IMAGING ELECTRON FLOW, INTERFERENCE, AND
INTERACTIONS IN HIGH-MOBILITY TWO-DIMENSIONAL
ELECTRON GASES
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF APPLIED PHYSICS
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Michael Pemberton Jura
June 2009
UMI Number: 3364520
INFORMATION TO USERS
The quality of this reproduction is dependent upon the quality of the copy
submitted. Broken or indistinct print, colored or poor quality illustrations
and photographs, print bleed-through, substandard margins, and improper
alignment can adversely affect reproduction.
In the unlikely event that the author did not send a complete manuscript
and there are missing pages, these will be noted. Also, if unauthorized
copyright material had to be removed, a note will indicate the deletion.
UMI*
UMI Microform 3364520
Copyright 2009 by ProQuest LLC
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
ProQuest LLC
789 East Eisenhower Parkway
P.O. Box 1346
Ann Arbor, Ml 48106-1346
© Copyright by Michael Pemberton Jura 2009
All Rights Reserved
I certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degree
of Doctor of Philosophy.
(David Goldhaber-Gordon)
Principal Adviser
I certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degree
of Doctor of Philosophy.
--J?/'
(Yoshihisa Yamamoto)
I certify that I have read this dissertation and that, in my opinion, it
is fully adequate in scope and quality as a dissertation for the degree
of Doctor of Philosophy.
(George. Papanicolaou)
Approved for the University Committee on Graduate Studies.
in
iv
Abstract
This thesis investigates spatially-resolved electron transport through high-mobility
two-dimensional electron gases (2DEGs) at low-temperature (T < 4.2 K). Scanning
gate microscopy (SGM) is used to image electron flow emanating from a quantum
point contact (QPC) into a 2DEG, hosted in a GaAs/AlGaAs heterostructure. Two
factors important in determining electron trajectories in 2DEGs are researched: disorder and electron-electron scattering. Furthermore, electron interferometry based on
the SGM imaging technique is characterized and used to investigate these two factors.
GaAs-based 2DEGs have extremely low levels of disorder, with mean free paths
ranging from microns to hundreds of microns at low-temperature. Previous SGM
experiments showed that disorder, although present at very low levels, still greatly
impacts the spatial structure of electron flow on length scales much shorter than
the mean free path; disorder causes electrons to flow along narrow branches. By
studying samples with mean free paths ranging over an order of magnitude, this
thesis shows how varying levels of disorder affect electron flow. Furthermore, this
thesis demonstrates that the branches are surprisingly stable to changes in initial
conditions of injected electrons. The formation of branched flow previously had been
understood classically, but the newly-observed stability to changes in initial conditions
requires a quantum-mechanical explanation.
Interference of wave-like electrons traveling along different paths causes interference fringes in images of electron flow. One source of electron paths is impurity
scattering, and impurity-induced fringes were observed throughout all previous SGM
images of electron flow. A technique to determine locations and densities of impurities in 2DEGs using interference fringes is demonstrated. This thesis reports a lack
v
of impurity-induced fringes in samples with mobility higher than those previously
imaged. By imaging electron flow in one of the same high-mobility samples at lower
temperatures than previously reported (350 mK), interference fringes from a different
mechanism are observed. These fringes are due to an interferometer formed between
the QPC and SGM tip, similar to an optical Fabry-Perot interferometer. New, more
complex spatial interference patterns are observed close to the QPC. Interference
effects and their destruction are used to investigate dephasing.
Electron-electron (e-e) interactions are the dominant source of inelastic scattering
and dephasing for electrons in clean 2DEGs at low-temperature. The e-e scattering
rate is adjusted by injecting electrons above or below the Fermi energy of the 2DEG.
At low injection energies, effects specific to e-e scattering in 2D are found: because of
the confined phase space compared to 3D, electrons are scattered by small angles. At
high injection energies, an unexpected effect is observed: the differential conductance
through the system is increased by moving the SGM tip into the electron flow and
thereby backscattering current. This effect can be explained by the injected electron
beam scattering with a highly non-equilibrium distribution of electrons in a localized
region of 2DEG near the QPC. The e-e scattering rate between injected electrons and
this non-equilibrium distribution is measured.
VI
Acknowledgements
There are many people I would like to thank for helping me during my time as a
graduate student at Stanford. First of all, I would like to thank my research adviser, David Goldhaber-Gordon. David is an exceptional mentor and has an amazing
ability to understand and communicate physics. David's scientific rigor, emphasis
on fundamental understanding, and carefulness pervade our lab culture. Not only
does David inspire the highest scientific standards, but he genuinely cares about the
success and well-being of his students on a personal level. As one would guess from
his own personality, David has assembled a research group of especially friendly and
intelligent people. I have had the pleasure of working in this fantastic group over the
past six years.
Mark Topinka was a post-doc from whom I learned a great deal. Mark is not only
an expert in SGM, having pioneered the technique as a graduate student himself, but
also an all-around excellent experimentalist. Mark knows how and when to devote a
lot of time to solving a problem carefully, but he also knows when a quick (i.e. duct
tape-based) solution suffices. With his ability to get instruments to work combined
with impressive physical insight, Mark contributed significantly to my development
as a scientist. Plus, Mark always made the lab a fun place to be.
Mike Grobis was another post-doc with whom I had the pleasure of interacting in
lab as well as collaborating on a scientific level. Mike became involved in the analysis
of our SGM experiments, particularly the electron-electron scattering experiments.
Mike, with his outgoing and curious attitude, is always eager to discuss science, and
I've truly appreciated our discussions. Numerous conversations with him were critical
for developing our understanding of some of the counter-intuitive experimental results.
VII
Lukas Urban, a graduate student at Princeton, and his adviser, Ali Yazdani,
visited Stanford for several months during 2004. With their expertise in scanning
probe systems, they helped us construct the SGM. Despite the frustrations with
an instrument that sometimes refused to function properly, I still fondly remember
working late into the night with Mark and Lukas. By tirelessly wiring, machining,
and epoxying, we succeeded in operating the SGM as a low-temperature AFM by a
certain date, and as agreed upon, David shaved his beard.
Adam Sciambi joined the lab two years after me. As he easily learned the ropes of
making low-temperature measurements, he helped characterize some of our samples,
as well as contributing to our knowledge of how slowly cooling samples stabilizes
gating. I believe Adam's fingerprints are still on one of our best samples (literally).
Adam quickly started an independent project and, now with Matt Pelliccione, is
making significant progress towards implementing what was once just a scanning
probe daydream: the virtual scanning tunneling microscope.
Andrei Garcia, also starting two years after me, took the reins of the SGM as I
finished my experiments. He quickly absorbed all things SGM, and his participation
made conducting experiments even more enjoyable. Andrei has since been joined by
two enthusiastic and bright post-docs, Markus Konig and Matthias Baenninger. I
have greatly enjoyed working with these three to start their new SGM experiments.
Although the SGM sometimes misbehaves, I have no doubt they will use the SGM
to successfully study exciting physics.
When I first joined the group, there was already a group of excellent students:
Lindsay Moore, Charis Quay, Hung-tao Chou, and Ron Potok. They made the lab
what it is today, both in terms of physical infrastructure and the creation of a comfortable, intellectual atmosphere. They were all excellent teachers, always willing to
explain everything from basic physics to the details of electronics. They set an early
example for me; their work ethic and thirst for physical understanding are still a
model to me. I'm additionally grateful to Hung-tao for showing me around Taipei
during one of my trips there.
I started working in David's lab at the same time as two other students, Joey
vin
Sulpizio and Ileana Rau, and it has been a great experience to work and learn alongside both of them. Joey injects a good deal of humor into the lab and often leads
the charge into discussions on basic physics, while Ileana tends to keep discussions
level-headed. I have had the pleasure to work in the lab at the same time as several talented post-docs: Sami Amasha, Benjamin Huard, John Cumings, and Silvia
Liischer. Observing the extent of their expertise has always left me encouraged that
graduate school leaves one infused with a great deal of knowledge. I have also enjoyed
interacting with the other graduate students who joined the lab after me: Kathryn
Todd, Alex Neuhausen, and Nimrod Stander. They have all made excellent additions.
There have been a number of undergraduates who worked in the lab over the years,
including Dennis Lo, Patrick Gallagher, Yani Zhai, and Jeremy Hiatt, and it has been
good having them contribute.
I am grateful to Hadas Shtrikman, Loren Pfeiffer, and Ken West for providing
us with high-quality, high-mobility 2DEGs. Once we identified heterostructures with
nice QPCs, these samples were used for the bulk of our research. Mike Stopa provided
the code used for electrostatic simulations (SETE), as well as giving us advice on
using the code. We've had several illuminating and useful discussions with theorists,
including Rick Heller, Greg Fiete, Yoseph Imry, Yuval Oreg, Dganit Meidan, Daniel
Loss, and Steve Kivelson.
The Stanford-IBM Center for Probing the Nanoscale has created a great community of scanning-probe-related researchers. I really enjoyed collaborating with Dan
Rugar at IBM Almaden and Martino Poggio, a post-doc there at the time, on a
project to use a QPC as a detector of cantilever motion. For a graduate student, it
was great to be exposed to a different type of research setting. Martino is an excellent
experimentalist and was able to really run with a lot of our ideas. His friendly and
dedicated attitude made collaborating a pleasure. I've enjoyed communicating with
everyone in the nanoelectronics theme group, including Nahid Harjee, Beth Pruitt,
and Keji Lai. Nahid, a graduate student in Beth's group, is working to fabricate a
new generation of SGM tips, which may enable new experiments in nanostructures.
I've had the opportunity to discuss many of the issues with Nahid, and I'm always
IX
impressed by his interest in a range of subjects, including physics as well as fabrication techniques. There is also a good community of physicists in the McCullough
building basement, including the Moler, KGB, and Shen labs. I'm grateful for the
opportunities to discuss physics with them as well as the fun we've had outside the
basement.
I'd like to thank the other members of my reading committee, Yoshi Yamamoto
and George Papanicolaou, for providing their perspectives. I am further grateful to
them, Mike McGehee, and David Miller for asking insightful questions during my
oral defense. There are several other people at Stanford who make the gears run
smoothly and aided me with administrative, procurement-related, or other issues:
Laraine Lietz-Lucas, Paula Perron, Claire Nicholas, Kyle Cole, Tobi Beetz, Roberta
Edwards, Corrina Peng, Carol Han, Droni Chiu, Mark Gibson, and Larry Candido.
For sample fabrication, Tom Carver and James Conway were helpful, and the staff at
the Varian machine shop assisted in the construction of various parts for the SGM.
I would like to thank my many friends who have made graduate student life
at Stanford such a fun experience. In particular, I enjoyed living with my former
roommates: fellow physicists, gentlemen, and scholars, Guillaume Chabot-Couture,
Nick Cizek, and Jon Schuller. Other people with various physics associations who
have made the years lively are Eugene Motoyama, Jamie Mak, Praj Kulkarni, Naoko
Kurahashi, Ann Erickson, Tommer Wizansky, Sara Gamble, Mike Minar, and Geert
Vrisjen. Rather than attempting to list all the other friendships for which I am
grateful, I hope it is sufficient to say that many friends have been important over the
years.
I finally would like to thank my family for their endless support. My parents,
Martha and Michael, have always encouraged me and done everything they can for
me. In part due to their own backgrounds, they helped nurture my own scientific
interests. I would also like to thank my parents-in-law, Jack Hua and Ann Yu. After
spending more time with these wonderful people over the past year, I see where my
wife, Ying, gets many of her amazing qualities. Jack and Ann have welcomed me into
the family; they have been extremely hospitable as I sometimes stayed at their house
in San Francisco during the last year when I was mostly living in Boston with Ying.
x
Finally, I would like to thank Ying. She is extremely supportive and understanding.
I find her intelligence and dedication to her own work (residency is no small task)
inspiring.
XI
Contents
Abstract
v
Acknowledgements
vii
1 Introduction
1
1.1
Motivation and Introductory Concepts
1
1.2
Transport in the Ballistic Regime
4
1.3
Interference and Phase Coherence
5
1.4
Thesis Outline
8
2 Experimental Setup and Background
3
10
2.1
2DEG Growth and Disorder
10
2.2
SGM Imaging Technique
14
2.3
Example Data
16
2.4
Branches: Small-Angle Scattering
17
2.5
Interference Fringes
19
2.6
Related Scanning Probe Research
25
Disorder
27
3.1
Branch Length
28
3.2
Stability of Branches
30
3.3
Detecting Impurities
41
xn
4
Electron Interferometry
47
4.1
Spatial Variation of Interference Fringes
49
4.2
Checkerboard Interference Patterns
53
4.3
Interferometer Control and Dephasing
59
4.4
Prospects for Studying Electron Interactions
63
5 Electron-Electron Scattering
6
64
5.1
Technique to Measure Electron-Electron Scattering
65
5.2
SGM Measurements of Electron-Electron Scattering
72
5.3
Electron-Electron Scattering Model
77
Conclusions and Future Directions
90
6.1
Summary
90
6.2
Future Experiments: Nanostructures
93
6.3
Future Experiments: New Materials
95
A SGM Construction and Procedures
A.l Positioning
98
98
A.2 Sample
104
A.3 Electronics
107
A.4 Vibrations
118
A.5 Sample Behavior Due to Cooling and Attocubes
120
A.6 Image Processing and Drift Correction
121
A.7 Determining the Tip-to-QPC Distance
121
A.8 Fabrication Steps
122
Bibliography
129
Xlll
List of Tables
3.1
Properties of Three 2DEG Samples
xiv
28
List of Figures
1.1
Different Types of Scattering
5
1.2
Two Path Interference
7
2.1
2DEG Schematic
13
2.2
2DEG Disorder Schematic
14
2.3
SGM Geometry
16
2.4
SGM Imaging Technique
17
2.5
Electron Flow Data
18
2.6
Branch Formation Due to Small-Angle Scattering
20
2.7
Two Mechanisms for Interference Fringes
22
2.8
Interference between Two Paths with Different Lengths
23
2.9
Thermal Averaging
24
3.1
Electron Flow in Samples with Different Mean Free Paths
29
3.2
Analysis of Branch Length
30
3.3
Model Disorder Potential
33
3.4
Stability of Single Classical Electron Trajectories
34
3.5
SGM Images of Branch Stability
36
3.6
Classical Simulations of Branch Stability
38
3.7
Quantum Simulations of Branch Stability
40
3.8
Classical Simulations Taking into Account Spatial Width
41
3.9
Classical Simulations of Different Injection Energies
42
3.10 Interference Fringes in Three Samples with Different Mobilities . . . .
44
3.11 Spatially Detecting Impurities
45
xv
3.12 Schematic of Impurity Density
46
4.1
Appearance of Interference Fringes at Low Temperature
50
4.2
Calculated Thermal Averaging
51
4.3
Decay of SGM Signal Away from QPC
52
4.4
Disappearance of Interference Fringes
53
4.5
SGM Images While Changing Transmission of QPC
55
4.6
Interference Mechanisms at Different QPC Transmission Coefficients .
57
4.7
G In and Out of Electron Flow
60
4.8
Changing Electron Wavelength (Energy) and Interferometer Length .
61
5.1
Electron-Electron Scattering with a Fermi Circle
67
5.2
Electron-Electron Scattering Length
69
5.3
Imaging Electron-Electron Scattering with SGM
71
5.4
Schematic of Finite Injection Energy
72
5.5
Images of Electron Flow at High Injection Energy
74
5.6
Dependence of SGM Signal on Injection Energy
75
5.7
Distance Dependence of SGM Data and Calculations
78
5.8
Calculated Scattering Cross Section with the Fermi Circle
79
5.9
Mechanism Causing Increase in Differential Conductance
81
5.10 Non-Equilibrium Distribution of Electrons Near the QPC
82
5.11 Effective Temperature Calculation
84
5.12 Effective Temperature Model Calculations
87
5.13 Comparison of Data to Effective Temperature Model
88
5.14 Distribution of Electrons
89
A.l Scanning Gate Microscope
99
A.2 Positioner Wiring
103
A.3 Sample Layout
106
A.4 Sample Wiring
108
A.5 Experiment Inputs and Outputs
Ill
A.6 Isolator and Adder Circuits
112
xvi
A.7 Tip Circuitry
114
A.8 Piezotube Circuitry
117
A.9 Conductance Measurement Circuitry
117
A.10 Gate Circuitry
118
xvn
xviii
Chapter 1
Introduction
1.1
Motivation and Introductory Concepts
How electrons travel through materials is of great importance practically and theoretically. The ability to understand and control electron flow in semiconductors has
resulted in a vast portfolio of electronic devices that have transformed the world. Most
modern semiconductor devices can be understood with simple approximations for how
electrons flow through materials. Generally quantum mechanics is only needed to explain band structure; the most important result is the existence of energy bands in
which no electronic states exist (the bandgap). Beyond band structure, it is often sufficient to think of electrons traveling diffusively and behaving like classical particles.
These approximations work especially well for large devices, but as semiconductor devices are made smaller (to increase speed, decrease power consumption, and decrease
cost), these approximations break down.
When the approximations of classical and diffusive transport fail, electronic devices display a rich set of physics. The breakdown of these approximations pose
problems in conventional electronic designs, but if harnessed, there is potential to
construct more sophisticated, useful applications. For example, modern flash memory devices can rely on quantum tunneling. Clean semiconductor-based devices at
low temperature have provided physicists a medium in which to study a variety of
intriguing phenomena; some well-known examples include the integer and fractional
1
2
CHAPTER 1.
INTRODUCTION
quantum Hall effects [1, 2]. There is also considerable effort to harness quantum
mechanical effects for practical purposes; quantum computing might vastly increase
computing power [3] and spintronics might increase the speed and efficiency of devices
[4, 5].
Out of both scientific curiosity and an eye towards possible future applications, we
therefore seek to better understand electron transport through materials in a regime
in which the classical and diffusive approximations break down. In particular, GaAsbased heterostructures have proved to host nearly ideal two-dimensional electron gases
(2DEGs) because of the cleanliness with which the materials can be grown and the
resulting remarkably high mobilities. At room-temperature, GaAs-based 2DEGs are
used as fast transistors, and at low-temperature, at which mean free paths of 100's of
fim can be achieved [6], they serve as the basis for research on exotic electronic states
[2, 7, 8], electron interactions in nanostructures [9, 10], and quantum computing
prototypes [11]. This thesis therefore focuses on understanding electron transport
through GaAs 2DEGs at low-temperature because of their record as widely-studied
materials. Their low levels of disorder allow us to focus on intrinsic electron behavior,
and insights gained about these materials can impact other uses.
The technique we use to investigate transport is scanning gate microscopy (SGM)
[12, 13]. Most experiments on 2DEGs are performed just with transport measurements; that is, voltages and currents are applied and measured with a few other
parameters to adjust, such as magnetic field and gate voltages. While these transport measurements have revealed a great deal about the physics of 2DEGs, they are
in some sense blind to how electrons organize and flow. By using scanning probe
techniques, we complement existing transport measurements with direct spatial information about how electrons move through these materials.
The general questions with which we are concerned include: (1) What do electron
trajectories actually look like over short length scales when they are not moving
diffusively? The characteristic distance over which an electron retains its momentum
is the mean free path lm, and transport over a length much greater than lm can be
considered diffusive. We are interested in what determines lm and transport on length
scales shorter than lm, the ballistic regime. (2) How does quantum mechanics affect
1.1. MOTIVATION AND INTRODUCTORY
CONCEPTS
3
electron trajectories? Most obviously quantum mechanical electrons carry a phase
and can interfere like waves. Electrons do not retain a well defined phase indefinitely
because of interactions with the environment. The characteristic distance over which
an electron retains its phase is the dephasing length 1$, and interference only occurs for
electron paths shorter than 1$. We therefore are concerned with interference between
different electron paths as well as what determines 1$. We also find that quantum
mechanics affects electron trajectories in other ways.
At low-temperature, the mean free path lm in 2DEGs is determined by disorder
in the crystal; electron-phonon scattering is weak at the temperatures with which we
are concerned (4.2 K and below). lm is typically 10's of nm for semiconductors at
room-temperature, but is 1 to 100's of [im for GaAs 2DEGs at low-temperature. As
discussed further in Chapter 2, this disorder can come from both randomly placed
intentional dopants in the crystal as well as unintentional dopants (impurities). Thus,
we are concerned with characteristics of the disorder potential and how it affects
electron trajectories; conversely, we are also interested in what images of electron
flow can tell us about the disorder potential and by extension how less disordered
2DEGs might be grown.
For clean 2DEGs at low-temperature, 1$ is determined by electron-electron scattering [14, 15, 16, 17, 18]. An electron's strongest interaction with its environment is
with other conduction electrons. Electrons are charged and scatter off of each other,
but an amazing insight derived from Fermi liquid theory is that because of Pauli
exclusion and the filling of the Fermi circle, the electron-electron scattering rate is
comparatively low. Although the typical densities of 2DEGs translate into a typical
separation between electrons of only 10's of nm, the typical electron-electron scattering length /e_e is generally 10's of [im [14, 16]. Still, scattering with the Fermi circle
is the dominant mechanism through which an electron interacts with its environment
and loses its phases. Thus, we are concerned with the electron-electron scattering
rate and how it affects phase coherence.
4
CHAPTER 1.
1.2
INTRODUCTION
Transport in the Ballistic Regime
In a metallic system, conduction electrons must move past positively charged atoms
of the host material. An important result of band-structure considerations is that if
these ions form a perfectly periodic lattice, electrons can move unimpeded with some
constant velocity. However in the real world, electrons scatter due to imperfections
in the lattice, which can come from lattice vibrations at finite temperature (phonons)
or other defects (for example, a different type of atom at a lattice site). Thus, to
understand electron transport, we must understand how electrons are scattered.
The very simple Drude model is often able to explain a variety of electron transport
phenomena. Electrons travel unimpeded for some characteristic time r before suffering a collision (for example, with a phonon). After a collision, electrons emerge with
a statistical distribution of momenta and energies, and the distribution is completely
determined by local properties, such as temperature. Most importantly, collisions
randomize the direction of electrons. The mean free path is simply the distance
between collisions: lm = VT, where v is the velocity of electrons. More generally,
lm is the momentum relaxation length (i.e. a momentum correlation length), and
in this Drude model, lm is also the characteristic length between scattering events.
This model often works so well because macroscopic phenomena occur in the diffusive
regime, on length scales longer than lm, and the details of individual scattering events
do not matter.
The mobility /x of a sample is proportional to r and is defined as the ratio of
the drift velocity to the applied electric field. In contrast to the v defined above
(the actual velocity of electrons), the drift velocity is the average velocity at which
electrons move through a material after suffering many collisions and having their
directions randomized. This definition gives \x = er/m where e is the electron charge
and m is the effective mass (in GaAs m = 0.067m0 where m0 is the bare electron
mass). Thus, \x is a measure of the cleanliness of a sample.
As discussed in Chapter 2, fabrication of samples with £m's that are very long
(> 10 jim) at low-temperature is readily achieved. It is possible to investigate transport on a scale shorter than lm, and the microscopic details of the scattering become
1.3. INTERFERENCE
AND PHASE
Hard Scattering
COHERENCE
5
Small-Angle Scattering
Figure 1.1: Different Types of Scattering. 3 samples can have the same mean free
path lm, but very different types of scattering. lm is the characteristic distance over
which an electron retains its momentum. The blue background represents a scattering
potential, and the red line denotes a classical electron trajectory through that potential, (a) Sparse hard scatterers significantly change the momentum of an electron
upon collision, (b) There is now a relatively denser concentration of softer scatterers.
Compared to the scattering events in (a), there is less chance for direct back-scattering
(through an angle of n) and more chance for forward scattering (an angle closer to
0). (c) There is now a small-angle scattering potential. An electron's momentum is
continually being deviated by small amounts.
important. As depicted in Fig 1.1, we can imagine different samples with the same lm,
but very different types of scattering [19]. For any type of scattering, lm is a measure
of how far an electron travels before it loses memory of its original momentum; lm
is a momentum correlation length. In one extreme, the hard scattering regime, electrons travel in straight lines until rare collisions change their momentum significantly
(through an angle on the order of n). In the opposite extreme of small-angle scattering, electrons are constantly changing directions (by angles closer to 0). If we examine
transport on a length scale shorter than lm, these two types of scattering create very
different looking trajectories and the resulting flow has very different properties.
1.3
Interference and Phase Coherence
We must recognize that electrons are quantum mechanical particles. A crude approximation is to consider a quantum mechanical particle as a collection of classical
6
CHAPTER 1.
INTRODUCTION
particles with distributions in position and momentum that correspond to the uncertainty in position and momentum. A better approximation is semiclassical where each
trajectory carries phase and the different trajectories interfere like waves. One issue
we consider in this thesis is the difference between quantum mechanical and classical
flow through the same scattering potential. For example, consider a scattering site
with potential energy less than the energy of an electron (the Fermi energy Ep). The
scattering site cannot directly backscatter a classical electron. However if the scattering site is spatially confined and short compared to an electron's Fermi wavelength
\p, the site directly backscatters part of the wavefunction of a quantum mechanical
electron (s-wave-like scattering). This quantum-mechanical scattering can affect how
electrons travel through a scattering potential.
Another defining quantum mechanical effect is the interference of electrons. The
simplest example is the interference between two paths, analogous to Young's famous
double-slit experiment with light. As depicted in Fig. 1.2, if an electron can take
two (or more) paths between two points, the probability of finding an electron at the
final point depends on the relative electron phase accumulated along the two paths.
Putting this into more mathematical terms, the (un-normalized) wavefunction at
the origin point is <p(x = 0) = <fi(x = 0) + ^{x
= 0) and at the destination
point (p(x — 1) = (fi(x = I) + (f2{x = I), where / is the length of the paths. The
probability of finding the electron at the destination point is \ip(x = l)\2 = \<pi(x =
l)\2 + \(f2(x = l)\2 + 2Re{ipl(x = l)ip2{x = /)}. If electrons simply behaved like
classical particles we would expect the probability to be \ip\{x = l)\2 + \(fi2{x = l)\2Thus, 2Re{(pl(x = l)(f2(x = I)} is the quantum-mechanical interference term that
depends on the relative phase between tp\ and ip2Two-path experiments have been performed and the probability of finding an
electron at a detector (the destination point) is seen to oscillate as a function of the
phase between the two paths. One method of experimentally adjusting the phase
between the two paths is with a perpendicular magnetic field between the two paths
(the Aharonov-Bohm effect) [20]. It is also possible to change slightly the length of
one path or the local wavevector k along one path. With no magnetic field and a
slowly-varying k(x), the phase accumulated along each path is simply 4> = fQdx
k(x).
1.3. INTERFERENCE
AND PHASE
COHERENCE
7
Destination
/
Origin
Figure 1.2: Two Path Interference. An electron can travel along two paths from
an origin to a destination. If the path length is shorter than the phase coherence
length, I < l^, the probability of finding an electron at the destination depends on
the interference of the two paths.
Previous experiments have used a local electrostatic gate over one path to change k(x)
in a small region and hence vary the phase [17]. Other widely studied phenomena due
to the interference of multiple paths are weak localization and universal conductance
fluctuations [21, 22]. Weak localization is caused by time-reversed paths that both
start and end at the injection point; these two paths constructively interfere at the
injection point and increase the resistance through a sample above what would be
classically predicted. Universal conductance fluctuations are due to the interference
of many paths through a sample; as the relative phases of these paths are changed,
with a magnetic field or gate voltage for example, the conductance through the sample
fluctuates in a reproducible, but random manner.
We next ask how long can we make / and still observe interference (see Ref.
[23] for more treatment of the following discussion). Once we make I greater than
some characteristic distance 1$, because of interactions with the environment, we
expect interference to disappear. Now including the state x °f the environment (with
coordinates 77), we write the initial wavefunction as {tfi(x = 0) + <^2(£ =
fyj^xiv)-
If
an electron along each path interacts with the environment in different ways, we then
8
CHAPTER 1.
INTRODUCTION
write the wavefunction at the destination as <fi(x = I) <S> Xi(v) + ^ ( ^ = 0 ® X2(v)Now the probability of finding the electron at the destination requires integrating
over the environmental coordinates (which are not observed): \ipi(x = l)\2 + \^2{x =
l)\2 + 2Re{ipl(x = l)<p2{x = I) Jdr]xl(f])X2(v)}•
The interference term now includes
a factor of f drjxl{T))X2{v)- It is clear that if the two paths leave the environment
in the same state, the interference term is unchanged. However, if the two paths
put the environment into orthogonal states, the interference term disappears and
the probability of finding the electron is just the classical result. This is simply the
standard quantum mechanical "which-path detector": we can observe interference
effects of an electron traveling along a superposition of different paths only if we
do not detect which path is taken. Another equivalent point of view examines the
effect of an interaction on the phase of the electron traversing the paths. When the
environment has been put into orthogonal states depending on which path is taken
by the electron, we can think of there being an uncertain phase relationship between
the two paths.
In this thesis, we are interested in observing interference effects as well as understanding further what limits the dephasing length 1$. Elastic scattering off disorder
leaves the state of the environment unchanged and so does not cause dephasing. As
discussed further in Chapter 5, the dominant source of dephasing in the regime we
study is scattering with other electrons. Thus, we are also interested in measuring
the electron-electron scattering rate.
1.4
Thesis Outline
In this thesis, we use SGM to investigate higher mobility 2DEGs at lower temperatures than were probed in previous experiments imaging electron flow. High-mobility
2DEGs at low-temperature often show the most interesting behavior in other transport measurements [2, 7, 8], and thus it is important to spatially understand electron
flow and organization in these materials. Our images of electron flow in high mobility
samples demonstrate few or no features due to impurities in the sample, allowing us
to concentrate on the intrinsic behavior of electrons in 2DEGs. Imaging samples at
1.4. THESIS
OUTLINE
9
lower temperatures allows us to observe new phase-coherent interference effects.
Chapter 2 of this thesis covers the experimental setup and procedure. It shows
example data and, as background for further chapters, explains the basic causes of
certain features. It presents different mechanisms for electron interference which are
used to study 2DEGs in further chapters.
Chapter 3 examines the role of disorder in guiding electron flow through 2DEGs. It
examines flow in samples with mean free paths ranging by over an order of magnitude.
It shows a property of electron flow through a small-angle scattering disorder potential
which requires a quantum mechanical explanation; namely, branches of electron flow
are more stable to changes in initial condition than one would predict classically.
This chapter also demonstrates the use of electron interference as a spatial detector
for impurities in the sample. Results from this chapter are reported in Ref. [24].
Chapter 4 explores a previously unobserved mechanism for interference: multiple
reflections between the SGM tip and injection point (the quantum point contact),
similar to an optical Fabry-Perot interferometer.
is demonstrated.
