QUANTUM COHERENCE AND INTERACTIONS IN QUANTUM
DOTS
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
Ileana Georgeta Rau
August 2011
© 2011 by Ileana Georgeta Rau. All Rights Reserved.
Re-distributed by Stanford University under license with the author.
This work is licensed under a Creative Commons AttributionNoncommercial 3.0 United States License.
http://creativecommons.org/licenses/by-nc/3.0/us/
This dissertation is online at: http://purl.stanford.edu/tc600cr9197
ii
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, Primary 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.
Ian Fisher, Co-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.
Steven Kivelson
Approved for the Stanford University Committee on Graduate Studies.
Patricia J. Gumport, Vice Provost Graduate Education
This signature page was generated electronically upon submission of this dissertation in
electronic format. An original signed hard copy of the signature page is on file in
University Archives.
iii
iv
Abstract
The behavior of electrons in solid state systems is determined by the interaction
of their charge and spin degrees of freedom with each other and with the degrees
of freedom of their environment. Whether the interactions manifest themselves by
modifying some of the systems properties such as the effective mass in the Fermi
liquid picture, or more directly by suppressing transport in the Coulomb blockade
model, depends on the details and complexity of the system. This thesis investigates
two cases: the Fermi liquid behavior in a system with exchange interactions (the
spin
1
2
Kondo model) and effect of Coulomb interactions on the phase coherence of
electrons in a quantum dot with single mode leads.
The first experiment tests the Fermi liquid theory prediction of a quadratic power
law dependence of the electron scattering rate on energy in the non-equilibrium
regime. We measure transport though a lateral GaAs/AlGaAs quantum dot that
acts as an artificial magnetic impurity coupled to a single reservoir and find that
the low energy conductance obeys universal scaling with temperature and bias with
a quadratic exponent as expected for the single channel Kondo state. This single
particle picture fails when a second independent channel is added and the quantum
correlations lead to non-Fermi liquid behavior.
To understand how the short range Coulomb repulsion affects phase coherence
we measure the quantum correction due to the weak localization of electrons in a
quantum dot coupled to a reservoir via perfectly transmitting quantum point contacts.
We extract the dephasing time and observe that it continues to increase down to the
lowest temperatures, in accordance with the predictions from Fermi liquid theory and
in contradiction with previous experiments in zero-dimensional structures. When
v
the phase coherence time of the electrons becomes large enough, we observe the
effects of the long range Coulomb repulsion as Coulomb blockade emerges. This
was previously assumed to be characteristic of transport through quantum dots with
tunneling quantum point contacts (QPCs). We show that despite the fully open
QPCs of our device, the coherent backscattering of electrons at zero magnetic field
is responsible for Coulomb charging effects and estimate that the residual charge
quantization at the lowest temperature is 1/3 electron charge via charge sensing
measurements.
vi
Acknowledgements
I have been very lucky to have some amazing Physics teachers and professors over
the years. First, I would like to thank David, my PhD adviser, from whom I have
learned an incredible amount. David is an exceptional physicist and researcher, he
thinks about science in a very precise and carful way that allows him to focus on what
is essential about a concept, and explain it in such a way that is is clear, correct, and
understandable at the same time. I believe that this is the most valuable lesson that
anyone, regardless of major or occupation can teach or learn. I am very thankful for
having had David as an example and a mentor. In addition, I want to thank David
for his encouragement over the years and in particular, for his availability to talk to
me and offer advice or suggestions when I needed them.
I want to thank my undergraduate adviser, Gabe for his encouragement and support. I had never done any experiments before coming to I.W.U. and his enthusiasm
for Physics and for experiments was contagious. I will always remember the time I
spent there, in lab and in Physics class, with fondness.
I would also like to thank my high school Physics teacher, Prof. Smaranda Zaharie,
for the many, many hours spent solving Physics problems, for putting up with my
requests for which Physics topics to study and for lending me her Physics books,
which I still have. Without that time, I highly doubt that I would have become a
physics major in the first place.
For the projects in this thesis, we collaborated with several physicists. I am
grateful to Yuval Oreg for his discussions with us and for his willingness to explain
and to answer my questions, to Piet Brouwer for his help with the formulas for N = 1,
and to Hadas Shtrikman for the heterostructure used in our devices. I would also like
vii
to thank my thesis/defense committee: Mac Beasley, Ian Fisher, Steve Kivelson and
George Papanicolaou.
There are many people I met during my time as a graduate student, whose friendship and support have been extremely important. In (mostly) chronological order,
I want to thank the students in the DGG lab. Lindsay, Charis and Hung-Tao have
been the best group of senior graduate students. Their knowledge about Physics,
the lab and their willingness to help and discuss have made my first years in lab a
pleasure. It has been a lot of fun, as well as extremely interesting, and sometimes
hilarious, to work in lab and take classes with Mike J. and Joey. I have really enjoyed
being able to talk about science and about experimental details with Andrei, Markus,
Kathryn, Mark, Benjamin, Adam, Nimrod, Matt, Matthias, Francois, Michael as well
as everyone else past and present in the DGG group. I have to thank many members
of the DGG lab, as well as some members of the Moler lab, for help with Helium
transfers. In particular, I would like to thank Andrei and Markus for the entertaining
company during lunch time.
Over the course of my PhD I have worked closely with Ron Potok, Mike Grobis
and Sami Amasha. Initially, I met Ron when I was rotating in the Moler lab and I
was surprised to find out that Ron was in lab at almost all hours of the day (and
sometimes of the night). After I joined David’s lab and started working with him, I
realized why. Ron had already done a huge amount of work to get the dilution fridge
and the devices ready for the 2CK experiment. He was very knowledgeable about
quantum dot physics and very driven and I learned a lot from him over the next few
years: how to use a dilution refrigerator, how to measure quantum dots and how to
troubleshoot noise. I also learned that the dilution fridge works better if it is called
“the baby”, that chickens need to be sacrificed to the dilution fridge gods to keep the
system up and running, and that the most frustrating experiment can be fun when
you have the right lab-partner.
Mike came to the DGG lab with a rather different background than mine. His
expertise with vacuum equipment and instrument troubleshooting was extensive and
I gained very useful experience by working with him. He is very good at trying things
out, and finding a way to fix or troubleshoot something even when there is very little
viii
information available. Because of his different background, Mike had an alternative
way to think about concepts, which was very interesting and I learned a lot from
talking about physics with him.
I have been extremely lucky that Sami joined the group and decided to work on
the dilution fridge project. He is an exceptional physicist and a very considerate
and conscientious lab partner. He is very knowledgable about quantum dots and
mesoscopic physics, but also about physics in general. I always learn something new
or find a better way of understanding a topic when I talk to him about science.
He thinks about things very carefully and in detail, and is not afraid to question
assumptions or to try to find a different way to approach a problem. I have learned a
lot from him about how to decide what the possible ways to proceed with a task are
and how to identify the best option among them. I want to thank him for sharing his
knowledge and expertise with me, for entertaining random discussions about science
and for his help with everything from Igor commands to understanding a specific
topic. Finally and most importantly, Sami, thank you for your friendship and your
advice.
I would like to thank Andrei, Julie and Lisa for the fun times dancing, for the
occasional Saddlerack outing and for the interesting discussions inside and outside the
lab. I am very happy that I came to Stanford and met Naoko, Annica, Eugene, Jess,
Tony, Rich, Alex B., Katalin, Beena and Katie. Their friendship and company have
made graduate school much more fun. I would like to thank Hendrik for driving me to
lab and picking me up at the oddest hours of the night, for helping me transfer Helium
when no one else was around, for his support in my decision to come to Stanford and
for his decision to join me here. Seeing California with him was an adventure.
I especially want to thank Kiran for four memorable years as roommates, for her
friendship all through graduate school, for listening to me and encouraging me when
things were not working out and for being happy with me when things were going
well. I have also been fortunate to have Alison and Michelle as my friends for over a
decade. They are two of the smartest people I know and their advice and opinions
have been very helpful over these years. I have a special thank you for Lindsay, Ophir
and Chewy who have been my friends, welcomed me into their home and introduced
ix
me to a different world when I needed it most. I am particularly grateful to Wong
for sharing some of his calm and patience with me and for giving me a more positive
perspective on research and academia.
And finally, I would like to thank my family and friends at home: Vasilica Tolciu,
Reli Florescu and Corina Chipei, who have been my teachers and my friends, my
cousin Radu and my aunts Mariana and Marie. Having them share their life experience and advice with me provided me with wisdom and perspective that I would
not have had otherwise. My grandma is the most amazing woman I know and I am
deeply grateful for her presence in my life. Thank you to my parents who not only
supported and encouraged me all these years, but are also responsible for my interest
in Physics: my dad, from whom I learned that being careful is one of the most important qualities of an experimentalist (when I got (slightly) shocked while investigating
the electrical installation of the house), and my mom, who was the first to explain
to me what electrons and holes are, where electrical and mechanical properties of
different materials come from, or how Foucault’s pendulum can show that the earth
is rotating around its own axis.
x
Contents
Abstract
v
Acknowledgements
vii
1 Introduction
1
1.1 The Transition from Quantum to Classical Behavior . . . . . . . . . .
1
1.2 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3 Interactions with the Environment
5
. . . . . . . . . . . . . . . . . . .
1.4 The Kondo Effect: an Example of a Correlated Quantum State
. . .
9
1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2 Quantum Dots as a Model System
16
2.1 The Two Dimensional Electron Gas (2DEG) . . . . . . . . . . . . . .
17
2.2 The Quantum Dot . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.3 From a Closed to an Open Quantum Dot . . . . . . . . . . . . . . . .
25
3 Quantum dots: Coulomb Blockade
29
3.1 The Charging Energy . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.2 Coulomb Blockade Diamonds . . . . . . . . . . . . . . . . . . . . . .
34
4 Quantum Dots: the Kondo Effect
39
4.1 The Single Channel Kondo Effect in a Quantum Dot . . . . . . . . .
41
4.2 The Non-equilibrium Single Channel Kondo Effect . . . . . . . . . . .
47
4.3 Impurity Quantum Phase Transitions . . . . . . . . . . . . . . . . . .
56
xi
4.4
Tuning to the 2CK Point in a Double Quantum Dot . . . . . . . . . .
5 Quantum Dots: Perfectly Transmitting QPCs
62
67
5.1
Open Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
5.2
The Quantum Dot Conductance at N = 1 . . . . . . . . . . . . . . .
73
5.3
The Dephasing Time in a Quantum Dot at N = 1 . . . . . . . . . . .
86
5.4
Mesoscopic Coulomb Blockade in a Quantum Dot at N = 1 . . . . . .
91
Bibliography
98
xii
List of Tables
2.1 Characteristic parameters of the 2DEG . . . . . . . . . . . . . . . . .
19
2.2 Characteristic parameters of the quantum dots . . . . . . . . . . . . .
24
xiii
List of Figures
1.1
The Anderson model . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.2
The Kondo effect in bulk . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.1
The band structure of AlGaAs and GaAs . . . . . . . . . . . . . . . .
18
2.2
Band bending in AlGaAs/GaAs . . . . . . . . . . . . . . . . . . . . .
20
2.3
SEM picture of the quantum dots . . . . . . . . . . . . . . . . . . . .
24
2.4
QPC energy levels
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.5
QPC conductance quantization . . . . . . . . . . . . . . . . . . . . .
27
3.1
Quantum dot electrical circuit . . . . . . . . . . . . . . . . . . . . . .
30
3.2
Total energy and charge of the quantum dot . . . . . . . . . . . . . .
31
3.3
Transport through the quantum dot . . . . . . . . . . . . . . . . . . .
32
3.4
Measurements of Coulomb blockade peaks . . . . . . . . . . . . . . .
33
3.5
Coulomb blockade diamond diagram . . . . . . . . . . . . . . . . . .
34
3.6
Measurements of Coulomb blockade diamonds . . . . . . . . . . . . .
35
3.7
Coulomb blockade thermometry . . . . . . . . . . . . . . . . . . . . .
37
4.1
Kondo conductance in a quantum dot . . . . . . . . . . . . . . . . . .
43
4.2
Bias and temperature dependence of the Kondo conductance . . . . .
45
4.3
Temperature evolution of the Kondo bias peak and valley conductance
48
4.4
G0 and TK across the Kondo valley . . . . . . . . . . . . . . . . . . .
49
4.5
Temperature and bias scaling on the Kondo plateau . . . . . . . . . .
51
4.6
Bias exponent and coefficients on the Kondo plateau . . . . . . . . .
52
4.7
Single channel Kondo conductance scaling . . . . . . . . . . . . . . .
55
xiv
4.8 The two channel Kondo phase diagram . . . . . . . . . . . . . . . . .
59
4.9 The two channel Kondo double quantum dot system
. . . . . . . . .
63
4.10 The Kondo conductance through a three-leaded dot . . . . . . . . . .
64
4.11 Finite bias measurements of the two single channel Kondo states . . .
65
5.1 The QPC conductance plateaus . . . . . . . . . . . . . . . . . . . . .
74
5.2 The center of the N = 1 plateau . . . . . . . . . . . . . . . . . . . . .
74
5.3 Universal conductance fluctuations at N = 1 . . . . . . . . . . . . . .
76
5.4 Coulomb blockade oscillations at N = 1 . . . . . . . . . . . . . . . . .
77
5.5 The evolution of the Coulomb diamonds from N ≪ 1 to N > 1 . . . .
78
5.6 The charging energy at N = 1 . . . . . . . . . . . . . . . . . . . . . .
79
5.7 Changing the shape of the quantum dot . . . . . . . . . . . . . . . .
80
5.8 The conductance average vs. magnetic field and temperature . . . . .
83
5.9 The conductance variance vs. magnetic field and temperature . . . .
85
5.10 The temperature dependence of the dephasing time . . . . . . . . . .
87
5.11 Characteristic time and energy scales for the quantum dot and the 2DEG 90
5.12 Coulomb blockade oscillations vs gate voltage and magnetic field
. .
92
. . . . . . . . . .
93
5.14 Temperature dependence of the MCB oscillations . . . . . . . . . . .
94
5.15 The charge sensing procedure . . . . . . . . . . . . . . . . . . . . . .
95
5.16 The charge sensing signal at N ≪ 1 and N = 1 . . . . . . . . . . . .
96
5.13 Magnetic field dependence of the MCB oscillations
xv
xvi
Chapter 1
Introduction
1.1
The Transition from Quantum to Classical Behavior
For the past 100 years physicists have used quantum mechanics to explain and predict
a variety of phenomena. Despite its widespread use and proven predictive capabilities, the question of how to really understand quantum mechanics is still unanswered.
Several interpretations have been suggested over time. Among them are the Copenhagen interpretation of quantum mechanics proposed by Niels Bohr, von Neumann’s
idea of a collapse of the wavefunction, hidden variable approaches and some more
exotic suggestions such as Everett’s many world interpretation. The problem with
interpreting the laws and concepts of quantum mechanics is that our perception of
the world around us, filled with definite (“classical”) outcomes, is distinctly different from the probabilistic character of the quantum world. In quantum mechanics
any linear superposition of states corresponds to a possible state. In contrast, most
things we interact with day to day are in a definite state. Here is a simple example
to illustrate this:
A classical system can move from one position to another, but at any given time
its position is well defined. However, a quantum system, described by a superposition
of all possible positions, has a position that is not well-defined. There is an obvious
1
2
CHAPTER 1. INTRODUCTION
problem in applying quantum mechanics to classical objects: a train cannot be both
here and there or a (somewhat famous) cat cannot be described as a superposition of
being dead and being alive.
