experiments, the ability to adjust theresistance in the regime between 15 kΩ and 100 kΩ is often associated with the presenceof impurities near the junction [8–10]. Finally, in one case it was possible to adjust thethe resistance from about 11 kΩ to 10 MΩ with a large, repeatable, jump in resistance
between 11 kΩ and about 150 kΩ (figure 7.4). This type of repeatable jump in resistanceis generally associated with an uncontaminated gold APC [9, 10].
In most cases, attempts to increase the junction resistance RAP C from below 100 Ω
to above 20 kΩ by straining the junction results in a sudden jump in resistance from lessthan 1 kΩ to greater than 10 GΩ (figure 7.2). Decreasing the strain then results in asudden jump from greater than 10 GΩ to about 100 Ω. In order to produce a device that
I can use to sensitively detect nanomechanical motion (about 20 kΩ < RAP C < 1 MΩ),I repeatedly open and close the junction. After many cycles, the the junction resistanceeventually jumps from closed to less than 1 MΩ (instead of > 10 GΩ) or the junction
resistance decreases smoothly from open to a resistance less than 10 MΩ instead ofjumping to RAP C < 1 kΩ. In general, the number of times that I needed to cycle thejunction before obtaining a useful resistance increased over time.
This behavior is in contrast to the expected behavior of a gold junction [8]. Straining a gold junction usually results in the formation of a chain of gold atoms with aresistance close to half the quantum of resistance, RQ /2 = h/(2e2 ) = 12.9 kΩ. When
this chain breaks, the resistance of the junction usually jumps to RAP C > 100 kΩ. At
these higher resistances, gradually increasing the strain results in a gradually increasing

219
10

10

R

APC

(Ω)

8

10

6

10

4

10

2

Plunger Displacement (μm)

10

0

10

20

30

40

50

10

20
30
Time (min)

40

50

10
5
0
−5
0

Figure 7.2: The junction resistance (top) and plunger displacement (bottom) as a function of time during a typical attempt to create a junction with a useable resistance
20 kΩ < RAP C < 1 MΩ. Increasing strain typically causes a jump from low resistance
RAP C < 1 kΩ to high resistance RAP C > 10 GΩ (red). The noise in the resistance measurement (chapter 5) places an upper limit of about 10 GΩ on the measurable junction
resistance. After breaking the junction, decreasing the strain typically causes a jumpfrom high resistance RAP C > 10 GΩ to low resistance RAP C < 1 kΩ (blue). Occasionally, the junction resistance will not jump suddenly to low resistance but instead
will decrease controllably as the strain is decreased (at 17 min and 48 min). I use thisbehavior to create a junction that can be used to detect nanomechanical motion. Morerarely, the junction resistance will jump from a low resistance to a useable resistance as
the strain is increased.

resistance with only small discontinuities. The discontinuities are usually attributed
to small rearrangements of the gold atoms near the junction. Similarly, decreasing thestrain of a high resistance junction will decrease the resistance RAP C with only small dis-

220
continuities until, in a typical junction, the resistance jumps from from RAP C > 100 kΩ
to close to half the quantum of resistance RQ /2. Continuing to decrease the strain will
typically cause the conductance GAP C = 1/RAP C to increase in steps that are initiallyequal to about half the quantum of conductance GQ = 1/RQ . The decrease in resistancebecomes continuous at lower resistances.
Two possible explanations for the deviation of the observed behavior from theexpected behavior are the incorporation of other contaminants into the junction anddifferences between the nanostructure used to suspend and strain the junction in thisexperiment and previous experiments. The presence of contaminate atoms or moleculeschanges the mechanical properties of the junction [8, 49, 52–54, 57]. The junction described in this chapter (figure 7.1) was cycled between opened and closed hundreds oftimes in a cryogenic vacuum and also exposed to the atmosphere five times in order toadjust the electrical resonance circuit (chapter 5). While the junction was always closed
RAP C < 50 Ω when not in vacuum, the exposure will still increase the contaminantsaround the junction. Contaminates (especially helium atoms) around a gold junction
are generally believed to stabilize the junction and allow junction resistances between
20 kΩ and 100 kΩ [8–10]. However, different types of contaminates or the presence of alarge number of contaminates incorporated into gold near the junction could lead to abrittle junction which breaks instead of forming a stable atomic contact. The expected
increase in contamination over time can also explain the observed increase in the typicalnumber of breaking and closing cycles needed before achieving a useful junction.
This atypical behavior could also be caused by the nanomechanical structure(figure 7.1). This structure is more mechanically pliant than most support structures [8]to create a lower mass mechanical system and achieve mechanical resonance frequencies
around 100 MHz. This compliant structure enables more sensitive force detection,
however it is also more difficult to concentrate the strain at the junction (chapter 4)
when breaking a low resistance junction RAP C < 100 Ω. In general, any strain in the221
structure is released when the junction breaks which increases the size of the junction
gap after breaking. This process could result in an abrupt transition from a closed
junction to an open junction with a large gap, as observed (figure 7.2), if the strain in the
structure is large. After breaking the contact, decreasing the size of the junction’s gap
will eventually result in the atoms in the junction jumping into contact. A sufficiently
complaint structure could have a small effect on the size of the gap at which the atoms
jump into contact and could also effect the size of the contact after the jump. The
nanomechanical structure will effect the junction’s behavior under strain, however thiseffect should not change with time which makes it unlikely that the nanomechanicalstructure is the only cause of the observed breaking and closing cycles.

7

RAPC (Ω)

10

6

10

5

10

4

10

0

0.5
1
1.5
Plunger Displacement (μm)

2Figure 7.3: The junction resistance RAP C is plotted as a function of plunger displacement. The position of the plunger is increased (blue) and decreased (red) causing, inthis case, the junction resistance to sweep between 20 kΩ and > 1 MΩ with only small
discontinuous jumps. The small jumps are due to the rearangment of atoms near thejunction. The ability to control the resistance between 20 kΩ and 100 kΩ, as in thisfigure, is often thought to be a sign that there are contaminants near the junction.

222
When the junction resistance does not jump between less than 1 kΩ and greaterthan 1 GΩ, it is usually possible to continuously sweep the junction resistance between
20 kΩ and 10 MΩ by applying and removing mechanical strain (figure 7.3). There areusually only small, unrepeatable, jumps in resistance as a function of strain. This behavior is not expected in clean gold junctions, where the junction resistance is expected
to jump between RAP C = RQ /2 = 12.9 kΩ and RAP C > 100 kΩ. A common explanation for the ability to adjust the junction resistance between 20 kΩ and 100 kΩ isthe presence of contaminants around the junction [8–10]. This explanation is consistent
with the nonlinear I-V curves observed at resistances up to about 1 MΩ, which havefeatures that could be due to the vibrational modes of contaminants. The features inthese nonlinear I-V curves depend on the junction resistance and will be discussed in
more detail in section 7.5.
In one case, I did observe the behavior expected from a clean junction. I observe aplateau in the junction resistance as a function of strain at RAP C ≈ RQ /2, likely due tothe formation of a chain of gold atoms, and then an abrupt jump in resistance to greaterthan 1 MΩ as the chain breaks. As I reduce mechanical strain, the junction resistancedecreases with small hysteretic features until the resistance jumps from about 100 kΩ
to RQ /2. Initially the dependence of the resistance on mechanical strain is different
every time the resistance is swept from RQ /2 to greater than 10 MΩ and back to RQ /2.
The dependence of resistance on strain becomes repeatable, though still hysteretic, after
more than fifty cycles (figure 7.4). This type of “training” behavior has been analyzed
in similar gold junctions [9, 10] and is associated with a clean, uncontaminated gold
contact.
The junction resistance’s dependence on plunger position can be used to determine
the length scale λ that controls the probability of electrons tunneling across the junction.
At large junction resistances where electrons tunnel across a gap, the resistance RAP C

223
8

10

RAPC (Ω)

7

10

6

10

5

10

4

10

0

0.25

0.5
0.75
1
1.25
Plunger Displacement (μm)

1.5Figure 7.4: The junction resistance RAP C is plotted as a function of plunger displacement. After “training” the junction [9, 10], sweeping the plunger over 1.5 µm results ina repeatable hysteretic loop in resistance between 11 kΩ and 200 MΩ. As the positionof the plunger is increased from 0 µm (blue), there is a plateau where RAP C increases
slightly from 11 kΩ to 13 kΩ which is likely due to the formation of a chain of gold
atoms. There is an abrupt jump to 10 MΩ at 0.9 µm as the chain breaks, after which
the resistance increases exponential due to plunger displacement. When decreasing thestrain (red), the resistance decreases exponentially as the plunger is retracted until theatoms forming the junction jump to contact from 200 kΩ to 11 kΩ at 0 µm. There
are two small, repeatable, discontinuities at 0.25 µm and 0.75 µm when the strain isdecreased (red); these discontinuities do not significantly effect the exponential dependence of resistance on plunger displacement (the slope of the red line is constant). While
ideally a “trained” junction would not have any discontinuities, the constant exponential dependence supports the idea that the discontinuities are due to the rearrangementof atoms away from the junction rather than a change in the atoms forming the junctionor the presence of a contaminate. This repeatable hysteretic loop is not typical; I wasonly able to successfully train a junction one time (shown here).

of the junction depends exponentially on changes x in the size of the gap
2x
RAP C (x) = R0 exp
λ


(7.1)

In the simple case where the strain at the top surface of the chip is homogeneous, the224
size of the gap x is related to motion of the plunger u by
x
3tD
= 2
u
L

(7.2)

where t is the thickness of the chip, D is the length of the suspended nanostructure, and
L is the distance between the supports holding the chip (chapter 4). The length scale
λ is determined from the changes in junction resistance RAP C (u) caused by plunger
motion u
λ=2u
ln RAP C (u) − ln R0



3tD
L2


(7.3)

The length scale λ of the junctions discussed in this chapter cannot be determinedfrom RAP C (u) because the strain at the top surface of the chip is not homogenous. The
chip has been permanently deformed and the strain is larger near the center of the chip,
where the nanostructure is located, than near the supports. The ratio x/u 6= 3tD/L2
is unknown, and so λ cannot be determined from equation 7.3.
If the gold junction is free of contaminates, then the dependence of RAP C onplunger displacement u can be used to determine the ratio x/u of changes in the gap x
to plunger displacement u. When a gold junction has a large resistance RAP C > 1 MΩ
and is free of impurities then the length scale is expected to be λ = 0.1 nm [8], consistent√
with the bulk work function of gold φ = 5 eV (λ = h̄/ 2me φ). By assuming λ = 0.1 nm,
the actual ratio
x
λ
=
u
2



u
ln RAP C (u) − ln R0

−1
6=

3tD
L2

(7.4)

can be determined from RAP C (u). If I assume that the junction is not contaminatedwhen training behavior is observed (figure 7.4), then the ratio x/u = 2.1 × 10−4 =5(3tD/L2 ) is five times larger than would have been predicted assuming homogeneous
strain. Since the chip has undergone plastic deformation and the strain is concentrated
near the nanostructure, this increase in the ratio x/u is to be expected.