Control over the interferometer
Dephasing in clean samples is measured using this interference
mechanism. Applications to studying electron interactions in nanostructures are also
considered. This chapter's results are reported in Ref. [25].
Chapter 5 investigates in detail electron-electron scattering. It compares results to
the calculated electron-electron scattering rate and previous experiments. It reports
a result that is initially counter-intuitive: when injecting electrons into the 2DEG
.-.-*- Viirt-l-i
£»T-ir» v i r i o n
mATrinrr
+ V10 Q P A / T
4" 1 r~* lTlfi-v f V l O
Ctt
CllCltilVvO,
H A W V IJLXEi
UJ.J.VJ W V j f A V i
U±LV AJ.J.UW
l-LlK^ll
o l O n f mTt
UllV^ ^ i V U U l U U
QmTiT
1 1 V It
QTirl
r£»fl £»r»"H Ti O" £• I P f f m n Q
t^JJ.J.V-1 l ^ l l ^ - ^ U i i x ^
v> j. v^ VJ vn. *j J.AU
backwards increases the differential conductance through the system. This result is
explained due to electron-electron scattering with a highly non-equilibrium distribution of electrons near the injection point. Results from this chapter will be reported
in Ref. [26].
Chapter 6 concludes the thesis, discussing implications of this research and giving
an outlook for future research directions. It also briefly describes results from Ref. [27]
in which we used a QPC as a detector of the motion of a scanning probe cantilever.
Appendix A documents more details on the construction of the SGM and its
operation.
Chapter 2
Experimental Setup and
Background
2.1
2DEG Growth and Disorder
The capability to grow very high mobility GaAs 2DEGs, achieved roughly 30 years
ago, opened the possibility to build and study electronic devices in which intrinsic
electron behavior is evident and less clouded by defect scattering. 2DEGs have been
used to study a variety of remarkable electronic states, including the integer and fractional quantum Hall effects and other complex interacting states at high magnetic
field [2, 7, 8]. 2DEGs have also been used as clean systems in which to study interference effects, such as weak localization, universal conductance fluctuations, and the
Aharonov-Bohm effect. Additionally, a great deal of research has centered on nanostructures created in 2DEGs and the non-trivial ways in which electrons organize in
these structures [9, 10]. Proposals aim to use 2DEGs to host quantum computers;
these fall under two general categories: spins in quantum dots [28] and topological
states in the fractional quantum Hall regime [29]. While research has extended into
other materials with different properties (such as higher spin-orbit coupling or higher
effective mass), GaAs 2DEGs still have the highest mobilities, and their growth and
fabrication is extremely well understood.
The key development in GaAs 2DEGs to achieving high mobilities is the physical
10
2.1. 2DEG GROWTH AND
DISORDER
11
separation of the conduction electrons from the ionized dopants, which act as electron
scatterers. In conventional doped semiconductors, the conduction electrons reside in
the same space as the ions which donated them. The structure of a 2DEG is depicted
in Fig. 2.1. The cross-section in Fig. 2.1a shows the heterostructure grown one atomic
layer at a time by molecular beam epitaxy. The heterostructure changes abruptly from
GaAs to Al x Gai_ x As (with x ~ 0.3). AlGaAs has a larger bandgap than GaAs, and
the bandgap difference can be controlled via the Al alloy concentration x. Fig. 2.1b
shows the energy Ec of the bottom of the conduction band as a function of depth
z into the 2DEG wafer. Normally the Fermi energy EF lies within the bandgap of
the material, but as depicted in Fig. 2.1c, Si donors are added to the AlGaAs layer
and donate electrons. However, the lowest energy electronic states are located in the
GaAs, not AlGaAs, so electrons move to the GaAs interface. Fig. 2. Id shows the
self-consistent energy diagram where donated electrons reside in a triangular potential
well at the interface between GaAs and AlGaAs, forming the 2DEG. The bending of
the bands comes from the electric field generated by the positively charged Si donors.
Electrons in the 2DEG are attracted to the interface by the Si donors, but do not
extend significantly into the AlGaAs layer because of the higher bandgap. It should
be noted that some electrons are donated to the top GaAs layer, but they reside in
non-mobile surface states.
In Fig. 2. Id, the wavefunction ty(z) of electrons in the z-direction is depicted
trapped in the confining triangular potential well. Some 2DEGs are created with
another AlGaAs layer below the 2DEG, forming a confining square well for electrons. However, all 2DEGs used in this thesis are at a single GaAs/AlGaAs interface. The confinement energy along the z-direction is high enough such that at
low-temperature, only one sub-band is occupied; that is, the energy difference between
different electronic states along the z-direction is larger than the thermal energy or
the Fermi energy. This means that there is no motion in the z-direction. The electrons are effectively confined to motion in 2-dimensions, the x-y plane. The density
n can range from ~ 109 to ~ 1012 cm~2, and for the samples which we measure,
n is typically ~ 2 x 1011 cra~2. This corresponds to a typical Fermi wavelength of
XF = <sj2ir/n ~ 56 nm and Fermi energy of EF = h2kF/2m
= irh2n/m ~ 7.1 meV.
CHAPTER 2. EXPERIMENTAL
12
SETUP AND
BACKGROUND
At low temperatures, the thermal energy (kBT = 360 /ieV at 4.2 K) is much smaller
than EF, so the electron gas is degenerate. We can think of the electrons occupying a
Fermi circle with wavenumber radius kF with a thin band of partially occupied states
around the circle's edge, with energy width ksT.
The positively charged Si donors are randomly placed in the crystal and create
potential scattering sites for electrons. Unlike standard doped semiconductors, the
charged donors are far from the conduction electrons (around ~ 50 nm away in our
samples) and therefore create much weaker scattering sites for electrons. Not all Si
atoms ionize, but at low temperature in well-behaved samples, the charge state does
not change over time. At room temperature though, the charge state can fluctuate.
On different cool-downs, therefore there can be different charge configurations on
the Si donors and different scattering potentials for electrons. If the structure is
cooled slowly, allowing the charge state on the donors to reorganize to find a local
minimum energy state, the potential felt by electrons is smoothed [30]. The charge
on the donors repels and becomes more evenly distributed. Charge is actually anticorrelated; a charge on some Si dopant makes it less likely to find a nearby Si atom
ionized.
A schematic for a disorder potential in a 2DEG is shown in Fig. 2.2. The low,
bumpy background is a small-angle scattering potential created by the Si donors above
the 2DEG. As discussed further in Chapter 3, for these bumps we expect the typical
width to be on the order of the separation between the donors and 2DEG (~ 50 nm,
which happens to be around Xp) and the typical height to be ~ 10% EF [13]. The
large spike in the middle is a hard scattering site created by an impurity that is close
to the 2DEG. The relative concentration of these hard scattering sites depends on
how cleanly the heterostructure was grown and the concentration of impurities.
Later in this chapter, we discuss how these types of scattering affect electron flow,
and in Chapter 3, we present further results related to disorder scattering. We next
turn to the experimental technique for imaging electron flow.
2.1. 2DEG GROWTH AND
DISORDER
13
-100 nm
(C)
GaAs
AIGaAs
•^+ + + + + -^ +
Impurity *!^
-100 nm
Si donors
^j&eJ
GaAs
Figure 2.1: 2DEG Schematic, (a) Cross-section of GaAs/AIGaAs heterostructure
before Si dopants are added; no 2DEG is present, (b) The bottom of the conduction
band Ec as a function of distance z into the heterostructure. AIGaAs has a larger
bandgap than GaAs, and Ec is higher in the AIGaAs region. The Fermi energy
Ep lies in the bandgap. (c) Si donors are added to the AIGaAs layer, donating
electrons. The 2DEG resides at the GaAs/AIGaAs interface, separated by a distance
Zd ~ 50 nm from the Si donors. The large physical separation between the positively
charged Si donors and the 2DEG enables high mobilities. Unintentional impurities
(gray) incorporated into the crystal can end up much closer to the 2DEG, creating
stronger scattering sites for electrons, (d) With positively charged Si ions, a triangular
well is formed at the GaAs/AIGaAs interface. At low-temperature, electrons only
have enough energy to occupy the lowest sub-band, with wavefunction ^(z) along z
denoted.
14
CHAPTER 2. EXPERIMENTAL
SETUP AND BACKGROUND
EF| ^ ^ ^ H ^ ^ B ^ B U
~ 50 nm
IJ^^^^^^W^^
Figure 2.2: 2DEG Disorder Schematic. A schematic of the potential felt by electrons.
The low, bumpy background is a small-angle scattering potential created by the Si
donors far from the 2DEG. The tall spike in the center is a hard scattering site created
by an impurity that happens to be near the 2DEG at that location.
2.2
S G M Imaging Technique
In this section, we present the concepts necessary to understand the workings of our
SGM experiment. For more experimental details about the actual construction and
operation of the SGM, consult Appendix A.
Because the 2DEG is generally around ~ 100 nm below the surface of the sample,
the well-known technique of scanning tunneling microscopy (STM) cannot be used;
the 2DEG is much too far from the metallic tip to allow tunneling between the two.
We therefore use a metallic tip to simply gate the 2DEG in the technique of scanning
gate microscopy (SGM). M. Topinka and the Westervelt group pioneered the use of
SGM to image electron flow in 2DEGs [12, 13, 31, 32, 33, 34, 35, 36, 37]. Related
scanning probe research is discussed at the end of this Chapter.
To image electron flow with SGM, we measure how scanning the SGM tip alters
transport across the 2DEG sample, as depicted in Fig. 2.3. We measure the differential conductance G = dl/dV
with lock-in techniques. This is accomplished by
measuring the current driven by an oscillating voltage VAC (smaller than or comparable to the thermal energy,
CVAC
^ kBT). Dividing the resulting oscillating current
by the small VAC therefore gives us a measure of G. A larger DC voltage V^c can
be applied across the sample. Unless specified otherwise, VDC = 0- VDC becomes
2.2. SGM IMAGING
TECHNIQUE
15
important in Chapters 4 and 5 to inject high energy electrons.
A quantum point contact (QPC) [38, 39] is created by fabricating a metal splitgate on the surface and applying a negative voltage Vg to these gates. By making
Vg negative enough, we deplete the 2DEG underneath and laterally, leaving just a
narrow channel with width comparable to A^. As a side note, the 300 nm lithographic distance between the gates is chosen to be a few times A^ plus the width
of lateral depletion, which, for each gate, is roughly the surface-to-2DEG distance.
This quasi-ID channel of width comparable to XF is the QPC, which separates two
regions of 2DEG. Because of its narrowness, the QPC dominates the resistance across
the sample. The conductance across the sample is proportional to the transmission
probability for electrons through the QPC. Although there are many interesting features of QPCs, some of which SGM may be well suited to investigate (as discussed
in Chapters 4 and 6), for now we simply use the QPC as both an injector and detector of electrons. Unless stated otherwise, we set G of the QPC to the first plateau
of conductance, 2e2/h (where e is the electron charge, h is Planck's constant, and
the 2 comes from spin degeneracy), allowing through only one mode confined in the
y-direction.
We position a metallic SGM tip ~ 30 nm above the surface of the sample, near
the QPC [40]. Like the fixed surface gates, applying a negative voltage Vtiv to the
tip also creates a depletion region in the 2DEG below. By measuring how transport
changes as we scan this depletion disk, we can determine the spatial distribution of
electron now out of the QPC, as depicted in Fig. 2.4. The depletion disk scatters
electrons. If electrons injected from the QPC are scattered by the tip back through
the QPC, as depicted in Fig. 2.4c, G is reduced because the effective transmission
coefficient through the QPC is decreased. In contrast, if the tip does not disrupt
electrons flowing out of the QPC, as in Fig. 2.4a, G does not change. By scanning
the tip and measuring G as a function of position, we can therefore map electron
propagation. We plot AG(x, y) = G(x, y) - G(x0, y0) where G(x0, y0) is a background
conductance at a location (x0,y0) at which the tip does not interrupt electron flow
(generally G(x0,y0)
= 2e2/h).
That is, AG is the change in conductance produced
by introducing the scattering effect of the tip. AG is normally negative, and |AG|
16
CHAPTER 2. EXPERIMENTAL
SETUP AND
BACKGROUND
Figure 2.3: SGM Geometry. We measure the differential conductance G across the
2DEG (green) by applying a small oscillating voltage VAC and measuring the oscillating component of the driven current / . A DC voltage VDC can also be applied across
the sample. A QPC is formed by applying Vg to metallic surface gates (orange),
depleting the 2DEG below (black). The QPC dominates the resistance across the
sample. The SGM tip (orange) creates a scannable depletion disk (black).
indicates the strength of electron flow at the location.
We note that, as in Fig. 2.4b, if the tip scatters electrons, but not back through
the QPC, there is no change in G because the electrons still stay in the same region of
2DEG. Because all the resistance is in the QPC, G only substantially changes if the
transmission coefficient for electrons to move from one side of the QPC to the other
changes. Theoretical simulations have shown this SGM imaging technique accurately
reproduces the underlying electron flow [13, 41, 42].
2.3
Example Data
Data taken using this technique are shown in Fig. 2.5. Electrons flow out from the
QPC at the bottom, up towards the top and sides of the image. One striking feature
is that current flows along narrow branches rather than spreading out evenly as one
might expect from analogy to light diffracting out of a narrow slit. These branches
were seen previously and explained as the result of small-angle scattering [13]. The
cause of these branches is discussed in Section 2.4. Another strong feature is the
2.4. BRANCHES: SMALL-ANGLE
(a)
AG = 0 4 ^
SCATTERING
(b) AG = 0 j g ^
17
(c)
AG < 0
^ ^
Figure 2.4: SGM Imaging Technique. Electron trajectories are denoted with blue
arrows, (a) If the SGM tip (black depletion disk) does not scatter electrons injected
from the QPC, G does not change, (b) If the tip scatters electrons, but not back
through the QPC, G still does not change, (c) Only in this case is there a decrease
in G, AG < 0. Electrons are reflected by the tip back through the QPC.
fringes denoted with a purple arrow. Mechanisms that cause fringes are explained in
Section 2.5. Previous images of electron flow were decorated with fringes throughout,
but in this image, fringes are only visible in certain locations. In Chapters 3 and 4,
we investigate further the use of fringes as a tool for locally detecting impurities as
well as measuring phase coherence.
2.4
Branches: Small-Angle Scattering
The distance from the QPC to top of the image in Fig. 2.5 is about 4 [im and the
mean free path is lm = 13 \im (measured in a Hall bar configuration). Therefore, the
entire image and all branches are on a length scale significantly shorter than the mean
free path. These branches were observed previously and were explained as the result
of electron propagation in a small-angle scattering disorder potential [13, 41, 42]. All
the bumps and dips in the disorder potential act like imperfect lenses, which focus
different electron trajectories into narrow branches, sometimes referred to as caustics.
It was previously understood that small-angle scattering was an important source of
scattering in 2DEGs [19], but this striking bunching of electrons into branches was
completely unanticipated before the first SGM images of electron flow [13].
Fig. 2.6 shows how we can understand branch formation as a classical effect. Fig.
2.6a shows a small-angle scattering disorder potential that approximates the disorder
18
CHAPTER 2. EXPERIMENTAL
SETUP AND
BACKGROUND
AG
(2e2/h)
0.01
Figure 2.5: Electron Flow Data. Two striking features are the strong branches of
current flow and the fringes denoted with a purple arrow. The approximate locations
of the depletion regions from the QPC gates are denoted schematically in black at
the bottom of the image. The data were taken at 4.2 K and with the QPC biased to
the second plateau, G = 4e2/h.
2.5. INTERFERENCE
FRINGES
19
potential for the sample in Fig. 2.5. In Fig. 2.6b, we present the results of a purely
classical (not semiclassical) simulation for particles moving through the disorder potential. Thousands of trajectories with varying initial conditions are calculated and
plotted. The color corresponds to the density of trajectories lying at a certain location. The trajectories' initial conditions have a distribution in initial position and
angle corresponding to the approximate width and angle of emission from the QPC.
For more details on how we create an approximate disorder potential and simulate
electron flow, see Chapter 3.
The simulation of classical particles moving through a small-angle scattering disorder potential reproduces the presence of the strong branches observed in experiments.
Thus, the formation of branches can be understood as a classical lensing effect. In
Chapter 3, we examine characteristics of the branches in samples with different mean
free paths. We also examine the branches' stability to changes in initial condition, a
property which we find requires a quantum mechanical explanation.
2.5
Interference Fringes
We next examine mechanisms that can cause fringes, such as those in Fig. 2.5. Fringes
occur because of the quantum interference of wave-like electrons traveling along at
least two paths from the same origin to the same destination [41]. Changing the tip
position causes these two paths to accumulate different phases, and therefore moving
the tip oscillates the imaging signal through cycles of constructive and destructive
interference. In Fig. 2.4c, for the SGM imaging technique described thus far, there is
only one path that completes the roundtrip from QPC to tip and back; both the origin
and destination are the QPC. The one path alone cannot account for interference
effects, and as discussed in Chapter 3, indeed we do not observe interference fringes
when there is only one path.
In order to account for the interference fringes, therefore there must be another
roundtrip that starts and ends at the QPC. Fig. 2.7 shows two possible mechanisms
for the creation of a second path (red), in addition to the path directly backscattered
by the tip (blue). In Fig. 2.7a, Mechanism # 1 is depicted: there is another hard
20
CHAPTER 2. EXPERIMENTAL
SETUP AND
BACKGROUND
Figure 2.6: Branch Formation Due to Small-Angle Scattering, (a) A small-angle
scattering disorder potential that approximates the disorder potential in the sample
in Fig. 2.5. The height of the dips and bumps is typically ~ 10% Ep, so electrons are
scattered only at small-angles, (b) Results of a classical simulation of electron flow
through the disorder potential in (a). In the simulation, many trajectories are injected
with different initial conditions corresponding to the width and angular spread of
quantum mechanical electrons (see simulations in Chapter 3). The color represents
the density of trajectories at given point. The classical simulation reproduces the
presence of the strong branches seen in data.
2.5. INTERFERENCE
FRINGES
21
scatterer (gray) in addition to the tip. As discussed previously in this chapter, this
hard scatterer can come from a charged impurity atom near the 2DEG, for example.
Like the tip, this hard scatterer can completely back-reflect a significant amount of
electron flux. The length of the red path is constant. The length of the blue path
changes with the tip position. As the tip moves away from the QPC by XF/2, the blue
path length increases by Xp, causing a full cycle of interference. Thus, interference
fringes in SGM images are expected to be spaced by Xp/2. We can make an analogy
between Mechanism # 1 and an optical Michelson interferometer, in which a movable
mirror changes the length of one path while the length of a second path is fixed. As
discussed further in Chapter 3, all previous images of electron flow in 2DEGs were
decorated everywhere with fringes due to Mechanism # 1 .
In Fig. 2.7b, Mechanism # 2 is depicted: now the second path (red) is caused
by multiple reflections between the QPC and tip. In the second path, an electron
is injected from the QPC, reflects off the tip, off the QPC gates, off the tip again,
and then is finally retransmitted through the QPC. This mechanism is similar to an
optical Fabry-Perot interferometer; the tip acts like the movable mirror at the end
of a cavity, in which a wave can multiply reflect. We omit paths with even more
reflections because of thermal averaging, to be discussed shortly, and each reflection
causes a reduction in electron flux, to be discussed further in Chapter 4. As with
Mechanism # 1 , we expect fringes spaced by A^/2. Moving the tip away from the
QPC by XF/2 increases the path length of the blue path by Xp and the red path by
" " ? •
We also need to consider whether interference is visible between the two paths
for each mechanism. The dephasing length l^ is generally ~ 10 \im or longer in our
samples at low-temperature. Thus, even for typical distances between the QPC and
tip of L ~ 1 — 2 fim, two roundtrips (such as the red 4L path in Fig. 2.7b) are still
shorter than 1$. Thus, 1$ is generally not a limit to observing interference effects for
the normal conditions of our experiment. However, there is a limit to the difference
in path length for the two paths.
As depicted in Fig. 2.8, we can now consider whether interference is visible between two paths with lengths ^ and l2. In order to observe interference, in addition
22
CHAPTER 2. EXPERIMENTAL
SETUP AND
BACKGROUND
Figure 2.7: Two Mechanisms for Interference Fringes. In both cases, the blue path
depicts the standard SGM imaging path: electrons injected by the QPC are reflected
back through the QPC by the tip. The distance from the QPC to the tip is L. In order
to observe interference fringes, there needs to be a second path in addition to the blue
path, (a) Mechanism # 1 : an impurity near the 2DEG creates a hard scatterer (gray),
which also scatters electrons back through the QPC. The distance from the QPC to
the impurity is L/. (b) Mechanism # 2 : an electron can take multiple roundtrips
between the QPC and tip.
to the requirement that l\ < 1$ and l2 < l<j> (which are generally satisfied for our
samples), there is also a limit on the difference |Ii — l2\, to avoid thermal averaging
[43].
At finite temperature, electrons involved in transport have a distribution of energies, with width comparable to ksT around Ep", see Fig. 2.9. Different energy
electrons have different wavelengths and accumulate phase at different rates, as depicted in Fig. 2.9c. This distribution of electrons that start in phase have a spread in
relative phase of 1 radian after traveling the thermal length LT = h2 / (^TrmApkBT).
That is, AkLT = 1 where the spread in wavevectors Ak is ksT = H2kAk/m.
For a
typical sample with XF ~ 50 nm, LT ~ 400 nm at 4.2 K and LT ~ 5 [im at 350 mK.
Thus, for the regimes with which we typically deal, LT < l<t>- In optics, the "coherence
length" of a laser is set by a similar finite line-width of emitted wavelengths.
In Fig. 2.8 each path contains a thermal distribution of electrons. If l\ = l2 (or
more generally \l\ — h\ < LT), interference is visible because all electrons are still
coherent. Although there is a spread in phase for the electrons on path 1, the phase
difference between the two paths is the same for any electron with energy within the
thermal distribution. Only when \h - l2\ > LT does the spread in relative phases
2.5. INTERFERENCE
FRINGES
23
Destination
k
Origin
Figure 2.8: Interference between Two Paths with Different Lengths. When the two
paths interfering have different lengths, interference is only visible when the difference
in path length \l\-h\ is short enough. The limit to the path length difference comes
from thermal averaging at finite temperature.
between the two paths become comparable to 1 radian. Thus, when \lx — l2\ >
LT, although each electron is still coherent and interferes, the interference pattern
averages away because we are measuring a thermal distribution. Chapter 4 includes
experiments and more calculations on thermal averaging.
We now consider this thermal averaging limit when applied to the interference
fringe Mechanisms in Fig. 2.7. In both Mechanisms, the path directly backscattered by the tip (blue) has length 2L. In Mechanism # 1 in Fig. 2.7a, the impurity
backscattered path (red) has length 2Lj. Thus we expect interference to be visible for
\L — Lj\ < LT/2. TO observe interference from Mechanism # 1 , the tip-to-QPC distance must be well matched to the impurity-to-QPC distance, to within the thermal
length. Thus, even at high temperatures (short LT), we expect to see interference
fringes far from the QPC, provided there is an impurity also at that distance from
the QPC. In Chapter 3, we see how interference fringes allow us to locally detect
impurities. In Mechanism # 2 in Fig. 2.7b, the double roundtrip path between the
QPC and tip (red) has length 4L. Thus, the difference in path length is 2L, and we
expect interference to be visible for L < LT/2. Interference fringes due to this Mechanism should only be visible close to the QPC, within LT/2. By measuring at lower
temperatures (longer LT), these fringes should become visible farther away from the
QPC.
All previous SGM images of electron flow displayed interference fringes throughout
24
CHAPTER 2. EXPERIMENTAL
(a) f(E)i
SETUP AND
BACKGROUND
Fermi Function ~^B '
^E
(b) f (E)A involved in Transport
0
Figure 2.9: Thermal Averaging, (a) The well-known Fermi function f(E) (describing
the occupation of different energy states). All energy states up to the Fermi energy
EF are occupied, with some partially occupied states in an energy window ~ kBT
around EF. (b) By applying a low-bias across the sample, we measure transport of
electrons with a distribution in energy that looks like the derivative of f{E). That is,
by applying a small energy difference dE = edV across the sampie, we measure net
transport f(E + dE) - f(E) oc f'(E). At T = 0, electrons involved in transport have
energy exactly EF, but at finite temperature, transport electrons have a distribution
of energies, (c) Different energy electrons accumulate phase at a different rate. Low
energy (red) electrons accumulate phase more slowly than high energy (blue) electrons. If all electrons start in phase at some point (x = 0), their spread in phases
becomes comparable to 1 radian after traveling the thermal length LT.
2.6. RELATED SCANNING PROBE
RESEARCH
25
due to Mechanism # 1 . In Chapter 3, we show that by using cleaner samples with
a low density of impurities, Mechanism # 1 does not occur. In Chapter 4, we show
that by using the same clean samples but at lower temperature, Mechanism # 2 does
occur, allowing us to probe phase-coherent effects in clean samples.
2.6
Related Scanning Probe Research
SGM has been used previously to spatially map electron flow in 2DEGs [13] and cyclotron flow bent by a magnetic field [37, 44]. SGM was also used to image the modes
responsible for quantized conductance through a QPC [12, 45]. Interference fringes
were examined in a purposely fabricated interferometer [35] and used to measure
the local electron wavelength (and 2DEG density) [32]. More broadly, SGM has been
used to investigate the disorder potential landscape and interference in InGaAs-based
QPCs [46, 47], as well as being used to tune the potential near a GaAs-based QPC
so that the QPC exhibits interesting transport features [48].
There is considerable interest in mapping electron wavefunctions inside confined
structures, especially when electron-electron interactions are important in which case
the organization of electrons can be quite complex. SGM was used to investigate
wavefunctions inside large quantum billiards [49] and there have been attempts at
imaging electronic states in small quantum dots [50, 51]. The major obstacle to
imaging wavefunctions in quantum dots is that the size of the potential perturbation
from the tip can be comparable to or larger than the electronic wavefunction. It is
thus difficult to probe individual parts of an electron's wavefunction in very small
structures without a different tip-sample geometry. The SGM maps the location of
the quantum dot and acts with long-range capacitive coupling like other fixed gates.
There have been related efforts to understand the potential perturbation induced by
an SGM tip at a quantum dot [52]. SGM has also been used to locate quantum dots
in carbon nanotubes [53] and nanowires [54]. There has been success in using SGM
to measure transport and the wavefunction-squared in Aharonov-Bohm rings [55, 56].
In addition to scanning tunneling microscopy, which has been used extensively to
probe 2D surfaces, other scanning probe techniques have achieved great success. We
26
CHAPTER 2. EXPERIMENTAL
SETUP AND
BACKGROUND
mention here a few prominent examples on 2D systems similar to those we measure. A
scanning capacitance probe was used to investigate potential fluctuations in a 2DEG
in the quantum Hall regime [57]. A single-electron transistor (SET) fabricated on
the end of a scanning probe tip sensitively measures electrostatic potential. This
scanning SET has been used to study the quantum Hall [58] and fractional quantum
Hall [59] regimes in 2DEGs. It has also been used to investigate potential fluctuations
in graphene [60], a new 2D system that has gained a lot of recent attention for its
unique band structure and transport properties [61, 62].
Chapter 3
Disorder
In part because of their extremely low levels of disorder, GaAs 2DEGs show a wealth
of remarkable electronic states and serve as the basis for fast transistors, research on
electrons in nanostructures, and prototypes of quantum computing schemes. In this
chapter, we study how disorder affects the spatial structure of electron transport in
2DEGs on length scales shorter than the mean free path lm. We are interested in
how features of electron flow that we introduced in the previous chapter, branches
and fringes, depend on the specifics of the disorder potential as well as what they tell
us about disorder in the sample. A detailed spatial picture of the disorder potential
[57, 58] may help to understand why exotic electron organization emerges in some
2DEGs and not others. The ability to understand the source of disorder in our samples
is important for growing 2DEGs with ever weaker disorder or even tailored disorder
[63].
To investigate the role of disorder, we image electron flow in three samples with
mean free paths lm ranging by over an order of magnitude. In the first section, we
characterize the branches in these three samples and find that the branches in each
sample remain straight over a distance roughly proportional to lm. In the second
section, in one sample we investigate the stability of the branches to changes in the
initial conditions of injected electron. We find that the branches are more stable
than classical mechanics would predict alone and their stability requires a quantum
mechanical explanation. Finally in the third section, we report a striking difference
27
28
CHAPTER 3.
Sample
density, n (1011 cm~2)
mobility, [i (106 err? jVs)
mean free path, lm (/im)
distance from surface to 2DEG (nm)
distance from donors to 2DEG, Zd (nm)
A
4.3
0.14
1.5
57
22
B
2.1
1.7
13
68
25
DISORDER
C
1.5
4.4
28
100
68
Table 3.1: Properties of Three 2DEG Samples, measured in a Hall bar configuration. Sample A was grown by the commercial grower IQE (wafer HTR STR 1 SI, run
114855; "IQls" in Goldhaber-Gordon laboratory nomenclature). Sample B was grown
by H. Shtrikman (wafer CBE296; "HS1" in Goldhaber-Gordon laboratory nomenclature). Sample C was grown by L. N. Pfeiffer and K. W. West (wafer 12-16-03.2;
"LP6" in Goldhaber-Gordon laboratory nomenclature).
in the strength of interference fringes in the three samples. In the highest mobility sample, there are no observable interference fringes, indicating very little or no
impurity-induced hard scattering. We demonstrate how the presence of interference
fringes can be used to locally detect impurities in samples. Ref. [24] contains the
results presented in this chapter.
3.1
Branch Length
In Table 3.1, we summarize the properties of the 3 samples in which we map electron
flow. The lowest mobility sample, Sample A, has mobility \i = 0.14 x 10° cmf jVs
and mean free path lm = 1.5 am. The cleanest sample, Sample C, has mobility and
mean free path that are over an order of magnitude larger, u = 4.4 x 106 cm2/Vs
and lm = 28 jim. In Fig. 3.1, we show electron flow in the 3 samples. The flow varies
from twisted and diffusive in Sample A to straight, smooth branches in Sample C.
In Fig. 3.1, we can see that as lm increases, so does the length over which branches
remain straight, as we expect. As a way to characterize the straightness of branches,
we average the distances along branches between all observable pairs of points where
one branch splits into two. We calculate this average distance lb between branch points
in 5 images each for samples B and C (the samples with well defined branches). We
3.1. BRANCH
LENGTH
29
(c)
(b)
Sample B
i^ = 13um
Sample C
In = 28 ^m
AG
(2e2/h)
AG
(2e2/h)
0.01
0.01
SampleA
(2^h)
^ , = 1.5(im
0.01
I
-0.19
200 nm
5»m
-0.19
20 nm
°
-0.11
Figure 3.1: Electron Flow in Samples with Different Mean Free Paths. The experiments, and all those performed in this chapter, are conducted at 4.2 K. Each figure
is labeled with the sample and its mean free path lm. In Sample A, the shortest mean
free path sample, the electron flow is twisted and diffusive, whereas in Sample C, the
longest mean free path sample, there are straight, smooth branches.