The “measurement problem” is the general name given to the following conundrum: if quantum mechanics is the right theory to describe everything, then why do
we not always observe these superpositions? How do you convert a quantum state into
something with a definite outcome? The reason for the name “measurement problem”
is that this question becomes particularly puzzling when one considers measuring a
quantum system. The result of the measurement is a classical definite result. So
one can either postulate that the measurement apparatus is now entangled with the
quantum system (von Neuman interpretation), and as such, is in a superposition of
states, just like the quantum system, or that it is separate from the quantum system
and should be treated classically (Copenhagen interpretation).
An artificial and ultimately unsatisfying distinction that would circumvent this
problem is to postulate that quantum mechanics only describes the quantum world
which is separate from our “everyday world”. The difficulty then shifts to deciding
what belongs in the quantum world and what belongs in the classical world. Initially,
quantum mechanics was applied to microscopic objects: photons, electrons, etc.,
which prompted an identification of the microscopic with the quantum. Since most
macroscopic objects obeyed classical mechanics and had robust, non-probabilistic
properties such as position and momentum, the macroscopic was equated to the classical world. The theoretical prediction and experimental observation of quantum
states in macroscopic systems such as superconducting states that involve macroscopic numbers of electrons and extend over macroscopic distances, or macroscopic
cantilevers that behave as quantum harmonic oscillators makes this distinction inadequate. Thus it is clear that the border between quantum and classical does not
correspond to the border between the macroscopic and the microscopic.
A different way to categorize what belongs to the quantum and what belongs to
the classical world is based on the difference in coupling to the environment [1]. When
a system is isolated from the environment all possible superpositions of its states are
allowed. However, this situation is not realistic: most systems are coupled to their
1.2. DECOHERENCE
3
environment and can interact with it. It turns out that with the right interaction, a
quantum system can be made to look classical. This is called decoherence (for a recent review of research on the theory of decoherence and its applications see [2]). The
question of whether decoherence can solve the measurement problem, offer a mechanism for what looks like a collapse of the wavefunction and explain the quantum
to classical transition, is and has been debated since its introduction [1, 3]. While
the meaning of decoherence and its implications for understanding quantum mechanics have not been fully explored, from the experimental point of view decoherence
is a measurable quantum phenomenon that offers a way to explore the fascinating
consequences of quantum nature.
1.2
Decoherence
The easiest way to think of quantum coherence is in terms of an interference experiment. Classical particles incident on a double slit have a probability Pi to pass through
slit i, where i = 1, 2. The total probability that it hits the screen at a position A can
be expressed as the sum over the individual probabilities: PT (A) = P1 (A) + P2 (A).
A quantum system is described by a wavefunction and as such it will pass through
both slits. Consequently, the probability to hit the screen is given by the square over
the sum of the probability amplitudes a1,2 = |a1,2 |ei(ωt−kx) corresponding to the path
though each slit: PT (A) = |a1 (A) + a2 (A)|2 = P1 (A) + P2 (A) + 2|a1 ||a2 | cos θ. θ
depends on the position of point A with respect to the slits and on the wavevector
k, and P1,2 = |a1,2 |2 is the classical probability to have taken the path through slit 1
or 2. The probability for the quantum system contains an extra term that describes
the phase relationship between the paths taken through the two slits compared to
the classical system. This term encompasses the quantum correlations and will be
referred to as the quantum interference term.
In general, quantum systems are described by a coherent superposition of states
whose phase relationships lead to quantum correlations and to the presence of this
extra quantum interference term. When one tries to detect which state the quantum system is in, the quantum correlations are disturbed and the interference term
4
CHAPTER 1. INTRODUCTION
disappears. Alternatively, this interference term can also be suppressed by certain
interactions of the quantum system with its environment. This process is called decoherence. When the quantum system and the environment become entangled due to
their interaction, the phase relationship between the states involved in the coherent
superposition is defined for the whole combined system, and not for the individual
subsystem. As a result measurements of the combined system will show the extra
term due to quantum coherence, while those involving the subsystem will not.
From the point of view of the mathematical description of such a system, the
reduced density matrix that describes the probabilities for each of the possible states
of the quantum system will have rapidly decaying off diagonal matrix elements (in
some basis) [4]. A more intuitive way to think about the effect of the quantum
correlations between the system and its environment is to say that because of the
interaction, the environment “measures” the state of the system. In analogy to the
double slit experiment described above, this selects one specific state and suppresses
the interference term.
Sometimes decoherence is also called dephasing or decorrelation and the distinction between these terms is not exactly agreed on in the literature. For the purposes
of this thesis, dephasing and decoherence will be used interchangeably.
Decoherence can be quantified in terms of a reduction of the interference term
with respect to the classical term, but it does not simply refer to any reduction
of the interference pattern visibility. For example, covering one of the slits in the
interference example will cause the interference pattern to disappear, but this is not
due to decoherence. Similarly, averaging over different “noisy” realizations of the
system (e.g. thermal distribution of energies) can suppress the interference pattern
but is not decoherence.
Decoherence can be present in a system with or without dissipation: due to their
interaction, the environment obtains “which path” information about the quantum
system. This can happen without necessarily exchanging energy, but in order for
decoherence to take place, an inelastic process involving some degree of freedom of
the quantum system and the environment (e.g. a spin-flip) has to take place [5]. This
will be discussed in more detain in the next section.
1.3. INTERACTIONS WITH THE ENVIRONMENT
1.3
5
Interactions with the Environment
To study the effect of decoherence on electron transport in a solid, one has to be able
to calculate or measure the visibility of the quantum interference term. The condition
required to observe an interference pattern is that phase coherence be maintained over
time and distances comparable to characteristic transport scales such as system size,
Fermi wavelength and elastic mean free path. This condition is fulfilled in mesoscopic
devices with one or more confined dimensions such as two dimensional electrons gases,
quantum wires, or quantum dots, which show quantum interference effects such as
Aharonov Bohm oscillations [6], universal conductance fluctuations [7], and weak
localization effects [8].
The simplest picture of electron transport in a metal neglects interactions with
the environment. In the Drude model, electrons are treated as free (they do not interact with ions or impurities other than by elastic collsions) and independent (between
collisions electrons do not feel the electrostatic interaction with other electrons). The
Sommerfeld model takes one step further and accounts for the Pauli exclusion principle by using the Fermi-Dirac distribution for the electron velocity. A picture that
takes both the quantum decription into account as well as the electrostatic interaction
with the underlying atomic lattice is the description of electrons in terms of Bloch
waves: these are stationary solutions to Schroedinger’s equation with a periodic potential. The semiclassical model combines the Sommerfeld model with the description
in terms of Bloch waves by constructing a wavepacket of Bloch levels with a spatial
extent that is smaller than the length over which externally applied fields vary. For
a detailed treatment see ref. [9].
All of these models assume non-interacting electrons. Since electrons interact via
Coulomb repulsion, this assumption is not valid. Landau showed that a system of
interacting electrons can be approximated by a system of nearly independent quasiparticles with the same charge and momentum as the electron, obeying the exclusion
principles, but with modified parameters such as effective mass and magnetic moment. This is the Fermi liquid picture and it makes it possible to use the results of
the above mentioned models, as long as the electrons are replaced with the Landau
6
CHAPTER 1. INTRODUCTION
quasiparticles. These quasiparticles are sometimes referred to as “dressed” electrons
because they can be thought of as electrons screened by the interactions. In general,
this thesis uses the word electron to refer to these screened electrons.
Despite its limitations, the Drude-Sommerfeld model can be used to derive key
features of electron transport in metals. The effects of elastic scattering are accounted
for by the introduction of a collision or relaxation time. This is a measure of the
disorder of the system and affects the conduction. Electrons move at the Fermi
velocity vF but in thermal equilibrium their direction of motion is random so there is
no average net current or velocity.
Under the influence of an electric field, electrons acquire a net drift velocity vD ,
which depends on the electric field, but also on how disorder affects the momentum of
the electrons. Elastic collisions do not change the momentum of the electron, but they
can randomize its direction. The average time it takes for the electron to undergo
enough collisions to randomize its direction of motion is referred to as the momentum
relaxation time or elastic time τe . The distance it can travel in this time is the mean
free path le . This length scale can refer to the average distance between impurities
if they are hard scatterers, or can stretch over several collisions events in the case of
small angle scattering.
Assuming that an electron with mass m moving in an electric field E obeys Newton’s laws between collisions:
m
~
dv
~
= eE,
dt
(1.1)
~ e /m. The proportionality factor
their average drift velocity is given by v~D = −eEτ
between the drift velocity and the electric field is called the mobility µ = eτe /m.
From the net current density ~j = nev~D and for electric transport in the Ohmic
~ we can express the conductivity of the system in terms of the the
regime (~j = σ E)
density n and the mobility µ of the material alone:
ne2 τe
σ=
= neµ.
m
(1.2)
In this model elastic scattering determines the conductivity of a system. Elastic
1.3. INTERACTIONS WITH THE ENVIRONMENT
7
scattering events, for example collisions with static defects of the lattice, such as
crystal dislocations or charged impurities, as well as other electrons lead to a finite
conductivity.
Inelastic scattering events are responsible for decoherence. Inelastic scattering
events that lead to decoherence are collisions with scatterers with an internal degree
of freedom such as phonons, magnetic impurities and other electrons. The efficiency
of inelastic scattering in destroying phase coherence is enhanced by elastic scattering.
The latter cannot destroy the phase coherence but it can modify the quantum interference term. Similar to the elastic time, the inelastic or decoherence time τφ is the
time it takes an electron to lose its phase information to the environment (to change
its quantum state). Calculating the relation of the decoherence time and the conductivity of the system depends on the dimensionality and the relative length scales of
the system [10].
If the size of the size L of the system is smaller than the mean free path, transport
through the system is independent of the impurities and the Drude conductivity becomes irrelevant. The conductance is determined only by the transmission probability
T through the system and is described by the Landauer formula G = (e2 /h)T . This
is the regime of ballistic transport. In order to understand which processes dominate,
and what the resulting effects are, we have to compare several energy scales. Some
of these requirements can be relaxed at finite temperature, where the timescale to
compare to is the smaller of the thermal length or system size, but in general the
following conditions are valid:
1. λF < le : the Fermi wavelength has to be smaller than the mean free path for
the system to be a good conductor.
2. τcross > τe : the time it takes the electrons to cross the system has to be larger
than the elastic mean free time for transport to be diffusive. In this case τcross =
L2 /D where D = vF2 τe /2 is the diffusion constant.
3. τcross < τe : the time it takes the electrons to cross the system has to be smaller
than the elastic mean free time for transport to be ballistic. In this case τcross =
L/vF .
8
CHAPTER 1. INTRODUCTION
4. τφ ≪ τcross , τe : if the decoherence time is smaller than all the timescales in
the system, the conductance is determined by whether transport is ballistic or
diffusive.
5. τφ ≥ τcross , τe : if the decoherence time is comparable to some or all of the
transport times in the system, the conductance will be modified by the extra
presence of the quantum interference terms.
There are other time scales that are relevant such as the Ehrenfest time or the ergodic
time that will be discussed in chapter 5.
Depending on the elastic and inelastic scattering events that are present, the
dependence of the decoherence time on energy, temperature and bias can look very
different. Calculations of electron-electron dephasing in clean (ballistic) systems show
that as the energy ∆ of the electron-quasiparticle with respect to the Fermi level increases, the phase space available for scattering processes increases, and the dephasing
time decreases. At finite temperatures, electrons with energy kB T with respect to
the Fermi energy are involved in transport and can decohere. Depending on the dimensionality of the system, the dephasing time as a function of temperature has the
following dependence [11, 12]
1
τφe−e
∝







T2
for 3D
2
T ln(T ) for 2D
T
(1.3)
for 1D
The total momentum of the electron system is conserved in these scattering processes,
and as such, they do not directly affect the net current. However, they affect the
quantum interference corrections to the conductance. As a result the decoherence
time will be important in any system where interference is possible.
The theory of electron-electron dephasing in disordered conductors was developed
by Altshuler, Aronov, and Khmelnitsky who showed that the Fermi liquid description
is applicable in low dimensions and in the presence of disorder [13, 14].
Because the the inelastic scattering in 3D and in the presence of disorder is dominated by large energy transfer processes, the decoherence time has the same power
1.4. THE KONDO EFFECT: AN EXAMPLE OF A CORRELATED QUANTUM STATE9
law dependence on temperature as in the ballistic case [14]. In lower dimensions,
processes with small energy transfers become important and the dephasing time has
the following dependence on energy/temperature:
1
τφe−e

2

for 3D

 T
∝
T
for 2D


 T 2/3 for 1D
(1.4)
The definition of the decoherence time and the precise relationship to the microscopic parameters of the system and to the interaction is a delicate subject. Depending on the case, one can think of the dephasing time as equal or proportional to the
quasi-lifetime of the single particle state, but this is not a priori obvious [15].
As any well-behaved quantum system, an electron moving through any material
will take all possible paths from point A to point B. The phase relationship between
the possible paths will determine the interference term that is the quantum correction
to the classical Drude conductivity. The presence of disorder (elastic impurities),
leads to the localization of the electrons at certain impurities due to the constructive
interference of some closed trajectories. The resulting reduction of the conductance
of the system is called weak localization and can be used to study the coherence time
in disordered systems. In ballistic systems, the quantum interference manifests itself
as fluctuations of the conductance from the value predicted by Landauer formula.
How the size of these fluctuations can be used to extract the dephsaing time will be
explained in more detail in chapter 5.
1.4
The Kondo Effect: an Example of a Correlated
Quantum State
A system in which coherent quantum properties lead to a dramatic departure from
classical behavior is that of a metal containing very few magnetic impurities. At
relatively high temperatures, the resistivity of a metal is dominated by electronphonon scattering. As the temperature is lowered the scattering rate decreases. At
10
CHAPTER 1. INTRODUCTION
low enough temperatures, electron-phonon scattering becomes insignificant and the
resistivity saturates at a finite value determined by scattering from defects in the
crystal lattice.
In the 1930s measurements of Au cooled below 10 K sometimes showed a resistivity
rise rather than the predicted saturation as the temperature was lowered further [16].
This effect remained unexplained until the 1960s when further experiments established a correlation between the low-temperature resistivity rise with the presence of
dilute magnetic impurities in the metal [17]. The presence of the magnetic impurities
dramatically affects the transport properties of the metal at low temperatures where
quantum coherence is maintained.
The Kondo Hamiltonian
The strong experimental evidence that magnetic impurities give rise to an enhanced
resistivity led J. Kondo to calculate the effect that magnetic impurities which can
scatter electrons have on the resistivity of a metal. The s-d Hamiltonian described
the exchange interaction between an impurity spin and conduction electrons:
Hs−d =
X
†
ǫks ψks
ψks + J
X
i
k
~i ,
~σi · S
(1.5)
~i is the ith impurity spin, ~σi is the spin of the conduction electrons at the
where S
location of the ith impurity, and ψ and ψ † represent the annihilation and creation
operators for the conduction electrons with given momentum k and spin s. Kondo
used perturbation theory on the s-d model and showed that an antiferromagnetic interaction J leads to a logarithmic rise in electron-impurity scattering with decreasing
temperature. This explained the observed rise of the resistivity at low temperatures.
The Anderson Hamiltonian was originally proposed to describe a magnetic impurity atom in a metal [18]:
HA =
X
k
†
ǫks ψks
ψks + Un↓ n↑ +
X
ǫds d†s ds + V ψs† (0)ds + h.c. .