225

7.2

Changes in APC Resistance Effect Nanomechanical Modes
The nanomechanical resonance frequency and damping depend on the resistanceof the junction and strain on the nanostructure. At many resistances, the resonancefrequency is not stable and often jumps between two distinct states. The cryostat
also needs to be mechanically isolated to avoid low frequency (100 Hz) changes in themechanical resonance frequency.
As an example, I measure the mechanical resonance frequency and damping as theresistance of the APC is increased from 200 kΩ to 2 MΩ (figure 7.5). This junction had
been “trained,” as described in the previous section (figure 7.4). In general, increasing
resistance and gap size results in a decreasing mechanical resonance frequency. Thisdecrease is not surprising, since the elastic connection between the two sides of thejunction is expected to become weaker as the size of the gap increases. Discontinuouschanges in junction resistance, usually associated with the rearrangement of atoms inthe junction, also cause discontinuous changes in the mechanical resonance frequency. In
this case the damping constant also decreases with increasing resistance of the junction.
This is consistent with a decreasing elastic connection, since any damping due to thejunction should decrease when the elastic connection due to the junction decreases
because the forces on the junction will be smaller. The mechanical resonance frequency
and damping can also depend on the bias voltage across the junction and on the cryostat
temperature (as discussed in chapter 6).
At some resistances and junction configurations, the resonance frequency anddamping constant are not stable. Usually the resonance frequency jumps between twovalues, sometimes on the time scale of a second and sometimes much faster. Increasing
the bias voltage can also make the resonance frequency unstable and, eventually, causes
the junction resistance to fluctuate. In some cases the junction resistance is unstableeven at low bias voltages; this is usually accompanied by fluctuations in the mechanical

61
60.9

2
1.5

Plunger Motion

1
.5

0.15
0.2
0.25
Plunger Displacement (μm)

0.3

2
1.5

Plunger Motion

1
.5

60.8
60.7
0.1

.3

0.15
0.2
0.25
Plunger Displacement (μm)

RAPC (MΩ)

55
50
45
40
35
30
25
0.1

0.3

.3

RAPC (MΩ)

Resonance Freq (MHz)Resonance FWHM (kHz)226

Figure 7.5: The mechanical damping, i.e., Full Width at Half Maximum (FWHM,
top, blue), and mechanical resonance frequency (bottom, red) are plotted versus theplunger displacement. This data was taken using the “trained” junction (figure 7.4)
while increasing the plunger displacement and junction resistance (black). In general,a junction with a larger gap is expected to have both a larger resistance and provide aweaker elastic connection. This expectation is consistent with both the smooth decrease
in mechanical resonance frequency with smooth increase in junction resistance and thegeneral trend of decreasing mechanical damping with increasing resisance. The mechanical resonance frequency is unstable at resistances below 300 kΩ and at the discontinuity
RAP C ≈ 1.4 MΩ; the instability causes the large FWHM at RAP C ≈ 1.4 MΩ as well asa non-lorentzian line shape.

damping and mechanical resonance frequency.
It is also necessary to isolate the cryostat from vibrations in the floor and cables(figure 7.6). When the cryostat is not isolated, the nanomechanical resonance frequency
oscillates at frequencies >50 Hz. This oscillation is damped by placing the dewar on

227

Figure 7.6: Pictures of the mechanically isolated cryostat. The weight of the cryostat is
supported by four springs (bottom left, bottom right) which mechanically isolates thecryostat from most vibrations > 50 Hz. Additional damping material is placed on topof the springs. The cryostat is also attached to the ceiling (top right) as a precautionary
measure. Finally, all of the cables connected to the cryostat are supported and damped
by either bungee cord or surgical tubing (top left, top right.

springs and supporting any cables connected to the cryostat with surgical tubing or
bungee cords. The springs compress by l = 9 mm when weighted by the cryostat,
creating a mechanical resonance with a resonance frequency
1
f0 =
2π

r

g
= 5 Hzl(7.5)

Cryostat vibrations at frequencies much greater than f0 = 5 Hz are damped by the presence of the springs and absorbent material on top of the springs. Using this mechanicalisolation, it is possible to observe stable nanomechanical resonance frequencies.

228

7.3

APC Resistance Effects Displacement Sensitivity
The length scale λ/χ and the nanomechanical displacement sensitivity also de-pend on the junction resistance RAP C . At each junction resistance, I use the Brownian motion of the nanomechanical structure and the response to an electrostatic drive(chapter 6) to calibrate the displacement measurement and determine both λ/χ and thedisplacement sensitivity. I observe the expected increase in the length scale (chapter
6) at lower resistances RAP C < 500 kΩ and a length scale consistent with the work
function of bulk gold at resistances RAP C > 500 kΩ. The best displacement sensitivityp√in comparison to the standard quantum limit, Sx = 4.4 SxSQL , is obtained at a
resistance RAP C = 60 kΩ.
The displacement measurement is calibrated using the Brownian motion of thenanostructure at a single temperature and an electrostatic drive. It is not possible touse the Brownian motion at multiple temperatures, as described in chapter 6, because
the needed > 10 K change in cryostat temperature also changes the junction resistance.
Instead, I measure the 5 K Brownian motion of the nanostructure using a small microwave voltage across the junction. Because the measurement is weak, the backaction
force and change in the local temperature due to power dissipated at the APC are small
compared to 5 K and I can use the nanomechanical motion at this single temperaturepoint to calibrate the displacement measurement (see chapter 5). I simultaneously calibrate the electrostatic force FC applied to the nanostructure using a voltage VC on
a nearby electrostatic gate. The measured displacement in response to the calibrated
electrostatic force is then used to determine the length scale λ/χ (see chapter 6), displacement sensitivity, and backaction of a displacement measurement which uses a large
voltage across the APC.
Using this calibration, I measure the length scale λ/χ at junction resistances
between 80 kΩ and 1 MΩ (figure 7.7). At large resistances RAP C > 500 kΩ, the length

229

0.7
0.6

λ (nm)

0.5
0.4
0.3
0.2
0.1
0
0

200

400
600
800
APC Resistance (kΩ)

1000

1200

Figure 7.7: The length scale λ/χ for the 65 MHz mode is plotted as a function of APC
resistance. As expected, the length scale increases from the expected value of 0.1 nm as
the resistance of the junction is decreased. There is no data between 600 kΩ and 1 MΩbecause the the resonance frequency was not stable. The different data points were also
not taken as part of a continuous sweep in resistance, but instead were taken duringmultiple sweeps from 100 kΩ to above 1 MΩ while attempting to find resistances withstable mechanical resonance frequencies.

scale λ/χ ≈ 0.1 nm is consistent with electrons tunneling through a barrier equal to√
the work function of bulk gold φ = 5 eV (λ = h̄/ 2me φ, see chapter 3), as in astandard STM. As the resistance is decreased below 500 kΩ, the length scale increases
to a maximum, in this case, of λ/χ = 0.6 nm.
The best displacement sensitivity is achieved by optimizing the combination of asmall length scale, large junction bias, and small contribution to the measurement noisefrom the amplifier (figure 7.8). When measuring the displacement of the nanostructure
shown in figure 7.1, that optimum was achieved using a microwave bias voltage VAP C =47 mV and a junction with a resistance RAP C = 61 kΩ and length scale λ = 0.5 nm.

230

1

0

VAPC=47 mV

Shot
Noise

10

2

Sx (fm /Hz)

10

Tcryo = 4.7 K

Amplifier
Noise
−1

10

Imprecision
−2

10

61

61.5
62
Mechanical Frequency (MHz)Figure 7.8: The spectral density of the fluctuations in the measured displacement (red)and a Lorentzian fit (black) are plotted as a function of frequency near one of thenanostructure’s mechanical resonances. The Lorentzian peak in the noise is due to the
Brownian motion of the beam at Tcryo = 4.7 K. The Lorentzian peak is on top of a whitebackground; this apparent motion is the measurement imprecision (Sx = 0.08 fm2 /Hz =20SxSQL for VAP C = 47 mV) due to a combination of shot noise (green) and amplifier
noise (grey). This measurement has the best displacement sensitivity compared to thestandard quantum limit achieved using an APC displacement detector. I use a single
sideband measurement with a microwave bias voltage VAP C = 47 mV and a junction
with a resistance RAP C = 61 kΩ and length scale λ = 0.5 nm.

Using a single-sideband measurement, the best displacement imprecision Sx = 0.0841 ±
0.0001 fm2 /Hz is 20 times the imprecision at the standard quantum limit, SxSQL =h̄/mω0 γ. This imprecision is dominated by the shot noise of tunneling electrons, thefundamental source of noise in the measurement, but a single sideband measurement
only collects half of the voltage signal (chapter 5). The imprecision Sx could be decreased

231
by a factor of two if it were possible to change the measurement electronics and performa double sideband measurement.
I observe a backaction force much larger than required by quantum mechanics,
SF = 3750 ± 50 aN2 /Hz. If the backaction force is caused by momentum kicks fromtunneling electrons, each electron would have to deliver a momentum equal to 23 times
the fermi momentum, pe = 23pF . The total imprecision, in quanta, is Ntot = 365 ± 5which is 730 times the total imprecision at the standard quantum limit (chapter 6).
Finally, this displacement measurement can be compared to the Heisenberg constraint√
and the product of the imprecision and backaction Sx SF = 168h̄ is 168 times larger
than required by quantum mechanics.
The standard deviations in the previous two paragraphs are derived from statistical variations and do not include two important sources of systematic error. First, thisanalysis is sensitive to any differences between the measured temperature of the cryostat and the local temperature of the nanostructure at low voltage bias. This differenceis expected to be less than 10% which would result in less than a 10% change in thevalue of Sx as well as a larger effect on SF . Second, the quoted value of the backaction
force SF includes any increase in the local temperature of the beam caused by the large
voltage bias. The actual value for SF could therefore be smaller than stated above,
though this effect is expected to be small because the increase in the local temperatureobserved at Tcryo = 5 K in chapter 6 was relatively small.

7.4

APC Detector Backaction
I attempt to understand the observed backaction force in more detail using threedifferent methods. First, I measure the unintentional feedback caused by the backaction
force. This feedback could be caused by a force proportional to voltage or current, but
the observations are more consistent with a backaction force proportional to current.
Second, I attempt to determine whether the backaction force is correlated with the shot

232
noise of tunneling electrons. I place an upper limit on the magnitude of any correlated
backaction force consistent with electron momentum kicks pe < 50pF . The previously
observed backaction is included within this limit, so more accurate measurements areneeded to determine whether the backaction force is correlated with shot noise. Finally,
I examine the dependence of the backaction force on the dc voltage across the junction.
I observe increased backaction and peaks or dips in the differential resistance at approximately the same dc voltage. The peaks and dips in the differential resistance arethought to be caused by the vibrational modes of contaminates near the junction whichsuggests that the backaction could also be increased by the presence and influence ofcontaminates.