CHAPTER 3.
Sample B
/m = 13|im
^ = 480nm
(a)
DISORDER
Sample C
/ = 28 nm ^ = 740 nm
(b)\
\ /
V
i i
i
\
V
I
«/
1
i
)
•r
' /
l
i
i i
1
1
200 nm
Figure 3.2: Analysis of Branch Length. Samples B and C from Fig. 3.1 with branches
marked with purple dashed lines. For each sample in 5 images, we calculate lb, the
average distance between observable branch points where 1 branch splits into 2. lb
increases with lm.
find lb = 480 nm in Sample B and lb — 740 nm in Sample C; thus, we find that lb
grows with lm. Fig. 3.2 shows the branches in Fig. 3.1 marked with purple dashed
lines. In Chapter 2, we saw how branches form due to small-angle scattering. Now lb
gives us a way to further characterize the disorder potential in addition to lm. Both
hard scattering and small-angle scattering set lm, but lb is related to the small-angle
scattering component.
3.2
Stability of Branches
We next examine another property of the branches: their stability to changes in
initial conditions. Experimentally we vary the injection position and angle. With
simulations, we examine the stability of branches to these changes as well as changes
in initial energy. As discussed later, the small-angle scattering disorder potential is
3.2. STABILITY
OF BRANCHES
31
classically chaotic. Previous experiments have inferred information about electron
trajectories in chaotic systems from transport information [64], but SGM allows us to
directly image the electron flow. The hallmark of chaotic systems is that two trajectories injected with slightly different initial conditions exponentially diverge. Before
we report our experimental results, we first present simulations that give intuition
about the sensitivity to initial conditions.
We perform the stability experiments in Sample B, so we are interested in simulating electron flow in this sample. With an approximate disorder potential, we
can simulate a classical trajectory through this potential by simply time-stepping
Newton's second law. To simulate quantum mechanical flow through the potential,
we time-step the Schrodinger equation for an injected wavepacket. We are therefore
interested in creating an approximate disorder potential [19, 42, 65, 66, 67, 68].
Simulations of electron flow require a good assessment of the disorder scattering
potential. One strategy for calculating the disorder potential, which we do not fully
adopt, would entail calculating the potential perturbation in a metallic sheet (the
2DEG) induced by a point charge above the sheet (by a distance Zd by which the
donors are separated from the 2DEG; see Fig. 2.1). It is easy to show that this
potential perturbation scales as ~ l/(f 2 + z^)3^2 where r is the distance away from
the charge along the plane parallel to the 2DEG. This potential can be convolved
with a random distribution of charged donors, with the same sheet density as that of
electrons in the 2DEG. In Sample B, Zd = 25 nm so the full-width at half-maximum
(FWHM) of the potential perturbation created by a single charge is 40 nm. When
this perturbation from a single donor is convolved with a random donor distribution,
the resulting disorder potential has correlation length 50 nm. It should be noted that
this procedure assumes that electrons with wavelength \F are able to screen charge
over distance zd; Thomas-Fermi screening should only be valid for Xp < zd, but we
are actually closer to A^ ~ Zd- This procedure also ignores other charged donors
(which might donate charge to surface states) or potential perturbations created by
inhomogeneities of Al concentration in the AlGaAs layer. However, the biggest shortcoming of such a procedure is that it ignores spatial correlations among the charge
states of the donor atoms.
32
CHAPTER 3.
DISORDER
As discussed in Chapter 2, the donor atoms are randomly placed in the AlGaAs
layer of the 2DEG, but they do not all ionize and the ionization is therefore not
random. If all the donor atoms ionize, the charge is randomly distributed. However
if only a small fraction of the donor atoms ionize, charges can redistribute to avoid
each other [30, 63, 69]. Allowing donor charge correlations therefore causes the potential perturbations at the 2DEG to have smaller height and different correlation
length (charge at the donor layer is actually anti-correlated, but there can also be
longer range correlations). We assume that the correlation length does not change
significantly. For the model disorder potential which we adopt, we use the correlation length calculated above, but we need to adjust the amplitude of potential-energy
fluctuations. We assume that the mean free path is set by small-angle scattering (a
reasonable assumption given the low density of hard scatterers, see next section). We
therefore create a disorder potential with correlation length determined by Zd and
height of potential-energy fluctuations adjusted so that the quantum-mechanically
simulated lm matches our experimentally measured value. Thus, the two parameters
that characterize the small-angle scattering disorder potential, correlation length and
standard deviation of potential-energy fluctuations, are determined by two experimentally known parameters, Zd and lm. Fig. 3.3 shows a portion of our approximate
disorder potential for Sample B.
Given this disorder potential, we next examine the stability of single classical
trajectories. As shown in Fig. 3.4, we calculate the trajectory of a single classical
electron injected with the Fermi velocity vp at varying initial points separated by
2 nm. This shift in initial condition is small compared to the correlation length of
the potential, 50 nm. As can be seen, once the trajectories are ~ 1 /im away from
the injection point, the trajectories look completely different from one another. In
a classically chaotic system, two trajectories initially separated by a small amount
5X0 exponentially diverge, at intermediate times following the equation: |<W(£)| ~
|<5X0|est where s is known as the Lyapunov exponent. At very short times, the two
trajectories can converge or diverge and thus the average separation |<W(£)| does not
change. At very long times, when the two trajectories have diverged such that they
are in uncorrelated parts of the potential, they continue to diverge, but no longer
3.2. STABILITY
OF BRANCHES
33
Figure 3.3: Model Disorder Potential. The approximate disorder potential for Sample
B used in our simulations. Based on the 2DEG parameters of zd = 25 nm and lm =
13 /j,m, our model potential has correlation length 50 nm and potential fluctuations
with standard deviation of 0.08 Ep.
exponentially. The traditional Lyapunov exponent s can be converted to a Lyapunov
length II = VF/S. TO calculate II, we classically simulate many pairs of trajectories
initially separated by a very small displacement and calculate the average exponent
of the rate of their divergence. We find lL f» 300 nm for our approximate disorder
potential. This can be seen in Fig. 3.4; once different trajectories have traveled a
few times II, the paths look completely different. Because our SGM images extend
several //m's from the QPC, we would naively expect the position of the branches to
be very sensitive to the initial injection conditions.
To experimentally change the initial injection condition, we apply unequal potentials to the two QPC gates, putting as much as a 500 mV offset between the two gates
[31, 33]. In order to estimate how this gating affects electron flow, we use SETE, a
three-dimensional Schrodinger-Poisson solver written by M. Stopa [70]. SETE first
determines the profile of the conduction band edge along the growth direction of
the heterostructure by solving the Schrodinger equation in one dimension, and then
self-consistently calculates the potential and electron density at the 2DEG from the
Poisson equation. We simulate quantum mechanical electron flow through the resulting gate potential. If the QPC confining potential was simply an infinite square well
34
CHAPTER 3.
DISORDER
Ax = 0 nm
Ax = 2 nm
Ax = 4 nm
Ax = 6 nm
Origin
Figure 3.4: Stability of Single Classical Electron Trajectories. Four classical trajectories of single electrons through the approximate disorder potential are simulated. The
initial position of each trajectory is shifted by 2 nm from the previous trajectory, a
small displacement compared to the correlation length of the potential, 50 nm. The
black circles (separated by 200 nm) in this and upcoming figures are a reference grid.
The trajectories have diverged into random parts of the potential once they are 1 jim
away from the origin.
3.2. STABILITY
OF BRANCHES
35
potential at the narrowest point of the constriction, we would expect the width of the
QPC to be half the bulk Fermi wavelength XF/2 = 28 nm. However, inside the QPC
the density is suppressed (increasing A) and the confining potential is not infinite and
square (meaning the electron wavefunction extends into regions of classically forbidden potential). At the narrowest point of the constriction, we find electrons have a
probability density (|^| 2 ) with FWHM of 50 nm and they are emitted out of the
QPC with a distribution of angles with FWHM of 19°. By applying unequal gate
voltages to two QPC gates, we estimate we shift the angle of injection by ±5° and
the center of the injection point by ±30 nm. Thus, the full possible experimental
shift of injection point is 60 nm, significant compared to the correlation length of
the potential (50 nm) and the width of the QPC (50 nm, related to the bulk Fermi
wavelength 56 nm).
Fig. 3.5 shows the experimental flow patterns in Sample B when shifting the initial
conditions of injected electrons. Contrary to the expectations that the flow pattern
should be very sensitive to the initial conditions, the experimental flow pattern is
remarkably immune to changes in electron injection position and angle. As we shift
the QPC potential, electrons flow along the same set of branches, with the only
difference being changes in the relative strength of flow along each branch. Fig. 3.5ac show flow when the QPC is biased to the first plateau of conductance, 2e2/h, and
in Fig. 3.5d, the QPC is biased to the second plateau, 4e 2 //i, allowing through a
wider spread of electrons. Fig. 3.5a-c show images in which the QPC gates push the
electron flow to the left, center, and right, respectively. Comparing the images in Fig.
3.5, it is remarkable that the central branches 4 am away from the center of the QPC
(denoted with a brown arrow in Fig. 3.5a-d) deviate by less than 10 nm. When the
side branches appear (blue in Fig. 3.5a,d; green in Fig. 3.5c,d), they always remain
in the same position as well.
In order to understand this unexpected stability, we turn to our simulations of
branched electron flow. In Chapter 2, we showed how simulations of purely classical
flow through a small-angle scattering disorder potential can produce branches. Those
simulations used the disorder potential described here and a distribution of initial
trajectories corresponding to our SETE estimates of the injection conditions. For
36
CHAPTER 3.
• First plateau
Ax = -30 nm
Ax = 0 nm
DISORDER
Second plateau
Ax = +30 nm
Ax = 0 nm
Figure 3.5: SGM Images of Branch Stability. The thick red arrows denote the initial
average position and momentum of injected electrons, (a)-(c) As before, the QPC is
biased to the first plateau of conductance, 2e2/h. (d) The QPC is now biased to the
second plateau of conductance, 4e 2 //i, allowing a wider spread of injected electrons.
A
PP
"trostaticall
V£-l
\aJ
—30 nm and —5° compared to (b). (c) The injected electrons are shifted by +30 nm
and +5° compared to (b). The central branches, marked with brown arrows, are
amazingly stable despite the change in initial conditions. The side branches, marked
with blue or green arrows when visible, have positions that are extremely stable as
well.
\ a I
j - J.X^ m.xc ¥*_••••
V^J.±Vj± C .±ZJV^V^
~ •'
J.JL V O l l l l l
-• •
" '
UllV^ ±J.XJ^V^IJV^VJ. U l ^ ^ l l U l l O
U j
3.2. STABILITY
OF BRANCHES
37
now, we assume a single injection point (instead of a distribution of initial points)
with some angular spread of initial angles, corresponding to the angular spread from
a QPC. We come back later to the case where there is a distribution of initial injection
points as well.
Fig. 3.6 shows classical simulations in which the original injection point is shifted.
As we saw previously, a single trajectory (red in Fig. 3.6a,b,d,e) is very unstable on
the scale of our images to a small shift of just 2 nm. However, the overall simulated
branched flow pattern (blue) demonstrates remarkable immunity to a shift of initial
position by 2 nm.
This stability is in accord with previous theoretical work on
stability in chaotic systems. Whereas individual trajectories are unstable, the overall
flow pattern of many trajectories (or manifold), can be much more stable to variations
in initial conditions [71, 72, 73]. However, although classical mechanics can explain
branch formation, it cannot reproduce the full experimentally observed stability of
flow patterns. When shifted by 60 nm compared to the original injection point, the
classically-predicted flow pattern is very different. Classical mechanics cannot fully
explain our experimentally observed stability.
We find that a quantum mechanical treatment is necessary to reproduce the observed experimental robustness, as shown in Fig. 3.7. For the quantum mechanical
simulations, we add the disorder potential and the gate potential determined from
SETE. We then time evolve a wavepacket with energy spread equal to
3.5/CBT
(the
FWHM of the derivative of the Fermi function) to account for effects of thermal averaging. In Fig. 3.7, we plot the time-averaged |\I/|2. Our simulations do not include
the effect of the tip. The positions of most of the branches (red dashed lines) in the
simulated quantum flow pattern deviate by less than 10 nm even several ^m's away
from the QPC. There is less than 5% overlap between the 2 initial injected electrons'
probability densities (|^| 2 ). Not only diffraction out of the QPC but also quantum
scattering (s-wave-like) off the disorder potential steadily increases the spatial overlap between the two differently injected beams as they move away from the QPC,
providing an additional source of stability. Thus, previously branches could be understood as a classical phenomenon, but in order to fully understand branch stability
to changes in initial condition, we must use quantum mechanics.
38
CHAPTER 3.
Ax = 0 nm
Ax = 2 nm
Ax = 60 nm
(b)
(c!
200 nm
Ax = 0 nm
DISORDER
200 lim
Ax = 2 nm
Ax = 60 nm
Figure 3.6: Classical Simulations of Branch Stability. An ensemble of many trajectories is injected into the disorder potential, vv njii a n a i i g , uicii o p i c a u o i n i i i c i i l u t i i d t
emitted from a QPC. Blue represents the density of trajectories at a location, (d)-(f)
show the same plots as (a)-(c) but on a log scale so that less-weighted branches are
also visible. In (a),(b),(d), and (e), the red line tracks the middle-most trajectory.
When shifting the injection point by 2 nm (compare (b) to (a) and (e) to (d)), an
individual trajectory is very unstable to the change, but the overall flow pattern looks
remarkably similar. However, when shifting the injection point by the experimental
amount of 60 nm (compare (c) to (a) and (f) to (d)), the flow pattern is unstable to
this change.
TTTT4-1I
nyi
rtM«nlnv
C'*" V V *GQ'~'
Q I W I I I A » I
3.2. STABILITY
OF BRANCHES
39
This striking branch stability is important in applications which steer or guide
electrons in the ballistic regime. For example, it is often assumed that electrons move
inside large quantum dots (or quantum billiards) ballistically; it is assumed that by
changing a gate voltage, the paths through the dot can be randomized. However, a
more careful analysis would take into account that electrons flow along branches and
that these branches are stable to changes to initial condition. It may be necessary
to change the gate voltage by more than expected in order to force electrons onto
a new, randomized path. Other examples include electron-optics analogies. There
are experiments and experimental proposals that use electrostatic gates to reflect and
bend electron paths. In these cases, it is also important to consider electron flow along
branches which are relatively fixed in position. Rather than shifting the position of
branches, we merely shifted the relative amount of current flow along each branch.
An SGM technique to image electron flow bent by a magnetic field has been recently
developed [37]. This technique presents another method for testing the sensitivity of
branched electron flow to slight deviations by a magnetic field.
We have shown that there is very little spatial overlap (less than 5%) between the
initial injected wavefunctions-squared in the quantum mechanical simulations. However, one might be concerned that the difference between the quantum mechanical
and classical cases is simply that the quantum electrons have some spatial width;
in the quantum case, there is a spatial distribution of injection positions, effectively
making the two injection conditions more similar. In Fig. 3.8, we show more classical
simulations in which the now pattern is the result of an ensemble of classical trajectories with a distribution in initial position corresponding to the initial |\I/|2 of the
quantum case. At each of these positions, an angular spread is also injected. When
the injection point is shifted by 60 nm, the resulting branch pattern is completely
different. Thus, we can conclude that the experimentally observed stability is indeed
due to quantum properties rather than just some initial spatial width of the electronic
wavefunction.
The experimentally observed stability to changes in initial conditions is striking,
but in some sense, all previous images of electron flow were sensitive to less strenuous
tests of stability to changes in initial conditions. As discussed in Chapter 2, at
40
CHAPTER 3.
Ax = 0 nm
(a)
DISORDER
Ax = 60 nm
(b)
•
•
f
60 nm
200 hm
(d)
</>
Ax = 0 nm
Ax = 60 nm
Ficrure 3.7 Quantum Simulations of Branch Stability A n uantum wave^acket is
injected into the disorder potential and |\I/|2 is plotted. c)-(d) show the same plots
as (a)-(b) on a log scale. When shifting the injection point by 60 nm (compare (b)
to (a) and (d) to (c)), the position of most branches remain stable between the two
simulations. The positions of some of these branches are denoted with red dashed
lines. The inset in (b), shows cuts of |\I/|2 from (a) and (b), demonstrating less than
5% overlap.
3.3. DETECTING
IMPURITIES
41
(t3>
y
\
200 nm
Ax = 0 nm
Ax = 60 nm
Figure 3.8: Classical Simulations Taking into Account Spatial Width. The injected
classical electrons now include a distribution in initial position corresponding to the
initial probability density of the quantum case. When shifting the injection point by
60 nm, the branching pattern is unstable.
finite temperature we measure the transport of electrons with a thermal distribution
of energies. If the flow pattern were highly unstable to changes in initial energy,
each energy electron would form a different branching pattern, and experimental
images of electron flow would show highly blurred flow patterns looking like an even
diffraction pattern. Instead, the branches observed in all SGM experiments rely on
the stability of branches to changes in initial energy. In Fig. 3.9, we examine the
stability of classical electron flow from an injection point to changes in initial energy
(EF = 7.5 meV) by ±1.75 fcBT = ±0.6 meV. We find that the branch patterns look
fairly similar and are somewhat stable to such a change. Thus classical mechanics
is able to explain the stability to this smaller change in initial condition, a change
in initial energy. By shifting the injection position, we were able to apply a more
strenuous test of stability and found quantum mechanics was necessary to explain it.
3.3
Detecting Impurities
Having explored one feature of electron flow that depends on quantum mechanics,
we now turn to another intrinsically quantum mechanical effect, the interference of
42
CHAPTER 3.
DISORDER
>?•
•
\
200nrrr
AE = -1.75kBT
AE = +1.75 kBT
Figure 3.9: Classical Simulations of Different Injection Energies. When the injection
energy of classical electrons is changed by ±1.75 kBT, the resulting classical flow patterns (blue) look fairly similar. However, the middle-most trajectory (red), like before,
is still highly unstable to such a change in initial condition. Classical mechanics is
therefore able to account for the stability of branches to the different initial conditions
present in all SGM experiments; that is, electrons start with different energies due to
temperature.
wave-like electrons and the resulting fringes in SGM images. Chapter 2 showed an
example of an SGM image with fringes and discussed two Mechanisms that can cause
interference fringes. Because all experiments in this chapter have been conducted at
4.2 K, LT/2 is 260 nm, 180 nm, and 150 nm for samples A, B, and C, respectively. All
images of electron flow begin farther away from the QPC than these distances: the tipto-QPC distance L always satisfies the relation L > LT/2. As discussed in Chapter 2,
LT is the distance over which a thermal distribution of electrons become out of phase.
Therefore, interference fringes due to Mechanism # 2 , involving multiple reflections
between the QPC and tip, are not visible. At these relatively warm temperatures, we
need only worry about Mechanism # 1 , involving a strong scatterer in addition to the
tip, such as an impurity near the 2DEG. For Mechanism # 1 fringes to be visible, an
impurity backscatters electron flow back through the QPC, and the impurity-to-QPC
distance Lj needs to be close to L, to within
±LT/2.
Fig. 3.10 shows electron flow in the same three samples as before, but we have
zoomed-in to focus on the interference fringes. Sample A, the lowest mobility sample,
3.3. DETECTING
IMPURITIES
43
shows interference fringes throughout the flow pattern. All previously observed SGM
electron flow images also showed interference fringes throughout the flow pattern.
Sample C, the highest mobility sample, does not show any observable interference
fringes. Fig. 3.10c shows a representative portion of flow with no fringes. Sample B,
the intermediate mobility sample (though still high by most standards), only shows
interference fringes in a few, rare locations, and Fig. 3.10b shows one of these fringe
appearances.
We can understand this relative strength of interference fringes in SGM images
as a result of different concentrations of unintentional impurities in the sample, as
depicted in the insets in Fig. 3.10. Sample A, the dirtiest, has a high enough concentration of impurities such that, for any tip location, there is always a hard scattering
site within the electron flow at a distance L/ where L — LT/2 < LI < L + LT/2.
Furthermore, the relatively strong visibility of interference fringes (the ratio of the
interference amplitude to average signal) indicates that the flux directly backscattered by impurities is comparable to the flux backscattered by the tip. In contrast in
Sample C, the cleanest, there are no observable interference fringes. This indicates
there are no hard scattering sites within the electron flow, and there is no source
of backscattered electron flux comparable to the tip. In Sample B, when we do find
interference fringes, they are observable over roughly a length L? away from the QPC
along a branch. Thus, there must be a hard scatterer at the location where we observe interference fringes or at a similar distance away from the QPC, but along a
different branch of flow. In practice, when there are interference fringes on different
branches at the same distance from the QPC, usually one branch has the strongest
fringes. This indicates that the presence of the tip on that branch strengthens the
scatterer: the long-tailed potential perturbation from the tip adds to the impurity
hard scattering site, creating an even stronger scatterer. Thus, we can often identify
on which branch a hard scatterer lies.
We can use these images of electron flow to spatially detect impurities and estimate the density of hard scatterers, something which is difficult using transport
measurements alone. Fig. 3.11 shows an image of electron flow from Sample B, in
which dAG/dy is plotted for easy identification of fringes (y is vertical direction, as
44
(a)
CHAPTER 3.
Sample A
6
2
0.14x10 cm /Vs
T,p
t A
\\jLf
(b)
Sample B
H=1.7x106cm2/Vs
Tip*
J
(c)
S a m p l i l?
n = 4.4 x 106 cm9
DISORDER
Tip*
HoSRsI
gHl
^^B
200 nm
H
Hb~
41
AG
(2e2/h)
0.01
200 nm
-0.20
Figure 3.10: Interference Fringes in Three Samples with Different Mobilities, (a)
Sample A, the lowest mobility sample, shows interference fringes throughout the
image due to the high concentration of hard scatterers. (b) Sample B, the intermediate
mobility sample, shows interference fringes only in a few sparse locations, such as the
one shown here. This indicates a sparse sprinkling of hard scatterers that cause
interference fringes at that location, (c) Sample C, the highest mobility sample, does
not show any observable interference fringes. There are no detectable hard scatterers
within the electron flow.
denoted). In 5 images each for Samples B and C, we count 7 and 0 hard scattering
sites respectively. The hard scatterer must be located within the electron flow in
order to observe interference fringes. Thus, to estimate the area over which we are
sensitive to the presence of hard scatterers, we calculate the area in our scans where
the flow signal is 50% of more of the local maximum signal for the nearest branch.
By dividing the number of hard scatterers by this area over which we are sensitive to
their presence, we can thus estimate their density. In Sample B, we estimate a density of nH.s. ~ 3//um2. In Sample C, we put a rough upper limit of nH.s. < 0.5//im 2 .
There are too many hard scatterers in Sample A for us to be able to individually
locate them, and so we cannot estimate their density. Fig. 3.12 shows what these
densities of hard scatterers translate into on the spatial scale of the original images
of electron flow. We stress that we only know the location of impurities when they
are in the electron flow (3 in Sample B, 0 in Sample C). All other locations of the
impurities in the image have been placed at random to give a flavor of the density of
impurities in these samples.
3.3. DETECTING
IMPURITIES
45
Figure 3.11: Spatially Detecting Impurities. An image of electron flow from Sample
B with d/\G/dy plotted for identification of interference fringes. The fringes indicate
the presence of an impurity in the electron flow at that distance from the QPC, and
we mark the approximate location of the impurity with a red X.
In this chapter we have seen in much more detail how branches are due to smallangle scattering and the formation of fringes is due to hard scattering. The presence
of interference fringes gives us a way to estimate the density of impurities in these
samples. It may be possible to extend our technique to accurately measure the density
of impurities in a sample and determine what limits the mobility of a 2DEG. For
example, by opening the QPC to a higher conductance, allowing through a very wide
spread of electron flow, it may be possible to test a very large area of 2DEG for the
presence of hard scatterers, allowing for better statistics with fewer samples. It may
also be important to use this technique on a sample calibrated with a known density
of impurities to determine how close an impurity must be to the 2DEG to act as a
hard scatterer. With these techniques, it may then be possible to determine whether
unintentional impurities or intentional Si donors limit the mobility in the highest
mobility samples. This is still an open question for the highest mobility samples.
Determining what indeed limits the mobility opens avenues to creating even higher
mobility samples. There is strong incentive to obtain the ability to grow even less
disordered samples because each previous increase in 2DEG mobility has resulted in
46
CHAPTER 3.
(b)
DISORDER
Sample C
n H s < 0.5/nm2
(a)
0
Sample B
nHS -3/nm2
O
AG
(2e2/h)
AG
(2e2/h)
0.01
0.01
o
o
I
It-
200 nm
._»,^B^W
200 nm
-0.19
-0.11
Figure 3.12: Schematic of Impurity Density. Hard scatterers, denoted as black and
green circles, are placed in the images of electron flow to match the density estimated
from interference fringe measurements. Only the location of the 3 hard scatterers
in the electron flow in (a) are actually known. The locations of the other hard
scatterers are placed at random according to the measured density in (a), or the
inferred maximum density in (b).
the discovery of some new, exotic physics in other contexts.
Chapter 4
Electron Interferometry
Interference effects lead to some of the most widely studied transport phenomena at
low-temperature, including the Aharonov-Bohm effect, weak localization, and universal conductance fluctuations. Interferometers have been constructed in 2DEGs
[74, 75] and other materials, including carbon nanotubes [76]. In Chapter 2, we described how interference leads to fringes in SGM images. In Chapter 3, we analyzed
interference fringes due to reflections off impurity-created hard scattering sites, Mechanism # 1 described in Chapter 2. In this chapter, we report an interferometer due
to multiple reflections between the QPC and tip, similar to an optical Fabry-Perot
interferometer, Mechanism # 2 in Chapter 2.
All previously reported fringes in SGM images of electron flow were obtained
at temperatures above 1 K and were due to Mechanism # 1 . In Chapter 3, we
did not observe interference fringes due to Mechanism # 1 in Sample C because of
the scarcity of impurity scattering. By imaging electron flow in Sample C at lower
temperature (350 vaK instead of 4.2 K), we now observe Mechanism # 2 fringes. That
is, we recover fringes in clean samples at lower temperatures, allowing us to spatially
probe phase-coherent properties, such as local electron wavelength. We report new
spatial interference patterns different from those previously observed. We describe
how temperature, electron wavelength, interferometer length, and reflectivity of the
QPC affect the interferometer. When injecting electrons above or below the Fermi
energy, we observe effects of dephasing. At the end of the chapter, we discuss how
47
48
CHAPTER 4. ELECTRON
INTERFEROMETRY
understanding interference effects may be important for studying electron interactions
in nanostructures. The results in this chapter are reported in Ref. [25].
Mechanism # 1 and Mechanism # 2 have very different origins and temperature
dependencies. In Mechanism # 1 (shown in Fig. 4.1a), one path of the interferometer
is created by the SGM tip and the other by impurities or a reflector gate [35]. In
the latter case, the reflector gate acts like impurities, but creates an even stronger
hard scatterer. Thus, the reflector gate setup is a variation on Mechanism # 1 . The
reflector gate creates a second strong path backscattered through the QPC (red in
Fig. 4.1a) in addition to the path backscattered by the tip (blue in Fig. 4.1b). As
discussed in Chapter 2, interference fringes due to Mechanism # 1 are visible when the
tip-to-QPC distance L is within LT/2 of the impurity-to-QPC distance L/. LT is the
thermal length described in Chapter 2, and it becomes shorter at higher temperatures.
Interactions with the environment cause dephasing, which leaves each electron with
an ill-defined phase. However even without dephasing, because we measure transport of a thermal distribution of electrons (each of which has a well-defined, though
energy-dependent, phase), in order to observe interference effects we must satisfy the
condition \L — Lj\ < LT/2. Interference fringes due to Mechanism # 1 can be observed
at any distance from the QPC, as long as there is an impurity at a similar distance
away. A great deal can be learned about fringes from Mechanism # 1 , such as the
local electron wavelength [32], but they are not visible in high mobility samples. In
high mobility samples, in order to observe fringes, it is possible to introduce a reflector
gate, but this has the disadvantage of interrupting the intrinsic flow from the QPC
as well as being fixed in position.
Now we report interference fringes due to an interferometer similar to a FabryPerot interferometer, Mechanism # 2 (denoted in Fig. 4.1b). In addition to the
roundtrip path between the QPC and tip (blue in Fig. 4.1b), there is a path with
two roundtrips between the QPC and tip (red in Fig. 4.1b). As discussed in Chapter
2, these fringes should only be observed when L < LT/2.
By using SGM in a 3He
cryostat (more information in Appendix A), we now have access to temperatures down
to 350 mK where LT/2 is long enough such that we can observe these fringes. In our
sample, LT/2 = 320 nm at 1.7 K, and LT/2 = 1.6 \im at 350 mK. To calculate these
4.1. SPATIAL VARIATION OF INTERFERENCE
FRINGES
49
L T 's, we have used an SGM measurement of the electron wavelength A/2 = 38 nm,
which is discussed later. Fig. 4.1c shows an image of electron flow in Sample C at
1.7 K. Fringes due to Mechanism # 1 are not visible because there are no impurities.
Fringes due to Mechanism # 2 are not visible because the sample is too warm; the
imaging area starts farther than ~ 320 nm away from the QPC (L > LT/2).
Fig.
4. Id shows an image of electron flow over the same area at 350 mK. Fringes due to
Mechanism # 2 are now visible at the bottom of the image, close to the QPC, because
this region now satisfies L < LT/2. All further images presented in this Chapter are
obtained at 350 mK.
4.1
Spatial Variation of Interference Fringes
We next explore thermal averaging and how interference fringes disappear as a function of distance L away from the QPC. We can define the visibility of interference
fringes as v = \AGosc/AGavg\,
where AG osc is the amplitude of oscillations and AGavg
is the average signal. In cases where AG oscillates between some minimum value
and 0, v = 1. For two paths with magnitudes (fii and ip2 and no loss of interference,
v = 2|(/?i||(£2|/(|^i|2 + l^l 2 )- Fig- 4.2 shows an idealized calculation of how two paths
of equal magnitude interfere as function of L at 350 mK. Electrons have wavelength
A/2 = 38 nm and oscillations with period A/2 appear, like those in a Fabry-Perot
interferometer. Because of thermal averaging, v decreases away from the QPC; the
average signal remains the same, but the oscillations decrease in amplitude over the
characteristic distance LT/2.