(1.6)
s
The first term represents the the kinetic energy of the electrons in the reservoir. U
1.4. THE KONDO EFFECT: AN EXAMPLE OF A CORRELATED QUANTUM STATE11
E
double
occupancy
V
EF
U
d
single
occupancy
electron
reservoir
magnetic
impurity
Figure 1.1: Schematic representation of the Anderson model indicating the parameters of the Hamiltonian (eq. 1.6).
is the charging energy and accounts for the repulsive Coulomb interactions on the
impurity site. The third term is the quantized energy of the localized electrons in a
single spin-degenerate state where the d’s are the creation and annihilation operators
of the impurity with ns = d†s ds . V is the hybridization between the conduction
electrons and the impurity electrons and describes co-tunneling on and off the local
site. These parameters are shown schematically in fig. 1.1.
The coupling between the electrons localized at the impurity (with a discrete
spectrum) and the conduction electrons (with a continuous spectrum) leads to a
finite lifetime of the electrons at the impurity. The interaction parameter U ensures
the doubly-degenerate site can be occupied by just a single electron. If the Coulomb
repulsion is weak, two electrons will occupy the site which means that in the simplest
case the groundstate has total spin 0. If it is strong, only one electron can occupy
the impurity level so total impurity spin is 21 . In this case, the site acts as a magnetic
impurity, and should exhibit the Kondo effect. Schrieffer and Wolff [19] showed that
indeed the s-d model is equivalent to the Anderson model in the limit to which local
charge fluctuations can be neglected (strong electron repulsion).
The ground state of the Kondo Hamiltonian is a spin singlet in which the spin of a
12
CHAPTER 1. INTRODUCTION
localized electron is matched with the spin of delocalized electrons to yield a net spin
of zero. The characteristic energy of this state, referred to as the Kondo temperature
is not simply proportional to the antiferromagnetic coupling strength J:
TK = De1/(Jν)
(1.7)
where D is the conduction electron bandwidth and ν the thermodynamic density of
states of the conduction electrons.
When T > TK , Kondo’s results, which were based on perturbation theory, start
to break down. Below the Kondo temperature the repeated impurity spin-flips and
the corresponding response of the Fermi sea lead to complex many-body dynamics
which produces a logarithmic divergence. In 1975 K. Wilson developed a new renormalization group (RG) technique that was able to solve this problem [20]. The RG
calculations showed that at temperatures below a characteristic Kondo temperature
TK , a magnetic impurity forms a singlet with the surrounding conduction electrons.
Despite this success in determining the ground state and the thermodynamic properties of a magnetic impurity in a metal, accurate calculation of transport properties
over a broad range of temperatures required the development of numerical RG techniques [21]. A more detailed theoretical description of the Kondo effect is given in
ref. [22].
Unlike in the single channel Kondo case where at energies well below TK the
conduction electrons around the spin
1
2
impurity behave as a Fermi liquid, a new
non-Fermi liquid state was predicted to occur in a system that is Kondo coupled to
multiple screening channels [23]:
HM CK =
X
k
†
ǫksα ψksα
ψksα + J
X
α
~
~σα · S.
(1.8)
This multichannel Kondo state along with Fermi liquids, Luttinger liquids, fractional
quantum Hall systems, and disordered systems with Coulomb interactions are the
only known classes of metals.
1.4. THE KONDO EFFECT: AN EXAMPLE OF A CORRELATED QUANTUM STATE13
The properties of the multichannel Kondo system at low temperatures can be calculated using boundary conformal field theory [24, 25] or renormalization group theory
(NRG). The two-channel Kondo model has been used to explain the experimentally
observed specific heat anomalies in certain heavy fermion materials [26, 27, 28] as
well as transport signatures in metallic nano-constrictions [29, 30].
Because the Kondo effect is a property of magnetic impurities in a host metal,
it should be observable experimentally in a variety of systems. Besides the original
measurements on Au, the Kondo effect appears in transport through tunnel junctions containing impurities [31, 32], single molecules or carbon nanotubes coupled to
electron reservoirs [33, 34], and in scanning tunneling microscope measurements of
surface adatoms [35, 36], and metal complexes [37]. In addition, the Kondo effect has
been observed in a variety of quantum dot systems that will be discussed in chapter 4.
Decoherence of Kondo State
The bound state between the impurity spin and the conduction electron spins can be
visualized as a ”screening cloud”, shown schematically in fig. 1.2. In a metal, electrons
close to the Fermi level are able to scatter because the occupied and unoccupied states
coexist in an energy range kB T around the Fermi level. Thus, the Kondo interaction
predominantly affects the electrons at the Fermi surface and the Kondo singlet appears
in spectroscopic measurements as a narrow resonance located at the Fermi energy.
For electrons with Fermi velocity vF , the characteristic size of the Kondo cloud
and thus the spatial extent of the quantum correlations, is given by [38]:
χK ≈
~vF
.
kB TK
(1.9)
For typical semiconductor quantum dots in the Kondo regime the size of the Kondo
cloud is ≈ 1µm. However, the slow exponential decay of the spin polarization in the
electron gas is modulated at distances greater than λF from the local moment by
oscillations and polynomial decay [39], and charge rearrangement due to the singlet
formation occurs at length scales on the order of λF . As a result, the spatial properties
of the Kondo state have been hard to measure.
14
CHAPTER 1. INTRODUCTION
Figure 1.2: The spins of conduction electrons screen the localized impurity spin (blue)
leading to a larger crossection for scattering than in the absence of the magnetic
impurity.
Perturbations such as temperature or bias affect the formation of the singlet
state: weak perturbations (E < kB TK ) can partially suppress the Kondo effect, while
stronger perturbations (E > kB TK ) eventually destroy the Kondo singlet. This can
happen because the Kondo singlet decoheres due to thermal fluctuations or because a
new state, either a non-Kondo state (in case of an applied magnetic field) or another
Kondo state (in case of an asymmetry in coupling to different reservoirs) becomes the
ground state of the system.
Various processes can lead to decoherence of the Kondo singlet. The Kondo singlet
state is a coherent superposition of spin-flip tunneling processes between conduction
electrons and the local state, which can be destroyed by non-Kondo processes such
as emission or absorption of a phonon or photon. At zero temperature and bias,
no phase space is available and delocalized electrons within χk of the impurity can
form the Kondo singlet. With increasing temperature, the broadening of the Fermi
distribution of the leads increases the available phase space for non-Kondo processes.
This disturbs the Kondo correlations. Similarly, finite bias also increases the phase
space and leads to a suppression of the Kondo effect. When the energy scale of the
1.5. OUTLINE
15
perturbation exceeds the binding energy of the singlet state, the Kondo singlet will
decohere completely.
1.5
Outline
Chapter 2 describes the two-dimensional electron gas, the quantum point contact
and the quantum dot from the point of view of electronic transport through low
dimensional systems and introduces the quantum dots used for the experiments in
this thesis. Chapter 3 explains the basic features of transport in the Coulomb blockade
regime that are used to characterize the energy scales of the measured quantum dots.
Chapter 4 describes the conductance though a quantum dot in the Kondo regime,
presents measurements of non-equilibrium Fermi liquid behavior, and discusses how to
tune a double quantum dot through a quantum phase transition. Chapter 5 explains
transport though a quantum dot with perfectly transmitting quantum point contacts.
Measurements of the decoherence caused by electron-electron interactions are shown
and the reappearance of Coulomb blockade features, despite the fact that the quantum
point contacts are not in the tunneling regime, is investigated.
Chapter 2
Quantum Dots as a Model System
As illustrated in the case of the Kondo effect, electrons can exhibit complex behavior
depending on the interactions between various subsystems of a material (electrons,
phonons, ions, etc.). Studying the mechanism that underlies some of these more exotic
groundstates in bulk materials such as high temperature superconductors or heavy
fermion systems is complicated. The reason is that the microscopic properties of bulk
materials are changed by altering a material’s structure or chemical composition, or
by changing an external parameter such as the applied pressure or magnetic field.
This does not just tune a single microscopic parameter of the material, but also
affects the other parameters or even the effective Hamiltonian of the system. As a
result, quantum phase transitions that are predicted to occur as a function of some
interaction parameter in these systems are difficult to study because the different
phases of the transition often belong to two different, although related, materials.
Advancements in lithography technology lead to the fabrication and measurement of the first artificial atoms in patterned GaAs heterostructures [40, 41] in the
late 1980s. Over the past 25 years, further developments in nano-fabrication, low
temperature cooling techniques, as well as the development of new computational
tools, have made it possible to design increasingly complex artificial nanostructures.
A single quantum dot acts as an artificial atom and can be used to model a localized magnetic moment [42, 43, 44]. Multi-dot systems coupled to distinct electron
reservoirs can be designed to imitate conventional materials.
16
2.1. THE TWO DIMENSIONAL ELECTRON GAS (2DEG)
17
The advantage offered by these artificial structures over bulk materials is the fact
that their individual properties such as energy spectrum, magnetic moment, coupling
to the environment, and the spatial distribution of the wavefunction are independently
tunable using gate voltages, magnetic field, or voltage bias. This makes it possible to
study the interplay between single spins and conduction electrons, the transport of
heat through low dimensional structures, charge/spin transport through atoms and
molecules [45, 46, 47, 48], and the transport properties of impurity quantum phase
transitions. The disadvantage is that it is difficult to measure thermodynamic quantities such as specific heat or magnetic susceptibility. As such, the insight gained by
studying artificial structures is complementary to the conclusions reached by studying
bulk materials.
This chapter will describe the GaAs heterostructures and the associated two dimensional electrons gas (2DEG), explain how it can be used to form a lateral gated
quantum dot, and explain how the quantum dot is tuned from a closed to an open
quantum system via the quantum point contacts that connect it to the 2DEG.
2.1
The Two Dimensional Electron Gas (2DEG)
The experiments presented in this thesis use a modulation doped GaAs/Alx Ga1−x As
heterostructure. This consists of AlGaAs on top of a GaAs substrate, with a thin
layer of GaAs deposited on the surface of AlGaAs. Because GaAs and AlGaAs have
almost the same lattice constant, the scattering at the interface where the 2DEG
resides is low and the 2DEG mobility is much higher than in other systems. The
band structures for the two materials are shown in fig. 2.1.
The AlGaAs is doped with Silicon atoms which act as n-type donors. Due to the
fact that the donors raise the Fermi level in AlGaAs above the conduction band of
GaAs, electrons will move from AlGaAs into GaAs and leave positively charged Si
ions behind. The Fermi levels of the two materials have to align at the interface and
thus, depending on the Si doping, a triangular potential well forms exactly between
AlGaAs and GaAs. This bending of the conduction and valence bands is illustrated,
not to scale, in fig. 2.2. At low temperatures, the electrons do not have enough energy
18
CHAPTER 2. QUANTUM DOTS AS A MODEL SYSTEM
GaAs
Al0.3Ga0.7As
vacuum level
3.74 eV
4.07 eV
EC
donor level
1.42 eV
1.8 eV
acceptor level
EV
Figure 2.1: Side by side comparison of the separate band diagrams for AlGaAs and
GaAs.
to exit the well and are trapped inside. The eigenenergies of the discrete states allowed
in a potential well with
V (z) =
(
e 2 ne z
2ǫ0 ǫr
for z > 0
∞
for z = 0
(2.1)
are given by:
Ezj =
9π 2
32m
~e2 ne
ǫ0 ǫr
2 ! 13
3 2
(j + ) 3 ,
4
j = 0, 1, 2...
(2.2)
where ǫr is the dielectric constant and ne is the electron density in the material.
The density of states (including spin degeneracy) in 2D is a constant for energies
lying between two subbands, with discontinuous jumps at the energy of each subband.
The temperatures used for the experiments described in this thesis are low enough
that only the first subband is occupied. By measuring the electron density (via the
Hall effect) and using the density of states to relate the Fermi wavelength to the
p
density λF = 2π/ne we can determine the Fermi energy of the 2DEG electrons.
For reference, the density of states g(E), for a system effective mass m∗ , with a
band minimum Epot for the 3D case and quantized energies Ei for the lower dimensions
2.2. THE QUANTUM DOT
19
electron mass m∗e
density ne
mobility µe
0.067 me
2e11 cm−2
2e6 cm2 /Vs
Fermi wavelength λF
Fermi velocity vF
Fermi energy EF
elastic time τe
elastic m.f.p. le
q
2π
ne
h
m∗ λF
h2
2m∗ λ2F
m∗ µe
e
56 nm
190000 m/s
7.5 meV
vF τe
70 ps
15 µm
Table 2.1: Characteristic parameters of the 2DEG
(square well confinement) is:
g(E) =







1
2π 2
2m∗ 2
~2
m∗
p
E − Epot
Σ Θ(E − Ei )
π~2 i q
m∗
1
Σ 2(E−E
π~ i
i)
for 3D
for 2D
(2.3)
for 1D.
Because the ionized donors are spatially separated from the 2DEG, the electrons
can be approximated as free electrons with a modified effective mass of 0.067me .
This system has a number of desirable properties such as a low density, a large Fermi
wavelength, high mobility, and a large mean free path. As a consequence, transport
on length scales comparable to the Fermi wavelength, the mean free path or the
coherence length can be investigated in this or in lower dimensional sub-systems such
as quantum point contacts or quantum dots.
The 2DEG used for the quantum dots measured in this thesis has been described
in [45] and its parameters are summarized in table 2.1.
2.2
The Quantum Dot
Quantum dots are isolated regions of a material where electrons are confined. The
quantum dot can be viewed as a particle in a box: spatial confinement leads to energy
quantization. Although the energy scales are very different, a quantum dot can be
20
CHAPTER 2. QUANTUM DOTS AS A MODEL SYSTEM
+
+
+
+
EC
0.2 eV
EF
EV
Si doped
AlGaAs
undoped
AlGaAs
GaAs
substrate
Figure 2.2: The triangular potential well at the interface between AlGaAs and GaAs
forms because the positively charged Si donor atoms create an electric field and the
Fermi level has to be discontinuous across the interface between the two materials. At
low temperatures where the discreteness of the spectrum becomes apparent, only the
first subband is occupied if the electron density (and consequently the Fermi level) is
appropriately tuned.
2.2. THE QUANTUM DOT
21
thought of as an artificial atom, with electrons arranging themselves in subshells. The
condition for observing energy quantization is that the energy level spacing has to be
larger than the broadening of the energy levels due to tunneling out of the dot.
The confinement of the electrons on the quantum dot also leads to an enhancement
of the Coulomb repulsion that manifests as charge quantization. For the charge on
the dot to be quantized, the energy associated with increasing the number of electrons
on the dot by one has to be larger than the energy associated with the time to tunnel
off the dot by means of the Uncertainty Principle. This means that the lifetime of the
electron on the island should be large. Exactly how large is an important question that
will be discussed in chapter 5. An intuitive way to explain the charging energy is by
analogy to a capacitor: once the quantum dot is charged with one electron, Coulomb
repulsion will keep other electrons from tunneling onto it, unless these electrons can
pay the additional energy cost.
There are two ways in which a 2DEG in a semiconductor heterostructure can be
used to form a quantum dot. A lateral quantum dot uses lithographically-defined
metallic gates deposited on the surface of the heterostructure to deplete the 2DEG.
Approximately 50 electrons are confined in a 100 nm diameter droplet. They are
coupled via tunable single mode quantum point contacts to extended sections of the
2DEG, which serve as leads. It is also possible to etch away sections of the 2DEG
to define the electron droplet. In this geometry, the conductance though the point
contacts often becomes immeasurably small before the dot is entirely emptied.
Vertical quantum dots are are formed by etching a double barrier heterostructure.
This allows the formation of a small, well-coupled, few-electron quantum dot with
wide lead-dot contacts that contain several partially-transmitting modes. They allow
the conductance of the dot to remain measurable all the way down to the last electron
but are not easily tunable.