7.4.1

Unintentional feedback caused by the backaction force

Part of the backaction force can be proportional to nanomechanical displacement,
creating a feedback loop. I use the observed effect of the feedback loop to distinguishbetween two phenomenological models for the backaction force: models where the force
is proportional to either voltage FBAV = σV VAP C or current FBAI = σI IAP C . The
correct model of APC backaction should account for both the unintentional feedback
and the directly observed backaction force. At all APC resistances, the data is bestaccounted for by a backaction force proportional to current. If this force were due tothe momentum of tunneling electrons pe = eσI , the momentum kick pe delivered byeach electron would be dependent on RAP C and would vary between 40 times the Fermi
momentum pF and the fermi momentum pF . At some APC resistances, this momentum
is consistent with the backaction force observed in electromigrated devices (chapter 6)
where pe ≈ 30pF .
The backaction force can create an unintentional feedback loop (figure 7.9). The
measurement of nanomechanical displacement requires electrically biasing the APC
which creates a current IAP C through the APC and a voltage VAP C across the APC,

233

VAPC

a)

Vb
RAPC

b) Feedback with FBA =
x(

)=

m

F

(

F(

2

+

+

FBA =

2
0

)

)+i

FBA I

I I APC

Za

I

I APC

dRAPC
x
dx
2
dRAPC
= RAPCdx

x

RAPC =RAPC
APC

I APC =

Vb
RAPC + Z ac) Feedback with FBA = V VAPC
F( )xx( ) =
m

F

(

+
FBA =

2

2
0

+

)+i

FBA V

V VAPCRAPC

Vb( RAPC + Z a )2

dRAPC
x
dx
dRAPC
2
= RAPCdx
RAPC =RAPC

APC V = Vb RAPC + R
APC
APC
RAPC + Z a

Vb Z a( RAPC + Z a )2

Figure 7.9: (a) circuit including a voltage bias Vb , junction resistance RAP C , and measurement circuit impedance ZA . (b,c) Feedback loop assuming a backaction force proportional to either current (b) or voltage (c). Force (red) causes nanomechanical displacement (orange) which changes the resistance of the APC (green). The changing
resistance also changes the current flowing through the junction (b, blue) and the voltage across the junction (c, blue) which closes the feedback loop by creating a change inthe backaction force.

234
where RAP C = VAP C /IAP C . If the backaction force is proportional to IAP C or VAP C ,then the applied bias will apply a force to the nanostructure. Because nanomechanicaldisplacement changes RAP C , a portion of the backaction force can be proportional todisplacement which creates a feedback loop. The magnitude of the feedback depends
on both the electrical circuit surrounding the APC (figure 7.9a) and whether there isa backaction force proportional to IAP C or VAP C (figure 7.9c and d, respectively). For
example, if the impedance Za of the electrical circuit at the mechanical resonance frequency goes to zero then the APC is voltage biased and VAP C is independent of RAP C .
On the other hand, IAP C will change as nanomechanical motion changes RAP C . In thiscase, a backaction force proportional to current will create feedback and a backaction
force which is proportional to voltage will not create feedback. In this experiment theresistance of the measurement circuit Za < 30 kΩ and is less than a third of the APC
resistances considered in this subsection. It is possible for either a backaction force
proportional to current or voltage to create feedback. I consider this feedback loop tobe “unintentional” because I do not intentionally apply a force to the nanostructure
which is proportional to the measured displacement.
Assuming a backaction force proportional to current FBAI = σI IAP C , the feedback due to a dc bias voltage Vb will shift the mechanical resonance frequency and a
voltage bias near twice the mechanical resonance frequency will create parametric gain.
From chapter 5 and the circuit diagram in figure 7.9, the current IAP C through the
APC due to a voltage bias Vb cos(ωb t) at a frequency ωb is
IAP C (t) =Vb cos(ωb t)
2χ
RAP C− Vb cos(ωb t)x(t)RAP C + Za
λ (RAP C + Za )2(7.6)

If there is a force backaction force proportional to current, there will then be a force
proportional to displacement. The effect of this backaction force is found by explicitly
including it in the differential equation which describes the nanomechanical motion x

235

Displacement (pm)

1.2

dc VAPC=+37 mV

dc VAPC=+37 mV

1
0.8
0.6
0.4
0.2
0

62

62.2 62.4 62.6 62.8
Frequency (MHz)63

Figure 7.10: The nanomechanical displacement in response to a 1 mV bias voltage isplotted as a function of the frequency of the bias voltage when there is also a dc voltage
VAP C = 37 mV (green) or VAP C = −37 mV (blue) across the junction. The feedback
created by the backaction force shifts the frequency of the mechanical resonance. Thechange in the magnitude of the response is due to the presence of an electrostatic force(chapter 6), which interferes constructively (green) or destructively (blue) with thebackaction force. The electrostatic force is proportional to dC/dx = 0.5 nF/m and issmall compared to the backaction force.

in response to other forces FVb cos(ωb t)
2χ
RAP C− xVb cos(ωb t)= mω02 x + mγ ẋ + mẍ (7.7)
RAP C + Za
λ (RAP C + Za )2


2χ
RAP C
Vb cos(ωb t)
2
F + σI= mω0 + σI Vb cos(ωb t)
x + mγ ẋ + mẍ (7.8)
RAP C + Za
λ (RAP C + Za )2
F + σIThe feedback created by a dc voltage (ωb = 0) bias shifts the mechanical resonancefrequency ω0 → ω0 + ∆I
∆I =σI Vb 2χRAP C
2ω0 m λ (RAP C + Za )2(7.9)

when ∆I  ω0 . An example of this shift in mechanical resonance frequency is shown236
in figure 7.10).
A voltage bias Vb cos(ωb t) applied at about twice the mechanical resonance frequency (ωb ≈ 2ω0 ) creates parametric gain and effectively decreases the width γ ofthe resonance (figure 7.11). The parametric gain is used to determine σI . As shownin equation 7.8, the voltage bias Vb cos(ωb t) will pump the mechanical spring constantmω02 at about twice the mechanical resonance frequency when ωb ≈ 2ω0 . The solution
to equation 7.8 when ωb ≈ 2ω0 is complicated (see appendix D), but becomes simple
when ωb = 2ω0 and the displacement x(t) is due to a force applied at the mechanicalresonance frequency F (t) = F0 cos(ω0 t). In this case, the parametric gain results in
3

2

Sx (fm/Hz )10

parametric drive:
VAPC=23 mV
at 115.1 MHz
210
at 119.1 MHz

1

10

0

10

59.4 59.45 59.5 59.55 59.6 59.65 59.7
Frequency (MHz)Figure 7.11: The spectral density Sx of the nanostructure’s Brownian motion is plotted versus frequency with a VAP C = 23 mV pump applied at 115.1 MHz  2ω0 /2π(blue) and 119.1 MHz ≈ 2ω0 /2π (red). The 115.1 MHz pump has little effect on thenanomechanical motion (blue), however the feedback from the 119.1 MHz ≈ 2ω0 /2π
pump results in parametric gain and significantly increased displacement (red).

237
increased displacement−2F0
x(t) =
m



2χ
RAP C
2γω0 − σI Vbmλ (RAP C + Za )2−1
sin(ω0 t)(7.10)2χ
RAP C
when 2γω0 > σI Vb mλ. An example of this parametric gain is shown in figure(RAP C +Za )22χ
RAP C
7.11. When 2γω0 ≤ σI Vb mλthe mechanical system will begin to regenera(RAP C +Za )2

tively oscillate. The proportionality constant σI is determined from the threshold bias
voltage VbR at which the mechanical oscillator begins to regeneratively oscillate
σI =2γω0 mλ (RAP C + Za )2
2RAP CVbR χ(7.11)

A backaction force FBAV = σV VAP C proportional to voltage can also cause feedback. In comparison to the model above of FBA = σI IAP C , the feedback due to thismodel has a very different dependence on the circuit impedance Za . From chapter 5
and the circuit diagram in figure 7.9, the voltage VAP C across the APC due to a voltage
bias Vb cos(ωb t) is
VAP C =

Vb cos(ωb t)RAP C
2χ RAP C Za
+ Vb cos(ωb t)x(t)RAP C + Za
λ (RAP C + Za )2(7.12)A small part of VAP C is proportional to displacement x(t) which, because of the backaction, causes a force proportional to displacement. This force can be explicitly includedin the differential equation that describes the nanomechanical motion x(t) in response
to other forces F :2χ RAP C Za
Vb cos(ωb t)RAP C
F + σV+ Vb cos(ωb t)x(t)=RAP C + Za
λ (RAP C + Za )2mω02 x + mγ ẋ + mẍ (7.13)
F + σVVb cos(ωb t)RAP C=RAP C + Za
2χ RAP C Za
2mω0 − σV Vb cos(ωb t)
x + mγ ẋ + mẍ (7.14)
λ (RAP C + Za )2

A dc voltage (ωb = 0) bias creates a feedback loop that shifts the mechanical resonancefrequency ω0 → ω0 + ∆V
∆V =σV Vb 2χ RAP C Za
2ω0 m λ (RAP C + Za )2(7.15)

238
when ∆V  ω0 . When the voltage bias is applied at twice the mechanical resonancefrequency (ωb = 2ω0 ), this parametric drive will cause the nanomechanical oscillator toregeneratively oscillate when the bias voltage Vb > VbR . This voltage VbR can be used
to calculate the proportionality constantσV =2γω0 mλ (RAP C + Za )2
RAP C Za
VbR 2χ

(7.16)

Given the bias voltage VbR at which regenerative oscillation begins, the two models
make different predictions for the backaction force FBA created by a voltage bias Vb
applied near the mechanical resonance frequency. When RAP C > Za , the model of a
backaction force proportional to current predicts a force FBAI on the nanostructure thatis larger than the force FBAV predicted by a model of the backaction force proportional
to voltage by a factor of Za /RAP C . If the backaction force is proportional to voltage
then
FBAV = σV VAP C =2γω0 Vb mλ RAP C + Za
Za
VbR 2χ

(7.17)

I infer the magnitude of the backaction force FBAV at different resistances (figure 7.12,
red data) caused by a voltage bias Vb = 1 mV from the observed onset of regenerative oscillations at a pump voltage VbR . On the other hand, if the backaction force is
proportional to current thenFBAI = σI IAP C =2γω0 Vb mλ RAP C + Za
RAP C
VbR 2χ

(7.18)

I also infer the magnitude of the backaction force FBAI at different resistances (figure
7.12, green data) caused by a voltage bias Vb = 1 mV from VbR . These two models
predict different backaction forces FBAV 6= FBAI .
The backaction force FBA is directly determined by measuring the displacement inresponse to a voltage applied across the APC. A bias Vb cos(ωt) applied at frequencies ωnear the nanomechanical resonance frequency ω0 creates a backaction force FBA cos(ωt)

239inferred from feedback,F proportional to VAPC

1

Force (pN)10inferred from feedback,F proportional to IAPC

0

10

−1

10

0

F measured directly
200

400
600
800
APC Resistance (kΩ)

1000

1200

Figure 7.12: Backaction force due to a voltage bias Vb = 1 mV at different junction
resistances. The backaction force is inferred from the onset of regenerative oscillations at
a pump voltage VbR using a model FBAI = σI IAP C (blue data) or using a model FBAV =σV VAP C (red data). The backaction force is also determined directly by measuring thenanomechanical motion in response to Vb (green data). The model FBAI = σI IAP C is abetter explanation for the observed force and feedback at all junction resistances.

I determine the magnitude of FBA by measuring the frequency-dependent displacementx(ω) =FBAm (ω02 − ω 2 ) + iγω(7.19)

using the APC detector and the calibration described in section 7.3. I plot the directly
observed backaction force FBA created by a bias voltage Vb = 1 mV in figure 7.12 (greendata).
The model of a backaction force proportional to current is a better explanation forthe observed unintentional feedback and backaction force than a model of a backaction
force proportional to voltage. In figure 7.12, the force inferred from feedback using
these two models is compared to the directly observed backaction force (green data).

240
The model of a backaction force proportional to current (blue data) is more consistent
with the observed force than the model than a model of a backaction force proportional
to voltage (red data) at all junction resistances.

Electron Momentum (pe/pF)

inferred from feedback
1

10

measured directly

0

10

0

200

400
600
800
APC Resistance (kΩ)

1000

1200

Figure 7.13: The electron momentum pe = σI /e needed to explain the directly measured
backaction force (green) and the backaction force inferred from feedback (red) in units of
the Fermi momentum pF at different junction resistances. At large resistances RAP C >400 kΩ the inferred electron momentum is similar to the electron momentum pe ≈30pF needed to explain the backaction observed in electromigrated devices (chapter 6).
However, the resistances RAP C of the devices measured in chapter 6 were RAP C <50 kΩ; the electron momentum inferred from the directly measured backaction (green)
is significantly smaller than 30pF when RAP C < 400 kΩ.