The simple model in Fig. 4.2 underestimates the degree to which fringes fade as
a function of L. The two paths in the experiment do not have magnitudes which are
constant or equal. Fig. 4.3a shows an image of electron flow, in which there is one
strong branch that persists over a long distance. Fig. 4.3b shows how the magnitude
of the SGM signal AG decreases with increasing L. Recall AG < 0 indicates the
strength of current flow. At each distance L, we plot (in blue) the minimum of the
scan line AGmin (i.e. the strongest SGM imaging signal) on a log-log scale. Assuming
a power law AG ~ La, a linear fit (green) on this log-log scale gives a = —0.99 ±0.05.
50
CHAPTER 4. ELECTRON
1.7 K
350 mK
^BagOOnm
% f c 200 nm
INTERFEROMETRY
+0.01
<
-1.00
Figure 4.1: Appearance of Interference Fringes at Low Temperature. Fringes are
visible when a second path (red in (a) and (b)) interferes with the path scattered by
the tip back through the QPC (blue in (a) and (b)). (a) Mechanism # 1 for interference
fringes, also described in Chapter 2. An impurity creates a hard scatterer that reflects
electron flux back through the QPC (red path), (b) Mechanism # 2 : an electron is
emitted from the QPC, reflects off the tip, off the QPC gates, off the tip again, and
then is finally retransmitted through the QPC (red path), (c) Electron flow at 1.7 K
shows no interference fringes because there are not enough impurities for Mechanism
# 1 to occur. The sample is too warm for Mechanism # 2 to be visible, (d) Electron
flow at 350 mK shows interference fringes at the bottom due to Mechanism # 2 .
4.1. SPATIAL VARIATION OF INTERFERENCE
FRINGES
51
Distance from QPC, L darn)
1.0
k
2
2.0
-1.0l
Figure 4.2: Calculated Thermal Averaging. Interference between two paths of equal
magnitude is plotted as a function of L. As shown in the inset, the signal is a simple
oscillation, the amplitude of which decays slowly. Thermal averaging causes v to
decrease over the characteristic distance LT/2. The Moire pattern is a graphical
artifact of the oscillations compressed into a small width.
Previously, values of a fell in the range — 1 > a > — 2 with a closer to —2 [77]. This
scaling was explained as a result of electrons flowing with some angular spread, both
out of the QPC and reflected off the tip. For electrons flowing with a distribution
of angles, the current density, or electron flux, scales as 1/L at large distances L.
Thus, the electron flux emitted from the QPC that arrives at the tip scales as 1/L.
Similarly, electrons are reflected off the tip with some angular distribution; compared
to the amount arriving at the tip, the flux that returns to the QPC also scales as
1/L. The SGM signal AG is proportional to the total electron flux returning through
the QPC, and without interference effects, AG1 is thus proportional to the product
of two factors of 1/L. AG oc 1/L2. However, in our case in such a high mobility
sample, the electron flux emitted from the QPC stays along a narrow branch for a
longer distance, and so does not decrease with L. Our scaling in Fig. 4.3b shows that
only one factor of 1/L enters the SGM imaging signal, due to reflections off the tip
that cause electrons to flow out at a distribution of angles. This 1/L decay is only a
rough characterization and can vary for different flow patterns (i.e. different devices
or different cool-downs). We have chosen to analyze a pattern with flow concentrated
along one strong branch. In addition, even within a single flow pattern, due to details
of the geometry and underlying disorder potential, AG can vary in a complicated,
52
CHAPTER 4. ELECTRON
(a)
AG
(2e2/h)
INTERFEROMETRY
(b)
+0.02
0.01
AG min = - 0 . 1 8 L " ^
CM
CM
-0.1
<
-1.0
0.3
1.0
3.0
Distance from QPC, L (urn)
200 nm
-0.96
Figure 4.3: Decay of SGM Signal Away from QPC. (a) Electron flow showing a single,
well-defined branch, (b) AG m m , the minimum (i.e. strongest) SGM signal AG, at
a given L from the QPC. The blue line is the data from (b), and the green line is a
fit assuming a power law dependence, giving AGmin = —0.18 (2e2/h) (1 / i m / L ) ° " .
The 1/L dependence comes from reflections off the tip with a distribution of angles,
which cause flux to decay as 1/L. The oscillations in AGmj„ for L < 1.0 fim are
the interference fringes. Other fluctuations are due to the branching details, which
depend on the disorder potential, in conjunction with the measurement details, which
can depend on the tip-QPC geometry.
non-monotonic manner, as seen in Fig. 4.3b. Consult Appendix A for the details on
how we determine the distance L.
One path of the interferometer (blue in Fig. 4.1b) decays as 1/L due to one
reflection off the tip. The other path (red in Fig. 4.1b) has an additional reflection
off the QPC gates and an additional reflection off the tip, each of which multiples the
signal by a factor that scales as 1/L. Thus, the second path should decay roughly as
1/L3. We can describe the first path as having flux, or current density, (a/L) and
the second path as having flux (b/L)3. The magnitudes of the paths are proportional
to the square root of the electron flux. We then expect the visibility should decay
as v =
2
^L)1(UL]33
2 and at
l a r S e L, v « | ^ | . That is, u decays as 1/L at longer
4.2. CHECKERBOARD
INTERFERENCE
PATTERNS
53
Distance from QPC, L (urn)
1.0
2
2.0
Model
Data
Distance from QPC, L (urn)
Figure 4.4: Disappearance of Interference Fringes, (a) Calculation showing the disappearance of interference fringes due to two paths with flux decreasing as 1/L and
1/L3 respectively, as well as thermal averaging, (b) Comparison of v from the model
in (a) and the data from Fig. 4.3a.
L without even taking into account thermal averaging. In Fig. 4.4, we calculate the
interference of two paths, including thermal averaging, and with flux decreasing as
described. We expect a ~ b ~ A and so for the calculation use a = b = A. Fig. 4.4a
shows the interference of the two paths and Fig. 4.4b shows v from this model (blue).
We compare the calculated v to the v extracted from the data. The black circles in
Fig. 4.4b are derived from the data in Fig. 4.3a by fitting a sine curve to local cuts
of interference fringes. The data follow the model well. There is scatter in the data
because the flux scattered from the tip varies with position in a complicated manner
not perfectly described by a power law. Although a long LT (i.e. low temperature) is
necessary to observe fringes from this mechanism, geometrical attenuation of reflected
waves ultimately limits the distance over which fringes are visible. We do not expect
the fringes to persist farther from the QPC at even lower temperatures.
4.2
Checkerboard Interference Patterns
We next explore previously unobserved interference patterns when imaging close to
the QPC. We also examine how the transmission coefficient T of the QPC affects the
interferometer. Fig. 4.5a shows electron flow at T = 1 (i.e. the conductance of the
QPC set to 2e2/h) and Fig. 4.5b shows flow at T = 0.5 (i.e. the conductance of the
54
CHAPTER 4. ELECTRON
INTERFEROMETRY
QPC set to e2/h). At T = 1 at close distances to the QPC (the bottom of the image
is about 300 nm away from the QPC), we observe a new checkerboard interference
pattern; red circles in the center of "squares" are an aid to the eye. Fig. 4.5c plots the
derivative dAG/dy of Fig. 4.5a for easier visualization. Fig. 4.5c shows the "squares"
as clear, individual locations; the darkest and lightest regions at the bottom do not
extend laterally as rings. At T = 0.5, the interference pattern does not show as strong
of a checkerboard pattern and instead has a more ring-like structure (denoted by red
lines in Fig. 4.5b). Fig. 4.5d plots the derivative of Fig. 4.5b.
In both Fig. 4.5b and 4.5d, vibrations are apparent at the bottom of the images
(the bottom ~ 100 nm of the image). At T = 0.5, the QPC conductance is very
sensitive to changes in potential at the QPC because the QPC is not at a plateau of
conductance. Thus, the conductance is very sensitive to the location of the tip. These
vibrations at the bottom of Fig. 4.5b & d may resemble the checkerboard pattern in
Fig. 4.5a & c, but the periodicity of the ripples at the bottom of Fig. 4.5b & d is
different from those in Fig. 4.5a & c. Furthermore, the periodicity of the ripples at
the bottom of Fig. 4.5b & d along the fast-scan axis (horizontally) corresponds to the
frequency of known vibrations (see Appendix A). Also, the location of ripples in Fig.
4.5b & d changes from scan line to scan line; this observation is evidence that the
ripples stem from vibrations of the system rather than interference of electron waves.
The two different types of interference patterns in Fig. 4.5 can be understood as
due to two different sets of interfering paths, as shown in Fig. 4.6. The first set of
paths, in Fig. 4.6a, creates the checkerboard pattern at T = 1.0. Now reflections off
both QPC gates are important because of the proximity to the QPC. Doubled-ended
arrows denote roundtrip paths. In Fig. 4.6a, the directly backscattered path (blue)
interferes with both paths reflected off the QPC gates (red). At certain distances from
the QPC (purple dashed lines), the two red paths destructively interfere, and there
is no net path amplitude with which the blue path interferes. As the tip moves away
from the QPC along a purple dashed line, there is no oscillation in the SGM signal
AG. However, when the tip is between the dashed purple lines, the summed red
paths have a net amplitude, and the sign of this amplitude changes on opposite sides
of a purple line. The blue path interferes with the net red paths. Fig. 4.6c shows the
4.2. CHECKERBOARD
INTERFERENCE
PATTERNS
55
Figure 4.5: SGM Images While Changing Transmission of QPC. (a) Electron flow at
T = 1. The "squares" of the checkerboard interference pattern are marked with red
circles, (b) Electron flow at T = 0.5. The pattern displays a stronger ring pattern;
the rings are denoted with red lines, (c) The derivative dAG/dy of (a) is plotted for
a different visualization of the interference pattern. The y-direction is denoted in (a),
(d) The derivative of (b), on the same scale as (c).
56
CHAPTER 4. ELECTRON
INTERFEROMETRY
calculated checkerboard pattern due to the interference of three paths, as depicted in
Fig. 4.6a. In this simple calculation, we assume: the paths all have equal magnitude;
the separation between reflection points off the QPC gates is d = 410 nm, estimated
from SETE code, described in Chapter 3; and A/2 = 32 nm, the wavelength implied
from the bulk density measurements.
One feature of the checkerboard interference pattern is that the spacing between
fringes along a line away from the QPC (along a line between purple lines in Fig. 4.6a)
can be longer than A/2. This is due to the geometry of three interfering paths that
becomes more important close to the QPC. This geometrical reflection effect does not
fully account for the measured fringe spacing which is longer than the half-wavelength
implied from bulk density measurements, to be discussed.
The lateral spacing w between "squares" in the checkerboard pattern is related to
the distance d between reflection points off the QPC gates and the distance L from
the tip to the QPC. We can approximate w « \\/L2
+ (d/2)2/(2d).
We measure a
local fringe spacing of A/2 = 43 nm, w = 55 nm, and L = 400 ± 50 nm (for more
discussion on the determination of L, see Appendix A). This gives us the estimate
d ~ 340 nm, in reasonable agreement with the d = 410 nm predicted by SETE
calculations. Alternatively if we scaled the size of the calculated fringes in Fig. 4.6 to
those measured in Fig. 4.5 (A/2 = 43 nm), we would estimate from our calculation
that the lateral spacing of fringes should be w = 47 nm, in reasonable agreement with
our measured w = 55 nm. This agreement helps confirm that indeed the checkerboard
pattern is due to two separate reflections off the QPC gates.
The second set of interfering paths, shown in Fig. 4.6b, is responsible for the
simple ring structure in Fig. 4.5b & d at T = 0.5. Now because T < 1, there is
direct reflection from the QPC (red). This red path interferes with the path directly
backscattered by the tip (blue). Rings of constant path length back to the QPC
(dashed green lines) are rings of constant phase for the blue path. Thus, when the
tip moves along a dashed green line, the two paths interfere with the same relative
phase. In Fig. 4.5b & d, we indeed observe a simpler ring structure. There may
still be some remnants of the checkerboard pattern in Fig. 4.5b & d because the
checkerboard mechanism, in Fig. 4.6a, can still occur. However, at T = 0.5 the red
4.2. CHECKERBOARD
INTERFERENCE
PATTERNS
57
Figure 4.6: Interference Mechanisms at Different QPC Transmission Coefficients.
Double-ended red and blue arrows denote roundtrip electron paths, (a) Interference
mechanism that creates the checkerboard pattern seen in Fig. 4.5a & c at T = 1.0.
There are two different reflections off the two QPC gates, both shown in red. (b)
Interference mechanism that creates the ring pattern seen in Fig. 4.5b & d at T = 0.5.
There is direct reflection from the QPC, denoted with the red path, (c) Calculated
checkerboard interference pattern for three paths of equal magnitude in (a). The plot
is the amplitude-squared of the complex sum of the three paths returning to the QPC.
A few dashed purple lines, like those in (a), have been added to denote locations of
the tip at which the red paths destructively interfere, (d) Calculated ring interference
pattern for the two paths in (b), both with equal magnitude.
paths in Fig. 4.6a are much weaker than the red path in Fig. 4.6b, and thus the
dominant interference mechanism is that shown in Fig. 4.6b. This explains the more
prominent ring structure.
We now address SGM measurements of the electron wavelength A which is related
to the electron density n through the relation n = 2it/\2.
Previous SGM experiments
demonstrated that the spacing of interference fringes is an accurate measurement of
n and A [32]. The local A can fluctuate because of density variations due to doping
inhomogeneities. Furthermore, n can be suppressed because of gating from the tip or
QPC gates [12, 32]. We also identified above a geometrical effect near the QPC that
58
CHAPTER 4. ELECTRON
INTERFEROMETRY
can make the interference fringe separation longer than A/2 at T = 1.0. We therefore
expect to find a measured A/2 that is different than that implied from the bulk density
A F / 2 = 32 nm. In the immediately preceding discussion, we were interested in the
local fringe spacing in the interference pattern in Fig. 4.5. Thus, we could measure
the local fringe spacing over just a few fringes, and indeed the measured fringe spacing
of 43 nm is significantly longer than Xp/2.
In further discussions in this chapter and the following chapter, unless noted
otherwise, we are interested in an estimate of the average 2DEG n and A in an area
stretching a few /Lira's away from the QPC. We measure an average fringe spacing of
A/2 = 38 nm over many fringes far from the QPC (~ 700 nm away and farther) and
at T = 0.5. In this measurement, therefore, there should be very little geometrical
effect, and thus the fringe spacing is a true measurement of n. The measured A/2 =
38 nm implies a density of 1.1 x 1011 cm"2. Previous SGM experiments also observed
fringes that implied a density as much as 30% lower than that measured for the bulk
[12, 32]. This lower density was attributed to partial depletion of the density due
to the QPC gates and tip. We expect gating from the QPC to have little effect on
density ~ 700 nm away because of screening by the 2DEG, which is 100 nm below
the gates. SETE indicates that n at ~ 700 nm away from the QPC gates should be
at least 95% of its non-gated value. The effects of the QPC gates, however, probably
do play a role in the significantly longer measured fringe spacing in Fig. 4.5, close to
the QPC.
We attribute most of the n suppression 700 nm away from the QPC to gating from
the SGM tip. The tip and related supporting cantilever structure can gate regions of
2DEG farther away because they extend perpendicularly away from the 2DEG wafer.
This metal farther away from the 2DEG is not screened as effectively. The amount of
gating due to the tip at these longer distances should be relatively slowly varying with
distance away from the tip. Therefore, when the tip is far from the QPC, movements
of the tip do not substantially change the gating on the 2DEG region between the tip
and QPC. It is also possible that n in the region studied is lower than that measured
for the bulk, even before any gating. At different locations on the 2DEG wafer, we
find different Hall measurements and differing voltages at which the 2DEG depletes
4.3. INTERFEROMETER
CONTROL AND
DEPHASING
59
(both related to the density of the 2DEG), indicating a non-uniform n across the
2DEG wafer.
4.3
Interferometer Control and Dephasing
We next change the injected electron wavelength (i.e. energy) and interferometer
length without moving the tip. We report that finite injection energy leads to dephasing. We can use the voltage Vg on the QPC gates to change L. Setting Vg
more negative extends the depletion region underneath the QPC gates and makes L
shorter. We now also use a finite source-drain bias VDC across the QPC shown in
Fig. 2.3. As discussed further in Chapter 5, this injects electron with energy —|e|Voc
above the Fermi energy of the 2DEG. Because of the negative charge of electrons,
negative VDC injects electrons with higher energy (shorter wavelength).
Fig. 4.7a shows an image of electron flow. In Fig. 4.7b, we measure G as a
function of Vg and VDC with the tip at the red circle in Fig. 4.7a, in the electron
flow 700 nm away from the QPC. The diamond-like features in
G(VDC,
Vg) are due
to transport through the QPC alone and are not due to the presence of the tip [78];
the diamonds stem from transport of individual modes with quantized conductance
through the QPC. Thus, to account for G features of the QPC alone, in Fig. 4.7c we
measure G with the tip at the green circle in Fig. 4.7a, out of the electron flow. In Fig.
4.7a, the location of the green circle is chosen so that the tip has similar capacitive
coupling to the QPC as it does at the location of the red circle. We subtract the data
in Fig. 4.7c from those in Fig. 4.7b to produce Fig. 4.8. We take the data in Fig.
4.7b & c alternating between the two tip locations for each Vg; any overall changes
in the QPC behavior over time should appear in both figures. Fig. 4.8 show AG, the
signal we plot in our SGM images, which is how G changes by moving the tip into
the electron flow.
Although we shift Fig. 4.7b slightly along Vg to account for slightly different
capacitive coupling between the tip and QPC in the two locations, there are still
some small differences. This subtraction produces an artifact which looks like the
derivative AG(Vg + AVg) - AG(Vg) oc dAG(Vg)/dVg.
This produces the outline of
60
CHAPTER 4. ELECTRON
(a)
J
INTERFEROMETRY
+0.I
^ • 2 0 0 nm -o.:
CD
<
-600 ^ ^ ^ ^
^ ^ ^ 1
| o
-1.1 0.0 1.1-1.1 0.0 1.1
V D C (mV)
V D C (mV) 0.0
Figure 4.7: G In and Out of Electron Flow, (a) Image of electron flow. The red
circle shows where G is measured in (b), in the electron flow 700 nm away from the
QPC. The green circle shows where G is measured in (c), out of the electron flow,
(b) and (c) The differential conductance G as a function of Vg and VDC- Vg controls
L, and VDC controls the energy of the injected electrons and their wavelength, (c) is
subtracted from (b) to produce Fig. 4.8, which shows AG (VDC, Vg), the change in G
due to the presence of the tip.
diamond-like features in Fig. 4.8 which are marked with a green arrow.
In Fig. 4.8, we observe strong diagonal features, which are marked with red
dashed lines. These diagonal features are due to the interferometer being at constant
phase 2kL = 2-Km, where k is the electron wavenumber and m is an integer; this is
the resonance condition of a Fabry-Perot interferometer. The interval between red
dashed lines along the Vg axis therefore corresponds to changing L by 7?/k = A/2
(with fixed A). This spacing between red dashed lines along the Vg axis is 65 mV,
implying that the QPC gate voltage extends the depletion region underneath the
gate at rate of 0.6 nm/'mV. This rate is in agreement with SETE calculations. The
interval between red dashed lines along the VDC
ax
is corresponds to changing k by
ir/L (with fixed L). This spacing between red dashed lines along the VDC axis is
420 \iV. This is in good agreement with a simple prediction that |eVoc| = AE =
h2kAk/m
= irh2k/(mL),
also giving VDC = 420 fj,V. We have thus demonstrated fine
control over the interferometer parameters.
We can also use finite injection voltage VDC to study dephasing. As discussed further in Chapter 5, increasing \VDC\ injects electrons above (or holes below) the Fermi
energy of the 2DEG. Increasing \VDC\ increases the electron-electron (e-e) scattering
rate (see Chapter 5 for the formulas), and so we expect dephasing at high |Voc|- In
4.3. INTERFEROMETER
CONTROL AND
-368
-600
DEPHASING
61
AG
(2e2/h)
0.00
0.0
V D C (mV)
0.27
Figure 4.8: Changing Electron Wavelength (Energy) and Interferometer Length. Interferometer length L is controlled by Vg, and the electron wavelength A is controlled
by VDC- The red dashed lines mark the constant phase condition of the interferometer. The end of the red dashed lines shows AVDC where the interference strength
is reduced to 50% of its maximum due to dephasing. The yellow dashed lines show
where we expect electron-electron scattering to reduce the interference strength to
50% of its maximum. The green arrow points to diamond-like features which are an
artifact of subtracting Fig. 4.7c from Fig. 4.7b.
62
CHAPTER 4. ELECTRON
INTERFEROMETRY
Fig. 4.8 we find that at high \VDC\ the diagonal features (marked with red dashed
lines) disappear because they depend on phase-coherent interference. To quantify the
energy dependence along VDC, for each VDC we look at AG oscillations as function
of Vg and fit a sine curve to these oscillations. The amplitude of these oscillations is
a measure of the interference strength; it can be thought of as the AGosc from our
definition of v or the interference term 2Re{ip\(x = l)ip2(x = I)} from Chapter 1.
We quantify the disappearance of the diagonal features by finding the two values of
VDC (positive and negative) at which the interference strength is reduced to 50% of
its maximum and finding the difference AVDC of these two values. The QPC adds
non-trivial features due to transport of different quantized modes, but we estimate
AVDC
= 920 fj,V. We mark where the diagonal features are reduced to 50% of their
maximum oscillation amplitude with short vertical red lines.
We now turn to the expected dephasing rate due to e-e scattering. At high
|VDC|>
the regime we experimentally address, a single e-e scattering event completely dephases the injected electron [17]. The blue path in Fig. 4.1b has magnitude of the
non-dephased component of its wavefunction that scales as v r ^ ^ where L e _ e is
the e-e scattering length, for which the formula is given in Chapter 5. The red path
in Fig. 4.1b has magnitude of the non-dephased component of its wavefunction that
scales as \/e~iL/Le~e.
Thus, the interference term scales as e-3L/Le-e_ That is, we need
to compare the average path length 2>L to the e-e scattering length Le-e- We calculate
from the formula 3L = /n(2)L e _ e (V D c) that the expected AVDC is 1250 fiV, in the
same range as our measured
AVDC-
The ln{2) comes from our definition that AV^c
is a reduction to 50% of the maximum interference strength. The yellow dashed lines
in Fig. 4.8 denote our expected AVDc- Thus, we have demonstrated a technique that
may be useful for studying the spatial dependence of dephasing. Most methods to
measure dephasing rely on transport measurements, such as the loss of weak localization due to the application of magnetic field. However, our technique to measure
dephasing complements existing transport methods because it can be used to study
dephasing at a range of distances directly in a single sample.
4.4. PROSPECTS FOR STUDYING ELECTRON INTERACTIONS
4.4
63
Prospects for Studying Electron Interactions
In addition to use as a tool for spatially measuring dephasing, the interferometer
demonstrated here may be useful for studying electron interactions in nanostructures,
such as the QPC which injects electrons. Some of the most interesting physics occurs
because of the complicated and unpredictable ways in which electrons interact with
each other in confined geometries. One phenomenon that has received considerable
attention and has still not been fully explained is the "0.7 structure" of a QPC
[79]. In addition to G plateaus at integer multiples of 2e2/h, which can be explained
in terms of transport of single-particle quantum mechanical modes with quantized
conductance, there is a "mini-plateau" (or shoulder) at 0.7 x 2e2/h (i.e. T = 0.7).
Many transport experiments have been conducted on this very simple nanostructure.
They indicate that multiple conductance mechanisms are involved and that spin and
many-electron effects play an important role. However, there are a limited number
of parameters that can be varied with such a simple nanostructure, and there is still
not full consensus regarding the full explanation.
There have been recent proposals to use fringes in SGM images of electron flow
to study electron interactions in nanostructures [80]. These theoretical proposals rely
on Friedel oscillations induced by the SGM tip, which cause the effective electron
density in the nanostructure to oscillate as the SGM tip is moved. However, to identify these types of fringes, it is necessary to distinguish them from fringes resulting
from interference effects described in this chapter. Thus, careful studies of the sources
of interference fringes are necessary before embarking on investigations of electroninteraction induced fringes. We have taken care to investigate the fringes when T < 1
because this is the regime in which electron-interactions occur in the QPC. Furthermore, the interferometer described in this chapter may provide another approach to
studying electron interactions. The ability to measure phase along branches of electron flow gives us a way of distinguishing different sources of conduction, even though
they may have overlapping spatial patterns. Thus, it may be possible to observe the
evolution of different conduction sources' spatial flow patterns as parameters, such as
T and temperature, are varied.
Chapter 5
Electron-Electron Scattering
Phase coherence is a fundamental aspect of quantum mechanics that leads to the
interference of electrons and, under certain circumstances, the formation of more
complex quantum states. In the previous chapters, we directly imaged interference of
electron waves. Interference also causes a variety of other widely studied phenomena
in low-temperature transport. To determine the length of paths that can contribute
to interference effects, it is therefore important to understand the length 1$ over which
electrons retain a well-defined phase. Knowledge of the dephasing length 1$ is necessary to understand and use quantum mechanical effects in devices. The dominant
dephasing mechanism for electrons in clean 2DEGs at low-temperature is electronelectron (e-e) scattering.
The e-e scattering rate l/r e _ e is also important from a fundamental level. r e _ e
is the quasiparticle lifetime in Fermi liquid theory, and because it is generally long
(compared to other time scales), we can think of excitations of the system as only
weakly interacting electron-like and hole-like quasiparticles. This is an amazing result
of Fermi liquid theory: in metallic 2D and 3D systems, although conduction electrons
repel each other and are spaced quite closely, electron-like quasiparticles can travel
long distances (i.e. for long times) without scattering with other electrons. For
example, the typical separation between conduction electrons in our systems is 10's
of nm whereas the typical e-e scattering length at low-temperature is 10's of lira.
In this chapter, we use SGM to spatially probe l/r e _ e between electrons injected
64
5.1. TECHNIQUE TO MEASURE ELECTRON-ELECTRON
SCATTERING
65
from the QPC and the 2DEG into which they are injected. We find that at low
injection energies, electrons scatter only by small angles, in agreement with previous
transport measurements [81]. At high injection energies, unexpectedly, the differential
conductance through the system is increased by moving the SGM into the electron
flow and scattering electrons backwards. We understand this increase in differential
conductance as a result of electron scattering with a highly non-equilibrium distribution of electrons near the QPC. Previous transport measurements found evidence
of 2DEG heating due to the injection of high-energy electrons [82]. Our spatially
resolved measurements indicate that the region of most excited 2DEG is close to the
injection point (within 1 jim). Furthermore, the evolution of high-energy electrons as
they are injected into a 2DEG, with which they can interact, is a complicated theoretical problem: the full solution requires solving the Boltzmann transport equation and
the e-e scattering rate with that electron distribution at each location. As discussed
further, there has been disagreement over the e-e scattering rate with even simple
thermal distributions of electrons. Our experiment addresses the e-e scattering rate
between injected electrons and a complicated non-equilibrium system. Our results
can guide theoretical work in the area. Results from this chapter will be reported in
Ref. [26].
5.1
Technique to Measure Electron-Electron Scattering
The characteristic energy A that an electron has with respect to the Fermi energy
EF is set by temperature kBT or an injection energy, as we saw in Chapter 4. As
discussed later, it is easy to argue that the e-e scattering rate scales as l/r e _ e ~ A 2 .
More careful calculations in 2D [15, 16] show that this scaling is modified by logarithmic corrections. Dephasing due to e-e scattering can occur either diffusively, where
1$ is longer than the disorder-determined elastic mean free path lm, or ballistically,
where 1$ < lm. In the diffusive regime, the dephasing rate 1/r^ ~ A [14]. Therefore
1/-T0 > l/r e _ e for low enough A, and at low A dephasing occurs in the diffusive
66
CHAPTER 5. ELECTRON-ELECTRON
SCATTERING
regime. In the diffusive regime, we speak of well-defined quasiparticles that, when
interacting with other electrons, retain their momentum but lose their phase. However, in the ballistic regime, which applies at higher A and is the regime we study, a
single e-e scattering event creates new quasiparticles, completely dephasing the scattered electron [17]. The scattering is inelastic and 1/r^ = l/r e _ e . In order to study
e-e scattering on length scales accessible with our SGM technique (a few /ira's and
shorter), it is necessary to raise the injection energy A [33]; thus, we are concerned
with dephasing in the ballistic regime.
The e-e scattering process with which we are interested is depicted in Fig. 5.1.
Electronic states with energy below EF = h2(kl+ky)/(2m)
are filled (green) and states
above are empty; this is the filled Fermi circle. An electron (dark blue) injected with
energy A above EF can scatter with electrons in the Fermi circle. Scattering processes must conserve energy and momentum, and satisfy Fermi statistics (no double
occupancy of states). The injected electron and the electron inside the Fermi circle
must scatter into two unoccupied states (light blue) outside the Fermi circle. This
scattering event leaves behind a hole (white circle) in the Fermi circle. Calculation of
the scattering rate involves integrating over the phase space available for the creation
of three quasiparticles: two electrons and one hole. However, full integrals only actually takes place over the creation of two quasiparticles, the third's properties being
completely determined by momentum conservation. In order to conserve energy and
obey Fermi statistics, only states within A of EF can be involved in scattering. Thus,
t u c p n a n c a j j a t c CLVCUICHJIC I U I t u c u c a n u u
u i c a u i q u a o i j j a i tiv_,ic ov^aico CLO i_i.
uc^auoc
the properties of two quasiparticles are free, the overall scattering rate therefore scales
as A 2 [83].
For an electron injected with energy A above a filled Fermi circle, the e-e scattering
rate was calculated to be [16]:
J__
when A < < h2kFQTF/m.
EF_
/_A\
QTF = 2m/(m0ea0)
= 27r/(32 nm) is the 2D Thomas-
Fermi screening wavenumber (a0 = 0.053 nm is the Bohr radius; e is the dielectric
5.1. TECHNIQUE TO MEASURE ELECTRON-ELECTRON
SCATTERING
67
i
/
•
/
Figure 5.1: Electron-Electron Scattering with a Fermi Circle. An electron (dark
blue) with energy A is injected above a filled Fermi circle (green). Note that A is
an energy whereas the coordinates are wavevectors. The injected electron can scatter
with electrons in the Fermi circle. The two electrons scatter into unoccupied electronic
states (light blue), outside the Fermi circle. A hole (white dot) is created in the Fermi
circle as the electron is excited above EF. Because of the filling of the Fermi circle,
the injected electron can only interact with those electrons within A of EF inside the
Fermi circle. The electrons can only scatter into states within A of Ep outside the
Fermi circle. This greatly restricts the phase space available for scattering and so
1/Te_e can be low for low A.