For a 2DEG with certain parameters, the energies associated with charging or with
energy quantization are determined by the size of the potential box. For a square
well of size L, the level spacing between the quantized levels is ∆ ∝
to a capacitor, the charging energy is EC ∝
1
.
L
1
.
L2
By analogy
Depending on the value of L, the
quantum dot is either in the metallic transport regime EC ≫ ∆ ≃ 0 and the density
22
CHAPTER 2. QUANTUM DOTS AS A MODEL SYSTEM
of states of the dot is continuous, or in the semiconducting regime where EC , ∆ > 0,
and the density of states of the dot has gaps. In addition, depending on how the size
of the dot compares to the characteristic length scales of the 2DEG and how large
its coupling to the leads (Γ) is, transport through the dot is determined by different
processes. Here is a summary of the different regimes that can be used to broadly
categorize quantum dots:
1. BALLISTIC (L < le ): electrons travel in a straight path inside the dot and
only scatter off the walls of the potential. If the irregularities in the potential
are smaller than the wavelength of the electrons, then electrons are simply
reflected back without any energy loss (specular scattering). The time to cross
the dot determines a characteristic energy scale called the ballistic Thouless
energy: ET h =
2
~vF
L
. If the motion of the electrons inside the dot is mostly
chaotic, transport though the dot at energies below ET h is independent of the
dot geometry and only dependent on the coupling to the leads.
2. DIFFUSIVE (L > le ): electrons scatter of the potential of impurities before
they scatter of the confining potential. If the disorder is weak, perturbation
theory can be used to describe the system. In large quantum dots that are
quasi-2D, disorder causes all single particle states to become localized. The size
of the system is characterized by the Thouless energy, given by: ET h =
2L2
vF τe
3. CHAOTIC: when the shape of the dot is irregular and the electrons bounce
around inside the dot several times, their motion is mostly chaotic. In a classical
system this means that there is an exponential sensitivity to initial conditions.
The statistical quantum fluctuations of a classically chaotic system are universal.
They are described by random matrix theory and can be studied in stadium
shaped quantum dots.
4. REGULAR: the shape of the dot leads to a confining potential (for example a
harmonic potential) where the single particle levels are arranged in shells and
well defined periodic orbits transport electrons through the quantum dot.
2.2. THE QUANTUM DOT
23
5. CLOSED (Γ ≪ ∆, U): in quantum dots that are well isolated from the leads,
charge quantization can be observed and transport properties are dominated by
the Coulomb repulsion between electrons.
6. OPEN (Γ ≫ ∆, U = 0): in quantum dots that are well coupled to leads, the
single particle levels are broadened and overlapping. These systems are well
described as non-interacting electron systems.
The quantum dots used in the experiments described in this thesis are shown in
fig. 2.3. They are lateral quantum dots defined by Au gate electrodes deposited on
the surface of a GaAs/AlGaAs heterostructure which has a 2DEG 68 nm beneath the
surface. The 2DEG parameters are given in table 2.1. The thin constrictions labelled
QPCs in fig. 2.3 connect the electrons in the dot to the electrons in the rest of the
2DEG. Depending on the size and location of the gates, they couple differently to the
quantum dot parameters. The gates labeled bp and sp primarily control the distance
of the dot levels to the Fermi level in the leads and thus the number of electrons on
the dot. The gates labeled sw1/sw2 and bw1/bw2 determine the coupling of each of
the dots to its leads. The gates labeled c1/c2 determine the interdot coupling and
have a strong effect on the area of the large dot. Gates sn1 and sn2 have been used
to form the small dot, but are not varied during the measurement. How the voltages
on the different gates affect the electrons on the dot can be quantified by associating
a capacitance with each gate and with the dot as described in chapter 3.
Both measured quantum dots are smaller than le of the 2DEG and thus, transport
inside the quantum dots is ballistic. The charging energies, level spacing, and total
number of electrons on the dot are listed in table 2.2. These parameters can be
measured experimentally as is explained in the next chapter. For consistency, they
can also be calculated using the area of the dot estimated from the SEM picture.
Because the 2DEG is separated by ≈ 70nm from the gates, the potential applied
to the gate will spread out by a similar distance in the plane of the 2DEG and the
quantum dots are smaller than the actual area inclosed by the gate contours in the
SEM picture. This consistency check works better for the large dot in fig. 2.3.
We measure the conductance though the quantum dots by using a 1 to 5 µVrms
24
CHAPTER 2. QUANTUM DOTS AS A MODEL SYSTEM
bp
bw1
c1
sn1
QPC1
sw1
sp
n
sw2
QPC2
1 μm
c2
bw2
sn2
Figure 2.3: The double quantum dot device used for the experiments in this thesis.
Both dots have been used for the experiment on the two channel Kondo effect (chapter 4 part 2) and for measuring the charge quantization in the large dot at N = 1
(chapter 5 part 2). The small dot was used for the out-of-equilibrium Kondo effect
measurement (chapter 4 part 1). The large dot was used for the dephasing time
measurements (chapter 5 part 1).
A
U
∆
Ne
ET h
ΓN =1
τdN =1
e2
C
2π~2
m∗ A
~v
√F
A
small dot
0.04 µm2
1 meV
100 µeV
≈ 30
80 µeV
large dot
2.6 µm2
150 µeV
3.1 µeV
≈ 4000
25 ps
0.75 ns
2∆
h
Γ
Table 2.2: Characteristic parameters of the quantum dots
2.3. FROM A CLOSED TO AN OPEN QUANTUM DOT
25
oscillating voltage with a frequency of 13, 17 or 97 Hz and measuring the resulting
current with a DL Instruments Model 1211 current pre-amplifier and a Princeton
Applied Research 124A lock-in amplifier. This AC oscillation is on top of a DC bias
voltage that is used to characterize the system at finite bias. The quantum dot is
cooled to 4 K with a positive bias of 180 to 200 mV applied to all gates. The positive
bias “pre-depletes” the gates, so that the quantum dot is already formed, albeit
very well coupled to the leads, even if all gates are set to 0 V. The measurements
are performed in a Kelvinox TLM 400 dilution refrigerator with a base temperature
Tbase ≈ 10mK where electrons can reach the lowest temperature of 13 mK [49].
2.3
From a Closed to an Open Quantum Dot
Electrons in quantum dots are coupled to the environment in two ways. They are
coupled to an environment that consists of the surrounding electrons, nuclear spins of
atoms, charged or magnetic impurities, and phonons inside the material. The coupling
to this environment is fixed by the parameters of the 2DEG and is only affected by
outside perturbations such as temperature and bias. It determines whether transport
through the dot is ballistic or diffusive.
Electrons are also coupled via quantum point contacts to the 2DEG reservoir.
The size of this coupling, Γ (the tunnel rate) can be tuned via gate voltages and the
conductance can be dramatically different depending on how Γ compares to other dot
energy scales. The coupling of the quantum dot to the 2DEG reservoirs via the QPCs
determines whether the quantum dot is called open or closed.
The quantum point contacts (QPCs) can be thought of as a 1D waveguide (ballistic
transport) whose transmission is tuned by gate voltages [10]. Electrons can move
along the x-axis in fig. 2.4 (a), but the wavevector component transversal to the QPC
is quantized and only electrons whose energies match the allowed subband energies,
schematically illustrated in fig. 2.4 (b), can be transmitted.
Thus, if the gate voltage is tuned such that the Fermi energy of the leads is below
the first subband, transport is exponentially suppressed. Electrons have to tunnel
through the barrier to reach the dot: most electrons are reflected and very few are
26
CHAPTER 2. QUANTUM DOTS AS A MODEL SYSTEM
(b)
(a)
Yy
E
VQPC1
E(Y)2
X x
EF
E(Y)1
E(Y)0
E(X)+E(Z)0
y
Figure 2.4: (a) Zoom in of the QPC region; electrons can travel in the x direction but
their motion in y is confined by the voltage on the QPC gates ( z is confined because
(y)
of the 2DEG). (b) Sketch of the allowed energy levels Ei inside the QPC; only the
subbands below the Fermi level can transmit electrons through the QPC.
transmitted. When the first subband reaches the Fermi level, the transmission of the
QPC is 100%: exactly one spin-degenerate mode of electrons can pass through the
QPC. As the gate voltage is further decreased and more subbands are pulled below
the Fermi level, additional modes open up as shown in fig. 2.5.
The conductance of the QPC is solely determined by the sum of the probabilities of
transmission of a state i in the dot to a state f in the leads (or vice-versa). The relation
between the conductance and the transmission is given by the Landauer formula
2 P
G = ms eh i→f |ti→f |2 where ms is the spin degeneracy. Thus, each fully transmitting
spin-degenerate mode contributes a quantum of conductance GQ = 2e2 /h, as can be
seen in the conductance in fig. 2.5. At zero temperature the conductance of the QPC
is a perfect staircase that washes out with increasing temperature.
The strength of the coupling between the dot and the leads Γ depends on the size
of the tunnel barrier. If Γ ≪ kB T, ∆, U then the quantum dot is isolated and the
discrete levels in the dot are not affected by the leads. This is called a closed quantum
dot. For larger Γ, the fact that the dot levels are coupled to the continuum density of
states in the reservoir leads to the broadening of the dot single particle energy states.
When the QPC is tuned to the first plateau, i.e.N = 1, where N is the number of
spin-degenerate modes in the QPC, its conductance is GQP C = 2e2 /h and the levels
on the dot are broadened such that Γ = ∆. The lifetime of the single electron state,
2.3. FROM A CLOSED TO AN OPEN QUANTUM DOT
27
E
E(Y)2
EF
E(Y)1
E(Y)0
2
GQPC (2e /h)
4
3
2
1
0
-800
-600
-400
VQPC1 (mV)
Figure 2.5: The conductance through the QPC increases in units of 2e2 /h with increasing gate voltage.
28
CHAPTER 2. QUANTUM DOTS AS A MODEL SYSTEM
or the dwell time of the electrons on the dot then only depends on the size of the dot.
The characteristic dwell times at N = 1 for the quantum dots studied in this thesis
are given in table 2.2.
The Coulomb Blockade model describes the behavior of a quantum dot in the weak
tunneling regime where GQP C1,2 << 2e2 /h. The model that describes the behavior
in the strong tunneling regime, where GQP C1,2 < 2e2 /h, is the Kondo model. The
model that describes the behavior when the QPCs are fully open and GQP C1,2 ≥ 2e2 /h
is random matrix theory. This thesis will describe measurements in the latter two
regimes.
Chapter 3
Quantum dots: Coulomb Blockade
The simplest model that describes quantum dots that are coupled to leads via tunneling point contacts is the constant interaction model. This model combines the
effects of the enhanced Coulomb repulsion with the presence of a quantized energy
spectrum and can be used to understand many of the features of electronic transport
through a quantum dot such as the evolution of the charge on the dot as a function
of gate voltage, the effect of finite applied bias on the conductance, and the tunneling
processes which contribute to transport.
3.1
The Charging Energy
I will first discuss a simpler case, that of a metallic dot (∆ = 0) and then consider the
modifications needed to account for the discrete density of states of the quantum dot.
Fig. 3.1 shows a schematic representation of a quantum dot tunnel coupled to source
(S) and drain (D) leads and coupled capacitively to both source drain and gates, the
latter generically denoted by the index G. Let the dot be neutral when there are no
voltages applied. As a function of the voltages applied (VR , VL and VG ) the charge
on the quantum dot QDOT will change. Conservation of charge and Kirchhoff’s laws
29
30
CHAPTER 3. QUANTUM DOTS: COULOMB BLOCKADE
dictate that:
−QDOT = −qD − qS − qG
qg
qD
−
−VG + VD =
CD CG
qS
qg
VG − VS = −
+
CS CG
(3.1a)
(3.1b)
(3.1c)
where qD,S,G is the charge accumulated on the capacitors connecting the dot to the
drain, source and gates.
CD
CS
VS
VD
dot, N
CG
VG
Figure 3.1: A quantum dot with N electrons is coupled electrostatically to source,
drain, and gate electrodes and tunnel-coupled to source and drain leads.
The total energy of the dot with charge QDOT is the sum of the energy stored in
the three capacitors minus the work done by the voltage sources:
Etotal (QDOT , QG ) =
(QDOT − QG )2
+ other terms(VD , VS , VG )
2C
(3.2)
where C = CD + CS + CG is the total capacitance of the island and QG = CD VD +
CS VS + CG VG is the total charge induced by the three voltage sources. Usually VS
and VD are small and QG can be thought of as the charge induced by the gate. QG
is a continuous quantity because VG is continuous. If the QPCs are in the tunneling
regime, then QDOT is discrete: the total charge on the dot has to correspond to an
3.1. THE CHARGING ENERGY
31
integer number of electrons Ne− . Disregarding the other terms, the total electrostatic
energy is:
Etotal (N) =
QG Q2G
e2 N 2
+ eN
+
2C
C
2C
(3.3)
The total energy of the dot as a function of the gate voltage is shown in fig. 3.2(a).
With increasing gate voltage, the number of electrons corresponding to the lowest
energy increases in discrete steps (fig. 3.2(b)), corresponding to the gate voltage
where
Etotal (N) = Etotal (N + 1).
(3.4)
(b)
(a)
2
1.0
>TODN<
U/latotE
0.5
0.0
-1
N=0
N=1
N=2
0
1
2
3
0
-2
-2
QG/e
0
2
QG/e
Figure 3.2: (a) At T=0, the lowest energy state of the quantum dot has different
numbers of electrons on the dot, depending on the gate voltage setting. (b) The
number of electrons on the dot increases by exactly one electron at specific gate
voltage values. The discrete steps at T = 0 (red trace) are smeared out at T>0
(black trace, with kB T = 81 U ).
The additional energy needed to add one electron to a dot with N electrons is:
δEtotal (N + 1) = Etotal (N + 1) − Etotal (N)
1 QG
)U
= (N + −
2
e
where U =
e2
C
(3.5)
is called the charging energy of the dot. Note that the classical charging
energy of a capacitor C with one electron is EC =
energy U of the dot.
e2
2C
and is different from the charging
32
CHAPTER 3. QUANTUM DOTS: COULOMB BLOCKADE
(b)
(a)
E
E
μN+1
U+
μS
μN
N+1
μD
U+
μN
electron
reservoir
N
quantum
dot
Figure 3.3: (a) Transport through the dot is forbidden when the Fermi energy of the
leads lies between the electrochemical potentials of the dot. (b) Transport through
the dot is allowed when a single particle level aligns with the Fermi level of the leads:
electrons can tunnel on and off the dot.
The constant interaction model accounts for the discrete energy spectrum of the
dot simply by adding the single particle energies ǫi to the total electrostatic energy
Etotal calculated above. The electro-chemical potential µN is defined as the energy
needed to add the N th electron to a conductor. If
E(N) =
N
X
i=1
then
ǫi +
e2 N 2
QG
+ eN
2C
C
CD VD + CS VS + CG VG
1
+e
.
µN = ǫN + U N −
2
C
(3.6)
(3.7)
Changing VG shifts the chemical potential of the dot with respect to the Fermi level
of the leads.
At low temperatures and small bias voltage eV compared to the charging energy
U, an electron can jump on or off the dot if and only if the chemical potential of the
dot aligns with the Fermi level of the leads. This is shown schematically in fig. 3.3.
Due to the finite charging energy U, most of the time the two are not aligned so the
number of electrons on the dot is fixed and transport through the dot is suppressed.