This model of a backaction force proportional to current is phenomenological andthe feedback data does not provide any insight into the microscopic origins of the force.
However, if the backaction force proportional to current is caused by the momentum
kicks of tunneling electrons then the momentum pe of each tunneling electron is within
an order of magnitude of the electron momentum needed to explain the backaction
force observed in the electromigrated junctions (chapter 6). I plot the momentum pe241in units of the Fermi momentum pF as a function of junction resistance in figure 7.13.
While the inferred electron momentum pe is similar to the electron momentum neededto account for the previously observed backaction (chapter 6, pe ≈ 30pF ) at resistances
RAP C > 400kΩ, the electromigrated devices in chapter 6 had resistances RAP C < 50kΩ.
At more comparable resistances, the electron momentum inferred from the directly
measured backaction force in the MCBJ device (green, figure 7.13) is less than a tenthof the momentum inferred in chapter 6. The devices in chapter 6 used different in APC
creation techniques, nanomechanical structures, and measurement techniques.

7.4.2

Correlations between shot noise and nanomechanical motion

A backaction force may create correlations between shot noise and nanomechanical motion. If the backaction force is proportional to the either the current IAP C
flowing through the junction FBAI = σI IAP C or the voltage VAP C across the APC
FBAV = σV VAP C then nanomechanical motion caused by the backaction force will becorrelated with the electrical shot noise at the nanomechanical resonance frequency. I
do not observe correlations between shot noise and mechanical motion. This places anupper bound on the backaction proportional to current or voltage, expressed in unitsof electron momentum pe = σI /e = σV /RAP C e, of pe < 50pF . This upper bound is notinconsistent with the previously measured backaction pe ≈ 30pF (chapter 6). In thismeasurement the noise is dominated by amplifier noise instead of the fundamental shotnoise. A more precise measurement of the correlation would require reducing the 5 K
Brownian motion by cooling the nanostructure and decreasing the amplifier noise.
Correlations between nanomechanical motion and electrical shot noise can bedetected using the measured voltage spectral density. I simultaneously apply a dc
voltage bias VAP Cdc and a microwave voltage bias VAP Crf at ωrf resulting in a voltage
signal due to nanomechanical motion at the nanomechanical resonance frequency ω0 andat ωrf ± ω0 . I use two electrical resonances to implement both a baseband measurement

242
of the signal at ω0 and single-sideband measurement of the signal at ωrf + ω0 (chapter
5). The backaction FBA = σI IAP C due to shot noise SV sn (ω) will result in fluctuationsin the displacement of the nanostructure with spectral density

SxBA (ω) = SV sn (ω)

σI
RAP C

2
m2

1
 2

(ω0 − ω 2 )2 + γ 2 ω 2(7.20)

Since the shot noise SV sn (ω) is equal to but not correlated with the shot noise in the
single-sideband measurement SV sn (ω + ωrf ), the signal due to SxBA adds incoherently
with the shot noise SV sn and the measured microwave voltage spectral density SV0 sn (ω +ωrf ) proportional to shot noise is
#
"
2
2
2
Za (ω + ωrf )
(σ
/R
)
|χV
/λ|
I
AP Crf
 AP2C +1(7.21)
SV0 sn (ω + ωrf ) = SV sn
RAP C + Za (ω + ωrf )m2 (ω0 − ω 2 )2 + γ 2 ω 2
On the other hand, near the mechanical resonance frequency the shot noise SV sn (ω) andthe displacement SxBA (ω) due to the backaction force are correlated. The two noise
sources interfere constructively on one side of the mechanical resonance and destructively
on the other side resulting in a measured microwave voltage spectral density SV0 sn (ω)
proportional to shot noise
SV0 sn (ω) = SV sn(σI /RAP C )(2χVAP Cdc /λ) +1
m (ω02 − ω 2 ) + iγω

22
Za (ω)RAP C + Za (ω)(7.22)

The correlations create a difference between the inferred displacement spectral density∆Sx using the baseband measurement and the single-sideband measurement
!
2
2
R
+
Z
(ω)
λ
a
AP C
∆Sx (ω) = SV0 sn (ω)− SV sn−Za (ω)
2χVAP Cdc!
2
RAP C + Za (ω + ωrf ) 2
λ0
SV sn (ω + ωrf )− SV snZa (ω + ωrf )χVAP Crf∆Sx (ω) =σI SV snλω2 − ω2 2 0

VAP Cdc RAP C χ m (ω0 − ω 2 )2 + γ 2 ω 2(7.23)(7.24)

explicitly removing the white background caused by the shot noise SV sn . The contribution from the amplifier, discussed below, is also subtracted.

243

1
0

x

ΔS (fm2/Hz)

2

−1
−2

2

ΔSx (fm /Hz)

0.3
0.2
0.1
0
−0.1
−0.2
−0.3

66
66.5
67
Mechanical Frequency (MHz)67.5

Figure 7.14: The difference ∆Sx (ω) between the magnitude of the displacement fluctuations Sx inferred from a baseband measurement and the displacement fluctuations
inferred from a single-sideband measurement as a function of mechanical frequency.
The displacement fluctuations are measured using a junction with RAP C = 80 kΩ anda length scale λ/χ = 0.2 nm. A dc voltage VAP Cdc = 42 mV and a microwave voltage VAP Crf = 50 mV are simultaneously applied to the junction and used to performthe baseband and single-sideband measurements, respectively. If there is a backaction
force proportional to voltage or current, then the correlation between shot noise andand the backaction force caused by shot noise will create interference in the baseband
measurement but not in the single-sideband measurement performed at microwave frequencies. Parameterizing the backaction force proportional to current in terms of theelectron momentum FBAI = (pe /e)IAP C , I plot the expected signal ∆Sx (ω) (equation
7.24) when pe = 50pF (red) and pe = −50pF (blue); the sign of pe is dependent on thedefinition of a positive current IAP C . The observed ∆Sx (ω) (black dots) is inconsistent with |pe | > 50pF . I also fit the observed data to equation 7.24 (green) and infer
pe = (−3 ± 7)pF not including sources of systematic error. The error in this measurement is large enough that it cannot distinguish between a correlated backaction force
and an uncorrelated backaction force; in this data, the on-resonance noise is dominatedby the nanostructure’s Brownian motion and the off-resonance noise is dominated byamplifier noise (see figure 7.15).

244
I plot ∆Sx (ω) in figure 7.14; due to the additional noise added by the amplifier andthe Brownian motion of the nanostructure, this measurement can only place an upper
limit on the size of any correlated backaction which is consistent with each electron
delivering a momentum kick pe < 50pF (red line). This measurement is performed
using a junction with resistance RAP C = 80 kΩ and length scale λ/χ = 0.2 nm. I
attempt to detect a backaction force that is correlated with the shot noise due to a dc
voltage VAP Cdc = 42 mV and a microwave voltage VAP Crf = 50 mV. The backaction ofprevious measurements can also be explained by electron momentums less than 50pF , so
this measurement of ∆Sx (ω) is not able to distinguish between a backaction correlatedwith shot noise from an uncorrelated backaction.
The amplifier and the Brownian motion of the nanostructure add additional noise
terms to SV0 sn (ω) and SV0 sn (ωrf + ω). In figure 7.15 I plot the voltage spectral density
as a function of the frequency of the nanomechanical displacement. At frequencies
away from the mechanical resonance, the noise in the measurement is dominated by
the amplifier noise (see chapter 5 for details). Near the mechanical resonance frequency
the noise in the measurement is dominated by the Tcryo = 5 K Brownian motion of
the beam. These noise sources dominate the noise in the measurement of ∆Sx (ω), so
a more precise measurement of the correlated backaction can be made by reducing thecontribution of the amplifier noise relative to the shot noise or reducing the temperatureof the nanostructure.
Two sources of systematic error in this measurement are the determination of thelength scale λ/χ and the unstable mechanical resonance frequency. The measurementof λ/χ was discussed in 7.3 and any systematic error will scale the inferred magnitudeof the backaction (equation 7.24). Small changes in the mechanical resonance frequency
could also create an effect on ∆Sx (ω) that is similar to the effect of correlations between
backaction and shot noise. My measurement of SV0 sn (ω) and SV0 sn (ωrf +ω) are separated
by minutes, so any shift in resonance frequency in that time would result in a ∆Sx (ω)

245
0

2

SV (arb V /Hz)

10

−1

10

Single Sideband
Measurement
Amplifier Noise

VAPCdc=42 mV
VAPCrf=50 mV
Tcryo = 5 K

BasebandMeasurement−2
Amplifier Noise
10

64

65
66
67
68
Mechanical Frequency (MHz)69

Figure 7.15: The voltage spectral density SV observed using both a baseband measurement (blue, cyan) and single-sideband measurement (red, magenta) are plotted versus
the inferred mechanical frequency. The voltage spectral densities are scaled so thatthe voltage noise spectral density due to on-resonance Brownian motion of the mechanical structure is equal to one. The junction has a resistance RAP C = 80 kΩ and alength scale λ/χ = 0.2 nm. A dc voltage VAP Cdc = 42 mV and a microwave voltage
VAP Crf = 50 mV are simultaneously applied to the junction and used to perform thebaseband(blue) and single-sideband (red) measurements, respectively. In both measurements, the total noise is dominated by the Tcryo = 5 K Brownian motion on-resonance.
Off-resonance, the noise is dominated by amplifier noise (cyan/magenta). The shotnoise contribution to the noise is equal to the difference between the amplifier noise(cyan/magenta), i.e., the observed spectral density when VAP Cdc = VAP Crf = 0 mV,
and the measurement noise (blue/red) off-resonance. In the single-sideband measurement, the amplifier noise (magenta) and the shot noise are not dependent on frequency,
because the microwave amplifier noise is flat and the microwave electrical resonancefrequency is broad > 50 MHz. On the other hand, the baseband amplifier noise isfrequency dependent around 66 MHz and the observed voltage fluctuations due to shotnoise are frequency dependent because the 66 MHz electrical resonance is only 2 MHz
wide. This electrical resonance filters the shot noise and signal due to nanomechanicaldisplacement. The measurement of ∆Sx (ω) can be improved by improving the electricalmeasurement so that the measurement noise is dominated by shot noise (see chapter 5)and by reducing the temperature of the nanostructure.

246
which changes sign at the average resonance frequency and decreases away from theresonance frequency; this is similar to the ∆Sx (ω) created by correlations. I attempt toidentify changes in the resonance frequency by alternating between measuring SV0 sn (ω)
and SV0 sn (ωrf + ω) multiple times.