68
CHAPTER 5. ELECTRON-ELECTRON
SCATTERING
constant, 13 for GaAs). Using transport measurements, this e-e scattering rate has
been verified [82, 18, 84] and shown to also cause dephasing [17]. The scattering rate
can be easily converted to a length le_e = vre_e where v is the velocity of the injected
electron. The injected electron has energy EF + A, so v = \f2(EF + A)/ra. As in
Chapter 4, we experimentally control A by applying a voltage VDC across the sample,
as depicted in Fig. 2.3 and Fig. 5.3; A = — |e|Vbc- Fig. 5.2 shows the calculated /e_e
for our sample, Sample C in Chapter 3 and the same sample as in Chapter 4. We
assume n = 1.1 x 10ncm~2 from the measurement of the fringe spacing in Chapter
4, giving EF = 3.9 meV. The asymmetry in Ze_e comes from the asymmetric effect
VDC has on the injected electron velocity v; more negative VDC means higher v. We
can thus adjust Ze_e with VDCWhen considering how the injected high-energy electrons lose energy, for the range
of injection energies we study (up to A = 4 meV) we can neglect other sources of
scattering, including plasmon emission and scattering with phonons. The threshold
energy Ac above which plasmon emission occurs was calculated to be (Eq. 17 in Ref.
[16]; see corrections in Ref. [18]):
l§V2me2EF
^c
w h p r p
A
=
\SV2J
{EF + A c ) 1 / 2
3ek
EF.
1/2
7T
COS
1
— I — arccos
3
3
EF + A
EF + Ar
3/4
(5.2)
A measure of the interaction strength, re is the
ratio of the interelectronic distance 1/y/nn to the effective Bohr radius a0emo/m =
10.3 nm. This gives Ac = 1.7 EF = 6.7 meV. The threshold injection energy above
which longitudinal optical (LO) phonons are emitted is 36 meV [85, 86]. For acoustic
phonons, the typical electron-phonon scattering length le-Ph is also significantly longer
than the e-e scattering length le-e. le-ph is typically estimated to be on the order of
100 lira for A « 1 meV [82, 87]. If we assume the electron-phonon scattering rate
scales as l/r e _ p/l ~ A 3 [88], for A = 4 meV, we still have l/r e _ e > > l/re-ph by an
order of magnitude.
SGM has been used previously to study e-e scattering and found results that
5.1. TECHNIQUE TO MEASURE ELECTRON-ELECTRON
SCATTERING
69
4.0
3.0
3 2.0
9
1.0
0.0
-2.0
0.0
2.0
VDC (mV)
Figure 5.2: Electron-Electron Scattering Length. The e-e scattering length Ze_e is
calculated according to Eq. 5.1. In order to decrease Ze_e to the lengths accessible
by our SGM imaging technique, a few /im's, the injection voltage VDC must be a few
100's of fiV.
followed Eq. 5.1 [33]. Here, by using a lower density 2DEG and injecting electrons at
higher energies, we find that a more complex description of the scattering is necessary
to explain our data. Fig. 5.3 depicts how we image e-e scattering using SGM. Fig.
5.3a shows the standard SGM imaging technique described in Chapter 2. When the
SGM tip interrupts electron flow and scatters electrons back through the QPC, the
differential conductance G is reduced: AG < 0 upon introduction of the tip into the
electron flow. G — dl/dV
is determined by applying a small oscillating voltage VAC
and measuring the resulting oscillating portion of the current / .
Fig. 5.3b depicts how we measure e-e scattering with SGM. The large bias VDC
across the sample changes the electrochemical potential of the two regions of 2DEG,
separated by the QPC, as depicted in Fig. 5.4. We show that the two 2DEG reservoirs
have different electrochemical potentials by giving them different colors (blue and
green). Electrons (blue) are injected into the right 2DEG (green), the one in which
we are interested, at the electrochemical potential of the left 2DEG (blue). This
injection energy is — |e|Vi)c above the electrochemical potential of the right 2DEG,
which we continue to label with energy EF.
We are interested in measuring the scattering rate of the injected electron (blue
70
CHAPTER 5. ELECTRON-ELECTRON
SCATTERING
in Fig. 5.3b and Fig. 5.1) with the 2DEG into which it is injected (green in Fig. 5.3b
and Fig. 5.1). If the injected electron scatters with an electron (yellow in Fig. 5.3b)
in the 2DEG, the injected electron can be deflected such that it no longer completes
a QPC-tip roundtrip. If the injected electron is not scattered back through the QPC,
G does not change. Thus, when l/r e _ e is high, electrons are not able to complete a
QPC-tip roundtrip and AG = 0. We expected that as l/r e _ e was increased, our SGM
images would fade to blank images of AG = 0. The degree of fading (loss of SGM
signal | AG|) at a location depends on l/r e _ e and how effectively an e-e scattering
event deflects an electron off the QPC-tip roundtrip path. As discussed later, it may
take one or several e-e scattering events to cause enough deflection to reduce |AG|.
The previous SGM experiment studying e-e scattering relied on the loss of electron
energy [33]. We are sensitive to changes in momentum.
If a single e-e scattering event deflects the electron off the QPC-tip roundtrip
path, the SGM image should fade over the characteristic distance L = le-e/2 (because
the total distance traveled on the roundtrip path is 2L). However, if the width of
the hard scattering site from the tip is small compared to the electron wavelength,
electrons reflect off the tip with a broad, circular wave distribution. In this case of
a broad distribution of angles, we might assume that an e-e scattering event does
not prevent tip-reflected flux from traveling back through the QPC. Then we are
just concerned with e-e scattering events over a distance L, not 2L. That is, in this
second approximation, e-e scattering can deflect electrons from making it to the tip,
but once an electron is reflected off the tip, it will travel back through the QPC.
Then, we expect the SGM image to fade over the characteristic distance L = le-e.
Fig. 5.4 shows the schematic energy diagram for the two 2DEG reservoirs separated by the QPC. The vertical axis is energy E. Blue depicts the left reservoir,
and green the right reservoir. The dashed parabola in the middle represents quasi-lD
electronic states in the QPC with a parabolic dispersion relation E ~ k\. The QPC
connects the two reservoirs. Electronic states moving through the QPC with +kx
originate in the left reservoir and are colored blue along the parabola. Conversely,
states moving with —kx originate in the right reservoir and are colored green along
5.1. TECHNIQUE TO MEASURE ELECTRON-ELECTRON
SCATTERING
71
Figure 5.3: Imaging Electron-Electron Scattering with SGM. (a) The standard SGM
imaging technique. The SGM tip scatters electrons injected from the QPC, back
through the QPC, reducing G. (b) Now the SGM imaging technique is affected by
e-e scattering. The injected electron (blue) scatters with an electron (yellow) in the
2DEG (green) into which it is injected. For the tip location in (a), when l/r e _ e is
high, the injected electron's QPC-tip roundtrip is disrupted. The injected electron is
no longer reflected back through the QPC and now AG = 0.
the parabola. There is no net current of electrons with energy below Ep, the electrochemical potential of the right reservoir: current carried by electrons moving in the
+kx direction is cancelled by the electrons moving in the — kx direction. Net electron
current, above EF, is drawn with an arrow, the length of which is proportional to the
amount of current flowing in that energy window.
When measuring G = dl/dV, we change the electrochemical potential of the left
reservoir by a small amount (light blue) and measure the resulting current (light blue
arrow). We consider the left electrochemical potential increasing by — |e|rfV, the bottom of the parabolic dispersion increasing by — \e\dV/2, and the right electrochemical
potential remaining constant. We ignore any changes to the parabolic dispersion relation that are a result of shifts in the shape of the confinement potential because of
how the QPC gate voltage is applied. An equivalent picture would be to consider the
parabolic dispersion fixed in energy, while the left electrochemical potential increases
by —\e\dV/2, and the right electrochemical potential decreases by \e\dV/2. Because
we are most interested in an electron's energy above EF when dealing with e-e scattering, it is easiest to consider the right electrochemical potential fixed, as depicted in
Fig. 5.4. Thus all the AC current flows in the window — |e|<iV" at an energy —|e|Vbc
above Ep. If the bottom of the parabolic dispersion is above Ep, which is not the
72
CHAPTER 5. ELECTRON-ELECTRON
Left Reservoir
QPC
SCATTERING
Right Reservoir
-|e|dV
•|e|VDC
-|e|dV/2
Figure 5.4: Schematic of Finite Injection Energy. The blue shows the left 2DEG
reservoir and green the right 2DEG reservoir. The light blue depicts the oscillation
in electrochemical potential on the left due to a small AC voltage change. The
dashed parabola represents the dispersion relation for quasi-lD states in the QPC that
connect the 2 reservoirs. The arrows denote net current. Their width is proportional
to the energy window in which the current flows, while the length is proportional
to the amount of current per unit energy. Thus, the total amount of current is
proportional to the area of the arrow. AC current refers to electron current in the
energy window — |e|cfX^, an energy —|e|Vbc above EF. DC current refers to electron
current in the energy window — |e|VDC, from EF to EF — \e\Vnccase when G = 2e2/h, then we need to consider changes in current due to the bottom
of the parabola moving.
5.2
SGM Measurements of Electron-Electron Scattering
We show the results of our e-e scattering experiment in Fig. 5.5. These images
are taken at 4.2 K, but similar effects are seen at 350 mK.
Fig. 5.5a shows an
image at VDC = 0.0 mV, where there is minimal e-e scattering. In Fig. 5.5b at
Vac = —1.0 mV, the signal is not as strong as that in Fig. 5.5a, as expected due
to e-e scattering. The green bars in the panels denote the length Ze_e/2 calculated
according to Eq. 5.1. We can see that the signal in Fig. 5.5b persists more than
a few times le-e/2 away from the QPC. This suggests that the rate of deflection off
the QPC-tip roundtrip path is not as fast as l/r e _ e ; that is, the injected electron
can be scattered by another electron and still complete the tip-QPC roundtrip. At
5.2. SGM MEASUREMENTS
OF ELECTRON-ELECTRON
SCATTERING
73
VDC = —1-75 mV in Fig. 5.5c, the SGM image has nearly completely disappeared.
The broad and faint circular region of AG < 0 (light blue) at the bottom of the
images in Fig. 5.4c and 5.4d is an artifact of the measurement; see Appendix A for
details. In Fig. 5.4d at VDC = —2.5 rriV, we observe something surprising. We see
the same flow pattern as in Fig. 5.4a, but with AG > 0 (red on the color scale).
Often we think of the differential conductance G as proportional to the transmission coefficient for electrons injected at —|e|VDC moving through the system. It is
therefore surprising that introducing the tip into the electron flow and thus backscattering electrons should increase G. We clarify that the total current / is never increased by moving the tip into the electron flow. We still seek to explain this AG > 0
feature because it does not conform to our initial expectations. At high injection bias
| VDC |; w e always observe this AG > 0 feature with the tip positioned on both sides of
the QPC, on different devices in the same sample, and in devices on Sample B from
Chapter 3.
We next seek to understand the energy dependence of the SGM signal AG. As
in Chapter 4, we measure AG at a certain location as a function of VDC and gate
voltage Vg. We measure
G(VDC>
V?) with the tip in the electron flow and out of
the electron flow, and then subtract the results. Fig. 5.6a shows a measurement
of AG(VDc,Vg) 1.0 iim away from the QPC at 350 mK.
As in Fig. 5.5, we see
that for |Vbc| near 0 mV, AG < 0; however for |VDC| t 1-5 mV, AG > 0. For
the lowest gate voltages Vg, the QPC is completely closed and there is no transport
through it. We see that AG mostly depends on the injection energy VDC, but there
is some dependence on Vg, which controls the transmission of electrons through the
QPC. In order to understand the features of the QPC, in Fig. 5.6b we plot the
derivative dG/dVg (sometimes called transconductance), measured with the tip out
of the electron flow. The clear diamonds in dG/dVg (marked with yellow dashed
lines) are due to transport of discrete modes through the QPC [78, 89]. We mark the
locations of these diamonds in Fig. 5.6a with yellow dashed lines as well. There are
features in Fig. 5.6a that are associated with the diamonds, denoted with yellow and
pink arrows (see Fig. 6.5a). Fig. 5.6c shows AG as a function of VDC, averaged over
Vg in Fig. 5.6a. Averaging over Vg averages away effects due to specific geometries,
CHAPTER 5. ELECTRON-ELECTRON
74
(a) v D C = 0.0
mV
c
( b ) v'DC
n r = -1.0mV '( )v DC =
•1.75
SCATTERING
mV (d) V D C = -2.5 mV +0.08
1
T = 4.2K
-0.0
CM
<D
1
•pOOnm
CD
<
00 nm
i200 nm
JfiGMam
J
-0.30
Figure 5.5: Images of Electron Flow at High Injection Energy. All images are on the
same color scale. Increasing \VDC\ increases l/re__e. The green bars denote le-e/2.
(a) The standard image of electron flow at VDC = 0.0 mV. (b) At VDC = —1-0 mV,
the strength of the SGM signal jAGj is weaker than in (a). This follows the general
expectation that e-e scattering will lead to a weaker SGM signal. However, there
is still signal at several times Ze_e/2. (c) At VDC = —1.75 mV, the SGM signal has
nearly completely disappeared, (d) At VDC = —2.5 mV, there is an unexpected effect:
moving the tip into the locations where there was electron flow at VDC — 0.0 mV
increases the differential conductance (AG > 0).
5.2. SGM MEASUREMENTS
OF ELECTRON-ELECTRON
V D C (mV)
SCATTERING
75
V D C (mV)
Figure 5.6: Dependence of SGM Signal on Injection Energy, (a) The SGM signal
AC 1.0 [im away from the QPC as a function of injection energy VDC and QPC gate
voltage Vg. At high \VDC\ there are strong AG > 0 features. The yellow dashed lines
show the QPC diamonds from (b). Features that are associated with these diamonds
are marked with yellow and pink arrows, (b) dG/dVg of the QPC, with the tip out of
the electron flow. The diamond patterns are due to transport of separate modes with
quantized conductance. The diamonds in (b) are marked with yellow dashed lines,
and these lines are placed at the same locations in (a), (c) AG from (a), averaged
over Vg from the bottom of lowest diamond to the top of the figure. This shows AG
just as a function of injection energy.
such as interference effects discussed in Chapter 4. We mark AG = 0 with a dashed
line because AG > 0 is the unexpected portion.
We next investigate A G ( V D C ) a t different distances L from the QPC. Changing
L controls the total time of interaction between the injected electron and 2DEG. Fig.
5.7 shows plots of A G ( V D C ) ) obtained in the same manner as Fig. 5.6c, at 0.70 /im,
2.1 fim, and 4.0 /im away from the QPC. For each plot, AG has been normalized to
its minimum value. However, there is no offset so that AG = 0 still has the same
meaning. VDC is not scaled. At all distances, we observe AG > 0 features. Strikingly,
the plots at all distances have very similar dependences AG(VDc); their widths in
76
CHAPTER 5. ELECTRON-ELECTRON
SCATTERING
VDC are nearly the same. This suggests that the e-e scattering mechanism responsible
for these AG features occurs within 0.70 fim from the QPC.
In Fig. 5.7b & c, calculations for our simple expectations are plotted. In Fig.
5.7b, we assume that a single e-e scattering event anywhere on the roundtrip path
of length 2L deflects the injected electron and results in a loss of SGM signal |AG|.
In this scenario, AG oc — exp(—2L/L e _ e ). In Fig. 5.7c, we assume that a single e-e
scattering event only on the path from the QPC to the tip, with length L, deflects
the injected electron enough for a loss of SGM signal. In this scenario, if an electron
reaches the tip, it reflects at a wide spread of angles and an e-e scattering event
does not prevent electron flux from arriving back at the QPC. We therefore plot
AG = — exp(—L/L e _ e ). In both of these simple models, we do not predict AG > 0.
Also, in these models the loss of SGM signal depends on distance away from the QPC:
at farther L there is more time for e-e scattering and more reduction of |AG|. At a
given VDc (which determines L e _ e ), for a longer L, |AG| = exp(—L/Le_e) is lower.
Finally, we note that at L = 4.0 [im (red curves) the experimental
AG{VDC)
(Fig.
5.7a) is wider than both calculated scenarios (Fig. 5.7b and Fig. 5.7c). We measure
the full-width at half-minimum (FWHM, i.e. the width at AG = —0.5 in normalized
units) of the data in Fig. 5.7a to be 1.41 mV. The FWHM of the models in Fig. 5.7b
and Fig. 5.7c are 0.58 mV and 0.86 mV, respectively. The FWHM of AG depends
on the rate at which SGM signal | AG| is lost by deflection off the QPC-tip roundtrip.
Because the data
AG(VDC)
&re wider than either calculated scenario, the loss of | AG|
in our data occurs more slowly than l/r e _ e used in the models (Eq. 5.1). That is, a
single e-e scattering event does not necessarily cause a loss of SGM signal. This can
be explained in terms of small-angle scattering. Injected electrons can experience e-e
scattering events but are only deflected by small angles and are thus able to complete a
QPC-tip roundtrip. As discussed later, due to the confined phase for e-e scattering in
2D, electrons are only scattered by small-angles ~ y/A/Ep
[81]. For a given VDC, our
loss of |AG| at L = 4.0 [im is lower than predicted in the calculations, evidence that
e-e scattering only occurs at small-angles. As discussed in Chapter 3, we have seen
spatial evidence that electron flow is stable to deviations (~ 5°) in injection angle.
The stability of the flow pattern to small angular deviations is further reason to believe
5.3. ELECTRON-ELECTRON
SCATTERING
MODEL
77
that an injected electron scattered by a small-angle (by another conduction electron)
can still complete the QPC-tip roundtrip. It is interesting to consider measuring the
ratio of l/r e _ e to the rate of loss of |ACj, providing information about the rate of
momentum change for injected electrons. However, this is not possible until we can
account for AG > 0 features.
We note that the density assumed in the calculations for Fig. 5.7b & c is from
the measurement of the fringe spacing in Chapter 4 (n = 1.1 x 10 n cm~ 2 ). If we
instead assumed the density from the bulk measurement (n = 1.5 x 1011cm~2), we
would calculate a slower l/r e _ e , widening the calculated curves
AG(VDC)-
However,
at L = 4.0 fjirn the FWHM of the calculated scenario would be 1.11 mV (under the
wider scenario, AG = — exp(—L/L e _ e )). The FWHM of this calculated scenario is
still narrower than our data FWHM of 1.41 mV.
5.3
Electron-Electron Scattering Model
We next seek to describe how AG > 0 features arise and understand what information
they provide about the system. For better understanding of scattering in this system,
Fig. 5.8 shows numerical calculations for the scattering cross section between an
injected electron outside the Fermi circle and electrons filling the entire Fermi circle,
as in Fig. 5.1. The injected electron (purple circle) is given A = 0.5 Ep (comparable
to VDC = —2.0 mV). Fig. 5.8a shows the scattering cross section for electrons inside
the Fermi circle, with which the injected electron can scatter. Fig. 5.8b shows the
cross section for electronic states outside the Fermi circle, into which the electrons
scatter. The calculation integrates over the three possible scattering states (one inside
the Fermi circle, two outside, as depicted in Fig. 5.1), while obeying Fermi statistics
and conserving momentum and energy, as described earlier. A simple kernel for the
interaction is assumed |l / (q')| 2 ~ l/{q2 + Q^F), where q is the momentum exchanged.
This simple form of the interaction is not completely correct (for one, it ignores
spin and dependence on frequency a>), but we use it for simplicity in our numerical
calculation. We are interested in which electronic states are involved in scattering,
and this almost entirely depends on obeying Fermi statistics as well as conserving
78
CHAPTER 5. ELECTRON-ELECTRON
(b)
Calculation, AG =-exp(-2L/L^)
T3
"~\
<1>
N
O
c
O
(c)
SCATTERING
Calculation, AG = -exp(-L/LeJ
N\
/ -0.70 urn
/ - 2.1 urn
— 4.0 nm
-0.5
/ / - 0 . 7 0 urn
/ - 2.1 nm
/ — 4.0 urn
w
-1.0
4.0 -4.0
-2.0
0.0
VDC<mV)
2.0
4.0 -4.0
-2.0
0.0
2.0
4.0
Figure 5.7: Distance Dependence of SGM Data and Calculations, (a) AG as function
of VDC a t various distances L from the QPC. AG = 0 is kept at the dashed line,
but each plot is normalized to is minimum value, setting AG(Vg = 0) = — 1. For
AG < 0, the amount of reduction in | AG] is proportional to the total amount of e-e
scattering, which scales as L/re^.e. For longer L, we thus expect more e-e scattering, and at a given Vpc, more reduction in AG. That is, for longer L, we expect
narrower curves. In contrast to this expectation, the curves follow each other well,
indicating that the important e-e scattering occurs closer to the QPC than our closest
measurement, 0.70 fim away. The curves all show similar AG > 0 features, (b) Calculated AG assuming that e-e scattering anywhere along the QPC-tip roundtrip path
prevents the injected electron from traveling back through the QPC. (c) Calculated
AG assuming that e-e scattering only along the initial QPC-to-tip path prevents the
injected electron from traveling back through the QPC.
5.3. ELECTRON-ELECTRON
SCATTERING
MODEL
79
Figure 5.8: Calculated Scattering Cross Section with the Fermi Circle. We numerically calculate the scattering cross section between an injected electron and the filled
Fermi circle of electrons. The injected electron has excess energy A = 0.5 EF (location denoted by purple dot), (a) The scattering cross section for electrons inside
the Fermi circle, with which the injected electron scatters, (b) The scattering cross
section for electronic states outside the Fermi circle, into which the electrons scatter.
momentum and energy.
As seen in Fig. 5.8a, the injected electron can scatter with any electron inside
the Fermi circle that has energy Ep — A < E < Ep, but it is most likely to scatter
with electrons on the opposite side of the Fermi circle, with wavevector kx ~ — \kp\.
This strong scattering between electrons with opposite momenta can be understood
in terms of momentum conservation. The closer to net momentum of 0 that the
two scattering electrons have, the more phase space is available for scattering on
all sides of the Fermi circle. In Fig. 5.8b, we see that the two electrons scatter into
states outside the Fermi circle with energy Ep < E < EF + A. However, the electrons
preferentially scatter into states that are close in momentum to the original wavevector
kx — \fTSkp.
This is the small-angle scattering described earlier. Electrons only
scatter by the angle ~ y^A/Ep
in 2D because of the confined phase space along the
outside of the Fermi circle. This small-angle e-e scattering does not occur in 3D; it
is a 2D effect. At higher injection energies when A ~ EF, the scattering angle is
large, and we can assume an e-e scattering event prevents the injected electron from
completing the QPC-tip roundtrip.
We next address the mechanism which can cause AG > 0. Based on the energy
diagram in Fig. 5.4, we show a mechanism in Fig. 5.9. Fig. 5.9a shows the same
energy diagram as Fig. 5.4, but now we have introduced the effect of the SGM tip.
80
CHAPTER 5. ELECTRON-ELECTRON
SCATTERING
Now some portion of the AC and DC current (light blue and dark blue arrows) is
reflected back through the QPC. The length of the arrows is proportional to the
amount of current. Only some of the current injected from the QPC is reflected back
through the QPC by the tip. In Fig. 5.9b, we now include the effects of e-e scattering
according to our initial expectations. Electrons injected at higher energies scatter
with the 2DEG and do not complete a QPC-tip roundtrip path. Thus, the arrows
become shorter at higher energies, and there is no AC current reflected back through
the QPC. However, at lower energies, closer to EF (the electrochemical potential of
the right reservoir), there is DC current reflected back through the QPC.
Normally our differential conductance measurement is not sensitive to DC current.
However, in Fig. 5.9c we show the effects of an interaction (black wavy line) between
the AC injected current and the DC injected current. If high-energy AC electrons
also deflect low-energy DC electrons off their QPC-tip roundtrip, then there is less
electron current flowing in the — kx direction, or equivalently more electron current
flowing in the +kx direction. By driving an AC current in a small energy window
(—|e|Voc to —lelVoc ~ leMV), in addition to that AC current, we also have more
net DC current flow through the QPC (in the energy window 0 to — |e|Vc>c)- Our
measurement of G is sensitive to signals oscillating in response to the AC current and
is thus sensitive to changes in current at any energy. G can hence be higher with the
tip present than without.
Fig. 5.10 shows a schematic in real space for the mechanism that causes AG > 0.
At the injection point, electrons have energy A above EF. These high energy electrons
scatter with electrons in the 2DEG, transferring energy to the 2DEG and creating
a non-equilibrium distribution of electrons near the 2DEG. As discussed below, we
roughly think of this region of 2DEG as "hot" and having a higher effective temperature Teff. However, the distribution of electrons near the QPC is not properly
described by a thermal distribution. The important point is that this non-equilibrium
distribution of electrons has more partially occupied states around EF, opening more
phase space into which e-e scattering can occur. Thus, the addition of AC electrons
pushes the distribution of electrons farther out of equilibrium in this region of 2DEG
5.3. ELECTRON-ELECTRON
SCATTERING
(a)
MODEL
81
tlk,
AC
DC
(b)
(c)
-
*
Figure 5.9: Mechanism Causing Increase in Differential Conductance. See Fig. 5.4
for an introduction to these energy diagrams, (a) Now the effect of the SGM tip
is included. Electrons are reflected back through the QPC, denoted by arrows in
the — kx direction, (b) The effects of e-e scattering are included. Electrons injected
with high energy scatter with the 2DEG and do not complete QPC-tip roundtrip
paths. At higher energies, there is less current returning through the QPC, denoted
by shorter arrows. There is no returning AC current, so AG = 0. (c) Now we include
a possible interaction between AC electrons and DC electrons, which causes AG > 0.
The addition of AC electrons is responsible for the scattering of DC electrons off the
QPC-tip roundtrip path. The net current is now due to not only AC electrons, but
DC electrons prevented from returning through the QPC.
82
CHAPTER 5. ELECTRON-ELECTRON
IA
f(E)
. I-kTeff
f(E)
SCATTERING
Ec
f(E)
Figure 5.10: Non-Equilibrium Distribution of Electrons Near the QPC. The Fermi
function f(E), describing the occupation of electronic states moving to the right, is
shown at various locations. Electrons are injected with excess energy A above EF at
the QPC. Due to e-e scattering, these high energy electrons transfer energy to the
2DEG, creating a non-equilibrium distribution, with many partially occupied states
near Ep- These partially occupied states allow a higher e-e scattering rate. The
addition of AC electrons increases the number of partially occupied states (increases
Teff) and therefore increases the scattering of DC electrons. The increased scattering
of DC electrons prevents them from completing the QPC-tip roundtrip path, and
causes AG > 0.
(increasing Teg), which more effectively scatters DC electrons. Therefore, AC electrons result in extra scattering of DC electrons, the DC electrons do not complete
the QPC-tip roundtrip, and AG > 0.
The full calculation of the evolution of the distribution of electrons as they are
injected from a QPC into a 2DEG is a complex problem. It involves solving the
Boltzmann equation as well as calculating the e-e scattering rate at each location
with a different distribution of electrons. We seek a simpler, approximate description
of the system in terms of an effective temperature Teff. We can then use the Teg at
different locations to calculate the e-e scattering rate for all injected electrons based
5.3. ELECTRON-ELECTRON
SCATTERING
MODEL
83
on previous results for e-e scattering with a thermal distribution. We emphasize that
our Teff should not be thought of as a real temperature; it is merely a way of tracking
the injected energy from the QPC and estimating the number of partially occupied
states around EF. It should therefore also be noted that the scattering rate with this
distribution of electrons should not necessarily follow the e-e scattering rates for a
thermal distribution. We aim to explain our data in terms of this type of scattering
and determine the scattering rate.
We show our scheme for calculating Teff in Fig. 5.11. We want to calculate the
energy per unit area u(L) of injected scattered electrons. This calculation involves determining the injected energy per electron and how it spreads throughout the 2DEG.
Electrons are injected with some voltage V, which is between 0 and VDc- Ignoring signs, the electron carries energy eV. We approximate that this injected energy
only contributes to Tes once the electron is involved in an e-e scattering event. The
probability of scattering at some distance r, shorter than L, per unit length dr is
e-r/ie-ejj_^
e-r/ie-e
This comes from assuming scattering is a Poisson process with rate ^—\
[s th e probability that an electron has not scattered by the time it has traveled
to r.
Next we estimate the area over which the injected energy is distributed. Assuming
transport at the pth plateau, the current in voltage window dV is 2e2p/h dV, so the
particle current is 2ep/h dV. The power injected is therefore 2e2p/h VdV.
This
power gets distributed over the "dynamic area": the area covered by the scattered
electron per unit time. The "dynamic area" covered is vW where v is the velocity
and W is the typical width of the scattered electron beam. We can write that W =
pX + 2(L — r) tan(^/2) where (L — r) is the distance traveled since scattering at r
and the angle of scattering is 0 = \J/\/EF.
We do not consider the beam broadening
from additional scattering events because they involve quasiparticles with typical
energy ~ A/3: the energy of the initial injected electron is shared among three
quasiparticles [82]. Therefore after the first scattering event, the particles with energy
A/3 scatter much more slowly and at narrower angles. We therefore have the integral
u(L) = ^ X.rroL fv=oDC e~rlle~eY^-JZZdensity of a 2DEG f f k2BT2s.
We
equate U(L)
to the mternal
ener
§y
84
CHAPTER 5. ELECTRON-ELECTRON
SCATTERING
Figure 5.11: Effective Temperature Calculation, (a) Schematic of how energy is
injected into the 2DEG. Injected high energy electrons (red) are scattered near the
QPC and spread at wide angles. Injected lower energy electrons (blue) are scattered
farther from the QPC, over a wider range of distances, and spread at narrower angles,
(b) Calculation for Te^ as a function of L for a variety of injection voltages VDC- I n
this calculation we assume all electron are injected at the l s i plateau. In order to
contribute to Teff, injected electrons must be scattered, so at L = 0, Tes = 0. Teg
grows as electrons are scattered. Because the scattered energy diverges with some
angle, Teff is peaked. For injection voltages \VDC\ > 1 mV, le-e < 1 l^m, so most of
the injected energy is deposited near the QPC.
Fig. 5.11a shows a schematic of how energy is injected into the 2DEG. High energy
electrons (red) scatter close to the QPC and the energy spreads at wider angles. Low
energy electrons scatter farther from the QPC and energy spreads at narrower angles.