3.1. THE CHARGING ENERGY
33
(b)
(a)
-3
40x10
0.10
2
)h/ e(G
2
)h/ e(G
20
0.05
U+
∆
U
0.00
-295
V
BP
-290
(mV)
-285
-110
-100
V
SP
-90
(mV)
Figure 3.4: When the QPCs are in the tunneling regime, sharp peaks in conductance
alternate with large regions of zero conductance as a function of gate voltage. (a)
When ∆ < T the distance between many successive peaks remains constant and is
proportional to the charging energy U. (b) When ∆ ≫ T the Coulomb blockade
peaks are not equidistant.
This is called Coulomb blockade. When the gate voltage is such that µD ≈ µD ≈ µN ,
the number of electrons on the dot can fluctuate by one and a peak in conductance
is observed. The distance between the peaks corresponding to the N th and (N + 1)th
electron traversing the dot is µ( N + 1) − µN = U + ∆N +1 . This corresponds to a
change in gate voltage e∆VG =
C
(U
CG
+ ∆N +1 ).
Fig. 3.4 shows experimental results from measuring the two dots in fig. 2.3 at 15
mK. The alternating pattern of peaks and suppressed regions of conductance as a
function of gate voltage is apparent in both dots. For the large dot where ∆ < kB T
the Coulomb blockade peaks are equidistant in gate voltage as seen in fig. 3.4(a). The
discrete energy spectrum is only observable in the small dot measurement (fig. 3.4(b)).
The left Coulomb blockade valley is slightly larger than the right Coulomb Blockade
valley. This is because when one electron fills an unoccupied single particle state in
the dot, the next electron can fill the same state just with opposite spin. As such, the
first electron has to have an energy U + ∆ to tunnel on the dot, while the following
electron only needs energy U. If ∆ > kB T , as is the case for the small dot (see
table 2.2), the extra energy required to occupy a new single particle level modifies
the distance between Coulomb blockade peaks.
34
CHAPTER 3. QUANTUM DOTS: COULOMB BLOCKADE
3.2
Coulomb Blockade Diamonds
At finite bias, transport through the dot can occur as long as the energy of the dot
level lies in the transport window eV . In most experimental cases, the bias is not
applied symmetrically to the source and drain: VD = 0, VS = VSD . How much each
voltage affects the electrochemical potential of the dot is determined by the ratio
αD,S,G = −
CD,S,G
.
C
Transport through the dot at VSD > 0 is forbidden (fig. 3.3) if:
µN < EF
(3.8a)
µN +1 > EF + eVSD .
(3.8b)
Similar inequalities hold for VSD < 0.
(a)
(b)
VSD
VSD
(U+
N-1
N
N+1
N+1)/
+2
N-1
VG
(U+
+1)
N
+1
VG
G
Figure 3.5: Finite bias and gate voltage measurements reveal Coulomb blockade
diamonds in the metallic case (∆ < kb T ) (a) and Coulomb blockade diamonds with
excited states present in the semiconducting case (∆ ≫ kB T ) (b). The region where
transport is forbidden and the conductance through the quantum dot is suppressed
is shaded in gray.
Using eq. 3.7, these conditions can be used to solve for VG as a function of the
number of electrons on the dot N and bias VSD . They delimit diamond-like regions
in gate voltage and bias where the conductance is zero, as illustrated in fig. 3.5. The
size of the blockaded region is determined by the charging energy U. In the case of a
3.2. COULOMB BLOCKADE DIAMONDS
(a)
35
200
(b)
2
G(e /h)
2
G(e /h)
200
0
µ
V
V
01x
0.4
DS
DS
40
3-
µ
(
0
0.6
-200
(
60
)V
)V
80
-400
0.2
20
-200
0
-350
-345
V
GATE2
(mV)
0.0
-600
-230
V
-220
SP
(mV)
Figure 3.6: When the QPCs are in the tunneling regime, diamond-like regions of
suppressed conductance (dark blue) occur as a function of gate voltage and source
drain bias. (a) CB diamonds measured in the large dot (∆ < kB T ) show only Coulomb
blockade features. (b) The small dot measurements (∆ > kB T ) show transport
through excited states at finite bias.
discrete level spectrum, transport at finite bias shows extra features. This is because
electrons can use either the ground state or the excited state to tunnel though the
dot when one of the excited states of the dot spectrum also lies in the transport
window. When an excited state crosses the source or the drain level, another peak in
conductance is observed. The indicated level spacing assumes a single particle picture
where µ( N + 1) − µN = U + ∆N +1 and µ( N + 2) − µN +1 = U + ∆N +2 .
This difference between the appearance of Coulomb blockade (CB) diamonds in
the metallic vs semiconducting regimes is illustrated in the measurements in fig. 3.6.
The left graph (a) shows finite bias transport in the large quantum dot. The peak
corresponding to the drain line is rather faint, but Coulomb blockade diamonds without any traces of excited states can be seen. Fig. 3.6(b) shows a zoom in of a diamond
in the small quantum dot with extra peaks that correspond to transport though the
excited states. The spectrum of the dot is indeed discrete with an average single particle level spacing for the first few excited states of the N electron dot ∆ ≈ 100µeV
(see table 2.2). Note that the charging energy U is much larger than the bias voltage
range in this measurement. Also note the difference in bias range of the y-axis of the
two graphs shows that the charging energy for the large dot is significantly smaller
than for the small dot.
36
CHAPTER 3. QUANTUM DOTS: COULOMB BLOCKADE
The discussion so far does not explain how temperature or coupling to the electrical
leads affects the quantum dot. To measure any of the data presented above, the
quantum dot is coupled to two leads. The Hamiltonian of the system then includes
the different subsystems and the interaction between them:
H = Hleft + Hright + Hdot + Htunneling + Hcharging
(3.9)
where Hlef t /Hright describes the non-interacting electrons with momentum k/q and
spin s in the left/right reservoir (source and drain), Hcharging is due to the Coulomb
repulsion and Htunneling describes electrons tunneling from the leads on and off the
dot and are given by:
Hleft =
X
†
ǫk lks
lks
(3.10a)
†
rqs
ǫq rqs
(3.10b)
ǫp d†ps dps
(3.10c)
k,s
Hright =
X
q,s
Hdot =
X
p,
2
e
(n̂ − QG )2
2C
X
X
L †
R †
=
Tkp
lks dps + h.c. +
Tqp
rqs dps + h.c.
Hcharging =
(3.10d)
Htunneling
(3.10e)
p,k
p,q
(3.10f)
where n̂ =
P
p
d†ps dps is the number operator of the electrons on the dot and T L,R
are the tunneling matrix elements between a dot state and the leads and are used to
define the coupling Γ of the dot to the leads.
Depending on the relation between ~Γ, kB T, ∆ and U different processes dominate
transport and the characteristic dependence of the conductance on temperature and
bias can be calculated. In the regime where ~Γ ≪ kB T ≪ ∆, U, the quantum dot
can be used as a thermometer. Because the coupling to the leads is small, the levels
in the dot can be thought of as delta-functions. On the Coulomb blockade peak, a
single level contributes to transport. The broadening of the peak is determined by
3.2. COULOMB BLOCKADE DIAMONDS
(a)
37
(b)
(c)
2
-3
20x10
0.3
10
T(mK)
∆Vsp (mV)
2
G (e /h)
100
0.2
2
10
0.0
-225
Vsp (mV)
4
3
0.1
0
7
6
5
-224
0
100
200
T (mK)
4
10
resistantce for R13 (Ohm)
Figure 3.7: (a) A thermally broadened Coulomb blockade peak. The fit yields a
width that corresponds to a temperature of 14.5 mK. (b) The evolution of the width
of a Coulomb blockade peak with temperature. (c) The correspondence between the
electron temperature (y-axis) and the resistance indicated by the Oxford dilution
refrigerator probe thermometer nr. 13 (x-axis).
the thermal broadening of the leads. The lineshape of the peak in gate voltage is
given by:
e2
G(VG ) =
4kB T
1
1
+
Γs ΓD
−1
−2
cosh
αG (VG0 − VG )
2kB T
(3.11)
where VG0 is the gate voltage coordinate of the peak and ΓS,D is the coupling of the
dot to the source and drain, respectively.
Fig. 3.7(a) shows a temperature broadened Coulomb blockade peak measured in
the small dot. By fitting a cosh−2 function to the peak, the width of the peak can be
extracted and is plotted as a function of temperature in fig. 3.7(b). The temperature
dependence of the width in fig. 3.7(b) allows an accurate determination of the lowest
electron temperatures of the system.
At high temperatures the electrons in a quantum dot are in thermal equilibrium
with the surrounding systems. As such, the temperature can be determined by measuring a calibrated resistor nearby. The resistor readings do not correspond to the
temperature of the electrons at low temperatures and a different method for determining the electron temperature has to be used. For reference, I am showing the
38
CHAPTER 3. QUANTUM DOTS: COULOMB BLOCKADE
conversion between the actual value of the dilution refrigerator resistor and the actual temperature of the electrons in the quantum dot. The electron temperatures
values have an error for the peak fit of ±10% at low temperature. This conversion
has been used throughout the experiments in this thesis to determine the electron
temperature.
Chapter 4
Quantum Dots: the Kondo Effect
In the late 1980’s it was noticed that a quantum dot with a net spin 12 , tunnel-coupled
to an electron reservoir, has the same Hamiltonian as a magnetic impurity inside a
metal [43, 44] and should thus exhibit Kondo physics. About a decade later, the
Kondo effect was experimentally observed in a lithographically defined quantum dot
containing an odd number of electrons [50] with the 2DEG electrical leads playing the
role of the metal. In bulk systems, the formation of the Kondo screening cloud at low
temperatures enhances the scattering of conduction electrons by the local magnetic
site and thus leads to a suppressed conductance. In quantum dots, the Kondo state
has the opposite effect: because transport is dominated by tunneling directly though
the site, the enhanced scattering at low temperature gives rise to an extra mechanism
for transport that enhances the conductance [21].
Quantum dots are particularly attractive for studying the Kondo effect because it
is possible to study the effect of a single impurity rather than a statistical average over
many impurity sites and because parameters of the Hamiltonian such as impurity-lead
coupling or the energy of the impurity relative to the Fermi level can be independently
controlled via gate voltages. Depending on the number and the type of degeneracies
of a quantum dot state, a variety of Kondo correlations can be measured. Here are
three examples of Kondo states that are not exclusively based on a spin
1
2
degeneracy.
1. When the electronic state of a system contains both orbital and spin degeneracy,
the same delocalized lead electrons that magnetically screen the unpaired spin
39
40
CHAPTER 4. QUANTUM DOTS: THE KONDO EFFECT
can also electrostatically screen the orbital degeneracy. The two degrees of
freedom give rise to an SU(4) Kondo effect, which has been observed in vertical
GaAs dots [51] and carbon nanotubes [52].
2. The ground state of a quantum dot with an even number of electrons is usually
a spin singlet at zero magnetic field, where each occupied orbital contains a pair
of electrons. The triplet state becomes the ground state if the exchange energy
gained for parallel spin filling exceeds the orbital energy difference between levels
[53]. Since the reservoir electrons are spin
1
2
they can only partially screen the
S=1 quantum dot, leaving behind a residual spin 21 . A low Kondo temperature
has been predicted for these systems [54] and experiments have not been able
to unambiguously demonstrate a pure spin-1 Kondo effect yet.
3. Another multiple degeneracy can be achieved using an applied magnetic field
to bring the singlet and the triplet state into degeneracy. The Kondo effect
associated with this four-fold degeneracy has been observed in transport measurements through vertical quantum dots [53], lateral quantum dots [55, 56],
and carbon nanotubes [57, 52, 58]. Depending on exact the details of this system, a two stage Kondo effect [59, 60] or a quantum phase transition can be
observed [61]
I will start this chapter with the microscopic description of the Kondo effect in
a spin
1
2
quantum dot coupled to a single reservoir of electrons. I will present the
expectations for the scaling of the conductance with temperature and bias as well
as the experimental findings for the scaling behavior in the non-equilibrium regime.
Finally, I will summarize the procedure for tuning a double dot system though a
quantum phase transition.
4.1. THE SINGLE CHANNEL KONDO EFFECT IN A QUANTUM DOT
4.1
41
The Single Channel Kondo Effect in a Quantum Dot
A localized state that is degenerate and partially filled, coupled to a continuum of
states is well described by the Anderson Hamiltonian. At low temperature an enhancement of the conductance due to the Kondo effect is expected to occur. In a
quantum dot, in the absence of additional degeneracies, the state with an odd number of electrons on the dot is doubly degenerate. In this case, it has a net spin
1
2
and is therefore equivalent to a magnetic artificial atom. The s-d Hamiltonian can be
used to describe the quantum dot coupled to leads:
~DOT
Hint = J~σcond · S
(4.1)
~DOT is the net spin of the quantum dot, while ~σcond is sum of spin operators for
where S
the conduction electrons and J is the antiferromagnetic exchange coupling. The reason
the interaction between the dot electrons and the lead electrons is antiferromagnetic is
because spin-flip cotunneling processes dominate in the odd Coulomb Blockade valley
at low temperatures. Coupling additional leads to the quantum dot modifies the total
coupling of the dot to the leads but does not affect overall Kondo behavior. As long
as Kondo processes can freely exchange electrons between each pair of reservoirs the
leads behave as a single collective reservoir and the traveling electrons are a linear
combination of the different lead electrons [62]. Only the linear combination that has
a non-zero DOS at the dot is coupled to the dot. The other linear combinations are
irrelevant.
The Kondo temperature (eq.1.7) introduced in section 1.4 can be rewritten in
terms of quantities that are more easily controlled and measured in quantum dot
experiments [63]:
TK ∝
√
ΓU e−
πǫd (ǫd +U )
ΓU
(4.2)
where ǫd is the energy of the dot state measured relative to the Fermi energy EF
of the leads and Γ is the rate for electron tunneling on and off the dot. The Kondo
temperature is a way to quantify the binding energy of the Kondo singlet but because
42
CHAPTER 4. QUANTUM DOTS: THE KONDO EFFECT
this state forms gradually, definitions can differ by a constant factor. The definition
used in this thesis is appealing from an experimental point of view: the Kondo temperature is the temperature at which the Kondo conductance has reached half of its
zero-temperature value (as extrapolated from measurements) [21, 50].
Since the spin
1
2
Kondo effect occurs only when the local site is singly occupied
(ǫd < 0 and ǫd + U > 0), the exponent in eq. 4.2 is negative, as expected from
equation eq. 1.7. TK is maximized when the quantum dot state is energetically close
to EF (ǫd ≈ 0) and strongly coupled to the leads (large Γ). However, near the charge
degeneracy points ( ǫd = 0 and ǫd + U = 0), the state is not localized anymore and the
Kondo model will break down. This regime of strong charge fluctuations is referred
to as the mixed valence regime [21]. The values of the charging energy (U) and level
spacing (∆) determine how high TK can be. In semiconductor quantum dots it is in
the range of 0.1 to 1K and around 10K in carbon nanotube quantum dots.
Conductance through a Kondo coupled quantum dot
The Kondo state can be probed by measuring the differential conductance though the
quantum dot G(V, T ) =
dI
dV
. The conductance is related to the transmission elements
of the dot’s scattering matrix. If one of the leads acts as a weak probe, then the
conductance is in fact probing the density of states of the dot. The presence of the
Kondo state manifests itself as a thin narrow resonance at the Fermi energy of the
leads. As a function of gate voltage, at zero applied bias, an enhanced conductance
in the odd valley, when the total spin on the dot is
1
2
appears.