7.5

APC Backaction and Tunneling Spectroscopy
I finally examine the effect of dc bias on the backaction force. In molecularelectronics experiments, changes in differential resistance as a function of dc bias voltage
have been used to observe the vibrational modes of atoms and molecules in atomic point
contacts (APCs) [8, 10, 40, 49, 50, 52–54, 57]. I observe similar changes in differential
resistance at bias voltages |VAP C | between 15 mV and 50 mV. At the same bias voltage, I
also observe increased backaction. One possible explanation for this increased backaction
is interaction between the nanomechanical mode and the vibration, created by tunneling
electrons, of nearby atoms or molecules.
Inelastic scattering by electrons can create additional channels for electron tunneling, changing the resistance of the APC. This discussion follows a more detailed
review of inelastic scattering in the context of experimental APCs contained in M.L.
Trouwborst’s thesis [10]. In this inelastic process, an electron scatters off of an impurity
exciting a vibration at frequency ωv . An energy h̄ωv is transferred from the electron to
the impurity, so this process only occurs when the electron energy e|VAP C | ≥ h̄ωv or
when kB T ≥ h̄ωv . At large resistances where the expected tunneling channel is mostly
closed (RAP C  RQ /2 = h/(2e2 ) = 12.9 kΩ) the presence of an additional channel
decreases the resistance of the APC. This effect will only appear when |VAP C | > h̄ωv /ecreating a step in the differential resistance RAP C as a function of dc bias VAP C at a
voltage bias ±h̄ωv /e. The observation of this type of symmetric step is called inelastic
tunneling spectroscopy (IETS).
At small resistances where the expected tunneling channel is mostly open (RAP C ≈247
RQ /2 = h/(2e2 ) = 12.9 kΩ), electrons scattering off of an impurity will increase theresistance of the APC. After interacting with the impurity and losing an energy h̄ωv ,the electron scatters into a lower energy state. The low momentum states in the forward direction on the far side of the APC are filled, so the electron will backscatter into
the low momentum states in the reverse direction on the near side of the APC. The
backscatter increases the resistance of the junction, and the observation of this effect iscalled point contact spectroscopy (PCS).
Experimentally, the step in differential resistance due to vibrational levels is often accompanied by a larger peak or dip in differential resistance RAP C at |VAP Cv | =h̄ωv /e [10, 40, 52]. In gold chains, vibration modes have been observed at voltages|VAP Cv | = 10 mV and 18 mV thought to be due to the transversal and longitudinal
vibrational modes in gold [152]. Some evidence of the vibrational modes of helium, alikely contaminate, have been observed at |VAP Cv | ≈ 15-20 mV [182] and the vibrationalmodes of other materials have also been observed [10, 191–194]. In many cases, peaks
and dips are observed at |VAP Cv | which are much larger than the step in resistance.
The microscopic mechanism that results in a peak or dip in resistance is unknown,
but most models posit an impurity that fluctuates between two positions with different
conductances [191–193].
Using the MCBJ shown in figure 7.1, I sometimes observe clear, symmetric peaks
and dips in the differential resistance RAP C as a function of dc voltage bias VAP C aswell as overall steps in RAP C . In figures 7.16 and 7.17 I plot the differential resistanceas a function of the voltage across the APC RAP C (VAP C ) at different plunger positions. I change the plunger position over 600 nm and increase the zero bias resistance
RAP C (0 mV) from 270 kΩ to 2.2 MΩ.
At resistances RAP C (0 mV) < 500 kΩ I observe a dip in resistance and a stepto smaller resistance with increasing voltage bias |VAP C |. At the lowest measured
RAP C =270 kΩ the dip and step occur at a dc bias voltage |VAP C | = 25 mV. As the

248

7

APC Resistance (Ω)

10

6

10

−60

−40

−20
0
20
APC dc Voltage (mV)

40

60

Figure 7.16: Differential resistance RAP C versus dc voltage VAP C across the junction as
the position of plunger is stepped in 30 nm increments over 600 nm. Also see figure 7.17
for a different representation of the same data. At low resistances there is a symmetricdip in RAP C [VAP C ]. As the junction is strained, increasing the resistance, the dip
appears at larger |VAP C | and the magnitude of the effect decreases. Simultaneously, apeak begins to appear in RAP C at a voltage |VAP C | that is slightly smaller than thedip. The |VAP C | location of the peak and the fractional size of the peak both increase
with increasing resistance. When there is a distinct dip or peak, there is also a smalldchange in the resistance across the feature.
with adip at VAP
C , there At low resistancesdd
is a small decrease in resistance RAP C |VAP C |  |VAP C | < RAP C |VAP C |  |VAP
C|when comparing the resistance at voltages |VAP C | significantly greater than and lessdthan VAP
C . Similarly, at higher resistances there is a small increase in resistance.
There is a smooth transition with increasing strain between a decrease in resistance atlarge voltages |VAP C | and an increase in resistance at large voltages which is similar to
the transition of the feature from a dip to a peak. There is also a small peak in RAP C
at zero bias.

0.6

Plunger (μm)

249color: RAPC(0)/RAPC(VAPC)

1.4

0.5

1.2

0.4

1

0.3

0.8

0.2

0.6

0.1
0
-60

0.4
-40

-20
0
20
40
APC dc Voltage (mV)

60

Figure 7.17: The differential conductance scaled by the conductance at zero bias,
RAP C (0)/RAP C (VAP C ), versus plunger position and dc voltage VAP C across the junction. See figure 7.16 for a more detailed discussion of the same data.

plunger is used to increase the zero bias resistance RAP C (0 mV), the magnitude of the
dip and the step become smaller and they occur at larger voltage biases |VAP C |. When
the APC resistance has reached 500 kΩ, the dip and step appear at |VAP C | = 33 mV
and have almost disappeared. The step to lower resistance is consistent with the modelof inelastic scattering and the energy scale e×25 mV is consistent with the vibrationalfrequency of possible contaminants as discussed above.
At resistances RAP C (0 mV) > 450 kΩ I observe a peak in resistance and a step tolarger resistance with increasing voltage bias |VAP C |. A small peak in resistance can beobserved at RAP C (0 mV) > 450 kΩ and inside the broader dip described in the previous
paragraph. At resistances RAP C (0 mV) > 750 kΩ the peak in resistance is accompanied
by a step to larger resistances. The magnitude of both the peak and the step grow with

250
increasing strain. The location |VAP C | of the peak and step also increases as the strain
increases to 50 mV when RAP C (0 mV) > 2.2 MΩ. This step to lower resistance is notconsistent with the model of inelastic scattering described above, which predicts a step
to lower resistance due to an additional channel for electron conductance.
Based on this measurement, it is difficult to determine whether the peak and dip
are due to a single inelastic channel which is modified by strain or due to two interferingchannels. The smooth change in the location VAP C of the feature is suggestive of asingle channel. However, a closer examination of the data when RAP C (0 mV) ≈ 500 kΩ
indicates that the dip and peak in resistance appear simultaneously at slightly different
voltages. In spite of this interesting behavior, I was unable to study this behavior inmore detail and determine the microscopic origin of the spectrum.
I also measure the backaction force as a function of dc voltage bias VAP Cdc andobserve peaks in the backaction and features (peaks or dips) in the differential resistanceat the same VAP Cdc . I measure the backaction force using APCs with three different
atomic configurations; that is, the junction was closed with a resistance < 100 Ω andthen a new useable resistance was obtained after opening and closing the junction multiple times (section 7.1). In figures 7.18, 7.19 and 7.20 I plot the characteristics of the
APC displacement measurement and the nanomechanical resonance as a function ofdc voltage VAP Cdc along with the different resistance RAP C . In figure 7.18 I observethat the differential resistance does not have peaks or dips indicating scattering andthe magnitude of the backaction, parameterized in units of the electron momentum pewhere FBA = (pe /e)IAP C , does not depend on the dc voltage bias VAP Cdc . On the other
hand, in figures 7.19 and 7.20 I observe peaks and dips, respectively, in the differential
resistance and an increase in the magnitude of the backaction force pe at the same dc
voltage bias VAP Cdc .
Finally, I measure the length scale λ/χ and the nanomechanical resonance characteristics as a function of VAP Cdc . I observe a clear change in the length scale λ/χ at

251
the same voltages VAP Cdc where there is a peak in the resistance (figure 7.19) and asmall peak in the length scale at the voltages VAP Cdc where there is a dip in the resistance (figure 7.20). The length scale λ/χ is constant when the differential resistance isfeatureless (figure 7.18). There is a small increase in the nanomechanical damping when
there is a dip the resistance (figure 7.20); in the other two cases the damping depends
smoothly on dc voltage |VAP Cdc |. In the data where there is a peak in resistance thedamping increases (figure 7.19), while the damping decreases when the resistance isfeatureless (figure 7.18).
In all three data sets, I observe changes in the resonance frequency due to the dc
bias voltage. Because of the unintentional feedback discussed in section 7.4.1, the dc
voltage bias can create antisymmetric changes in the mechanical resonance frequency,that is, changes proportional to voltage VAP Cdc . In these measurements, the dc voltage
creates a significant symmetric change (proportional to |VAP Cdc |) in the mechanicalresonance frequency which makes it difficult to detect feedback and can also cause
systematic errors. It is not possible to determine whether the asymmetric dependenceof the frequency on VAP Cdc is due to feedback or small asymmetries in the microscopic
processes which create the symmetric frequency shifts. However, in some cases thesymmetric frequency shift is small in comparison to the antisymmetric frequency shift;
in this case, the shift is likely due to feedback.
A possible explanation for the increase in backaction is the motion of an impurity.
The features in the differential resistance are commonly associated with vibrationalmodes of impurities, and any increase in that vibration due to an increase in the effective
temperature of the impurity [10, 195, 196] could create a random force which acts on
the beam creating backaction. Random motion of an impurity between two different
positions could also create a force on the beam and lead to a backaction force. Sucha backaction force could be larger than that expected from the momentum kicks oftunneling electrons. The data in figures 7.18, 7.19, and 7.20 suggests that the presence

252
of impurities and atomic motion could play a role in the backaction but further study
is necessary to understand that role.
One possible source of systematic error is a change in the temperature of thenanostructure caused by the increasing dc voltage bias. Since I am not using a thermal
sweep to calibrate the measurement, any increase in the local temperature of the nanostructure compared to the cryostat temperature will create the appearance of an increasein the backaction force. I observe this heating at large voltage bias (for example, figure 7.20 when |VAP C | > 30 mV), however the local temperature of the nanostructure
should monotonically increase with increasing voltage bias |VAP C |. Local heating of thenanostructure cannot explain both the increase and the decrease in the magnitude of
the observed backaction force.

198

25

189
180
10 20 30 40 50
207
198

0.55

189
180
10 20 30 40 50
207

RAPC (kΩ)

0

120

198

90

189

60
−50 −40 −30 −20 −10
64.75

0

180
10 20 30 40 50
207

RAPC (kΩ)γ/2 π (kHz)λ/χ (nm)

0

0.6
0.5
−50 −40 −30 −20 −10
150

253

RAPC (kΩ)

75−25−50 −40 −30 −20 −10
0.65

ω0/2 π (MHz)207

RAPC (kΩ)

2epe/(pF |p |)

125

64.7

198

64.65

189

64.6
180
−50 −40 −30 −20 −10 0 10 20 30 40 50
APC dc Voltage (mV)

Figure 7.18: As a function of dc voltage VAP Cdc across the junction, I plot the backaction
force parameterized in terms of electron momentum pe /pF (blue, left axis), the inferred
length scale λ/χ (green, left axis), the nanomechanical damping γ (red, left axis), andthe nanomechanical resonance frequency ω0 (cyan, left axis). I also plot the differential
resistance RAP C of the junction on each axis (right axis, black). The resistance RAP C
is determined both during the displacement measurement (black squares) with a 4 mV
microwave bias voltage and separately without an applied microwave bias (black lines).
The microwave voltage bias effectively averages these measurements over VAP Cdc ±4 mV(see chapter 5). The differential resistance RAP C does not contain any large peaks ordips, and both the effective electron momentum pe /pF and length scale λ/χ also havelittle dependence on VAP Cdc . Contrary to most expectations, in this case the damping
γ decreases with increasing voltage. There is also a small feature in the resonancefrequency as a function of VAP Cdc .