We numerically evaluate Teff as a function of L for various injection voltages VDC
m
Fig. 5.11b. The maximum Teg occurs within lum away from the QPC, and for
| VDC I > 2 raV, the maximum Teg is within 500 nm of the QPC. As |VDC| increases,
we also see that most of the increase in Tefj is within 1 fim of the QPC. This is
due to higher energy electrons scattering closer to the QPC and spreading at wider
angles. This peaked TeS explains why our data suggested that the majority of the e-e
scattering occurred near the QPC.
We next describe the e-e scattering of all the injected electrons, AC and DC,
with the distribution of electrons that we approximate as thermal. In Ref. [16], the
e-e scattering rate between an electron near EF and a 2DEG at temperature T was
calculated to be:
5.3. ELECTRON-ELECTRON
for A < < ksT «
SCATTERING
MODEL
85
Ep and the prefactor a = 1. However, there has been disagree-
ment over a. Tunneling spectroscopy experiments found they needed to use a = 6.3
in order to explain their results [90]. Following this tunneling experiment, theoretical
calculations suggested a = 7T2/4, and the discrepancy between 6.3 and 7r2/4 could
roughly be explained by another factor of 2 needed for the correct interpretation of
the experimental data [91, 92]. However, other theoretical calculations have found
[93] or used [94] a = 7r2. These theoretical calculations with a / 1 generally ignore
non-T dependent terms inside the square brackets in Eq. 5.3 because low-T is assumed. We recognize that a = n2 implies a l/r e _ e that may be unrealistically high.
Using a = 7r2, we calculate /e_e = 360 nm for our sample at 4.2 K. Given that we
have imaged electron flow over much longer distances than this at 4.2 K, we find
a = n2 unlikely and use a = 7r2/4 in our calculations. However, as discussed previously, because the electron distribution is not actually thermal, we recognize that
Eq. 5.3 with a = 7r2/4 may not properly describe the scattering in the system.
We numerically calculate AG in Fig. 5.12 and compare it to our data from Fig.
5.6. We compare it to the data 1.0 fxm away from the QPC because the expected
l/r e _ e according to Eq. 5.1 is still slower than the rate of |AG| suppression (the
experimental
AG(VDC)
curve is narrower than what would be calculated according to
Eq. 5.1). Therefore, we can add another e-e scattering mechanism to our calculations,
increasing the rate of |AG| suppression (narrowing the calculated AG(VDc)
curve)
and better matching our data. At farther distances L, we would need to account
for small-angle scattering at the lower injection voltages
|VDC|-
Comparison to data
1.0 nm away from the QPC is also a good choice because, as we saw in Fig. 5.11,
most of the non-equilibrium distribution of electrons is within 1.0 [im from the QPC.
We simply model transport through the QPC as having mode spacing (the half-width
of the diamonds in Fig. 5.12a) 2 mV; in the actual data, the mode spacing changes
with plateau (the mode spacing is roughly 2.5 mV for the first plateau and 1.9 mV
for the second plateau). Taking into account transport through all open modes, we
86
CHAPTER 5. ELECTRON-ELECTRON
SCATTERING
calculate l/r e _ e for all electrons in the voltage range between 0 and VDC according to
both Eq. 5.1 and Eq. 5.3 and using our Teff model described above. For each energy
electron, we then use the higher scattering rate as the scattering rate of electrons
off of the QPC-tip roundtrip path. This provides some self-consistency: the highest
energy electrons are scattered according to Eq. 5.1 because of their high A. These
then provide a non-equilibrium distribution with which lower energy electrons scatter
according to Eq. 5.3. We do not take into account small-angle scattering effects
because most of the features are at high energy, comparable to EF, such that the
angle of scattering should be high.
Fig. 5.12a & c show data from Fig. 5.6. In Fig. 5.12b, we show the results of
our calculation using a = 7r2/4. We see strong similarity between our data and the
calculation. There are AG > 0 features at high
\VDC\-
The yellow and pink arrows
denote AG features that are associated with the diamonds (yellow dashed lines), very
similar to those in the data. These can be understood as sudden changes in the ratio
of AC to DC electrons as a new mode is opened. In Fig. 5.12d, we average the
calculation in Fig. 5.12b along Vg. We see a similar energy dependence of AG (VDC)
between our data and the calculation. We plot both our data and calculation in
Fig. 5.13. Our calculation is not able to reproduce the asymmetry of the AG (VDC)
data curve. However, the similarity between our calculation and the data strongly
indicate we are measuring e-e scattering with a highly non-equilibrium distribution
of electrons near the injection point.
We have thus explained AG > 0 features and the distance dependence of
AG(VDC)'-
the highly non-equilibrium distribution of electrons is within ~ 700 nm of the QPC.
We additionally have an experimental measurement of the scattering rate of injected
electrons with this non-equilibrium distribution. We briefly touch upon the approximations made by assuming a Teff here. Fig. 5.14a depicts a Fermi circle according to
the Teff model. The dashed red circle denotes EF and the blue represents the occupation of electronic states. There are partially occupied states (light shades of blue)
around EF. At Teff = 0, the Fermi circle would be perfectly filled inside the red circle.
Fig. 5.14b depicts a schematic of the distribution of electrons that is closer to the
actual distribution. As seen in Fig. 5.8, for electrons injected in the +kx direction,
5.3. ELECTRON-ELECTRON
SCATTERING
MODEL
Calculation, a =
Data
87
TT2/4
-o.o
-3.0
0.0
VDC (mV)
3.0
-3.0
0.0
VDC (mV)
3.0
Figure 5.12: Effective Temperature Model Calculations. We use the Teg model described to calculate AG(VDc Vg) in (b), and compare it to data in (a) from Fig. 5.6.
(c) and (d) show AG(VDc) obtained by averaging AG over Vg in the graph above.
In the calculation we use a = 7r2/4 to calculate e-e scattering with a thermal distribution of electrons according to Eq. 5.3. We observe strong similarity between our
calculation and the data.
88
CHAPTER 5. ELECTRON-ELECTRON
SCATTERING
0.5r
§
0.0
•a
<D
.N
"5
e
o -0.5
O
<
-1.0
-2.0
0.0
V DC (mV)
2.0
Figure 5.13: Comparison of Data to Effective Temperature Model. Fig. 5.12c & dare
plotted on the same graph for easier comparison of the data to the calculation. The
calculation uses a = 7r2/4. The asymmetry of the data around VDc ~ 0.0 mV may
be due to interference effects discussed in Chapter 4. The asymmetry of the data at
high \VDC\ is not reproduced in our calculation. Using higher values of a produces
a calculated AG(VDC) which is narrower around VDC = 0 and has higher AG > 0
features at high |VDC|-
there should be more electrons moving in the +kx direction outside the Fermi circle
and more holes moving in the —kx direction inside the Fermi circle. Electrons are
scattered out of — kx states near the Fermi surface and into states on the +kx side.
The distribution has a wider energy range of partially occupied states (higher Teg) in
the ±kx directions and less in the ±ky directions.
Thus, the e-e scattering rate for a distribution like Fig. 5.14b can be different,
or at least have a different prefactor, than that in Fig. 5.14a. Furthermore, our
Teff model assumes the scattering rate is the same for an electron with kx = ±|A;|
where k is some wavevector \k\ > kF. However, it is clear that if this same electron
were scattering with the distribution in Fig. 5.14b, the scattering rate should be
considerably higher if the electron has kx = —\k\ compared to kx = +|/c|. When the
electron has kx = —\k\, it is farther in energy from the fully occupied states, and
thus there is considerably more phase space into which scattering is possible. This
indicates that the scattering rate may be considerably higher for electrons returning
to the QPC with kx < 0 rather than those being emitted initially towards the tip
with kx > 0. In our experiment, we measure the total scattering rate for an electron
5.3. ELECTRON-ELECTRON
SCATTERING
MODEL
89
Figure 5.14: Distribution of Electrons, (a) In our model, we roughly approximate
that the distribution of electrons is thermal, as in this panel, (b) A more realistic
schematic of the electron distribution when electrons are injected in the +kx direction.
Electrons collect in the +kx direction and they are scattered out of the Fermi circle
in the — kx direction. There is a wide range of partially occupied states in the +kx
and — kx directions.
traveling first with kx > 0 towards the tip and then back with kx < 0 towards the
QPC. The proper description of the non-equilibrium distribution of electrons away
from the QPC, and the e-e scattering rate with that distribution, is a difficult problem.
Our experimental measurement of the scattering rate can help guide theoretical work
on this problem.
Chapter 6
Conclusions and Future Directions
6.1
Summary
In this thesis, we investigated electron transport in high-mobility 2DEGs at lowtemperature. Using SGM, we obtained direct spatial information about electron flow,
information to which we would not otherwise have access using transport measurements alone. We researched two important factors that impact electron propagation
in these systems: disorder and electron-electron (e-e) scattering. Disorder determines
the electrons' mean free path lm. Even on length scales much shorter than lm, disorder is responsible for determining the trajectories traversed by electrons because
of small-angle scattering. E-e scattering is the dominant source of dephasing and
sets the dephasing length 1$, the distance over which an electron retains its quantum
mechanical phase. Thus, we have spatially investigated electron transport over distances shorter than two important length scales: lm and 1$. Because of extremely long
/ m 's and l^s (both on the order of 10's of fim), GaAs-based 2DEGs have allowed the
study of a rich set of physics. We have furthered our understanding of the microscopic
physical phenomena that set these lengths.
In Chapter 2 we explained the SGM technique and showed our typical imaging
data. Electrons flow along narrow branches, and we reviewed that, as was previously
understood, the formation of these branches is due to small-angle scattering off disorder. In Chapter 3, we imaged electron flow in three 2DEGs with lm ranging over an
90
6.1.
SUMMARY
91
order of magnitude. We found that, as expected, the branches remain straight over
a distance roughly proportional to lm.
In Chapter 3, we also found that the positions of the branches were unexpectedly
stable to changes in initial conditions of the injected electrons. Unlike the previous
understanding of the formation of branches, which only needed a classical explanation,
this branch stability required a quantum mechanical explanation. Compared to a
quantum mechanical electron, a classical electron scatters differently off a potential
scattering site that is sharp (compared to the electron wavelength \F) but weak
(compared to the electron energy EF).
The classical electron is only scattered by
small angles, whereas the quantum electron, while mostly scattered at small angles,
has some flux reflected at much larger angles due to s-wave-like scattering. This
difference between classical and quantum mechanics has important consequences for
propagation through a small-angle scattering potential, filled with bumps and dips
that are short but comparable in width to XF. This difference can contribute to the
stability we reported experimentally and in our simulations. Stability of branches
has important consequences for any application which relies on ballistic steering of
electrons. For example by using a gate, it may not be easy to bend an electron branch
smoothly through a range of angles to reach a detector (i.e. another QPC). On the
other hand, the amount of electron flux reaching a detector may be stable to small
perturbations of the system or injection conditions.
SGM images exhibit interference fringes, which can be used to probe both disorder
and e-e scattering. In Chapter 2 we presented two Mechanisms that can cause these
fringes. Mechanism # 1 is analogous to a Michelson interferometer and is due to
the presence of impurities in the sample. Mechanism # 2 is analogous to a FabryPerot interferometer, due to electrons reflecting multiple times between the QPC and
SGM tip. Although previously it was understood that both Mechanisms could cause
interference fringes, in practice all previous SGM images exhibited fringes throughout
due to Mechanism # 1 .
In Chapter 3, we showed that by imaging higher mobility samples with a low
density of impurities, Mechanism # 1 contributed minimally or not at all. There
were rare or no interference fringes due to Mechanism # 1 , giving us a method to
92
CHAPTER 6. CONCLUSIONS AND FUTURE
DIRECTIONS
estimate the density of hard scatterers in the 2DEG. This may prove a useful technique
for determining whether unintentional impurities limit the mobility in the highest
mobility 2DEGs; this knowledge could lead to the growth of even less disordered
samples.
In Chapter 4, we showed that we could recover interference fringes in the same
high mobility samples by imaging at lower temperatures (350 mK instead of 4.2 K);
these were due to Mechanism # 2 . We demonstrated new spatial interference patterns
(checkerboard patterns) different from those previously observed. We were also able
to explain how several factors affected the interferometer, including temperature,
interferometer length (controlled by both moving the tip and the QPC gate voltage
Vg), electron wavelength (controlled with the finite bias Voc across the sample), and
QPC reflectivity (controlled with Vg). We showed that these interference fringes could
also be used to measure dephasing (controlled by increasing the e-e scattering rate by
increasing
|VDC|)-
This method of measuring dephasing complements other methods
because it can be used at a variety of distances all within the same sample. As
will be discussed further, interference fringes may prove useful in studying electron
interactions in nanostructures.
In Chapter 5, we studied e-e scattering by measuring changes in the momentum
of injected electrons. We controlled the e-e scattering rate with
|VDC|)
we
|VDC|-
At lower
found evidence for e-e scattering at only small-angles, a consequence of
the confined phase space for scattering in 2D. At high \VDC\, we found a surprising
result: the differential conductance through the system G was increased by moving
the SGM tip into the electron flow and backscattering electrons. We found that
this effect could be explained by a localized region of highly non-equilibrium 2DEG
near the injection point. We explained our data in terms of an effective temperature
model. Our experimental measurements of the size of this non-equilibrium region of
2DEG, as well as the e-e scattering rate within it, may guide theoretical work on the
non-trivial problem of how the distribution of electrons evolves when injected from a
QPC out of equilibrium.
Not discussed in this thesis, we have also collaborated with Martino Poggio and
Dan Rugar at IBM Almaden to use QPCs as sensitive detectors of the motion of
6.2. FUTURE EXPERIMENTS:
NANOSTRUCTURES
93
cantilevers [27]. Because of their conductance's high sensitivity to electrostatic potential, QPCs have been used as detectors of nearby charge in nanostructures [95].
We used a QPC to measure the thermal mechanical motion of a cantilever at 4.2 K.
The QPC detector exhibited a displacement resolution of 10~12 m/Hz1^2,
comparable
to the resolution achieved by optical interferometry on cantilevers of a similar size.
By optimizing the geometry and 2DEG used, the QPC detector should be capable
of higher sensitivities, and in principle should be able to reach the quantum limit
on displacement detection [96, 97]. The QPC detector is also advantageous over traditional optical detection because it should not heat the cantilever significantly (a
problem even for low power lasers at temperatures below 1 K [98, 99]) and it can
measure oscillators smaller than the optical diffraction limit. Furthermore, because
the QPC is fabricated separately from the cantilever, it can be used with a variety
of cantilevers, specially designed to respond to different stimuli. Thus, while this
thesis has focused on using a scanning probe tip and cantilever to study electron flow
emitted from a QPC into a 2DEG, it is also possible to use the QPC to study the
motion of cantilevers and other oscillators.
6.2
Future Experiments: Nanostructures
While this thesis has focused on using SGM to understand electron propagation in
2DEGs, there is considerable interest in harnessing SGM to probe electron organization in nanostructures defined within 2DEG materials. Some of the most complex
and unpredictable physics occurs when electrons are forced to reside within confined
geometries.
One example discussed in Chapter 4 was the "0.7 structure" of a QPC [79]. The
QPC is one of the simplest nanostructures one can fabricate; in a QPC, electrons
are forced to move from one region of 2DEG to another through a short, narrow
quasi-ID constriction. The experimentally observed conductance plateaus at integer
multiples of the conductance quanta 2e2/h are easily understood in terms of single
particle quantum mechanics; each integer multiple of 2e2/h comes from a different
(spin-degenerate) ID mode that corresponds to a different wavefunction along the
94
CHAPTER 6. CONCLUSIONS AND FUTURE
DIRECTIONS
confining direction of the QPC. However, there is still not full consensus as to the
source of a conductance feature at 0.7 x 2e2/h.
Transport measurements indicate
that spin and many-particle physics are involved, but with such a simple structure,
there are a limited number of transport measurements to perform. SGM may prove
a useful probe of such a system in a number of ways.
First, there have been recent theoretical proposals to study interactions in nanostructures using fringes in SGM images [80]; these fringes are due to Friedel oscillations induced by the SGM tip changing the electron density in the nanostructure.
While the fringes studied in Chapter 4 are due to interference effects and not interactions, full understanding of interference fringes will be necessary before distinguishing
interaction-caused fringes. Second, SGM can distinguish different spatial patterns due
to different sources of conductance. For example, as studied previously, the different
modes responsible for conductance at integers of 2e2/h have different spatial flow patterns [12]. Our preliminary experiments do not reveal any obviously different spatial
flow patterns when the conductance is increased from 0.7 x 2e2/h to 2e2/h, though
subtle differences may be present. A more thorough exploration of the temperature
dependence of the phenomenon is required because the 0.7 x 2e2/h feature rises in conductance towards 2e2/h at lower temperatures. Furthermore, the interference fringes
we observe at lower temperatures give us a way of distinguishing between different
spatial flow patterns that may overlap or reside very close in space. Locally gating
one part of the flow pattern may suppress or enhance the "0.7" features and give
further information about its mechanism. The low temperatures anu temperature
control described in Chapter 4 (350 mK to 1.7 K) will be necessary in studying the
"0.7 structure".
As discussed in Chapter 2, there are also efforts to use SGM to image electronic
wavefunctions inside quantum dots (QDs). QDs are often referred to as "artificial
atoms" because electrons are localized inside an attractive potential and the number
of electrons inside the QD can be varied with gates. However, successively added
electrons to a QD rarely seem to fill in a manner predicted by single particle quantum mechanics: filling opposite spins of lower energy orbital states first and then
filling higher energy spatial states. Instead, because of Coulomb interactions, often
6.3. FUTURE EXPERIMENTS:
NEW
MATERIALS
95
successively added electrons have the same spin, indicating filling of different spatial states and more efficient avoidance of each other. Many-electron states may
even form. SGM may be able to provide a useful spatial picture of the electronic
wavefunctions-squared inside a QD. SGM images might elucidate how electrons organize in such nanostructures; such images could provide exciting physical maps of
many-electron states and how they change as more electrons are added. As discussed
in Chapter 2, the main obstacle to these experiments is that the size of the potential
perturbation from the SGM tip is of a comparable size or larger than the features of
electronic wavefunction being measured. We have collaborated with other members
of the Stanford-IBM Center for Probing the Nanoscale (including Nahid Harjee and
Beth Pruitt), a National Science Foundation Nanoscale Science and Engineering Center, to produce coaxial SGM tips. Such tips should produce much narrower potential
perturbations at the 2DEG, and they may enable studies of nanostructures.
6.3
Future Experiments: New Materials
Our SGM images of electron flow in GaAs-based 2DEGs have given us a great deal of
information about factors impacting electron transport. We are interested in pursuing
similar studies in other materials with different properties. While no other material
has achieved the same record high-mobilities of GaAs 2DEGs, they can have other
distinct properties.
GaN 2DEGs host electrons with higher effective mass and higher g-factor, both
of which should contribute to stronger electron interactions. GaN-based QPCs have
been fabricated and studied in the Goldhaber-Gordon laboratory [100]; these QPCs
also exhibited "0.7 structure". Imaging electron flow out of the QPC may reveal
different behavior from that of GaAs. GaN 2DEGs suffer from significantly lower
mobilities (often on the order of a few 105 cm2/Vs),
which appear to be dominated
by hard scattering from defects. Images of electron flow in GaN may allow us to
study trajectories when hard scattering is extremely important as well as allowing us
to better understand the disorder in this material.
96
CHAPTER 6. CONCLUSIONS AND FUTURE
DIRECTIONS
There is also interest in using SGM to study 2DEGs with high spin-orbit coupling, such as those defined in InGaAs or HgTe heterostructures. Such materials
might be used for spintronic applications. In conjunction with some of these proposals to separate electrons with different spins [101], SGM may be able to visualize
different spatial flow patterns with different spin polarizations. Furthermore, because
of spin-orbit coupling, different spin electrons can have the same velocity but different wavenumber k. Use of fringes in SGM images of electron flow might be able to
resolve different k. HgTe has been used to study the quantum spin Hall effect [102]
and presents opportunities to investigate topological states. SGM might be used to
probe edge states and scattering between them in the quantum spin Hall regime.
As discussed in Chapter 2, SGM may also be used to image electron flow in
graphene, as well as studying nanostructures within this novel material. Unlike other
heterostructures described previously, graphene is a 2D honeycomb lattice of carbon,
a single atomic layer of graphite. Electrons in this material have a unique band structure, with a linear dispersion relation and no bandgap; electrons act like relativistic
Dirac fermions [61, 62]. Because of these unique properties and analogies to other
fields of physics, graphene has received a great deal of attention recently. Using SGM
to image electron flow, we could learn how electrons spatially move through this material. Electron propagation should be quite different in graphene because there is
non-trivial angular dependence to scattering [103, 104]. Like other scanning probe
studies, SGM may also provide useful information about the disorder in graphene
and how to reduce it. Furthermore, there has been interest in fabricating nanostructures in graphene which may be studied by SGM. Graphene nanoribbons have been
fabricated and studied in the Goldhaber-Gordon laboratory; transport measurements
indicate that quantum dots (QDs) formed due to disorder play an important role
[105]. By capacitively coupling to the QDs with SGM, it may be possible to locate
different QDs within graphene and study their energetics.
In the last two cases of high spin-orbit coupling materials and graphene, the SGM
technique to image electron flow may need to be modified. As described in Chapter 2,
in the standard imaging technique we used a QPC to inject electrons as well as detect
6.3. FUTURE EXPERIMENTS:
NEW
MATERIALS
97
those directly backscattered by the tip (i.e. by an angle of 7r). However, in high spinorbit materials and graphene, complete backscattering is expected to be suppressed.
However, suppression of backscattering means that the standard SGM technique may
not function properly because there will be no signal from backscattered electrons
from the tip. The SGM imaging technique can be modified to include two QPCs, one
functioning as an electron injector and the other as a detector. The detector changes
signal (voltage) based on the amount of electron flux from the injector scattered by
the SGM tip, but not scattered by an angle of ir. The optimal geometry of the two
QPCs depends on the expected angular dependence of scattering in the material.
Spatially testing the angular dependence of scattering is another interesting aspect
of these materials to probe with SGM.
In this thesis, we have demonstrated the use of SGM as an important tool for
understanding electron propagation in high-mobility GaAs 2DEGs. It may prove
uniquely powerful for studying electrons in nanostructures and other exotic materials.
Appendix A
SGM Construction and Procedures
In this Appendix, we provide experimental details on the construction of the scanning
gate microscope (SGM) as well as more information about its operation. Many details
on similar systems can be found in the following references: [77, 106, 107].
A.l
Positioning
Fig. A.l shows a photograph of the SGM. The sample is at the bottom of the image
and is situated facing up. The apparatus above the sample (attocubes and piezotube)
is used to position the tip above the sample. As discussed below, the piezotube
performs fine scale positioning. All images presented were obtained by raster scanning
the tip with the piezotube. The attocubes perform coarse scale positioning, allowing
us to move the piezotube within reach of the QPC.
The piezotube (EBL # 4 piezoceramic tube) is purchased from Staveley (now
Olympus, www.olympus-ims.com) and operates by the piezoelectric effect. Application of a large voltage to one portion of the piezotube with respect to another causes
the tube to bend. There are 5 electrodes covering the hollow tube: 4 quadrants along
the outside (X+,X-,Y+,Y-) and 1 coating the inside (Z). In practice when we apply
a voltage to an X or Y quadrant, we apply the opposite voltage to the opposite quadrant; that is, if we apply +100 V to the X+ quadrant, our electronics automatically
apply —100 V to the X- quadrant. Thus, from our computer controls, we specify
98
A.l.
POSITIONING
99
Attocubes
Piezotube
5 cm
Sample
Figure A.l: Scanning Gate Microscope. The sample is normally positioned directly
beneath the tip. The wires connecting to the sample pass through the thin metal tube
on the left portion of the image. The piezotube and attocubes position the tip over
the sample. The piezotube performs fine scale positioning. The attocubes perform
coarse scale positioning, taking discrete steps along 3 axes. The attocubes allow us
to find a device on the sample surface.
100
APPENDIX A. SGM CONSTRUCTION
AND
PROCEDURES
3 voltages (X,Y,Z), which define coordinates. At low-temperature, the tube bends
about 24 nm/1 V, applied to the X or Y quadrants (and opposite voltage applied to
the opposite quadrant). For the safety of the wiring on the probe on which the SGM
is mounted, we limit the voltage applied to any wire to about ±200 V. Thus, our
total scan range (the window over which we can image) at low-temperature is around
10 um. At room-temperature, this scan range is about 3 times larger. Increasing
the voltage on the Z electrode extends the piezotube, bringing the tip closer to the
sample at about 2.8 nm/1 V at low-temperature. As discussed below, the resolution
with which the tip can be placed above the sample is limited by mechanical vibrations
of the system. These vibrations cause the tip to fluctuate typically by about 5 nm
horizontally.
The attocubes allow discrete steps in 3 axes of motion. The attocubes are purchased from Attocube Systems (www.attocube.com). The X and Y axes are ANPxlOlLT positioners, and the Z axis is a ANPzlOl-LT positioner. The attocubes are based
on a slip-stick positioning mechanism. A triangular voltage waveform is applied to a
piezoelectric actuator. A slow rise time of the voltage causes the piezoelectric actuator to move a platform (the "stick" phase), and a very fast drop in voltage causes
the piezoelectric actuator to slip past the platform (the "slip" phase). Thus, for each
application of a triangular pulse, the platform moves along the piezoelectric actuator by some step size. A step in the opposite direction can be accomplished by
reversing the waveform in time: first applying a fast rise in voltage and then a slow
drop. The length of the slow rise time component of the waveform is not important,
but for slippage to occur, the fast drop must occur quickly. The ANC150 attocube
controller purchased from Attocube allows the height of the voltage pulse to be set
(between 30 V and 70 V) while providing a fast drop time (~ 1 fis). For slippage,
the slew rate (in V/s) actually reaching the attocube piezoelectric actuator must be
large enough. The RC (low-pass) filtering along the wires to the attocubes must allow
through high enough frequencies. Because of the relatively high capacitance of the
attocubes (~ 200 nF at low-temperature), Attocube recommends less than 5 Vt total
resistance along the wires running to the positioners. As discussed below, we had to
modify some of the wiring on the probe to meet this low resistance requirement. At
A.l.
POSITIONING
101
low-temperature, along the X and Y axes, the typical step size is about 60 nm for a
voltage pulse with height 50 V. The range of motion of the attocubes is limited by
the physical constraints of the outer can that covers the SGM (not pictured in Fig.
A.l); this limits motion in the X and Y directions to 1 — 2 mm.
The SGM is mounted on the end of a probe which loads into top-loading Oxford
cryostats (www.oxford-instruments.com).
For the first experiments we performed,
we loaded the probe into a storage dewar of liquid 4He at 4.2 K. Later, we loaded
the probe into a 3He cryostat (Heliox 3 TL) with a base temperature of 350 mK.
When loaded in the 3He cryostat, the SGM and sample sit in liquid 3He. The hold
time of the cryostat (time at base temperature before re-condensing the 3He) is about
48 hours, which is significantly less than that of a differently-wired non-SGM probe.
As discussed below, there are copper wires (some of which are litz wire purchased
from MWS Wire Industries, www.mwswire.com) from the SGM to connectors directly
above (located closer to the 1 K pot in the cryostat). Replacing some of the copper
wires with resistive wires could significantly increase the hold time. On the other
side of these connectors, the probe was wired by Oxford with 4 looms (referred to
by their Customer number) with 24 wires each. The wires are resistive (~ 160 Vt at
low-temperature or room-temperature), but on one loom (Customer 4), there are 6
wires superconducting along part of their length and copper the rest (total of ~ 5 Vt
at low-temperature and ~ 50 Q at room-temperature). The total resistance along a
pair of superconducting wires, connected to both terminals of an attocube, is thus
too high for proper stepping of the attocubes. As depicted in Fig. A.2, we connected
several wires in parallel to meet the low resistance requirements.
Fig. A.2 shows the numbering of the different wires on the loom used to position
and measure the tip. The figure shows the connector located just above the SGM,
looking down on the plug side of the connector. As discussed below, there are 4
wires which measure the deflection of the tip and apply a voltage to it (Bias+, Bias-,
Sense+, and Sense-). There are 5 wires that position the piezotube (Piezo X+, Piezo
X-, Piezo Y+, Piezo Y-, and Piezo Z). At room-temperature some wires are grounded
(Gnd) and some floating (Fit, i.e. unconnected). All these wires, except those used
for the attocubes, run the length of the probe as a single resistive wire.
102
APPENDIX A. SGM CONSTRUCTION
AND
PROCEDURES
For the attocube wiring, we used the superconducting wires (red in Fig. A.2) for
low resistance. We joined the negative terminals all together (Atto-) at the connector
depicted to further lower the resistance. In the diagram, the left portion represents
the connector at low-temperature and the right portion represents the top of the
probe at room-temperature. We denote the displacer on the probe. To lower the
resistance even further, we connected the negative attocube line to a copper portion
of the probe just above the displacer; this electrical connection is carried through the
displacer by 24 wires from Customer 2. Thus, the attocube negative line is also the
probe ground, but this can be separated from the sample ground if necessary. The
resistance along the attocube negative line thus ends up being about 1 Q. For each of
the 3 positive attocube lines (Atto X+, Atto Y+, and Atto Z+), in parallel with each
superconducting wire, we connected 8 wires from Customer 1. The resistance of each
attocube positive line thus ends up being about 4 0 at low-temperature and 15 0
at room-temperature. The total resistance is low enough for successful operation at
either low-temperature or room-temperature using the 50 V setting on the attocube
electronics. Although the total resistance at room-temperature is larger than the
5 Q's recommended by Attocube, the positioners still function because of the larger
piezo-response of the actuators (i.e. more distance moved for a given voltage applied).
However, the resistance is close to being too large.
We originally used a different set of attocubes made out of macor. When cooling
these older, macor attocubes straight in liquid helium, we sometimes had difficulty
moving them. Often they would not step initially after cooling down, and even if they
stepped, sometimes they would not move past a certain point. In order to resolve this
issue, we added mass to each positioner with a thin piece of brass. Most Attocube
positioners are made from titanium, but we originally chose macor because it does
not superconduct at low-temperature (removing concerns about applying a magnetic
field) and the same-sized positioners are less massive (raising resonant frequencies of
the assembly and making it less prone to vibrations). However, because of the reduced
mass, the macor attocubes most likely had difficulty in the slip phase of motion.
Adding mass to the attocubes mostly resolved the problem, although sometimes
using up to 70 V on the attocube electronics was necessary. When cooling the SGM
A.l.