Fig. 4.1 shows how as a function of the occupancy of a dot, the low-temperature
conductance alternates between high (odd valley) and low (even valley) at zero bias,
and decreases away from zero bias. This alternating pattern in conductance values
for consecutive Coulomb Blockade valleys, that is consistent with the presence of the
Kondo effect in the odd but not in the even valley, was first observed in experiments in
lateral quantum dots [50, 64, 65]. One of the disadvantages of the lateral geometry is
that direct transport though the dot can become too small to measure before the dot
is emptied. This makes it impossible to determine the absolute number of electrons
on the dot.
4.1. THE SINGLE CHANNEL KONDO EFFECT IN A QUANTUM DOT
(a)
(b)
40
43
2.0
2
G(e /h)
1.5
1.6
1.2
V
DS
2
(
0
)h/ e( G
µV)
20
-20
1.0
even
odd
even
odd
even
0.8
0.4
-40
-220
-200
-180
V
SP
-160
-140
-120
0.5
-200
(mV)
-150
V
SP
(mV)
Figure 4.1: (a) The conductance though the dot is enhanced in non-consecutive valleys
at zero bias. The vertically tilted high conductance lines correspond to transport
through a resonant level. (b) Horizontal cut through (a) at VS D = 0 indicating the
even and odd Coulomb blockade valleys.
Experiments that showed that the enhancement was indeed in an odd valley were
preformed in vertical quantum dots [66] where the total electron number is determined by counting Coulomb blockade peaks as the quantum dot is emptied by an
increasingly negative gate voltage. An alternative way of counting electrons is to use
a quantum point contact (QPC) fabricated close to the quantum dot, to detect the
dot occupancy [67]. This makes it easier to determine the total number of electrons in
lateral dots. Even when transport through the quantum dot is undetectable in direct
transport measurements the QPC conductance is altered by the electrostatic potential of each electron added to the quantum dot and the exact number of electrons
on the dot can be counted. In small GaAs quantum dots the even-odd conductance
alternation due to spin- 12 Kondo is the rule rather than the exception. If the local
degeneracy is not spin based, this even-odd rule does not apply.
With decreasing coupling to the leads (Γ) the Kondo temperature is reduced
(eq. 4.2) and the high conductance for odd occupancy disappears. When it is lower
than the measurement temperature we recover the Coulomb blockade regime discussed in the previous chapter.
44
CHAPTER 4. QUANTUM DOTS: THE KONDO EFFECT
The effect of temperature, bias and magnetic field
So far we have discussed that the Kondo state is characterized by the even-odd alternation of conductance as a function of the dot occupancy. In addition, the Kondo
state has several other distinctive features: the enhanced conductance of a spin
1
2
Kondo dot has a characteristic temperature dependence, it is suppressed by applying
a finite bias across the dot, and is also suppressed by magnetic field. The magnetic
field splits the local spin degeneracy which can be recovered at finite magnetic field
by applying a bias voltage equal to the Zeeman splitting. This characteristic dependence of the conductance on external parameters (temperature, magnetic field,
voltage across the dot) can be calculated [68]. Even though the three perturbations
have similar effects on conductance, the mechanism is slightly different in each case
and will be discussed in more detail below.
Two regimes can be distinguished depending on whether the perturbations (kB T, eV,
and gµB B) are small or large compared to kB TK . In the high energy regime Kondo
transport shows a logarithmic dependence on energy which corresponds to Kondo’s
perturbative treatment:
G(x) ≈
1
ln2 (x/k
B TK )
x = kB T, eV, gµB B ≫ kB TK
(4.3)
At low energy, the system behaves as a conventional Fermi liquid, with modified numerical parameters [69]. Hence, for kB T, eV, and gµB B ≪ kB TK the Kondo
spectral function and the associated transport through a spin- 21 Kondo dot show a
quadratic dependence on external parameters:
G(x) ≈ G0 (1 − C
x 2
)
kB TK
x = kB T, eV, gµB B
(4.4)
where G0 = G(x = 0) and C are different for the three perturbations. This E 2
dependence of the Kondo spectral function, which reflects the E 2 scattering rate
of quasiparticles in a Fermi liquid, can be probed by varying temperature and by
applying a bias across the dot [50, 64].
Fig. 4.2 (a) shows that for small biases, the Kondo conductance peak measured
4.1. THE SINGLE CHANNEL KONDO EFFECT IN A QUANTUM DOT
(b)
(a)
2.0
1.8
G (e2/h)
1.2
G(e2/h)
45
1.0
0.8
1.6
1/ln2(T/TK)
1.4
1.2
G0-CVP
-40
-20
0
G0-BTP
1.0
0.6
20
40
101
102
T (mK)
Vbias(µV)
Figure 4.2: (a) For low bias, the Kondo resonance has a power law dependence on
bias. (b) Similarly, at low temperatures, the conductance follows a power law that
crosses over at higher temperatures to a logarithmic dependence.
in the small quantum dot falls off with V 2 . The characteristic T 2 dependence at
low temperatures that crosses over to a logarithmic temperature dependence as T
approaches TK can be seen in the conductance measurements shown in fig. 4.2(b).
There is no analytic expression connecting the two regimes but an empirical expression [70] that matches numerical renormalization group (NRG) calculations for spin
1
2
Kondo [21] over the entire temperature range describes the data remarkably well
and is often used to characterize experimental measurements:
G(x) ≈ G0 (
TK′ 2
)s
′ 2
2
T + TK
(4.5)
where s determines the slope of the conductance decrease and matches the slope found
in NRG calculations best for s = 0.22. Here TK′ =
defining TK such that G(TK ) =
G0
,
2
TK
,
(21/s −1)1/2
which is equivalent to
as noted earlier.
As temperature is raised, the conductance in the odd valleys decreases due to suppression of the Kondo effect while the conductance in the even valleys increases due to
thermal broadening of the quantum dot levels. At temperatures above TK , non-Kondo
conductance channels can develop [68], which lead to deviations from equation 4.5.
Some quantum dot devices exhibit parallel non-Kondo conduction channels. These
46
CHAPTER 4. QUANTUM DOTS: THE KONDO EFFECT
are accounted for by adding a temperature independent offset to eq. 4.5 but the
validity of using such an offset in analyzing Kondo behavior is not yet established.
An applied magnetic field also affects the Kondo state because it lifts the degeneracy between the two spin states on the quantum dot. The asymmetry between the
two spin states acts to suppress Kondo correlations. The splitting of the Kondo spectral function with higher magnetic field can be probed by measurement of differential
conductance as a function of bias [50, 64]
Conductance scaling
As mentioned above, temperature, bias and magnetic field affects the Kondo state
in a similar manner. As a consequence, the characteristic exponent of the lowest
order response is identical for the different perturbations. In addition, although the
coefficients of the lowest order response generally depend on system-specific energy
scales, these dependences can usually be eliminated by scaling each perturbation
relative to a characteristic energy. This allows a system that exhibits the Kondo
effect to be described by a single universal scaling function.
Universality refers to the fact that similar behavior can be observed in systems
that seem not to be related. Universal scaling laws describe phase transitions and
critical phenomena. For example, the evolutions of thermodynamic parameters of
certain liquid-gas and paramagnetic-ferromagnetic phase transitions are characterized by identical sets of critical exponents although the underlying forces (van der
Waals and magnetic exchange, respectively) are very different. Similarly, the conductance of Kondo systems in different materials collapses onto a single curve once
each conductance value is scaled appropriately using the Kondo temperature and zero
temperature conductance, TK and G0 , respectively [30, 71].
The generalized scaling relation between temperature and bias can be expressed
as:
(G(V = 0, T ) − G(V, T ))/G0
eV
= F(
)
α
CT
kB T
(4.6)
where F ( keV
) is a universal function that depends on the type of Kondo effect, but
BT
not on the exact details of the Kondo system at hand. The scaling constants G0 and
4.2. THE NON-EQUILIBRIUM SINGLE CHANNEL KONDO EFFECT
C are defined in eq. 4.4. The exponent for a spin
1
2
47
quantum dot is α = 2, but can
take on different values in more exotic Kondo systems [30, 45] which will be described
in the next section. Deviations from universality [72] are expected in certain regimes.
Universal scaling laws governing perturbations that drive a system out-of-equilibrium
and in particular those describing the non-equilibrium Kondo effect have been especially difficult to derive theoretically with high accuracy [73, 71, 74] Previous measurements of single channel Kondo behavior [30, 45] have shown that transport at
zero bias does indeed follow a universal scaling curve in temperature but the nonequilibrium regime has not been closely examined. In the following section we report
measurements of the universal scaling function of the non-equilibrium spin
1
2
Kondo
effect.
4.2
The Non-equilibrium Single Channel Kondo
Effect
In quantum dots, the non-equilibrium Kondo effect can occur when both leads participate in forming the Kondo state. When a bias voltage is applied between the
two leads, neither reservoir is in equilibrium with the quantum dot which drives
the Kondo state out of equilibrium. Theoretical work predicts that universal scaling behavior in temperature and applied bias is maintained in the non-equilibrium
Kondo regime [30, 71, 21] but disagrees about the values of the universal scaling
function coefficients and about how many system-specific scaling parameters are necessary [74]. Reliable low-energy scaling behavior has not been previously extracted
because experimental measurements have focused on the higher energy (eV ≈ kB TK )
regime [50, 64, 70, 75, 45].
This section describes measurements of non-equilibrium transport through a singlechannel spin
1
2
Kondo (1CK) quantum dot at low temperature and bias (see ref. [76]).
Fig. 4.3(a) shows the small dot and the measurement set-up used to determine the
out-of equilibrium conductance. The gate labeled VG tunes the dot to odd occupancy
and controls the energy ǫd of the singly occupied level. The characteristic energy
48
CHAPTER 4. QUANTUM DOTS: THE KONDO EFFECT
(a)
(c)
(b)
(d)
Figure 4.3: (a) The SEM image of the quantum dot and the measurement schematic
where the lead marked NC does not contribute to transport. (b) The zero bias enhancement of G(V,T) across an odd valley at T = 13mK. (c) Temperature dependence
of the Kondo plateau for T = 13 − 205mK at V = 0µV. (d) Temperature dependence
of the Kondo peak for T = 13 − 205mK at VG = −203mV
scales of this dot are given in table 2.2. The mean level spacing is ∆ ≈ 100µeV and
the bare charging energy is U ≈ 1meV. In the Kondo regime the actual value of U
is reduced by a factor of 5 − 10 due to the increased Γ necessary to achieve a large
enough Kondo temperature.
The enhancement of the conductance through the quantum dot at zero bias and
across an odd valley (−ǫd , ǫd + U > Γ) between −220mV and −188mV is shown in
Fig. 4.3(b). The zero bias conductance reaches a maximal value 1.75e2 /h in the
middle region of the Kondo valley. This region from −209mV to −199.5mV is called
the Kondo plateau and the mostly constant conductance indicates that the coupling
4.2. THE NON-EQUILIBRIUM SINGLE CHANNEL KONDO EFFECT
49
(b)
Figure 4.4: (a) The extracted values of G0 and TK across the Kondo plateau, using
eq. 4.5 over 13 − 35mK. (b) The scaled conductance versus T /TK for all measured
temperatures and for all gate voltage points across the Kondo plateau agrees very
well with the solid line given by eq. 4.5.
asymmetry is around 2:1 [73]. In the rest of the odd valley Kondo processes mix with
sequential tunneling processes and give rise to mixed valence behavior. The analysis
presented here is restricted to the conductance measured across the Kondo plateau.
As the temperature is increased from 13 mK to 205 mK the overall Kondo conductance across the odd valley decreases as shown fig. 4.3(c). Conductance as a function
of source-drain bias in the middle of the Kondo plateau shows a narrow peak centered at zero bias, that broadens with increasing temperature (fig. 4.3(d)). To get
G(T,V=0) at each point across the plateau we fit the Kondo peak in bias using a
parabola (see fig. 4.2(a)) and use the maximum of the parabolic fit as the zero-bias
conductance. This improves the accuracy of the fit by eliminating errors from the slow
drift (< 1µV per hour) of the input bias of the current amplifier. The bias range for
the fitting excludes the edges of the Kondo plateau (VG ≈ −209 mV and VG ≈ −200
mV) close to the Coulomb blockade peak where sequential tunneling contributes to
conductance.
We fit the conductance extracted in this manner to the empirical Kondo (EK)
form given by eq. 4.5 and extract the extrapolated zero temperature conductance G0
and the Kondo temperature TK across the whole Kondo valley. From fig. 4.4(a) we
observe that, as predicted by eq. 4.2, the Kondo temperature is larger when the level
50
CHAPTER 4. QUANTUM DOTS: THE KONDO EFFECT
resides close to Fermi level of the leads. In extracting the Kondo fitting parameters
we limit the temperature range for our fits to T < TK /4. In the middle of the plateau
this corresponds to T < 35 mK.
The scaling of the conductance with temperature across the Kondo plateau is
shown in Fig. 4.4(b). The solid line is the EK form and it accurately fits the data up
to T /TK ≈ 0.25 (40 − 60 mK). Above this limit, deviations between the data and the
EK form are above the experimental noise and are probably due to the emergence of
additional transport processes at higher temperatures. The fit at higher temperatures
can be improved by adding a temperature independent offset to the EK form as a
fitting parameter. Similar offsets have been used in previous experiments in quantum
dots and but because they are likely accounting for these additional high temperature
transport processes, this parameter should not be truly temperature-independent.
When an offset of ≈ 0.4 ± 0.1e2 /h across the plateau is included in the fit, the
extrapolated value for G0 and TK decreases by 20 − 30% across the Kondo plateau.
However, the quality of the fit below 40 mK becomes worse when using an offset:
the deviations between the fit and data in the offset fit are twice as large as in the
offset-less fit. Because we are interested in this low energy regime, we extract the
Kondo fitting parameters in the temperature range T < 35 mK ≈ TK /4 (for gate
voltages in the middle of the plateau) without adding an offset to the fits.
Choosing the appropriate fit range for determining scaling exponents is particularly important because the power law behavior of the conductance at low temperatures transitions gradually into the logarithmic dependence at high energy. Fitting too
far out in energy introduces artifacts from higher order terms. The empirical Kondo
form encompasses both the low energy and the high energy regime. To determine how
far the power-law regime extends, we consider the behavior of G0 − G(0, T ) vs. T /TK
and G(0, T ) − G(V, T ) vs. eV /kTK on log-log plots, as shown in fig. 4.5. We limit
the range to regions where the conductance traces follow quadratic behavior (dotted
line) to within the scatter of measured points (T /TK ≈ 0.11 and eV /kTK ≈ 0.5, as
noted by the vertical dashed lines in fig. 4.5).
To check that single channel Kondo, predicted to be a Fermi liquid, does indeed
follow a scaling law with quadratic exponents in bias and temperature we fit the
4.2. THE NON-EQUILIBRIUM SINGLE CHANNEL KONDO EFFECT
51
Figure 4.5: (a) The scaled Kondo plateau conductance versus the scaled temperature T /TK matches a quadratic (gray dashed line) for T /TK < 0.1. (b) The scaled
Kondo plateau conductance versus eV /kTK for T = 13mK matches a quadratic for
eV /KB TK < 0.4 (c) The scaled Kondo peak width vs T /TK across the Kondo plateau
fits a quadratic also over the whole temperature range but the vertical dashed line
marks the limit of the temperature range used to extract the scaling coefficients.
conductance in the low energy range, determined above, to the form:
G(T, V ) ≈ G0 − c′T (kB T )PT − c′V (eV )PV .
(4.7)
Here PT , PV are the exponents and c′T , c′V are the expansion coefficients that characterize the temperature and bias dependence. Although the choice of fit range is
determined by assuming single-channel Kondo behavior, unlike the EK form (eq. 4.5),
eq. 4.7 does not assume quadratic behavior at low temperature. Thus the values of
G0 , TK and the form of the conductance fit do not assume Fermi liquid behavior.