45

190

5

140
90
10 20 30 40 50
240

0.18

190

0.14

140
0

90
10 20 30 40 50
240

RAPC (kΩ)γ/2 π (kHz)0.1−50 −40 −30 −20 −10
46
42

190

38

140

34
−50 −40 −30 −20 −10
59.81

ω0/2 π (MHz)

0

0

90
10 20 30 40 50
240

RAPC (kΩ)λ/χ (nm)−35−50 −40 −30 −20 −10
0.22

254

RAPC (kΩ)

240

RAPC (kΩ)

2epe/(pF |p |)

85

59.78

190

59.75

140

59.72
90
−50 −40 −30 −20 −10 0 10 20 30 40 50
APC dc Voltage (mV)

Figure 7.19: As a function of dc voltage VAP Cdc across the junction, I plot the backaction force parameterized in terms of electron momentum pe /pF (blue, left axis), theinferred length scale λ/χ (green, left axis), the nanomechanical damping γ (red, leftaxis), and the nanomechanical resonance frequency ω0 (cyan, left axis). I also plot thedifferential resistance RAP C of the junction on each axis (right axis, black). The resistance RAP C is determined both during the displacement measurement (black squares)
with a 2 mV microwave bias voltage and separately without an applied microwave bias
(black lines). The microwave voltage bias effectively averages these measurements over
VAP Cdc ±2 mV (see chapter 5). The differential resistance RAP C contains a large peakat VAP Cdc = ±18 mV, and both the effective electron momentum pe /pF and length scaleλ/χ also have sharp features at VAP Cdc = ±18 mV. The damping γ increases slightly
with increasing voltage. There are also small features and an overall increase in theresonance frequency as a function of VAP Cdc .

25

63

0

58
53
10 20 30 40 50
68

0.54

63

0.51

58
0

53
10 20 30 40 50
68

RAPC (kΩ)γ/2 π (kHz)0.48−50 −40 −30 −20 −10
53
48

63

43

58

38
−50 −40 −30 −20 −1061.74

ω0/2 π (MHz)

0

61.65

0

53
10 20 30 40 50
68

RAPC (kΩ)λ/χ (nm)−25−50 −40 −30 −20 −10
0.57

255

RAPC (kΩ)

68

RAPC (kΩ)

2epe/(pF |p |)

50

63
58

61.56
53
−50 −40 −30 −20 −10 0 10 20 30 40 50
APC dc Voltage (mV)

Figure 7.20: As a function of dc voltage VAP Cdc across the junction, I plot the backaction
force parameterized in terms of electron momentum pe /pF (blue, left axis), the inferred
length scale λ/χ (green, left axis), the nanomechanical damping γ (red, left axis), andthe nanomechanical resonance frequency ω0 (cyan, left axis). I also plot the differential
resistance RAP C of the junction on each axis (right axis, black). The resistance RAP C
is determined both during the displacement measurement (black squares) with a 4 mV
microwave bias voltage and separately without an applied microwave bias (black lines).
The microwave voltage bias effectively averages these measurements over VAP Cdc ±4 mV(see chapter 5). The differential resistance RAP C has a dip at VAP Cdc = ±21 mV, andboth the effective electron momentum pe /pF and damping γ have features at VAP Cdc =±21 mV. The resonance frequency ω0 also depends on VAP Cdc with a larger rate ofchange near VAP Cdc = ±21 mV and the length scale λ/χ has a small feature near
VAP Cdc = ±21 mV.

Chapter 8

Conclusions and Future Directions

8.1

Conclusions
In conclusion, I have demonstrated a sensitive measurement of nanomechanicaldisplacement using an atomic point contact (APC). Two techniques were used to create
an APC which was coupled to a nanostructure. In both cases, the standard tools of
nanolithography were used to create a suspended gold nanostructure with a constriction
which was thin and narrow in comparison to the rest of the nanostructure. The device
was placed in the ultrahigh vacuum of a 4 K cryogenic system. In the first technique,electromigration, I passed a large current through the constriction and the large current
density effectively forced atoms away from the constriction. Eventually, all except two
atoms where removed and the flow of electrons through the constriction was dominatedby electrons tunneling through a vacuum gap between the two atoms. This technique
has the advantage that it is relatively easy to implement, but it is not possible to control
the resistance of the APC, that is, the width of the gap. The second technique usedmechanical strain to control the resistance of the constriction, creating a mechanicallycontrollable break junction (MCBJ). The applied strain caused the metal constriction
to become thinner and fracture on the atomic scale, creating an atomic point contact.
This technique required additional infrastructure to create strain by bending the substrate beneath the suspended nanostructure, but had the advantage that the widthof the APC gap was controllable and the configuration of atoms in the APC could

257
be changed, though not with microscopic control. I used this technique to measure
the APC properties as a function of APC resistance. In both cases, the motion of thenanostructure created additional strain on the APC because the APC was in the middleof the suspended nanostructure. The changing width of the APC gap created a change
in the resistance of the APC which I used to infer the nanomechanical displacement.
Having created an APC coupled to a nanomechanical structure, I used a microwave resonant circuit to increase the speed of the APC measurement. The filter
created by the combination of the APC, a high impedance device RAP C > 10 kΩ, andthe stray capacitance in the measurement circuit limited the bandwidth of earlier APC
measurements to less than 100 kHz. I used the same microwave technique that was used
to create the radio-frequency single electron transistor (RF-SET) to overcome this bandwidth limitation. The high impedance device was embedded in an LC resonant circuit.
Near the electrical resonance frequency, the LC circuit matched the APC resistance tothe resistance of the 50 Ω measurement circuit. The bandwidth of the measurement
is controlled by the quality factor of the resonant circuit. Using a 500 MHz resonant
circuit, I increased the bandwidth of the APC measurement to more than 30 MHz.
Kemiktarak et al [140] recently used this technique, which I described in reference [11],to create an RF-STM with the goals of decreasing STM image acquisition time, using
the STM tip as a local thermometer, and sensing nanomechanical motion.
Because of the large bandwidth of the microwave measurement and the low noiseof 50 Ω cryogenic microwave amplifiers, the noise in the APC displacement measurement
was dominated by the shot noise of tunneling electrons. The shot noise of tunneling
electrons is the fundamental source of noise in an APC measurement. Minimizing theeffect of other, less interesting, sources of noise is important because it is a prerequisite for performing a quantum limited measurement. In this thesis, the displacement
measurement was calibrated using the Brownian motion of the nanostructure at cryogenic temperatures. I measured displacement with a shot-noise limited imprecision

258
√

p
√
p
Sx = 0.29 fm/ Hz; this is equal to 4 times the imprecision SxSQL = h̄/mω0 γ at

the standard quantum limit where m is the effective mass of the nanostructure, ω0 theresonance frequency, and γ the damping.
I compared the APC displacement detector to a quantum limited detector by
simultaneously measuring the backaction force of the APC detector. The APC mea√√
surement created a noisy backaction force with a spectral density SF = 61 aN/ Hz;p√this is equal to 40 times the backaction SF SQL = h̄mω0 γ at the standard quantum
limit. The combination of the measurement imprecision Sx and the random motion
caused by the backaction force resulted in a total measurement uncertainty that is 27
ptimes the standard quantum limit 2h̄/mω0 γ. A quantum limited detector would op√
erate at the limit imposed by the Heisenberg uncertainty principle Sx SF ≥ h̄. In√
comparison, the imprecision-backaction product of the APC detector Sx SF = 168h̄is larger than required by quantum mechanics. Because the measurement imprecision
Sx was dominated by the fundamental source of noise in an APC measurement, theshot noise of tunneling electrons, the non-ideality of the APC detector is likely due toa backaction force SF in excess of that required by quantum mechanics.
The excess backaction force created by the APC measurement could have many
possible origins. For example, the mutual capacitance C between the APC electrodes
depends on the electrode separation x which results in an electrostatic backaction force
2
FC = (∂C/∂x)(VAP
C /2) where VAP C is the voltage across the APC. The presence oftrapped charge near the APC would result in a backaction force FV = σV VAP C where
the proportionality constant σV depends on the magnitude and location of the trappedcharge. As a final example, each tunneling electron imparts a momentum kick pe to thenanostructure resulting in a backaction force FI = (pe /e)IAP C . In a phenomenologicalmodel of the APC displacement detector it is the uncertainty in the magnitude of the
momentum kick pe which enforces the Heisenberg constraint.
While I cannot unambiguously determine the origin of this excess backaction force,

259
I was able to eliminate certain possible origins by investigating the properties of the backaction force in more detail. In two electromigrated APCs I observe a linear dependence
2
between the spectral density of the current fluctuations SI = SV /RAPC ≈ 2eIAP C andthe backaction force SF . This linear dependence is consistent with a backaction force
due to trapped charge FV or the momentum kicks of tunneling electrons FI , but inconsistent with an electrostatic backaction FC . To account for the observed backaction
force, each electron would have to deliver a momentum impulse of pe = pF × (32 ± 3),
for the first electromigrated device, or pe = pF × (34 ± 1) for the second electromigrated
device, where pF is the Fermi momentum of electrons in bulk gold. A simple phenomenological model of the backaction created by momentum kicks predicts that eachelectron should deliver a momentum kick approximately equal to the Fermi momentum.
In this context, the momentum impulse pe ≈ 30pF required to account for the APC
backaction seems implausibly large.
Using an MCBJ, I controlled the width of the vacuum gap between atoms thatform the APC and measured the backaction force at different APC resistances. I measured the unintentional feedback created by the APC backaction and directly measured
the magnitude of the backaction force. The combination of the APC and the measurement circuit formed an unintentional feedback loop. Nanomechanical motion changed
the resistance RAP C of the APC. The changing resistance modified the current IAP C
flowing through the APC and the voltage VAP C across the APC. The details of thiseffect depends on the measurement circuit; in this case, the impedance of the measurement circuit was smaller than RAP C so nanomechanical motion had a larger effect on
the current IAP C than on the voltage VAP C . A backaction force proportional to current
IAP C or voltage VAP C closed the feedback loop, creating a force on the nanomechanicalstructure that was proportional to displacement. At the same time, I directly measured
the magnitude of the backaction force by observing the nanomechanical response to a
voltage VAP C = IAP C RAP C across the APC at frequencies near the nanomechanical260
resonance frequency. The combination of the observed unintentional feedback and directly measured backaction force was better explained by a backaction force proportional
to current than a backaction force proportional to voltage FV at all APC resistances.
While these observations did not directly address the origin of the backaction force, I
can parameterize this backaction force proportional to current FI in terms of the momentum kick pe delivered by tunneling electrons. At different APC resistances, eachelectron would have to deliver a momentum impulse between pF and 40pF to accountfor the observed backaction force.
I also attempted to determine whether the noisy backaction force and the shotnoise of tunneling electrons were correlated. If the backaction force was due to theaverage momentum impulse pe delivered by each tunneling electron, then the shot noiseand backaction force would be correlated. This correlation measurement could only
place an upper limit on the size of any correlated backaction which is consistent witheach electron delivering a momentum kick pe < 50pF . The upper bound is so large
because in these measurements the amplifier regrettably contributed much more noise
than the electrical shot noise due to tunneling electrons.
Finally, I observed evidence of molecular vibrations in the APC using inelastic electron tunneling spectroscopy and simultaneously measured the backaction force.
Molecular vibrations can increase or decrease the differential resistance RAP C of the
APC when the vibrational energy h̄ω0 is equal to the bias energy e|VAP C |. When I
observed peaks or dips in RAP C at a bias VAP C , indicative of molecular vibrations, I
also observed a resonant increase in the backaction force at the same bias VAP C . Thisobservation suggests that the excess backaction may have been caused by the interactionof tunneling electrons with molecular vibrations.