POSITIONING
103
I Fit
Fit 2
3 Bias+
Bias- 4
5 Sense*
Sense- 6
7 Gnd
Gnd 8
9 PiezoX+
Fit 10
I I Fit
PiezoY+ 12
13 Piezo XFit 14
15 Fit
PiezoY- 16
Copper rod & probe
Displacer
1
17 PiezoZ
Fit 18
•
19 Atto-
AttoY+ 20
Ni
24 wires from Customer 2
r
}
1
1
1
21 Atto-
AttoX+ 22
• ^
23 Atto-
AttoZ+ 24
7^
/
8 wires from Customer 1
8 wires from Customer 1
7
7^
8 wires from Customer 1
Box connected
to Customer 4
Figure A.2: Positioner Wiring. The Cinch connector for tip, piezotube, and attocube
wires. This connector is located just above the SGM in a machined piece of G-10
fiberglass; it is cooled to low-temperature along with the SGM. The schematic is
denoted looking down on the plug side of the connector. There are 4 tip pins (Bias+,
Bias-, Sense+, and Sense-), 5 piezotube pins (Piezo X+, Piezo X-, Piezo Y+, Piezo
Y-, and Piezo Z), and 6 attocube pins (Atto X+, Atto Y+, Atto Z+, and 3 Atto-).
The other wires are grounded at room-temperature (Gnd) or left unconnected (Fit).
All the lines not explicitly drawn (1-18) are single resistive wires along the length
of the probe on the Customer 4 loom. The lines denoted in red (19-24) are the
superconducting wires on Customer 4. These are connected in parallel to resistive
wires from other looms (Customers 1 and 2). The negative attocube line (Atto-) is
connected to the probe just above the displacer. The connections at the right-most
portion of image are made at the top of the probe which resides at room-temperature.
104
APPENDIX A. SGM CONSTRUCTION
AND
PROCEDURES
down in the 3He cryostat, where the sample is cooled in a vacuum environment, we
never experienced problems moving the macor attocubes with mass added. However,
we still occasionally had problems with the old, macor attocubes when cooling the
SGM down in liquid 4He, where the sample is cooled in a gaseous He environment.
In this case, residual gases may form small amounts of ice which impede the attocube
motion. Thus, sometimes while cooling the SGM in liquid helium, we stepped the
attocubes periodically (every few seconds) to prevent ice formation. This generally
helped the attocubes step smoothly at low-temperature. We also sometimes used
homebuilt electronics to provide a triangular voltage waveform with a larger voltage
peak (200 V) and a faster voltage rise/drop (< 1 [is). These homebuilt electronics
used several 9 V batteries in series and gold pins connected by hand (clearly while
holding onto an insulating portion of the wires) to make very rapid electrical connection for the fast rise/drop of the waveform. However, this approach was cumbersome
because each attocube step had to be applied by hand. As discussed further below,
the large voltage spikes could capacitively couple into sample wires and alter the
behavior of a QPC on a cool-down. For these reasons, we largely avoided using the
homebuilt attocube electronics. We later switched to the titanium attocubes pictured
and discussed above. These attocubes were much stiffer, having higher resonant frequencies and reducing the amount of environmental vibrations affecting our system.
As of the writing of this thesis, we operate the SGM solely in the 3He cryostat, and
the titanium attocubes step properly.
A.2
Sample
The sample layout is shown in Fig. A.3. Devices are fabricated within a 500 fim x
500 fxm "scanning area". More details on fabrication steps can be found at the end
of this Appendix. In contrast to typical sample designs, all the gates move towards
just two edges of the chip, so that all bonding pads are just on two sides. This allows
the tip to approach from the other two sides safely without hitting any wirebonds.
On the right portion of the image, we schematically denote the approximate size and
orientation of the tip and cantilever chip. The tip protrudes perpendicularly from a
A.2.
SAMPLE
105
piezoresistive cantilever, which extends from the edge of a chip. This cantilever chip
is attached to a mount, which plugs into the end of the piezotube with pins from
Mill-Max (www.mill-max.com). The cantilever is close to parallel with the sample
surface; the cantilever is angled by 5° so that the tip is the closest portion of the
cantilever chip to the sample. We can also adjust the angle of the sample surface
because the sample is supported by spring-like pogo pins (purchased from Everett
Charles Technologies, www.ectinfo.com), as discussed below. Tightening screws on
the three corners of the sample carrier adjusts the angle of the sample surface so that
corners of the cantilever chip do not hit the surface before the tip does.
The "scanning area" includes topographic landmarks that allow us to determine
the location of the tip above the sample. That is, we use the tip as an atomic force
microscope (AFM) to measure the height of topographic features, and each landmark
is several pixels that designate a coordinate (x,y). We then step the attocubes so
that the scan range of the piezotube reaches the QPC. It should be noted that when
operating as an SGM, the tip is not in contact with the surface, and operation as
an AFM is necessary to locate the tip with respect to the QPC. The topographic
landmarks are separated by 10 fim because this is the piezotube scan range. The
topographic landmarks are fabricated at the same time as the QPC gates during
the electron-beam lithography step. Thus, we align the tip to the sample at roomtemperature by looking from the side with a microscope as the tip approaches the
sample. We make the "scanning area" 500 [im x 500 \im so that the area is large
enough that we can align the tip to it optically at room-temperature. We measure the
coordinate above which the tip is located at room-temperature (and make sure it is
indeed above the "scanning area"). After cooling the system down and re-approaching
the surface, the tip typically shifts coordinates by a few 10's of /urn's. We then step
the attocubes to move the tip towards a QPC to be studied.
We note that before cooling the system down, we retract the tip substantially
from the sample surface with the z-axis attocube. This protects the sample and
tip from each other during thermal contraction. However, we ensure that the z-axis
attocube is not fully retracted so that two surfaces do not freeze together during cooldown. To approach the surface, we extend the piezotube, checking for tip deflection
106
APPENDIX A. SGM CONSTRUCTION
AND
PROCEDURES
Figure A.3: Sample Layout. The GaAs chip is black, and surface metal is bright.
The chip is mounted on a square metallic bonding pad on an alumina sample carrier.
In principle this square bonding pad could act as a backgate, but for our samples it is
very far from the 2DEG. There are thin wirebonds from the sample chip to the bond
bands on the sample carrier. These wirebonds must all leave the sample chip from
two sides so that the tip can approach safely from the opposite sides. The general size
and orientation of the cantilever chip is depicted on the right. The tip is normally
positioned above the scanning area. Using topographic landmarks in the scanning
area, the attocubes are used to step the piezotube to within its scan range of a QPC.
that indicates the surface is contacted. If the surface is not found, we retract the
piezotube, step the z-axis attocube towards the sample, and then repeat the process.
For this procedure to safely find the surface with the piezotube and not step into it
with the attocube, the z range used for the piezotube must be greater than the z-axis
attocube step size. So that the procedure operates quickly, the z range used for the
piezotube should not be too much greater than the z-axis attocube step size.
Fig. A.4 shows the wiring for the sample. The sample wires run along the loom
Customer 3. Fig. A.4a denotes the numbering of wires on the plug side of the connector. These sample wires are connected to pogo pins upon which the alumina sample
carrier (purchased from Ceramag, www.netsv.com/ceramag) is held with screws. Fig.
A.4b depicts the torlon piece (purchased from Synergetix, www.synergetix.com) which
A.3.
ELECTRONICS
107
houses these pogo pins, looking down on the circular tor Ion piece from the sample
side. The three small circles arranged at the vertices of an equilateral triangle represent the tapped holes into which screws insert from above and hold the sample carrier
in place. The four small circles arranged at the vertices of a square represent the clear
holes through which screws insert from below to hold the torlon piece into the rest of
the SGM. Fig. A.4d shows the bottom of the sample carrier which is pressed against
the pogo pins. Notice that the wire numbers in Fig. A.4d are the mirror image of
those in Fig. A.4b so that the bottom of the pads on the sample carrier rest on the
correct pogo pins. We found that that the pogo pins generally made good electrical
contact to the sample carrier, but occasionally the position of the sample carrier had
to be adjusted to make electrical contact to all pogo pins. Fig. A.4c shows the other
side of the sample carrier, the side on which the sample is mounted. Wirebonds are
connected from the sample to the bondpads on the sample carrier. Thus, Fig. A.4c
shows how the metal bond pads wrap around the sides of the sample carrier from the
bottom depicted in Fig. A.4d.
A. 3
Electronics
We control the position of the tip with the piezotube. In order to determine the
location of the tip (both far away from the QPC when we are looking for topographic
landmarks, and close to the QPC when we are looking for the QPC gates themselves),
we use the system as an AFM. As we scan the tip across the surface, this involves
measuring the deflection on the tip and using feedback methods to adjust the height
of the tip in order to keep the deflection constant. As discussed below, we have two
homebuilt feedback electronics boxes which take input (such as deflection) and setpoint signals and adjust an output (such as tip height) until the input matches the
set-point.
We use the attocubes to position the tip near the QPC. We do not scan the tip on
the surface near the QPC or anywhere interesting flow may occur because touching
the tip to the surface can alter flow patterns measured afterwards. Instead, we find
the QPC by using the system as an SGM. We scan the tip high above the surface
108
APPENDIX A. SGM CONSTRUCTION
AND
PROCEDURES
Figure A.4: Sample Wiring, (a) The Cinch connector for sample wires, located just
above the SGM. This figure is depicted looking down on the plug side of the connector,
(b) The torlon piece which houses the sample-supporting pogo pins. This figure is
depicted looking down on the torlon piece, from the sample carrier side, (c) The top
of the sample carrier, the side on which the sample is mounted and wirebonded to
the metal pads, (d) The bottom of the sample carrier, which is mechanically pressed
against the pogo pins, making electrical contact.
A.3.
ELECTRONICS
109
(~ 400 nm) with a voltage applied to the tip and the QPC conductance set roughly
to a half-integer of conductance quanta (where its conductance is very sensitive to
changes in electrostatic potential). The location of the QPC can be easily determined
because the tip affects the QPC conductance G most strongly when the tip is directly
above the QPC. From previous AFM scans, we know the plane of the surface. We
can more precisely locate the QPC by scanning the tip above the surface at a height
where it will not touch the surface but still graze the QPC gates (~ 40 nm off the
surface; feedback can also be engaged so that if the tip touches a gate, it retracts
so that not too much deflection is endured). We can then use the system to image
electron flow as discussed in Chapter 2 where the tip is scanned ~ 30 nm above the
surface. Application of a negative voltage to the tip creates a depletion disk in the
2DEG below.
There is another effect which must be accounted for when imaging electron flow.
Not only does the tip create a depletion disk directly below it but it also couples
to the QPC with capacitance Ct. A negative voltage on the tip lowers G. We can
compensate for the tip voltage by increasing the gate voltage on the QPC, but as
the tip moves, Ct changes. Ct mainly depends on the distance between the tip and
QPC. Without changing the tip voltage or gate voltage, a scan and measurement of
G(x, y) shows features that are roughly circular due to partial depletion at the QPC
by the tip. To account for this changing Ct, an effect in which we are not interested,
we make what we call a "voltage guide". This "voltage guide" changes the QPC gate
voltage based on the tip position to cancel out any changing Ct effect. To make this
"voltage guide", we first scan the tip ~ 80 nm above the surface, in the same area
as the desired flow scan. We apply a voltage to the tip that creates a depletion disk
in the 2DEG when the tip is 30 nm above the surface, but with the tip 80 nm above
the surface, there is no depletion disk in the 2DEG below. However, Ct is nearly
the same whether the tip is 30 nm or 80 nm above the surface because the tip is
typically at least 100's of nm away from the QPC. That is, with the tip 80 nm above
the surface, we make the approximation that there is no depletion disk necessary for
imaging electron flow, but Ct is the same. During this scan 80 nm above the surface,
we use feedback to adjust the QPC gate voltage to keep G constant. We record this
110
APPENDIX A. SGM CONSTRUCTION
AND
PROCEDURES
feedback gate voltage as the "voltage guide". Then during the scan at 30 nm above
the surface, we adjust the gate voltage according to the "voltage guide" so that the
main effect of the tip is to introduce a depletion disk.
This "voltage guide" works best at distances farther from the QPC where Ct does
not change rapidly and where it is a better approximation to say that Ct does not
change if the tip is moved from 80 nm to 30 nm above the surface. At warmer
temperatures (4.2 K) the QPC conductance plateaus are not very flat, and it is
possible to feedback on G to make it an integer multiple of conductance quanta. We
make a single "voltage guide" for the G at which we wish to image electron flow. At
lower temperatures, because the QPC conductance plateaus flatten, it is not desirable
to use feedback to keep G constant on a conductance plateau because there is a large
range of gate voltages over which G is constant. Instead, to image electron flow at
an integer multiple of conductance quanta, we make two "voltage guides" for G at
half-integers of conductance quanta, above and below the desired integer value, and
then average them.
Fig. A.5 shows the main experimental inputs and outputs. We use two National Instruments cards (6733 and 6052E, www.ni.com) for digital-to-analog (output) and analog-to-digital (input) conversion. In addition, we control two Yokogawa (tmi.yokogawa.com) 7651 programmable DC voltage sources via GPIB (IEEE488) from the computer. The computer outputs include the piezotube coordinates
("X","Y", and "Z"). The voltage on the tip ("V tip ") is set either by the computer
or one of the Yokogawas. The computer outputs a set-point for the tip deflection
signal ("DeflFBSet"), which is used by feedback electronics to keep the tip in contact with the surface during AFM scans. The computer also outputs a set-point for
the QPC conductance ("GqpcFBSet"), which is used by the feedback electronics to
make the "voltage guide". We use two voltages to set the gate voltage on the QPC
("Vgfine" from the computer and "Vgcoarse" from a Yokogawa). Because a change of
the least significant bit in the computer output on the QPC gate voltage can produce
noticeable changes in the conductance, we divide "Vgfine" by 10 and then add it
to "Vgcoarse" (and then multiply the sum by 0.39). This division on the computer
output means that single bit changes do not produce noticeable changes in the QPC
A.3.
ELECTRONICS
111
Outputs
Comp.OutO - * - "X"
Comp.Outl - • "Y"
Comp.Out2 - » - "Z"
Comp.Out3
Comp.Out4 -*• "DeflFBSet"
Comp.Out5 - • - "Vgfine"
Comp.Out6 —*- "GqpcFBSet"
Comp.Out7 — "v DC "
- "V
Inputs
"ZFB" -+ Comp.InO
"Def!" -+ Comp.ln1
"Gqpc" —*• Comp.ln2
"V g FB»-~ Comp.ln3
Comp.DOutO -** "DeflFBOn"
Comp.DOutl —*• "GqpcFBOn"
JGPIB
YokogawaO —*• "V coarse"
Yokogawal ^ " V
T
"
Figure A.5: Experiment Inputs and Outputs. The position of the piezotube is output
with "X", "Y", and "Z". The tip voltage is output "V tip ". The DC injection voltage
across the QPC is "V D c" • The QPC gate voltage is the combination of "Vgfine" and
"Vgcoarse". The digital outputs "DeflFBOn" and "GqpcFBOn" engage the deflection
and conductance feedback boxes, respectively. Their set-points are "DeflFBSet" and
"GqpcSet". We measure the tip deflection "Defl" and QPC conductance "Gqpc". The
outputs from the deflection and conductance feedback boxes are input, respectively,
as "ZFB" and "V„FB".
conductance. Also outputted from the computer are the finite DC injection voltage
("VDC")
an
d digital outputs that turn on the feedback boxes for deflection and
conductance ("DeflFBOn" and "GqpcFBOn").
The computer reads as an input the tip deflection ("Defl"). Also input to the
computer is the output of the deflection feedback box ("ZFB"), the voltage applied to
Piezo Z to keep the tip deflection constant. The conductance of the QPC ("Gqpc")
is read. Finally, the computer reads the output of the conductance feedback box
("V g FB"), the gate voltage applied to keep G constant during the creation of a
"voltage guide".
In Fig. A.6, we show two useful circuits: an isolator and adder. The isolator
circuit is used to prevent ground loops forming when two outputs with a common
ground are both connected to the experiment. The op-amps running this circuit
112
APPENDIX A. SGM CONSTRUCTION
AND
PROCEDURES
(a)
r^VVV
vIN+ — A A / V i
v0UTVA + VB
v,N--rAAAH
M/W
VouT*
R = 25kn
Figure A.6: Isolator and Adder Circuits, (a) Isolator circuit used to prevent ground
loops. Separate pairs of batteries power each channel, so that the circuit ground is
different from the experimental ground until connected, (b) Adder circuit used to
add DC and AC excitations that measure the QPC conductance.
(and all of our homebuilt electronics) are powered by Yuasa 12 V lead acid batteries
(which can be purchased from many electronics vendors, such as Allied), so that the
ground of the circuit floats with respect to experimental ground until the two are
connected. We homebuilt a box with 8 isolator channels (each powered by different
pairs of batteries), and in future figures, we denote connections through an isolator
channel with the buffer symbol (triangle without +'s or —'s). We use the convention
t h a t a voltage uiiierellce is V J N
V
IN
IN-
-«.s uiscUsseu ueluW, tiie a u u e r IS USeu
to create the voltage that measures the QPC conductance: it adds the DC injection
voltage from the computer with an oscillating voltage from the lock-in amplifier.
Fig.
A.7 shows how we apply a tip voltage and measure the tip deflection.
The tip and piezoresistive cantilever are purchased from Veeco (PLNC cantilever,
www.veecoprobes.com, although no longer sold). We evaporate metal onto the tip
so that it conducts at low-temperature. During evaporation, we use a shadow mask
to protect most of the cantilever from metal (which could short the two sides of the
piezoresistor). We angle the cantilever so that it is about 35° off from being flat
compared to the evaporation source. We rotate the cantilever during the evaporation
A.3.
ELECTRONICS
113
to coat all sides of the tip. On a crystal monitor at about the same distance from the
source as the cantilever, we evaporate 10 nm of Cr and 5 nm of Au.
The tip and cantilever are sensitive to deflection due to the piezoresistive effect.
The cantilever has a resistance typically of ~ 2.5 kfl, and at low-temperature, we
find deflecting the tip increases the resistance by about 6.5 mVt/nm. To measure this
small change in resistance, we use a Wheatstone bridge, depicted in Fig. A.7a. The
other resistors in the bridge are chosen to have similar resistance to the cantilever,
and they are located just above the SGM (at the very top of Fig. A.l, covered in
white teflon tape). We apply a bias of —200 mV across the bridge. Application
of larger voltages (—500 mV) can raise the temperature of the 3He cryostat due to
resistive heating. We measure the voltage difference Sense in the bridge. As depicted
in Fig. A.7b, this Sense voltage is multiplied by 100,000 by two cascaded voltage
PARI 13 amplifiers. This Sense signal multiplied by 100,000 is the deflection signal.
When the tip is deflected, the cantilever increases resistance slightly, increasing the
Sense+ voltage, and increasing Sense and the deflection signal. The deflection signal
is viewed on an oscilloscope. It can also be converted to an audio frequency and
listened to on speakers. The deflection signal is also read by the computer; it is
filtered by a low-pass filter with a time constant of 140 /is to prevent false positives
when detecting the surface on the approach procedure. The deflection signal is also
used by the deflection feedback circuitry to be discussed.
The PAR113's must be cascaded because Sense is non-zero due to slightly different
resistances in the Wheatstone bridge. There is a maximum amplification possible for
the first PARI 13 so that it does not rail. One input to the second PARI 13 is adjusted
with a potentiometer and battery so that the input voltage difference is again close
to zero and can be amplified again. Thus, the actual value of Defl is meaningless, but
we are interested in relative changes. That is, we measure Defl when we know the tip
is un-deflected and then increases in Defl indicate deflection of the tip.
The voltage on the tip is set by Vt;p, coming from either the computer directly
or a Yokogawa, as depicted in Fig. A.7c. Vt;p is the voltage of the Bias- lead of the
Wheatstone bridge with respect to ground. Thus, the voltage on the tip is actually
around V tip - 150 mV.
The -150 mV comes from the bias across the bridge; the
114
APPENDIX A. SGM CONSTRUCTION
AND
PROCEDURES
(b)
Bias = -200 mV
5 Sense+
Comp.Out3 -
OR
Yokogawal •
scope
Comp.ln1
DeflFBIn
(to feedback)
Potentiometer Box for
balancing PAR
Figure A.7: Tip Circuitry, (a) Wheatstone bridge for detecting deflection of the
piezoresistive cantilever. V tip sets the voltage of the bridge with respect to ground,
(b) Amplification circuit for measuring the deflection signal, (c) If VtiP is output
from the computer, the voltage runs through an isolator channel. Alternatively, we
sometimes use a Yokogawa to generate V t i p .
voltage on the tip is roughly halfway between the voltage at Bias+ and the voltage
at Sense+. When scanning the tip over gates, VtiP can be adjusted so that the actual
voltage on the tip is closer to 0 V. It is wise to design the Wheatstone bridge in a
manner such that momentary grounding of the tip (due to intermittent contact with
a grounded gate) causes Defl to move in the same direction as actual deflection of
the tip (i.e. the tip pushing into the surface); this will cause the feedback circuitry
to react by retracting the tip rather than extending it (which may make even better
electrical contact to a grounded gate). For SGM imaging of electron flow, we adjust
Vtip until we have a strong signal of electron flow for the height at which we scan;
generally Vtip is around —2 V or —3 V.
A.3.
ELECTRONICS
115
Fig. A.8 shows the circuitry used to position the piezotube. We use four homebuilt
electronics boxes: XYZ, XY Correction, Feedback, and Feedback Control. More detail
on the actual construction of similar boxes can be found in Ref.'s [40, 77]. We also
use a commercially purchased high-voltage amplifier, an RHK HVA 900 (www.rhktech.com). The XYZ box takes X and Y inputs, and, using simple op-amp circuits,
the X and Y signals are buffered and also inverted, thus creating four voltages which
control the four quadrants of the piezotube. The XYZ box also adds the output of
the feedback box to the computer-output Z voltage. The XY Correction box accounts
for a non-ideality of the piezotube. Normally without the XY Correction box, as the
Z-voltage on the piezotube changes, the (X,Y) location of the tip shifts as well. The
XYZ Correction box takes as inputs (X,Y,Z) and outputs a corrected (X,Y) based on
the z-value. This correction is accomplished by adding in a fraction of Z to X and
Y; the fraction and polarity of the addition are controlled with potentiometers. The
RHK is a high-voltage amplifier that multiplies input voltages by 45. Before voltages
are input to the RHK, they pass through diode protection which limits the voltage to
roughly ±5.5 V and are then multiplied by 0.8 (accomplished by resistive dividing).
Thus, the outputs from the RHK are limited to about ±200 V; the voltages actually
applied to the piezotube are 36 times the outputs from the computer.
As discussed previously, the Feedback box takes input and set-point signals and
adjusts an output until the input matches the set-point. During AFM scans, we
set the set-point so that the tip deflects by a few nm's and is held in contact with
the surface. The feedback is accomplished through a simple integrator circuit that
outputs the integral of the error signal (difference between the input and set-point).
The polarity of the integration must be chosen properly so that the feedback changes
the output in the correct direction. The time constant of the integration can also be
changed by switching between internal resistors and capacitors. The time constant
should be small enough so that the feedback can react on the same time scale as
changes occur in the system. However, if the feedback time constant is made too
short, the feedback electronics respond faster than the system does and ringing can
occur. The Feedback Control box engages the Feedback box into the system. That is,
during SGM scans and many other times, we do not want feedback engaged and we
116
APPENDIX A. SGM CONSTRUCTION
AND
PROCEDURES
want full control over the tip height. The Feedback Control box determines whether to
add the output of the Feedback box to an actual experimental input based on a digital
signal (DControl). The output of Feedback Control is the sum of two inputs (Inl and
In2) and, if the DControl is high, a third input (Controlln). Thus, if DControl is
high, then the Feedback box output is passed through the Feedback Control box and
added into the Z voltage sent to the piezotube. If DControl is low, the Feedback box
output is not passed through the Feedback Control box and is not added into the Z
voltage sent to the piezotube. Thus, when DControl is low, because the Feedback
box output has no control over the system, the Feedback box's output will rail. An
important part of the Feedback Control box is to engage the feedback output slowly
enough so that a sudden jolt is not applied to the experimental setup (in this case,
the Z voltage). We want the feedback signal to change appropriately as the system
responds while feedback is engaged slowly. This slow engagement is accomplished in
the Feedback Control box by applying a voltage to a field effect transistor through a
resistor and capacitor, with appropriately long time constant (~ 50 ms).
In Fig. A.9 we show how we apply a voltage to the ohmic contacts of a sample
in order to make a conductance measurement. We use standard lock-in techniques to
measure the differential conductance. The lock-in amplifier is a PAR124A, and the
current pre-amplifier is an Ithaco model 1211. We add the oscillating (AC) reference
signal from the lock-in to "VDC" from the computer. This sum is divided by 1000
through a resistive divider, and the resulting AC + DC voltage is applied to an ohmic
contact of the sample. The current driven is amplified by the Ithaco and measured
by the lock-in; the lock-in is only sensitive to an input oscillating at the frequency of
its output oscillation.
The circuitry used to control the QPC gate voltage is shown in Fig. A.10. We
denote the output of two isolator channels with explicit + and - wires. The gate
voltage is the addition of a voltage from the computer ("Vgfine") and a coarse voltage
from a Yokogawa ("Vgcoarse"). "Vgfine" is divided by 10, then added to "Vgcoarse".
The sum is multiplied by 0.39 and diode protection is applied so that it is not possible
to apply a positive voltage to the gates much greater than about ~ 100 mV; a larger
positive voltage can cause permanent damage. The Feedback and Feedback Control
A.3.
ELECTRONICS
117
Xin
Yin
Zin
Computer
Xout
Yout
RTl = diode protection to ±5.5V
x0.8
Comp.OutO
Comp.Outl
Comp.Out2
XYZ
RHK
X
x+
Y
XY+
Y-
zi —| Y+ I—[-ran
Z2-
X
Y- —-TaH-
Z3—(t)— Z - * - G D -
Feedback
X+
x45
Z
XY+
Y-
,
10kQ
1
/ /A A A / — -
'5
(t° Piezotube)
Z
Feedback Control
Defl
Comp.Out4
("DeflFBSet")
Controlln-y—©DControl -----
Comp.DOutO
("DeflFBOn")
• Comp.InO
("ZFB")
Figure A.8: Piezotube Circuitry. In this and following figures, we use the convention
that crossed wires are not connected unless explicitly denoted with a circle over the
connection. The XYZ box buffers and inverts the X and Y inputs. It also adds
together a Z input from the computer and Feedback Control. The XY Correction
box changes X and Y based on Z to account for skew in the piezotube. The Feedback
box changes the Z voltage of the piezotube so that the deflection signal matches a
set-point (corresponding to a few nm's of tip deflection). The Feedback Control box
engages feedback into the system based on the digital control DeflFBOn.
Adder
Comp.OutT
To sample ohmics
("VDCO
Lockin. Signal
(~50mVAC)
Figure A.9: Conductance Measurement Circuitry. The Adder box adds the DC
finite injection voltage from the computer and the oscillating signal from the lock-in
amplifier. This resulting AC + DC voltage is divided by 1000 and drives a current
through the sample. The Ithaco current pre-amplifier measures the current driven
through the sample, and the lock-in amplifier outputs a signal proportional to the
oscillating component of the current.
APPENDIX A. SGM CONSTRUCTION
118
Comp.Out5
("V/ine")
AND
PROCEDURES
Comp.ln2
"Gqpc")
Lockin.Out
Comp.Out6 ("GqpcFBSet")
Comp.DOutl ("GqpcFBOn")
YokogawaO +
("V coarse") -
Controlln-/-©DControl- ---•
-Comp.ln3
("VaFB")
_ To sample
Figure A. 10: Gate Circuitry. The gate voltage is the combination of feedback output,
a computer voltage ("Vgfine"), and a Yokogawa voltage ("Vgcoarse"). The Feedback
box changes its output based on the conductance (measured from the lock-in) and
the set-point from the computer. The Feedback Control box engages the feedback
output into the circuit. "Vgfine" and, if feedback is engaged, the feedback output are
divided by 10 and added to the Yokogawa voltage. The sum is multiplied by 0.39,
passes through diode protection to prevent accidental application of large positive
voltages, and then is applied to the QPC gates.
boxes are the same type of electronics described previously for deflection feedback.
We divide the voltage from the computer so that changes of a single bit do not cause
noticeable changes in the QPC conductance. We are also able to change the voltage
applied from the computer much more quickly than the voltage from the Yokogawa
because the Yokogawa is slowly controlled through GPIB. Thus, for changing the gate
voltage based on the "voltage guide", as described above, we change "Vgfme".
A. 4
Vibrations
Environmental vibrations limit the resolution with which the tip can be positioned
over the sample. In our systems we have generally achieved position resolution of
about 5 nm. This position resolution is sufficient because it is considerably smaller
A.4.
VIBRATIONS
119
than the feature sizes in which we are interested (about half the Fermi wavelength,
~ 25 nm). When operating the SGM in a liquid helium dewar, we buried the dewar
in about 2000 lbs of sand in a box. The box rested on large springs, giving the massive
system a resonant frequency of about 1 — 2 Hz (corresponding to the simple up-down
mode). We also placed foam underneath the box to act as a vibration dampener. The
low resonant frequency helped isolate the system from ground vibrations. To remove
acoustic vibrations, we then closed the top of the system with a lead-lined wooden
box and coated the inside with 1 inch thick Sonex acoustic foam (MSO-1 panels,
www.sonexfoam.com; purchased from West General, www.westgeneral.com). The soft
and heavy lead blocked a good deal of outside sound while the acoustic foam helped
absorb it. We have since moved the SGM to operate in a 3 He cryostat. The cryostat
dewar is hanging in a pit from a granite slab which weighs about 1400 lbs. During the
experiment, the granite slab floats on air pistons (purchased from TMC Corporation,
www.techmfg.com). We placed six fiberglass panels inside the pit around the dewar
to dampen low-frequency acoustic noise. We also filled the pit with polyester-fill
(commonly used in stuffed animals and pillows) to dampen higher frequency acoustic
noise. Currently, the dominant frequency of vibrations is around 30 Hz, which we
believe come from acoustic resonances of the shielded room in which the cryostat is
located.
Often times vibrations can become problematic for our experiments because either
something about the SGM or environment changes. It is possible to measure vibrations by placing the tip on the surface (a feature with a slope is best for measuring
horizontal vibrations) and recording the deflection signal. A more sensitive method
to measure vibrations is to use the QPC as an electrical position detector. We bias
the QPC conductance to a half-integer of conductance quanta, where its conductance
is very sensitive to changes in electrostatic potential. We position a charged tip near
the QPC and vibrations of the tip change the effective gating on the QPC, resulting
in changes in the conductance. The QPC conductance can thus act as a measurement
of the tip position; this is the effect we used in Ref. [27] to sensitively measure the
deflection of a cantilever with a QPC. With a measurement of deflection versus time
120
APPENDIX
A. SGM CONSTRUCTION
AND
PROCEDURES
(and a Fourier transform of this trace), it is easy to determine which frequency vibrations are causing the most problems. Generally, the frequency of vibrations provides
many clues as to the source. Geophones can be used to track down vibrations in
the environment. When the SGM has larger vibrations than usual, it is important
to determine whether something inside the SGM is resonating due to background vibration noise (for example, objects inside are touching that should not be), whether
environmental vibrations are being transmitted into the SGM more efficiently (for
example, the vibration isolation is not working properly), or whether environmental
vibrations are worse (for example, a pump in lab is operating in a different location).