We extract PV by fitting G(T, V ) as a power-law in voltage for |V | < 7µV at
each temperature point below 20 mK. We find that PV is nearly constant across the
Kondo plateau with an average value of PV = 1.9 ± 0.15 (fig. 4.6). This is in good
agreement with the predicted single-channel Kondo exponent of 2.
Extracting PT is more difficult because the power law regime extends only up
to T /TK < 0.1 and for each gate voltage point, there are only a few temperature
points in this range, as seen in fig. 4.5(a). Fitting over this temperature range, and
only considering gate voltage points with at least five temperature points inside this
range, yields a mean value of PT = 2.0 ± 0.6 across the Kondo plateau. These fits
are consistent with temperature and bias sharing a characteristic exponent of 2, as
52
CHAPTER 4. QUANTUM DOTS: THE KONDO EFFECT
Exponent
30
3.0
2.5
(a)
2.0
1.5
Alpha
a
(b)
0 15
0.15
0.10
G
Gamma
(c)
1.0
05
0.5
0.0
-210
-205
VG (mV)
-200
-195
Figure 4.6: (a) Values of the bias Kondo scaling exponent PV across the Kondo
plateau. The horizontal dashed line shows the theoretically predicted single-channel
Kondo exponent, P = 2. (b) and (c) Values for the scaling coefficients α and γ
extracted across the Kondo plateau described in the text.
theoretically expected for the single-channel Kondo effect. The conductance should
be well-described by a two parameter (G0 and TK ) universal scaling function.
To determine to what extent the low energy non-equilibrium conductance G(T, V )/G0
is described by a universal scaling function, F (T /TK , eV /kTK ) and to extract the
characteristic scaling coefficients we assume that the zero bias conductance follows
the EK form as a function of temperature. For each point on the Kondo plateau we fit
the Kondo peak using a low bias expansion that is applicable over a wide temperature
range [77]:
G(T, V ) = GEK (T, 0)(1 −
1+
cT α
eV 2
)
)(
T 2
− 1)( TK ) kB T
cT ( αγ
(4.8)
4.2. THE NON-EQUILIBRIUM SINGLE CHANNEL KONDO EFFECT
53
The coefficient cT ≈ 5.49 is fixed by the definition of TK via eq. 4.5: G(T, 0) =
G0 (1 − cT ( TTK )2 ) The coefficients α and γ characterize the zero-temperature curvature
and temperature broadening of the Kondo peak, respectively, and are independent of
how TK is defined. We limit the temperature range for extracting the coefficients to
points at which the zero bias conductance follows the EK form (T /TK < 0.25).
This quadratic line shape for the tip of the Kondo peak G(T, V ) = G(T, 0)(1 −
(V /W (T ))2) is appropriate because we are interested in the quadratic energy regime.
The temperature-dependent width of the Kondo peak W (T )2 follows a quadratic of
the form aTK2 + bT 2 as shown in fig.4.5(c). At low temperatures the form of eq. 4.8
reduces to a universal scaling function expansion suggested by Schiller et al. [4]:
G(T, 0) − G(T, V )
T
eV
eV 2
T
eV 2
= F( ,
) ≈ α(
) − cT γ( )2 (
)
cT G 0
TK kB TK
kB TK
TK kB TK
(4.9)
Figures 4.6(b) and (c) show the extracted values of α and γ across the Kondo
plateau, fitted over the regime T /TK < 0.2 and eV /kTK < 0.4. The coefficients are
nearly constant across the Kondo plateau and have average values α = 0.10 ± 0.015
and γ = 0.5 ± 0.1 in the middle of the Kondo plateau. Both α and γ increase slightly
on the right side of the Kondo plateau (VG > −199.5 mV). This may reflect the
expected break-down of the universal scaling relation in the mixed valence regime
where charge fluctuations are present in addition to Kondo processes.
Other fitting functions that reduce to a quadratic at low bias match the data over
a wider bias range but the extracted coefficients remain nearly constant across the
Kondo plateau. This indicates that these coefficients are indeed universal. Two forms
that fit over wider ranges of bias are the Lorentzian G(T, V ) = G(T, 0)/(1 + ( WV(T ) )2 )
and the modified quadratic G(T, V ) = G(T, 0)(1 −
V2
),
(W (T )2 +Q(T )2 V 2
where W (T )
and Q(T ) are free parameters for each temperature and gate voltage point. The
average values of the coefficients across the Kondo plateau were α = 0.12 ± 0.015 and
γ = 0.7 ± 0.1 for both fits. The modified quadratic fits the experimental data over
the widest bias range (±15µV in the center of the Kondo plateau) but there is little
physical basis for using this function. A Lorentzian has been commonly used to fit
Kondo peaks in the experimental literature but theoretical work indicates that the
54
CHAPTER 4. QUANTUM DOTS: THE KONDO EFFECT
true peak shape is more complex.
The reported error bars correspond to 68% confidence intervals for the extracted
scaling exponents and coefficient. They account for statistical errors such as random
measurement noise as well as systematic errors from the uncertainty in temperature
calibration and from the choice of fitting range in bias and temperature. The error
bars for the average exponent and coefficient values across the plateau arise mainly
from the systematic errors, which do not average out across the plateau. The temperature values are based on the width of temperature broadened Coulomb Blockade
peaks and we assume the temperature error (±0.7 mK) at each temperature point is
random. The extracted exponents and coefficients depend slightly on the temperature
and bias fit ranges. This dependence is included in the error analysis by examining
the variation caused by changing the fit ranges by one data point in each direction.
Hence, the error bars reflect the effects of varying the bias fit range by ±13% and
the temperature fit range by ±20%. The error due to treating the Kondo peak as a
quadratic was discussed earlier.
The presence of an offset in fitting the EK form to our data modifies the coefficient
values but not the exponent values. As mentioned before, the scaling parameteres G0
and TK are smaller when an offset is included, which results in an overall decrease in
the extracted values of α and γ. The coefficients show the same qualitative behavior
across the Kondo plateau as seen in fig. 4.6 of the main text, but have average values
α = 0.08 ± 0.015 and γ = 0.24 ± 0.07. Since the G0 and TK values extracted by
including a temperature independent conductance offset do not accurately reflect
behavior in the low temperature limit, this effect is not included in the analysis.
To see that low energy transport through a Kondo dot in the Kondo regime
is well-described by the universal scaling function given by eq. 4.9, we plot the
scaled conductance 1 − G(T, V )/G(T, 0)/αV′ versus (eV /kTK )2 , where αV′ = cT α/(1 +
cT ( αγ ( TTK ))2 ), using the average values of α and γ across the Kondo plateau from
fig. 4.7. The conductance data across the Kondo plateau for T /TK < 0.6 collapse
onto a single universal curve for bias values up to (eV /kTK )2 ≈ 0.5.
Because previous experimentally reported measurements of the Kondo peak in
spin
1
2
quantum dots did not investigate the low bias power-law regime [14, 16, 22,
4.2. THE NON-EQUILIBRIUM SINGLE CHANNEL KONDO EFFECT
(b)
~
G/!V
G(e2/h)
0.5
1.5
1.75 (a)
55
1.5
0.4
1
0.3
05
0.5
1.25
0.2
0
1.0
0.1
-20 -10
0
10
V ( V)
20
-2
-1
0
1
±(eV/kTK)2
2
T/TK
Figure 4.7: (a) Conductance as a function of bias for 0 < T /TK < 0.6 for six gate
voltages across the Kondo plateau. (b) The scaled conductance versus (eV /kTK )2
using the average values of the extracted scaling coefficients in the middle of the
Kondo plateau. The solid line shows the associated universal curve described by
eq. 4.8.
24] and measured the full-width at half-maximum (FWHM) of the Kondo peak at
a fixed values of TK , a comparison to our results is informative but not precise. By
approximating the Kondo peak in ref. [75] as a Lorentzian we can estimate that
α ≈ 0.25 and γ ≈ 0.5 in the middle of the Kondo plateau for this experiment. From
transport measurements through magnetic impurities coupled to highly asymmetric
leads, we extract α ≈ 0.05 and γ ≈ 0.1 in ref. [77] and α ≈ 0.15 and γ ≈ 0.5 in
ref. [33] but these experiments probe the equilibrium rather than non-equilibrium
spectral function. These values are comparable to our measured values but the wide
variation underlines the importance of restricting the fitting range to low energies for
extracting meaningful coefficients.
Existing theoretical calculations for non-equilibrium transport through a spin
1
2
Kondo quantum dot are based on either the Anderson [74, 78] or Kondo models [71,
68], and focus mainly on determining α. The Anderson model (eq. 1.6) predicts
α ≈ 0.15 in both the strong coupling non-equilibrium [74] and equilibrium limits [68].
This indicates that the Kondo conductance we observe decreases with bias more slowly
than predicted by the Anderson model calculations. Other theoretical treatments [78,
79, 80] show a greater level of disagreement with our results.
56
CHAPTER 4. QUANTUM DOTS: THE KONDO EFFECT
One possible explanation for the difference between our results and the theoretically predicted values is that the non-equilibrium calculations overestimate how
quickly Kondo-processes diminish with increasing bias or that additional non-Kondo
transport processes, such as inelastic co-tunneling, add a component to the biasdependent conductance in our measurement. By extracting scaling parameters at
different dot (U, ∆) and coupling (Γ) parameters for identical TK and G0 values,
the contribution from such additional transport processes could be experimentally
investigated.
Measurements of non-equilibrium transport through a single-channel spin
1
2
Kondo
quantum dot are consistent with a quadratic power-law at low energies, as theoretically predicted for the single-channel Kondo effect. The conductance is well-described
by a universal scaling function with two scaling parameters: the Kondo temperature
TK and the zero temperature conductance G0 . The scaling coefficients α and γ are
constant along the Kondo plateau and generally agree with calculations using the
Anderson-model.
4.3
Impurity Quantum Phase Transitions
Quantum phase transitions (QPTs) are a class of phase transitions that occur at
absolute zero temperature when a parameter other than temperature is varied. Second
order (continuous) QPTs are driven by quantum fluctuations of the order parameter,
which have properties that are completely different from those of the familiar thermal
fluctuations. The inherent zero-temperature nature of the QPT makes it impossible to
observe directly. However, in the low-temperature limit, correlation lengths and times
diverge near the transition between two groundstates. These long-range correlations
influence the behavior of the system at finite temperature: near the quantum critical
point (QCP), a distinctive set of excitations can be accessed experimentally. These
excitations are collective, so that Fermi-liquid theory fails to describe the physics in
the quantum critical region. The behavior of the system as a function of external
parameters obeys scaling laws with non-trivial exponents that are determined only
by the universality class of the transition and not by the microscopic details.
4.3. IMPURITY QUANTUM PHASE TRANSITIONS
57
We generally think of second-order phase transitions (classical or quantum) as
requiring the thermodynamic limit of system size. However, for a special kind of QPT
involving a boundary (e.g. an interface or impurity) embedded in a bulk system,
only the degrees of freedom belonging to the boundary become critical, and the
thermodynamic limit is only required for the bulk part of the system. Boundary phase
transitions show the same fascinating quantum critical behavior as bulk transitions.
While the entropy at a bulk QCP vanishes at zero temperature, an impurity QCP
can have residual entropy: fluctuations are strong enough to preserve some of the
local degrees of freedom.
The two channel Kondo model
A generalization of the system described in the previous section, the two channel
Kondo (2CK) system, corresponds to the QCP in a boundary QPT between two
distinct single channel Kondo states. The two channel Kondo Hamiltonian:
H2CK = J1 σ1 · S + J2 σ2 · S,
(4.10)
describes a quantum impurity coupled to two reservoirs: J1 , J2 > 0 are the antiferromagnetic interaction between the local spin S and the spins of the reservoir electrons
σ1 and σ2 . When J1 = J2 the ground state is an exotic non-Fermi liquid state with
an overscreened local moment and a residual entropy at zero temperature. When
one channel is more strongly coupled the traditional Kondo screening behavior is recovered. This model has been used to explain the experimentally observed specific
heat anomalies in certain heavy fermion materials [26, 27, 28] as well as transport
signatures in metallic nano-constrictions [29, 30].
There are several theoretical proposals for the realization of the 2CK effect where
the local degeneracy is based on the charge degree of freedom. The earliest proposal
for observing the 2CK effect in semiconductor nanostructures [81] involved a large
semiconductor quantum dot coupled via single-mode point contacts to a reservoir.
At the charge degeneracy points of the dot, strong charge fluctuations are expected
to give rise to a 2CK effect [82, 83]: two successive charge states play the role of the
58
CHAPTER 4. QUANTUM DOTS: THE KONDO EFFECT
local two-fold degeneracy, and the spin-up and spin-down electrons of the reservoir
form the two independent screening channels. Due to conflicting constraints on the
size of the dot this specific proposal may not be experimentally realizable [84].
To overcome this difficulty a single resonant level can be introduced between the
large quantum dot and the reservoir. At the charge degeneracy point of the large
dot, and in the mixed-valence regime of the small dot, a 2CK effect with a non-Fermi
liquid fixed point is expected to occur [85]. Further analysis of the different parameter
regimes of this double dot system predicts several exotic effects such as a line of twochannel fixed points, a continuous transition from spin-2CK to charge-2CK effect [86]
and a SU(4) Kondo effect with a stable fixed point [87].
The two channel Kondo effect in a double quantum dot
A possible implementation of the spin based 2CK model in a quantum dot geometry
was proposed by [88] and has been measured in [45]. The schematic representation of
this realization is shown in fig. 4.8(a). The localized magnetic impurity is represented
by a small quantum dot containing an odd number of electrons. The conduction
electrons that screen this local moment belong to two reservoirs, shown in blue and
in red in fig 4.8. One of the reservoirs (blue) corresponds to the source and drain
leads (“left” and “right” electrons), which, although physically separated, form a
single effective reservoir [43]. The second reservoir (red) is a finite electron bath:
a much larger quantum dot with fixed electron occupancy. It constitutes a second
independent screening channel at low temperature when Coulomb Blockade forbids it
from exchanging electrons with the infinite reservoir.
The QPT that takes place as a function of the relative couplings of the two channels to the small dot is shown in fig. 4.8(b). For equal coupling, the electrons in the
red and the blue reservoirs compete to screen the spin
1
2
electron on the quantum
dot system forming the 2CK state. When one channel is more strongly coupled the
traditional Kondo screening behavior (1CK) is recovered. Thus the two groundstates
on either side of the transition are both the standard Kondo singlet state (left and
right regions in the phase diagram in Fig. 4.8(b)), with a different set of electrons
participating in the screening of the local moment in each phase. In the quantum
4.3. IMPURITY QUANTUM PHASE TRANSITIONS
(b)
(a)
large
quantum
dot
59
crossover
regime
2CK
regime
T, V
1CK
(large dot)
left
1CK
(two leads)
right
0, 0
J
Figure 4.8: (a) Cartoon of a quantum system that realizes the 2CK model. A spin
on the small dot d can influence transport from one blue lead to the other. The red
electron reservoir formed by a large quantum dot will act as another independent
screening channel for the small dot. (b) Phase diagram of the 2CK system, with
two 1CK phases on the two sides of the quantum critical region and the associated
crossover region between them. The vertical axis denotes any energy (temperature,
frequency, bias voltage).
critical region (center region in Fig. 4.8(b)) temperature fluctuations mask the channel anisotropy and the long range correlations of the 2CK groundstate dominate the
behavior of the system, resulting in non-Fermi liquid scaling laws.