261

8.2

Future Directions
There are a number of different questions that future experiments involving APC

displacement detectors can explore. An improved correlation measurement could conclusively demonstrate whether the backaction force is due to the effect of single electrons.
In addition, a simultaneous measurement of the directly applied backaction force, theunintentional feedback created by the backaction force, and the correlations between the
noisy backaction force and the shot noise of tunneling electrons would place additional
constraints on the possible physical origins of the backaction force.
An improved correlation measurement would require a lower noise measurement
circuit. There are three main approaches to improving the measurement circuit. First,
the measurement could be improved by using an amplifier which adds less noise. Second,
the bandwidth and noise characteristics of the measurement circuit could be improvedby using a more complicated matching circuit with multiple stages instead of a single
capacitance and inductance. Finally, the APC could be coupled to a nanomechanicalstructure with a larger mechanical resonance frequency ω0 . This change in frequency
could help the measurement circuit because it is easier to create low noise 50 Ω amplifiers
at higher frequencies. It could also improve the stability of the APC and potentially pushthe mechanical structure towards the interesting quantum regime where h̄ω0 ≈ kB T .
However increasing the resonance frequency would usually also increase the stiffness
k = mω02 of the structure and decrease the displacement caused by applied forces which
would make it more difficult to observe correlations. This undesirable scaling could be
mitigated if the increase in resonance frequency was accomplished by decreasing themass of the nanomechanical structure using, for example, novel fabrication techniques
and nanotubes or nanowires.
Future experiments involving APC displacement detectors could also explore thedifferences between gold APCs and APCs made out of other materials. The static me-

262
chanical and electrical properties of APCs are material dependent, so other materials
could result in a more stable atomic point contact. More interestingly, measuring thebackaction force in different materials could help to differentiate between the fundamental backaction due to the momentum impulse delivered by tunneling electrons and thematerial-specific properties. In addition, a material that is lighter than gold could be
used to create a more compliant mechanical system which is more sensitive to applied
forces.
Finally, future experiments could study in detail the interaction between trappedmolecules, tunneling electrons, and nanomechanical structures. The observations of
the inelastic electron tunneling spectrum and backaction force I performed indicateda possible link between molecular vibrations and the backaction force. However, I did
not purposefully trap a specific molecular species or study in detail the link between
trapped molecules and tunneling electrons. Such an investigation could lead to valuable
insight into the properties of trapped molecules and the backaction force. If the excess
backaction is due to molecular vibrations, this insight could be used to improve the
APC displacement detector by decreasing the excess backaction force.

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Letters, vol. 100, no. 13, p. 136801, Apr 2008.

Appendix A

Fabrication of Nanostructures on Gallium Arsenide Substrates• cleave gallium arsenide wafer into approximately 5 mm by 5 mm squares• Clean by rinsing with acetone, then isopropanol (IPA), then blow dry with dry
nitrogen

Fabricate Bondpads• spin a layer of resist on top of the chip: 950K PMMA A7 (7% anisole) at 4 krpm
for 30 sec• bake on a hot plate at 180◦ C for 5 minutes• Expose resist using NPGS on a JOEL6400 SEM with beam set to 35 kV (figureA.1a); small orange alignment marks at a magnification of ×1000, beam current of
10 pA, and exposure of 450 µC/cm2 ; yellow bond pads at a magnification of ×100,
beam current of 1600 pA, and exposure of 375 µC/cm2 ; large orange alignment marks
at a magnification of ×100, beam current of 1600 pA, and exposure of 350 µC/cm2 .
• develop by shaking chip using tweezers in developer ( 1 part MIBK to 3 parts
IPA) for 60 seconds then rinsing with IPA and blow dry with dry nitrogen• evaporate 10 nm of titanium then 95 nm of gold at 0◦ (that is, vertically)• liftoff by letting the chip sit in acetone for between 1 and 12 hours; at the endof the 12 hours and while chip is submerged in acetone, use a syringe to spray chip withacetone. In some cases, it is necessary to gently use the ultrasonic bath to remove all of

279
the material that was not deposited on GaAs. Then rinse with acetone, IPA, blow dry
with dry nitrogen

Fabricate Devices• spin a layer of resist on top of the chip: 950K PMMA A7 (7% anisole) at 4 krpm
for 30 sec• bake on a hot plate at 180◦ C for 5 minutes• Expose resist using NPGS on a JOEL6400 SEM with beam set to 35 kV at amagnification of ×1000 and beam current of 10 pA (figure A.1b); beam (red) at exposureof 300 µC/cm2 ; point (pink) at exposure of 260 µC/cm2 ; electrostatic gate (green) at
exposure of 350 µC/cm2 ; support structure (cyan) at exposure of 400 µC/cm2 ;• develop by shaking chip using tweezers in developer (1 part MIBK to 3 parts
IPA) for 60 seconds then rinsing with IPA and blow dry with dry nitrogen• using an electron gun evaporator at a pressure of about 6×10−6 Torr, evaporate
40 nm of gold at 0◦ (that is, vertically), 180 nm of gold at −27◦ from vertical, and 180 nmof gold at 27◦ from vertical.
• liftoff by letting the chip sit in acetone for between 1 and 12 hours; at the endof the 12 hours and while chip is submerged in acetone, use a syringe to spray chip withacetone. In some cases, it is necessary to gently use the ultrasonic bath to remove all of
the material that was not deposited on GaAs. Then rinse with acetone, IPA, blow dry
with dry nitrogen• etch in a solution composed of 5 parts 1 M citric acid to 1 part 70% H2 02 for
150 seconds. In order to properly etch the GaAs that was exposed to the electron beam,
I need to use a magnetic stir bar to stir the etch solution and hold the GaAs chip in
the moving etch solution. Then rinse with acetone, IPA, blow dry with dry nitrogen.
These nanostructures are stiff enough that critical point drying is unnecessary.

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(a)

(b)

μm

μm

Figure A.1: (a) NPGS pattern used to create bond pads (yellow) and alignment marks(orange). (b) NPGS pattern used to create a device with a nanomechanical beam (red)and constriction (between pink and red).

Appendix B

Fabrication of Nanostructures on Silicon Substrates• at NIST-Boulder: an approximately 100 nm thick layer of silicon oxide is thermally grown on a high-resistivity silicon wafer. Alignment marks (and large bond pads
which are not used with these devices) are created using the standard tools of opticallithography and the silicon wafer is cut into 5 mm by 5 mm squares using a dicing saw.

Fabricate Holes in Silicon Oxide• spin a layer of resist on top of the chip: 950K PMMA A7 (7% anisole) at 4 krpm
for 30 sec• bake on a hot plate at 180◦ C for 5 minutes• Expose resist using NPGS on a JOEL6400 SEM with beam set to 35 kV; expose
holes in silicon oxide at a magnification of ×1000, beam current of 10 pA, and exposureof 500 µC/cm2 (figure B.1a).
• develop by shaking chip using tweezers in developer ( 1 part MIBK to 3 parts
IPA) for 60 seconds then rinsing with IPA and blow dry with dry nitrogen• etch the exposed silicon oxide using a buffered oxide etch (BOE). The chip isheld using plastic tongs and is shook in the BOE for 2 minutes. Then dipped in one
beaker of de-ionized (DI) water, dipped in another beaker of DI water, rinsed in ISO,and blown dry with dry nitrogen.

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μm

(a)

μm

(b)

(c)

Figure B.1: (a) NPGS pattern used to create a mask to etch the silicon oxide; the area
inside the white lines will be exposed and etched. (b & c) NPGS pattern used to bondpads (b, entire writing field) and device with a nanomechanical beam (c, close up).
This structure is a nanomechanical beam (red) interrupted by a constriction (pink); theyellow and green beams/devices are used to test exposures.

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Fabricate Devices• spin layer of resist on top of the chip: PMGI SF9 at 3 krpm for 40 sec• bake on a hot plate at 180◦ C for 10 minutes; let cool for 5 min• spin another layer of resist on top of the chip: 950K PMMA A7 (7% anisole)at 4 krpm for 30 sec• bake on a hot plate at 180◦ C for 10 minutes• Expose resist using NPGS on a JOEL6400 SEM with beam set to 35 kV (figure
B.1b,c); nanomechanical beam (red) at a magnification of ×1000, beam current of 10 pA,
and exposure of 250 µC/cm2 ; location of constriction in beam (pink) at a magnificationof ×1000, beam current of 10 pA, and exposure of 50 µC/cm2 ; beam supports (white) ata magnification of ×1000, beam current of 10 pA, and exposure of 450 µC/cm2 ; larger
electrodes (white) at a magnification of ×500, beam current of 135 pA, and exposureof 425 µC/cm2 ; large bond pads (yellow) at a magnification of ×100, beam current of1320 pA, and exposure of 400 µC/cm2 .
• develop PMMA by shaking chip using tweezers in developer ( 1 part MIBK to
3 parts IPA) for 60 seconds then rinsing with IPA and blow dry with dry nitrogen• develop PMGI by shaking chip using tweezers in PMGI101 developer for 3
minutes, shake in beaker of DI water, then rinse with IPA and blow dry with dry
nitrogen• briefly clean exposed surface of chip using oxygen plasma in the reactive ion
etcher (RIE): 20 seconds with 50 sccm 02 and 300 W of power; these setting result in achamber at about 105 mT and a potential of 150 V.
• using an electron gun evaporator at a pressure of about 1×10−6 Torr, evaporate
400 nm of gold at 0◦ (that is, vertically), 30 nm of gold at −30◦ from vertical, and 30 nmof gold at −30◦ from vertical.
• liftoff by letting the chip sit in PG Remover at 70◦ C for 20 minutes. While thechip is submerged in acetone, use a syringe to spray chip with PG Remover. In some

284
cases, it is necessary to repeat these steps to remove all of the material that was notdeposited on silicon or silicon oxide. Then rinse with acetone, IPA, blow dry with dry
nitrogen• etch using SF6 gas in the RIE: 90 seconds with 20 sccm SF6 and 500 W ofpower; these setting result in a chamber at about 50 mT and a potential of 40 V.
• clean surface of chip using oxygen plasma in the RIE: 60 seconds with 50 sccm
02 and 500 W of power; these setting result in a chamber at about 105 mT and apotential of 200 V.