A.5
Sample Behavior Due to Cooling and Attocubes
We have observed that for some samples, especially those with high-mobility, cooling
the sample slowly results in more stable QPC traces. That is, if we take a trace
of conductance versus gate voltage several times, ideally in "well-behaved" samples
these traces are identical. However, often times there is an undesirable small amount
of drift in these curves. If we cool a sample slowly, we find the QPC traces are
more "well-behaved". The drift in QPC traces comes from charge fluctuations near
the QPC. Charge fluctuations make imaging electron flow difficult, especially if the
tip changes the electrostatic potential below. In principle, cooling the sample slowly
allows the donor charges to find a lower energy configuration, which should be more
stable. In practice, we cool our samples from room-temperature to a few K over at
least two hours.
We have found that stepping the attocubes can cause more drift in QPC traces
afterwards (until thermally cycling the sample). This problem is due to the highfrequency, high-voltage waveforms applied to the attocubes capacitively coupling into
the sample lines. We initially intended to solve this attocube-sample-wire problem
with a "grounding 2DEG", which would ground the sample wires to each other well
during attocube steps. However, we later avoided problems of QPC drift by slowly
cooling the samples and limiting the attocube step voltage to 50 V. Mentioned here
in case future samples also prove to be sensitive to attocube steps, the concept of
A.6.
IMAGE PROCESSING
AND DRIFT
CORRECTION
121
the "grounding 2DEG" was to ground all sample wires to each other well at lowtemperature. All sample wires would have a parallel connection to ohmic contacts on
another high-mobility "grounding 2DEG". During attocube steps, the sample wires
would all be grounded to each other through the "grounding 2DEG"; brief voltage
differences would not develop between sample wires. Then for normal sample operation, an energized gate on the "grounding 2DEG" would remove electrical connections
between all the ohmic contacts, allowing each sample wire to behave independently.
A.6
Image Processing and Drift Correction
Some of our experimental images of electron flow have been processed to remove
sudden changes in QPC differential conductance G that are unrelated to electron
flow. These sudden changes in G are presumably due to background charge noise
that changes G through the QPC, but we observe that the spatial flow pattern does
not change after a shift in G. As can still be observed in some of our images, these
sudden changes show up as horizontal lines (along the fast-scan direction) that have
an average conductance different from neighboring scan lines.
In Chapter 3 when we take several images for different injection conditions (Fig.
3.5), we correct for piezotube drift. After performing a scan over many /im's, due to
drift the piezotube does not return to the same location as before the scan, resulting
in the next scan being slightly offset. After each scan, we take a calibration scan
over a small area that is nominally identical (with the same QPC gate voltages). We
compare the calibration scans to determine the piezotube drift between each image
for a different injection condition. Compared to Fig. 3.5a, we shift Fig.3.5b-d by
0 nm, 23 ram, and 21 nm, respectively.
A.7
Determining t h e Tip-to-QPC Distance
We briefly describe here how we determine the distance L from the tip to the QPC.
We know the topographic distance between tip and the QPC gates from a distance
calibration of our tip positioning system, the piezotube. However, we need to know
APPENDIX A. SGM CONSTRUCTION
122
AND
PROCEDURES
the location of the depletion regions underneath the tip and gates. For the gates, we
use SETE (described in Chapter 3), which suggests the depletion region underneath
the gates is about ~ 600 nm wide. In Chapter 4, we also use SETE to estimate
the locations on the gate depletion regions at which electrons reflect; the reflection
path between the tip and gates should be perpendicular to the edge of the depletion
region. The reflection point off the QPC gates only changes significantly when the
tip is moved close to the QPC and the angle of reflection changes. The size of the
depletion region underneath the tip is dependent on the distance from the tip to the
2DEG. This distance is limited by the distance from the surface of the sample to
the 2DEG, 100 nm in Sample C (see Fig. 2.1), and the size of the depletion disk
is mainly dependent on the surface-to-2DEG distance because of the significantly
higher dielectric constant in GaAs compared to vacuum (or 3He) [77]. As discussed
in Chapter 2, the potential perturbation in a 2DEG due to a point charge a distance
z above scales as l/(r 2 + z2)3^2 where r is the distance along the x-y plane. Thus, for
z = 100 nm and a potential perturbation that we estimate roughly is l.hEp at r = 0
(it of course depends on Vup), we therefore expect a depletion disk 110 nm in diameter.
That is, from (z2)3/21.5EF/(rlepletion
+ z2)3/2
= EF, we derive rdepietion = 55 nm. We
estimate the uncertainty in L is ±50 nm based on our uncertainty in the size of
the tip-induced depletion disk, the location of the QPC depletion regions, and the
reflection points. There may also be some error in the calibration of the piezotube
on the order of ±2%, which does not matter for small distances, but can become the
dominant source of error at large distances (L > 2 »m).
A.8
Fabrication Steps
Here we summarize the steps taken for sample fabrication.
Overall Process
1. Mesa Etch (Photolithography)
2. Ohmics (Photolithography)
A.8.
FABRICATION
STEPS
123
3. Fine-featured Gates (Electron-beam lithography)
4. Course-featured Gates (Photolithography)
Photolithography
1. Resist application.
(a) Spin S1813 photoresist on chip: using the spinner in the sub-micron room
at 4,000 rpm's and spinning for 30 — 40 seconds gives a resist layer that
is 1.0 — 1.1 \im thick.
(b) Bake the resist onto the chip at 115° C for 2 minutes. Overheating or
letting the resist bake for too long will make the resist difficult or impossible
to lift-off later.
2. UV exposure with Karl Suss mask aligner.
(a) Test the UV intensity with the hand-held meter. Given that it is between
16 and 17, use an exposure time of 3 seconds. Overexposing the photoresist
will result in "overcut" (the opposite of undercut) resist profiles, making
lift-off much more difficult.
(b) When loading the mask, press down on the left side while tightening the
screws that hold the mask into place; this helps keep the mask more level.
(c) Pressing the chip gently into contact with the mask helps achieve better
alignment because the chip will not move as the UV lamp moves forward
during exposure. If good alignment is not necessary (mesa etch and ohmic
steps), it will suffice to leave the chip slightly below the mask: look from
the side for the slightest gap of light on all sides. This keeps the mask and
chip clean from each other, and there is no chance of cracking the chip.
Since bringing the chip into contact with the mask too hard can crack the
chip, 4 pieces of electrical tape underneath the chip will cushion it well so
that it is much more difficult to crack.
124
APPENDIX
A. SGM CONSTRUCTION
AND
PROCEDURES
(d) During exposure, resist the movement forward of the UV lamp with a
hand on a sturdy part so that it does not come into place as forcefully,
thus jostling the chip less.
3. Develop photoresist.
(a) Move the chip around gently in CD30 developer for 90 seconds, changing
the sides of the chip held by tweezers.
(b) Rinse the chip in water, sonicate it for 10 seconds (if there are no gates
yet), and then dry.
4. Evaporate Metal or Etch. For ohmics and coarse-featured gates, before evaporation swish the chip in NH4OH for 5 seconds and then dry. This removes
the oxide layer, so do this with as little delay as possible before loading into an
evaporator.
5. Lift-off.
(a) Let the chip soak in acetone for a few minutes. If all goes well, everything
intended to lift-off should come up easily with a little swishing in the beaker
or spritzing with a spray bottle. Check under a microscope while still in
acetone that lift-off is complete because once the chip is dried, it may be
difficult to properly lift-off afterwards.
Sonication to promote lift-off is
highly discouraged because this can remove desired features, especially if
they are tall and thin.
(b) Rinse in isopropanol (IPA) and dry.
Electron-beam (E-beam) Lithography
1. Resist application.
(a) Spin 2% 950k PMMA on the chip using the spinner in the fume hood in
Moore 089. Make sure to turn on the orange knob for nitrogen supply
A.8.
FABRICATION
STEPS
125
to the hood. Appropriate parameters to use are: Accel Time 0.5; Spin
Time 26; Speed 5000; Deccel Time 2.0. Make sure to press cancel after
each spin or the spinner will not start the next time start is pressed. In
order to avoid letting the resist dry on the chip in an uneven fashion, after
dropping the PMMA onto the chip, start the spinner as fast as possible.
This requires having one hand on the start button as the other drops
PMMA. The resulting PMMA should be about 90 nm thick.
(b) Bake the resist on the hotplate at 180° C for 2 minutes.
2. E-beam exposure with Raith.
3. Develop resist.
(a) Rinse in IPA.
(b) Move the chip around gently in a solution mixed with the volume ratio 1
MIBK (methyl isobutyl ketone) : 3 IPA for 45 seconds, changing the sides
of the chip held by tweezers.
(c) Rinse in IPA and dry.
4. Evaporate metal.
5. Lift-off: same as photolithography lift-off.
Mesa Etch
1. H2O rinse.
2. Etch with 240 H20 : 8 H 2 0 2 : 1 H 2 S0 4 . This achieves about a 2 k/s etch rate,
but the rate should always be tested on a junk chip whenever a new batch is
mixed. The heights can be measured with the profilometer in Ginzton. Etching
down to the 2DEG will ensure there is no more 2DEG there. Be especially
careful about etching too far if the 2DEG is grown on a conducting substrate.
It also may become more difficult to run gates up over the side of the mesa if
126
APPENDIX A. SGM CONSTRUCTION
AND
PROCEDURES
the etch is too deep. Etching roughly 100 - 200 nm is generally a generically
good target.
3. H20 rinse.
4. H20 rinse and sonicate.
Ohmics
1. Use the Innotec evaporator in CIS or another evaporator that can handle three
magnetic sources. Evaporate as fast as possible in order to minimize heating of
the chip. For Au, 5 — 8 A/s is possible. For Ge, be careful not to heat or cool
the source too quickly. In order of deposition:
• 50 A Ni
• 800 A Ge
• 1600 A Au
• 100 A Ni
• 400 A Au
2. Using the Ginzton rapid thermal annealer, anneal at 430° C for 30 seconds.
Cool the chip after the anneal by flowing gas at a high rate (about 90% of the
flow on the meter) until it cools to 250° C. It may also be possible to allow
more temperature uniformity in the chamber by flowing forming gas a little bit
slower while its heating up past 250° C (only 10 — 20% of the flow on the
meter).
Fine-featured Gates. Use the thermal evaporator to deposit 50 A Cr and then 150
A Au. Alternatively, use the e-beam evaporator to deposit 50 A Ti and then 150
A Au.
A.8.
FABRICATION
STEPS
127
Coarse-Featured Gates. Use the thermal evaporator and angled rotator to deposit
the sticking layer (Cr) and first bit of gold at a 50° angle, so that the gates run up
the side of the mesa. This technique was used when a citric acid etch was used and
the mesa was possibly undercut; it is probably not as important for a sulfuric acid
etch like now recommended.
1. 150 A Cr (angled): evaporate at 1 A/s (amounts and rates quoted as read on
crystal monitor).
2. 150 A Au (angled): evaporate at dial setting ~ 39 to get rate of ~ 5 A/s.
3. Wait 20 minutes. This prevents overheating of the resist and makes lift-off
easier. These waiting periods may not be necessary in an e-beam evaporator.
The sample gets much warmer in the thermal evaporator where it is exposed to
a large, bright, very hot object.
4. 500 A Au (not angled).
5. Wait 20 minutes.
6. 500 A Au (not angled).
7. Wait 20 minutes.
Other Notes
1. Never sonicate in a glass beaker; this can crack the chip if it is touching the
side. Use a polypropylene beaker instead. Glass beakers may be useful when
heating the sample during lift-off or when etching.
2. Always dry the chip well with nitrogen gas after being in liquid (the only thing
ever to be dried off should be IPA, water, or NH4OH). Water requires more
128
APPENDIX A. SGM CONSTRUCTION
AND
PROCEDURES
thorough drying: move the chip around more, lift it up some to dry off underneath, and rotate where the tweezers grip the chip to remove the water between
them and the chip.
Bibliography
[1] von Klitzing, K., Dorda, G. & Pepper, M. New method for high-accuracy
determination of the fine-structure constant based on quantized Hall resistance.
Physical Review Letters 45, 494-497 (1980).
[2] Tsui, D. C , Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Physical Review Letters 48, 1559 - 1562
(1982).
[3] Nielsen, M. A. Sz Chuang, I. L. Quantum Computation and Quantum
Informa-
tion (Cambridge University Press, 2000).
[4] Wolf, S. A. et al. Spintronics: A spin-based electronics vision for the future.
Science 294, 1488-1495 (2001).
[5] Zutic, I., Fabian, J. & Sarma, S. D. Spintronics: Fundamentals and applications.
Reviews of Modern Physics 76, 323-410 (2004).
[6] Willett, R. L., Hsu, J. W. P., Natelson, D., West, K. W. & Pfeiffer, L. N.
Anisotropic disorder in high-mobility 2d heterostructures and its correlation to
electron transport. Physical Review Letters 87, 126803 (2001).
[7] Lilly, M. P., Cooper, K. B., Eisenstein, J. P., Pfeiffer, L. N. & West, K. W.
Evidence for an anisotropic state of two-dimensional electrons in high Landau
levels. Physical Review Letters 82, 394 - 397 (1999).
[8] Mani, R. G. et al. Zero-resistance states induced by electromagnetic-wave excitation in GaAs/AlGaAs heterostructures. Nature 420, 646 - 650 (2002).
129
130
BIBLIOGRAPHY
[9] Sohn, L. L., Schon, G. & Kouwenhoven, L. P. (eds.) Mesoscopic Electron Transport. No. 345 in NATO ASI Series E (Kluwer, 1997).
[10] van der Wiel, W. G. et al. Electron transport through double quantum dots.
Reviews of Modern Physics 75, 1 - 2 2 (2003).
[11] Petta, J. R. et al. Coherent manipulation of coupled electron spins in semiconductor quantum dots. Science 309, 2180 - 2184 (2005).
[12] Topinka, M. A. et al. Imaging coherent electron flow from a quantum point
contact. Science 289, 2323 - 6 (2000).
[13] Topinka, M. A. et al. Coherent branched flow in a two-dimensional electron
gas. Nature 410, 183 - 6 (2001).
[14] Altshuler, B. L., Aronov, A. G. & Khmelnitsky, D. E.
Effect of electron-
electron collisions with small energy transfers on quantum localisation. Journal
of Physics C: Solid State Physics 15, 7367 (1982).
[15] Chaplik, A. V. Energy spectrum and electron scattering processes in inversion
layers. Soviet Journal of Experimental and Theoretical Physics 33, 997 (1971).
[16] Giuliani, G. F. & Quinn, J. J. Lifetime of a quasiparticle in a two-dimensional
electron gas. Physical Review B 26, 4421 - 8 (1982).
[17] Yacoby, A., Sivan, U., Umbach, C. P. & Hong, J. M. Interference and dephasing
by electron-electron interaction on length scales shorter than the elastic mean
free path. Physical Review Letters 66, 1938 - 41 (1991).
[18] Miiller, F. et al. Electron-electron interaction in ballistic electron beams. Physical Review B 51, 5099-105 (1995).
[19] Coleridge, P. T. Small-angle scattering in 2-dimensional electron gases. Physical
Review B 44, 3793 - 3801 (1991).
BIBLIOGRAPHY
131
[20] Webb, R. A., Washburn, S., Umbach, C. P. & Laibowitz, R. B. Observation of
h/e Aharonov-Bohm oscillations in normal-metal rings. Physical Review Letters
54, 2 6 9 6 - 9 (1985).
[21] Lee, P. A. & T. V. Ramakrishnan, T. V. Disordered electronic systems. Reviews
of Modern Physics 57, 287-337 (1985).
[22] Lee, P. A. & Stone, A. D. Universal conductance fluctuations in metals. Physical
Review Letters 55, 1622 - 5 (1985).
[23] Stern, A., Aharonov, Y. & Imry, Y. Phase uncertainty and loss of interference:
A general picture. Physical Review A 41, 3436-3448 (1990).
[24] Jura, M. P., Topinka, M. A., Urban, L., Yazdani, A., Shtrikman, H., Pfeiffer,
L. N., West, K. W. & Goldhaber-Gordon, D. Unexpected features of branched
flow through high-mobility two-dimensional electron gases. Nature Physics 3,
841 - 5 (2007).
[25] Jura, M. P., Topinka, M. A., Grobis, M., Pfeiffer, L. N., West, K. W. &
Goldhaber-Gordon, D. Electron interferometer formed with a scanning probe
tip and quantum point contact. Physical Review B (to be published).
[26] Jura, M. P., Grobis, M., Topinka, M. A., Pfeiffer, L. N., West, K. W. k
Goldhaber-Gordon, D. Spatially probed electron-electron scattering in a twodimensional electron gas. In preparation.
[27] Poggio, M., Jura, M. P., Degen, C. L., Topinka, M. A., Mamin, H. J.,
Goldhaber-Gordon, D. & Rugar, D. An off-board quantum point contact as
a sensitive detector of cantilever motion. Nature Physics 4, 635 - 8 (2008).
[28] Loss, D. & DiVincenzo, D. P. Quantum computation with quantum dots. Physical Review A 57, 120 - 126 (1998).
[29] Kitaev, A. Y.
Fault-tolerant quantum computation by anyons.
Physics 303, 2 - 30 (2003).
Annals of
132
BIBLIOGRAPHY
[30] Buks, E., Heiblum, M. & Shtrikman, H. Correlated charged donors and strong
mobility enhancement in a two-dimensional electron gas. Physical Review B
49, 14790-14793 (1994).
[31] Topinka, M. A., LeRoy, B. J., Westervelt, R. M., Maranowski, K. D. & Gossard,
A. C Imaging coherent electron wave flow in a two-dimensional electron gas.
Physica E 12, 678 - 683 (2002).
[32] LeRoy, B. J., Topinka, M. A., Westervelt, R. M., Maranowski, K. D. & Gossard,
A. C. Imaging electron density in a two-dimensional electron gas. Applied
Physics Letters 80, 4431 - 3 (2002).
[33] LeRoy, B. J. et al. Imaging coherent electron flow. Physics of Semiconductors
2002 169 - 176 (2003).
[34] LeRoy, B . J . Imaging coherent electron flow. Journal of Physics: Condensed
Matter 15, R1835 - R1863 (2003).
[35] LeRoy, B. J. et al. Imaging electron interferometer. Physical Review Letters
94, 126801 (2005).
[36] Topinka, M. A., Westervelt, R. M. & Heller, E. J. Imaging electron flow. Physics
Today 5 6 , 4 7 - 5 2 (2003).
[37] Aidala, K. E. et al. Imaging magnetic focusing of coherent electron waves.
Nature Physics 3, 464-468 (2007).
[38] Wharam, D. A. et al. One-dimensional transport and the quantisation of the
ballistic resistance. Journal of Physics C: Solid State Physics 21, L209 - L214
(1988).
[39] van Wees, B. J. et al. Quantized conductance of point contacts in a twodimensional electron gas. Physical Review Letters 60, 848 - 50 (1988).
[40] Eriksson, M. A. et al. Cryogenic scanning probe characterization of semiconductor nanostructures. Applied Physics Letters 69, 671 - 3 (1996).
BIBLIOGRAPHY
133
[41] Heller, E. J. & Shaw, S. Branching and fringing in microstructure electron flow.
International Journal of Modern Physics B 17, 3977 - 3987 (2003).
[42] Shaw, S. E. J. Ph.D. thesis, Harvard University (2002).
[43] Heller, E. J. et al. Thermal averages in a quantum point contact with a single
coherent wave packet. Nano Letters 5, 1285 - 92 (2005).
[44] Crook, R., Smith, C. G., Simmons, M. Y. & Ritchie, D. A. Imaging cyclotron
orbits and scattering sites in a high-mobility two-dimensional electron gas. Physical Review B 62, 5174 - 5178 (2000).
[45] Crook, R., Smith, C. G., Barnes, C. H. W., Simmons, M. Y. & Ritchie, D. A.
Imaging diffraction-limited electronic collimation from a non-equilibrium onedimensional ballistic constriction. Journal of Physics: Condensed Matter 12,
L167-L172 (2000).
[46] Aoki, N., Cunha, C. R. D., Akis, R., Ferry, D. K. & Ochiai, Y. Scanning
gate microscopy investigations on an InGaAs quantum point contact. Applied
Physics Letters 87, 223501 (2005).
[47] da Cunha, C. R. et al. Imaging of quantum interference patterns within a
quantum point contact. Applied Physics Letters 89, 242109 (2006).
[48] Crook, R. et al. Conductance quantization at a half-integer plateau in a symmetric GaAs quantum wire. Science 312, 1359-1362 (2006).
[49] Crook, R. et al. Imaging fractal conductance fluctuations and scarred wave
functions in a quantum billiard. Physical Review Letters 91, 246803 (2003).
[50] Pioda, A. et al. Spatially resolved manipulation of single electrons in quantum
dots using a scanned probe. Physical Review Letters 93, 216801 (2004).
[51] Fallahi, P. et al. Imaging a single-electron quantum dot. Nano Letters 5, 223226 (2005).
134
BIBLIOGRAPHY
[52] Gildemeister, A. E. et al. Measurement of the tip-induced potential in scanning
gate experiments. Physical Review B 75, 195338 (2007).
[53] Woodside, M. T. & McEuen, P. L. Scanned probe imaging of single-electron
charge states in nanotube quantum dots. Science 296, 1098-1101 (2002).
[54] Bleszynski, A. C. et al. Scanned probe imaging of quantum dots inside InAs
nanowires. Nano Letters 7, 2559-2562 (2007).
[55] Martins, F. et al. Imaging electron wave functions inside open quantum rings.
Physical Review Letters 99, 136807 (2007).
[56] Hackens, B. et al. Imaging and controlling electron transport inside a quantum
ring. Nature Physics 2, 826-830 (2006).
[57] Steele, G. A., Ashoori, R. C , Pfeiffer, L. N. & West, K. W. Imaging transport
resonances in the quantum Hall effect.
Physical Review Letters 95, 136804
(2005).
[58] Hani, S. et al. The microscopic nature of localization in the quantum Hall effect.
Nature 427, 328 - 332 (2004).
[59] Martin, J. et al. Localization of fractionally charged quasi-particles. Science
Qr»er
non
noo
tO(\C\A\
[60] Martin, J. et al. Observation of electron-hole puddles in graphene using a
scanning single-electron transistor. Nature Physics 4, 144-148 (2007).
[61] Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in
graphene. Nature 438, 197-200 (2005).
[62] Zhang, Y., Tan, Y. W., Stormer, H. L. & Kim, P. Experimental observation
of quantum Hall effect and Berry's phase in graphene. Nature 438, 201-204
(2005).
BIBLIOGRAPHY
135
[63] Siegert, C , Ghosh, A., Pepper, M., Farrer, I. k Ritchie, D. A. The possibility of
an intrinsic spin lattice in high-mobility semiconductor heterostructures. Nature
Physics 3, 3 1 5 - 3 1 8 (2007).
[64] Marcus, C. M., Rimberg, A. J., Westervelt, R. M., Hopkins, R F. & Gossard,
A. C. Conductance fluctuations and chaotic scattering in ballistic microstructures. Physical Review Letters 69, 506 - 509 (1992).
[65] Nixon, J. A. & Davies, J. H. Potential fluctuations in heterostructure devices.
Physical Review B 4 1 , 7929-7932 (1990).
[66] Spector, J. et al. Ballistic electron optics. Surface Science 263, 240-246 (1992).
[67] Stopa, M. Single-mode quantum wires. Physical Review B 53, 9595-9598
(1996).
[68] Grill, R. & Dohler, G. H. Effect of charged donor correlation and Wigner
liquid formation on the transport properties of a two-dimensional electron gas
in modulation 5-doped heterojunctions. Physical Review B 59, 10769-10777
(1999).
[69] Moore, L. S. & Goldhaber-Gordon, D. Low-dimensional physics: Magnetic
lattice surprise. Nature Physics 3, 295-296 (2007).
[70] Stopa, M. Quantum dot self-consistent electronic structure and the Coulomb
blockade. Physical Review B 54, 13767 - 83 (1996).
[71] Berry, M. V. & Balazs, N. L. Evolution of semi-classical quantum states in
phase-space. Journal of Physics A: Mathematical and General 12, 625 - 642
(1979).
[72] Cerruti, N. R. & Tomsovic, S. Sensitivity of wave field evolution and manifold
stability in chaotic systems. Physical Review Letters 88, 054103 (2002).
[73] Vanicek, J. & Heller, E. J. Uniform semiclassical wave function for coherent
two-dimensional electron flow. Physical Review E 67, 016211 (2003).
136
BIBLIOGRAPHY
[74] van Wees, B. J. et al.
Observation of zero-dimensional states in a one-
dimensional electron interferometer.
Physical Review Letters 62, 2523 - 6
(1989).
[75] Katine, J. A. et al. Point contact conductance of an open resonator. Physical
Review Letters 79, 4806 - 9 (1997).
[76] Liang, W. et al. Fabry-Perot interference in a nanotube electron waveguide.
Nature 411, 665 - 9 (2001).
[77] Topinka, M. A. Ph.D. thesis, Harvard University (2002).
[78] Kouwenhoven, L. P. et al. Nonlinear conductance of quantum point contacts.
Physical Review B 39, 8040 - 3 (1989).
[79] Thomas, K. J. et al. Possible spin polarization in a one-dimensional electron
gas. Physical Review Letters 77, 135 - 8 (1996).
[80] Freyn, A., Kleftogiannis, I. & Pichard, J. L. Scanning gate microscopy of a
nanostructure where electrons interact. Physical Review Letters 100, 226802 1 (2008).
[81] Yanovsky, A. V. et al. Angle-resolved spectroscopy of electron-electron scattering in a 2D system. Europhysics Letters 56, 709-715 (2001).
[82] Predel, H. et al. Effects of electron-electron scattering on electron-beam propagation in a two-dimensional electron gas. Physical Review B 62, 2057-64
(2000).
[83] Ashcroft, N. W. & Mermin, N. D. Solid State Physics (Brooks Cole, 1976).
[84] Schapers, T. et al. Effect of electron-electron interaction on hot ballistic electron
beams. Applied Physics Letters 66, 3603-5 (1995).
[85] Sivan, U., Heiblum, M. & Umbach, C. P. Hot ballistic transport and phonon
emission in a two-dimensional electron gas. Physical Review Letters 63, 992-995
(1989).
BIBLIOGRAPHY
137
[86] Dzurak, A. S. et al. Two-dimensional electron-gas heating and phonon emission
by hot ballistic electrons. Physical Review B 45, 6309-6312 (1992).
[87] Schinner, G. J., Tranitz, H. P., Wegscheider, W., Kotthaus, J. P. & Ludwig, S.
Phonon-mediated nonequilibrium interaction between nanoscale devices. Physical Review Letters 102, 186801 (2009).
[88] Mittal, A., Wheeler, R. G., Keller, M. W., Prober, D. E. & Sacks, R. N.
Electron-phonon scattering rates in GaAs/AlGaAs 2DEG samples below 0.5
K. Surface Science 361/362, 537 (1996).
[89] Cronenwett, S. M. Ph.D. thesis, Stanford University (2001).
[90] Murphy, S. Q., Eisenstein, J. P., Pfeiffer, L. N. & West, K. W. Lifetime of twodimensional electrons measured by tunneling spectroscopy. Physical Review B
52, 14825-14828 (1995).
[91] Zheng, L. & Sarma, S. D. Coulomb scattering lifetime of a two-dimensional
electron gas. Physical Review B 53, 9964-9967 (1996).
[92] Jungwirth, T. & MacDonald, A. H. Electron-electron interactions and twodimensional-two-dimensional tunneling.
Physical Review B 53, 7403-7412
(1996).
[93] Fukuyama, H. & Abrahams, E. Inelastic scattering time in two-dimensional
disordered metals. Physical Review B 27, 5976-5980 (1983).
[94] Gurzhi, R. N., Kalinenko, A. N. & Kopeliovich, A. I. The theory of kinetic effects in two-dimensional degenerate gas of colliding electrons. Low Temperature
Physics 23, 44 (1997).
[95] Field, M. et al. Measurements of Coulomb blockade with a noninvasive voltage
probe. Physical Review Letters 70, 1311-1314 (1993).
[96] Clerk, A. A., Girvin, S. M. & Stone, A. D. Quantum-limited measurement and
information in mesoscopic detectors. Physical Review B 67, 165324 (2003).
138
BIBLIOGRAPHY
[97] Clerk, A. A. Quantum-limited position detection and amplification: A linear
response perspective. Physical Review B 70, 245306 (2004).
[98] Bleszynski-Jayich, A. C , Shanks, W. E. & Harris, J. G. E. Noise thermometry
and electron thermometry of a sample-on-cantilever system below 1 Kelvin.
Applied Physics Letters 92, 013123 (2008).
[99] Mamin, H. J. & Rugar, D. Sub-attonewton force detection at millikelvin temperatures. Applied Physics Letters 79, 3358-3360 (2001).
[100] Chou, H. T. et al. High-quality quantum point contacts in GaN/AlGaN heterostructures. Applied Physics Letters 86, 073108 (2005).
[101] Khodas, M., Shekhter, A. & Finkel'stein, A. M. Spin polarization of electrons
by nonmagnetic heterostructures: The basics of spin optics. Physical Review
Letters 92, 086602 (2004).
[102] Konig, M. et al. Quantum spin Hall insulator state in HgTe quantum wells.
Science 318, 766 (2007).
[103] Katsnelson, M. I., Novoselov, K. S. & Geim, A. K. Chiral tunnelling and the
Klein paradox in graphene. Nature Physics 2, 620-625 (2006).
[104] Stander, N., Huard, B. & Goldhaber-Gordon, D. Evidence for Klein tunneling
in graphene p-n junctions. Physical Review Letters 102, 026807 (2009).
[105] Todd, K., Chou, H. T., Amasha, S. & Goldhaber-Gordon, D. Quantum dot
behavior in graphene nanoconstrictions. Nano Letters 9, 416-421 (2009).
[106] Erikkson, M. A. Ph.D. thesis, Harvard University (1997).
[107] LeRoy, B. J. Ph.D. thesis, Harvard University (2003).