Appendix C

Fabrication of Nanostructures on Phosphor Bronze/Polyimide
Substrates

Prepare Phosphor Bronze/Polyimide substrate• JILA machine shop: NIST-Boulder: an approximately 5 cm diameter circularpiece (wafer) of phosphor bronze 0.016” thick shim stock was polished for about 24
hours using a solution of colloidal silicon that are about 50-70 nm in size.
• clean wafer by rinsing with acetone, IPA, the blow dry with dry nitrogen• clean wafer by rinsing with acetone, IPA, the blow dry with dry nitrogen• spin a layer of adhesion promoter on wafer: place wafer on spinner, relatively
quickly drip about 3 mL of adhesion promoter VM-652 onto wafer, wait 20 seconds,
take about 2 seconds to turn up speed of spinner to 2 krpm, spin for 30 seconds• bake on a hot plate at 100◦ C for 1 minutes; wait for wafer to cool (about 2
minutes)• spin a layer of polyimide on wafer: place wafer on spinner, cover wafer withpolyimide PI-2611, make sure to remove any bubbles, slowly (taking about 90sec) turnup speed of spinner to 2 krpm, spin for 30 seconds, turn down speed of spinner overabout 2 seconds• pre-bake on a hot plate at 100◦ C for 1 minutes then on a hot plate at 180◦ C
for 1 minutes; wait for wafer to cool (about 2 minutes)• spin a layer of polyimide on wafer: place wafer on spinner, cover wafer with286
polyimide PI-2611, make sure to remove any bubbles, slowly (taking about 90sec) turnup speed of spinner to 2 krpm, spin for 30 seconds, turn down speed of spinner overabout 2 seconds• pre-bake on a hot plate at 100◦ C for 1 minutes then on a hot plate at 180◦ C
for 1 minutes;• bake in a tube furnace: ramp up to 200◦ C, bake for 20 minutes, ramp down toroom temperature (total process takes 2.25 hours, though the furnace was still at 84◦ C
when the wafer was removed).
• bake/cure in a vacuum furnace (at less than 5 × 10−5 torr): ramp up to 350◦ C
over 85 minutes, cure at 350◦ C for 30 minutes, ramp down over 3 hours• spin a protective (not used for patterning chip) layer of resist on top of the chip:
PMGI SF9 at 2.5 krpm for 40 sec• bake on a hot plate at 180◦ C for 10 minutes; let cool for 5 min

Fabricate bond pads using optical lithography• cut an approximately 5 mm by 5 mm square from the phosphor bronze/polyimide
wafer using bench shears in the machine shop (carefully to avoid bending, rather thancutting, the phosphor bronze).
• remove protective layer of PMGI by placing chip in PG remover at 70◦ C for
10 minutes; rinse with acetone, IPA, blow dry with dry nitrogen• spin layer of resist on top of the chip: PMGI SF9 at 3 krpm for 45 seconds• bake on a hot plate at 180◦ C for 10 minutes.
• PMGI layer sometimes has large bubbles at the corner of the chips; if so,
cut/scrape off bubbles with a razor blade• dehydration bake on a hot plate at 115◦ C for 1 minutes; let cool for 1 min• spin another layer of resist on top of the chip: S1813 at 4 krpm for 45 seconds• bake on a hot plate at 115◦ C for 2 minutes.

287• Expose in mask aligner for 8 seconds• develop by shaking chip using tweezers in RD6 developer for 45 seconds, shakein beaker of DI water for 20 seconds, blow dry with dry nitrogen• bake on a hot plate at 130◦ C for 5 minutes; let cool for 1 minute• develop placing chip in RD6 developer for 2.5 minutes, shake in beaker of DI
water for 20 seconds, blow dry with dry nitrogen• briefly clean and rough (supposed to make evaporated metal stick to polyimide)
exposed surface of chip using oxygen plasma in the reactive ion etcher (RIE): 90 seconds
with 50 sccm 02 and 300 W of power; these setting result in a chamber at about 115 mT
and a potential of 180 V.
• using an electron gun evaporator at a pressure of about < 3 × 10−6 Torr,evaporate 750 nm of titanium at 0◦ (that is, vertically), rotating wafer about the verticalaxis by 90◦ every 50 nm. Evaporate 750 nm of gold at 0◦ (that is, vertically), rotating
wafer about the vertical axis by 90◦ every 50 nm.
• liftoff by letting the chip sit in PG Remover at 70◦ C for 35 minutes. While thechip is submerged in acetone, use a syringe to spray chip with PG Remover. In somecases, it is necessary to use the ultrasonic bath to remove additional material that was
not deposited on polyimide. Then rinse with acetone, IPA, blow dry with dry nitrogen

Fabricate Devices• spin layer of resist on top of the chip: PMGI SF9 at 3 krpm for 40 sec• bake on a hot plate at 180◦ C for 10 minutes; let cool for 10 min• spin another layer of resist on top of the chip: 950K PMMA A7 (7% anisole)at 4 krpm for 30 sec• bake on a hot plate at 180◦ C for 10 minutes• Expose resist using NPGS on a JOEL6400 SEM with beam set to 35 kV (figureC.1); nanomechanical beam (red) at a magnification of ×1000, beam current of 10 pA,

288

μm

(a)

(b)

Figure C.1: (a & b) NPGS pattern used to connect to bond pads (a, entire writing
field) and create device with a nanomechanical beam (b, close up). This structure isa nanomechanical beam (red) interrupted by a constriction with supports (white) andconnections to bond pads (yellow).

and exposure of 1.5 nC/cm; beam supports (white) at a magnification of ×1000, beam
current of 10 pA, and exposure of 500 µC/cm2 ; connection to bond pads (yellow) at amagnification of ×500, beam current of 130 pA, and exposure of 550 µC/cm2 .
• develop PMMA by shaking chip using tweezers in developer ( 1 part MIBK to
3 parts IPA) for 50 seconds then rinsing with IPA and blow dry with dry nitrogen• develop PMGI by shaking chip using tweezers in XP101A developer for 2 minutes, shake in beaker of DI water, then rinse with IPA and blow dry with dry nitrogen• briefly clean and rough (supposed to make evaporated metal stick to polyimide)
exposed surface of chip using oxygen plasma in the reactive ion etcher (RIE): 40 seconds
with 50 sccm 02 and 400 W of power; these setting result in a chamber at about 110 mT
and a potential of 180 V.
• using an electron gun evaporator at a pressure of about 2×10−6 Torr, evaporate
80 nm of gold at +5◦ from vertical, 80 nm of gold at −5◦ from vertical, and 2400 nm of

289
gold at 0◦ (that is, vertically).
• liftoff by letting the chip sit in PG Remover at 70◦ C for 65 minutes. While thechip is submerged in acetone, use a syringe to spray chip with PG Remover. In somecases, it is necessary to repeat these steps to remove all of the material that was notdeposited on polyimide. Then rinse with acetone, IPA, blow dry with dry nitrogen• etch using SF6 and O2 gas in the RIE: 11 minutes with 10 sccm SF6 , 90 sccm
O2 , and 200 W of power; these setting result in a chamber at about 200 mT and apotential of 28 V.

Appendix D

Algebra Related to Unintentional Feedback

I am going to try and find an approximate solution for a parametric oscillatorẍ + γ ẋ + ω02 1 + h1 ei2ω1 t + h2 e−i2ω1 t = a1 eiω2 t + a2 e−iω2 t(D.1)

and assume experimentally relevant conditions ω0 ≈ ω1 ≈ ω2 . That is, I have a formulafor a parametric oscillator where the parameter is oscillating at about twice the mechanical resonance frequency and there is a drive force at about the mechanical resonancefrequency. I am now going to make the simplifying assumption that the mechanicaloscillator is high Q (that is, ω0 /γ  1) which implies that the response to the force isgoing to be small at frequencies away from ω0 . Therefore I am going to try a solution
x = A1 eiω2 t + A2 e−iω2 t + B1 e−i(2ω1 −ω2 )t + B2 ei(2ω1 −ω2 )t(D.2)

and solve for A1 , A2 , B1 , and B2 while ignoring any term which is oscillating at afrequency far from ω0 . In this case, that means dropping terms at frequencies (4ω1 −ω2 )and (2ω1 +ω2 ). So plugging the trial solution into the differential equation and dropping

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frequency terms as described results in:−ω22 A1 eiω2 t − ω22 A2 e−iω2 t(D.3)−(2ω1 − ω2 )2 B1 e−i(2ω1 −ω2 )t − (2ω1 − ω2 )2 B2 ei(2ω1 −ω2 )t+iω2 γA1 eiω2 t − iω2 γA2 e−iω2 t−i(2ω1 − ω2 )γB1 e−i(2ω1 −ω2 )t + i(2ω1 − ω2 )γB2 ei(2ω1 −ω2 )t+ω02 A1 eiω2 t + ω02 A2 e−iω2 t + ω02 B1 e−i(2ω1 −ω2 )t + ω22 B2 ei(2ω1 −ω2 )t+ω02 h2 A1 e−i(2ω1 −ω2 )t + ω02 h1 A2 ei(2ω1 −ω2 )t + ω02 h1 B1 eiω2 t + ω22 h2 B2 e−iω2 t= a1 eiω2 t + a2 e−iω2 tcollecting terms at the same frequency results in four coupled equationsω02 − ω22 + iω2 γ A1 + ω02 h1 B1 = a1

(D.4)


ω02 − ω22 − iω2 γ A2 + ω02 h2 B2 = a2

(D.5)


ω02 − (2ω1 − ω2 )2 − i(2ω1 − ω2 )γ B1 + ω02 h2 A1 = 0

(D.6)


ω02 − (2ω1 − ω2 )2 + i(2ω1 − ω2 )γ B2 + ω02 h1 A2 = 0(D.7)

now define
a± = ω02 − ω22 ± iω2 γ(D.8)

b± = ω02 − (2ω1 − ω2 )2 ± i(2ω1 − ω2 )γ(D.9)

h+ = ω02 h1(D.10)

h− = ω02 h2(D.11)

292
resulting in the solutions
A1 =a1
a+ − h+b−h−−a1 h−
a+ b− − h+ h−a2A2 =a− − h+b+h−

B1 =

B2 =

−a2 h+
a− b+ − h+ h−(D.12)(D.13)(D.14)(D.15)I am now going to decide that the parametric oscillation is going to take the form ofh0 sin(2ω1 t) = h1 ei2ω1 t + h2 e−i2ω1 t and therefore h1 = h0 /2i and h2 = −h0 /2i and if I
define h = h0 ω02 then I can rewrite my solutions asa1
A1 =a+ − (h2 /4b− )


−ih
a1
B1 =
2b− a+ − (h2 /4b− )a2
A2 =a− − (h2 /4b+ )


ih
a2
B2 =
2b+ a− − (h2 /4b+ )(D.16)(D.17)(D.18)(D.19)

Now to check this with the results of other people’s calculations, let’s assume a drivea sin(ω2 t) = a1 eiω2 t + a2 e−iω2 t and therefore a1 = a/2i and a2 = −a/2i and equations
−ia/2
A1 =
a+ − (h2 /4b− )


−ih
−ia/2
B1 =
2b− a+ − (h2 /4b− )
ia/2
A2 =
a− − (h2 /4b+ )


ih
ia/2
B2 =
2b+ a− − (h2 /4b+ )(D.20)(D.21)(D.22)(D.23)and if ω1 = ω0 and ω2 = ω0 then the solution to the differential equation is
x = (A1 + B1 )eiω0 t + (A2 + B2 )e−iω0 t(D.24)

293
where


ih
−ia/2
A1 + B1 = 1 −2b− a+ − (h2 /4b− )


−ia/2
ih
A1 + B1 = 1 −2(−iγω0 ) (iγω0 ) − (h2 /4(−iγω0 ))


−a/2
h
A1 + B 1 = 1 +2γω0 γω0 − (h2 /4γω0 )−a
A1 + B1 = (2γω0 + h)4(γω0 )2 − h2−aA1 + B1 =2γω0 − h(D.25)(D.26)(D.27)(D.28)(D.29)

and

ia/2
ih
A2 + B 2 = 1 +2b+ a− − (h2 /4b+ )


ih
ia/2
A2 + B 2 = 1 +2(iγω0 ) (−iγω0 ) − (h2 /4(iγω0 ))


−a/2
h
A2 + B2 = 1 +2γω0 γω0 − (h2 /4γω0 )−aA2 + B 2 =2γω0 − h(D.30)(D.31)(D.32)(D.33)and thereforex=−2acos(ω0 t)2γω0 − hwhich diverges when h = 2γω0 as expected.

(D.34)