For the 2CK model, one can obtain the full scaling function rather than just a
power law approximation valid only at low energies. The conductance scaling relations
presented below can be used to identify the 2CK quantum critical point in the double
quantum dot system of fig. 4.8.
Starting from the Anderson Hamiltonian, the expressions for the conductance
between the two blue leads as a function of external parameters can be derived [89].
60
CHAPTER 4. QUANTUM DOTS: THE KONDO EFFECT
The Anderson Hamiltonian for this system is given by
H =
X
†
εlkσ lkσ
lkσ +
†
εrkσ rkσ
rkσ +
kσ
kσ
+ Ec
X
X
c†kσ ckσ
+
εdσ d†σ dσ
X
εckσ c†kσ ckσ
kσ
+ Und↑ nd↓
kσ
+
X
kσ
†
†
tkl lkσ
dσ + tkr rkσ
dσ + tkc c†kσ dσ + h.c. .
(4.11)
The first three terms represent the electrons with spin σ and momentum k in the
†
†
left and right leads and large dot, respectively. lσk , (lσk
), rσk , (rσk
), and cσk , (c†σk ) are
the annihilation (creation) operators of a free electron in state k with spin σ, for the
left and right leads and the large dot, respectively. The next three terms account for
the the charging energy Ec of the large dot, the quantized energy of the small dot
electrons and the charging energy U of the small dot. We assume that the tunneling
matrix elements tkr , tkl , tkd which characterize the coupling of the small dot to the
two leads or to the large dot are independent of k, and that only one spin-degenerate
level exits in the quantum dot.
When the tunneling matrix element to one of the two leads is much smaller than
to the other and if the coupling to the leads (with the same density of states ν)
2
Γl(r) = πν tl(r) is independent of spin, the conductance through the system is:
G(V, T ) = ν G̃0
XZ
s=↑,↓
dεf ′ (ε − eV ) Im T σ (ε),
(4.12)
where T σ (ε) is the scattering T -matrix and G̃0 is a proportionality constant depend-
ing on the coupling to the leads Γl,r , which is assumed to be independent of spin.
Most two leaded devices exhibit single channel Kondo physics because electrons can
move between the leads which effectively form only one screening channel. This channel then competes with the second screening channel formed by the large dot. For
the specific expressions for the scattering matrix T from ref. [25]
"
r
#
1
eV
πT
G(V, T ) = G0 1 −
F2CK
2
TK2
πT
(4.13)
4.3. IMPURITY QUANTUM PHASE TRANSITIONS
61
The function F2CK is given in [89] and its asymptotes are
F2CK (x) ≈
(
c x2 + 1 for x ≪ 1
√
√3
x for x ≫ 1
π
(4.14)
where c = 0.748336. Using the zero bias conductance G(0, T ) =
1
G
2 0
1−
q
πT
TK2
and the finite bias expression from eq. 4.13 the scaling relation for the conductance
of a quantum dot in the 2CK regime can be expressed as:
2 G(0, T ) − G(V, T )
p
=Y
G0
πT /TK2
|eV |
,
πT
(4.15)
with the scaling function Y (x) = F2CK (x) − 1. For more detailed calculations see
references [89, 90]
An additional note about the single-channel Kondo case
The 1CK model can be viewed as the 2CK model with strongly assymmetric coupling
to the two channels. Above a characteristic temperature scale T∆ [91] (and Toth,
maybe von Delft too.) that depends on how large the asymmetry parameter ∆J =
JBLU E − JRED is, temperature obscures the asymmetry and the system exhibits 2CK
behavior. At temperatures below T∆ the 2CK system will cross over to regular single
channel Kondo behavior. The small dot will be Kondo screened by whichever channel
is more strongly coupled.
We can use the expression for the conductance in in the limit T, eV ≪ T∆ , |tl | ≪
|tr | or vice-versa, for the scattering matrix given by [25]:
G(V, T ) = G0
(
θ(∆J) − sign(∆J)
πT
T∆
2 "
2 #)
3 eV
1+
.
2 πT
and the zero bias conductance G(0, T ) = G0 (θ(∆J) − sign(∆J)
πT
T∆
2
(4.16)
) to obtain the
62
CHAPTER 4. QUANTUM DOTS: THE KONDO EFFECT
scaling relation for the conductance of a quantum dot in the 1CK regime:
3
1 G(0, T ) − G(V, T )
= sign (∆J)
2
G0
2
(πT /T∆ )
eV
πT
2
.
(4.17)
The scaling curves for the 1CK and 2CK implementations of the system in figure
fig. 4.8(a) were used to describe the quantum phase transition observed in ref. [45].
4.4
Tuning to the 2CK Point in a Double Quantum
Dot
Fig. 4.9 shows an SEM image of the double quantum dot device used to tune across
the QPT described in the previous section. The labels for the gates used are indicated
in fig. 2.3. The observed Fermi liquid scaling laws for asymmetric coupling of the blue
and red reservoirs to the quantum dot as well as the 2CK scaling law in the quantum
critical region have been reported in [45].
This section summarizes one procedure that can be followed to tune this device
to the 2CK point. The voltage on gate sp controls the number of electrons on the
small dot and is used to tune to odd occupancy. The gates c1 and c2 (labeled as such
in fig. 2.3) tune the coupling of the small dot to the large dot. The gates sw1 and
sw2 tune the coupling of the small dot to the leads. The voltage on gate sw2 is set
such that the coupling of the dot to the lower blue lead is negligible in comparison to
the coupling to the other lead. As a consequence, the coupling to the blue reservoir
is dominated by the coupling controlled by gate voltage sw1.
By changing the gate voltages on gates sw1, c1 and c2, the conductance of the
small dot in the three different regions of the phase diagram can be measured.
Fig. 4.9(c) shows that a difference of 10 mV in c1 gate voltage affects the relative
couplings of the two reservoirs to the dot enough to change the enhanced conductance (correspondsing to the formation of the 1CK state with the blue reservoir) into
a suppression (corresponding to the formation of the other 1CK state between the
quantum dot and the large dot).
4.4. TUNING TO THE 2CK POINT IN A DOUBLE QUANTUM DOT
63
(c)
(a)
(b)
path 2
path 1
path 3
μV)
Figure 4.9: (a) SEM image of the quantum dot indicating the finite reservoir in red
and the leads in blue. (b) Schematic representation of the current paths for the three
lead measurement. (c)The evolution of the conductance through the small dot as a
the coupling to the finite reservoir becomes larger than the coupling to the leads. The
blue traces showing the enhanced zero bias conductance indicate the Kondo state is
formed with the leads. The red traces showing a dip instead of a peak indicate that
the conductance through the dot is suppressed by the formation of the Kondo state
with the finite reservoir
We start by measuring the small quantum dot in a three lead geometry. This is
accomplished when little or no voltage is applied to gate n, leaving the large quantum
dot open. The three possible current paths in the three lead arrangement are shown in
fig. 4.9(b). By measuring the conductance matrix of this system, the relative coupling
of the three leads can be determined. This has been explained in detail in [49] and is
illustrated in fig. 4.10.
The conductance traces show two Kondo valleys and one even valley for VSP >
−320 mV and VSP < −260 mV. The two traces in each graph are measured simul-
taneously using one lead for the voltage source and the other two for measuring the
current. The total current through the dot in each graph is due to parallel transport
along the two paths, so we can just sum the conductance along the two paths. We
compare the total conductance for each of the three configurations and we see that
there is less transport though paths 1 and 3 than there is through paths 1 and 2 or 2
and 3. The fact that the total conductance is smaller when the voltage source is on
the lower blue lead where paths 1 and 3 originate, indicates that the coupling of this
64
CHAPTER 4. QUANTUM DOTS: THE KONDO EFFECT
(a)
1.0
(b)
path 1
(c)
1.0
path 2
path 2
path 3
path 3
2
2
)h / e( G
)h / e( G
2
)h / e( G
0.5
0.5
0.0
-400
-300
V
SP
0.0
-400
(mV)
1.0
path 1
-300
V
SP
0.5
0.0
-400
(mV)
-300
V
SP
(mV)
Figure 4.10: (a) The conductance along path 1 and path 2. The total Kondo conductance close to VSP = −300 mV is > 1e2 /h. (b) The conductance along path
1 and path 3.The total Kondo conductance close to VSP = −300 mV is < 1e2 /h.
(c) The conductance along path 2 and path 3.The total Kondo conductance close to
VSP = −300 mV is > 1e2 /h
lead to the dot is less than the coupling of the other two leads to the dot.
To determine the exact ratios of the couplings to the three leads, the conductance
in the Kondo valley has to be measured as a function of temperature. At more
negative gate voltages, regular Coulomb blockade is seen. The zero temperature
Kondo conductance through a two leaded dot is maximal when the dot is equally well
coupled to the two leads:
G0 =
where n =
1
2
4ΓS ΓD
e2
sin2 (πndot )
ΓS + ΓD
h
(4.18)
is the occupancy of the dot in the Kondo valley. If the Kondo temper-
ature is large, and the couplings are equal, the conductance in the Kondo valley is
2e2 /h. A smaller value of the conductance can be due to asymmetric coupling or to
a small Kondo temperature or to both. Several iterations will probably be needed to
identify what voltages on gates c1, c2 and sw1 give equal coupling.
After identifying the gate voltages for which one of the blue leads is tunnel coupled
and the other two leads (one red, one blue) are close to equally coupled and are each
able to form the 1CK state with the small dot, the large dot is formed by applying
voltage to gate n. Gate n is far enough from the small dot, that only minor retuning of
the small dot gate voltages should be needed. In this regime, after carefully accounting
4.4. TUNING TO THE 2CK POINT IN A DOUBLE QUANTUM DOT
(a)
(b)
40
65
40
2
2
G(e /h)
G(e /h)
20
0.6
µV)
µV)
20
(
(
0.4
0.6
0
V
DS
V
DS
0
0.4
-20
-20
0.2
0.2
-40
-40
-300
-280
V
SP
(mV)
-260
-300
-280
V
SP
-260
(mV)
Figure 4.11: Transport through the small dot shows charging peaks caused by the
addition of charge to the large dot superimposed on the Kondo resonance for two
different voltages on gates (c1, c2). (a) The enhanced zero bias conductance in the
region −300 mV < Vsp < −280 mV shows the Kondo resonance between the small
dot and the leads. (b) The suppessed zero bias conductance in the region −300 mV <
Vs p < −280 mV shows the Kondo resonance between the small dot and the large
dot.
for all gate capacitive effects, the small dot zero bias conductance should be very
sensitive to the voltage on the gates that control the coupling between the two dots.
Fig. 4.11 shows finite bias transport measurements of the small dot conductance
for c1= −340 mV (left) and c1= −320 mV (right) with the large dot formed. The
gate voltage range spans approximately one odd and two even valleys. As a function
of VSP the number of electrons on the small dot changes by 3. However, the gate sp
is capacitively coupled to the large dot as well. The vertical lines are due to charging
effects on the large dot: every time an electron is added to the large dot du to the
change in VSP , the effective voltage felt by the small dot changes discontinuously.
Over the swept voltage range the number of electrons on the large dot changes by 8.
Fig. 4.11(a) corresponds to the more negative voltage on gate c1 and shows a zero
bias enhancement in the gate voltage region around −290 mV. In the same region,
fig. 4.11(b) which corresponds to the large dot being better coupled to the small dot
compared to (a) shows a dip where the peak at zero bias used to be. This is how the
two single channel Kondo regions of the phase diagram can be identified.
By repeating these types of sweeps we narrow down the values for the appropriate
66
CHAPTER 4. QUANTUM DOTS: THE KONDO EFFECT
gate voltages on gates c1, c2 and sw1 (or sw2, depending on which one of the two
leads has been closed off) to couple the small dot state equally to the two reservoirs
and therefore form the 2CK state. Further tuning to reach the QCP is done using
gate bp, which controls the number of electrons on the large dot and thus the energy
difference between the small dot state and the large dot levels. If the gates c1, c2 and
sw1 have been carefully tuned, it is possible to observe the evolution of the Kondo
enhancement, through the critical point to the suppression of the Kondo state with
the leads as a function of the voltage on gate bp and measure the scaling behavior in
these three regimes [45].
Chapter 5
Quantum Dots: Perfectly
Transmitting QPCs
The low electron density of the 2DEG and the confined spatial dimensions of quantum
dots make them particularly well suited for studying electron interference effects. As
a result, fluctuations of the conductance as a function of magnetic field or bias can
be observed for both closed and open dots.
In the case of a closed dot, connected to leads via tunneling point contacts
(GQPC ≪ 2 e2 /h, N ≪ 1), the conductance is suppressed by Coulomb Blockade
and the effect of interference is to cause fluctuations of the Coulomb blockade peak
heights. These fluctuations are expected to be more pronounced at low temperatures
because the electron-electron interaction induced dephasing is expected to vanish at
T < ∆ for a ballistic system [92] or to decrease as T −2 for a disordered quantum
dot [93].
When many modes are open in the QPCs (GQPC ≫ 2 e2 /h, N ≫ 1) there is
no Coulomb blockade [94, 95, 96] and the conductance is dominated by the universal conductance fluctuations (UCFs) [97, 98]. This open regime is well described
by the semiclassical approximation and by random matrix theory (RMT) (for a review see [99]). The effect of dephasing on these fluctuations can be modeled as an
extra phase randomizing lead that connects to the dot (the fictitious voltage probe
model) [100]. This extra lead allows electrons to escape and then be re-injected with
67
68
CHAPTER 5. QUANTUM DOTS: PERFECTLY TRANSMITTING QPCS
a random phase acquired.
Most experiments have concentrated on the low N regime as the time the electron
spends on the dot is diminished at large QPC openings and an accurate determination
of the amount of dephasing occurring in the dot becomes difficult. The theories that
describe the closed and the open regime are not well applicable in this intermediate
regime and no explicit theories that describe two-leaded quantum dots with one open
mode in each QPC (N ≈ 1) exist. As a consequence, theories that describe the two
limiting cases have been adapted to this intermediate regime, with varying degree of
success.
RMT results without explicit Coulomb interactions [99] have described experi-
ments that investigated the conductance distribution functions and dephasing times
very well [97, 101]. However, the temperature dependence of the dephasing time
extracted from the dot conductance in this regime showed puzzling features: it did
not follow a T −2 law, as naively expected from Fermi liquid theory, and an apparent
saturation at low temperatures was reported [98, 102, 103, 104, 105, 106, 107].
The absence of Coulomb Blockade in these systems has been predicted for the
case of strong dephasing [94, 95, 96]. In the case of a phase coherent quantum dot,
the presence of Coulomb Blockade was predicted for this regime if one of the QPCs
is in the tunneling regime while the other is perfectly transmitting [108]. There is no
theory that describes a quantum dot with two perfectly transmitting QPCs.
Experiments in one leaded dots have confirmed both theoretical predictions. As
the conductance of one QPC is increased from GQPC < 2e2 /h to GQPC = 2e2 /h
and above, the charging energy is reduced [109, 110] and capacitance measurement
indicate that Coulomb Blockade oscillations decrease in amplitude [111] and become
indistinguishable from the noise [112]. A type of Coulomb Blockade that is the
result of interference and is referred to as Mesoscopic Coulomb Blockade (MCB)
was observed in a quantum dot with one perfectly transmitting lead [113].
Experiments in two-leaded dots observed no Coulomb Blockade [114] or only a
weak conductance oscillation with gate voltage [115, 116, 117] that matched the
Coulomb blockade peak spacing in the tunneling regime but did not investigate the
charging energy or confirm that charge is indeed quantized.
5.1. OPEN QUANTUM DOTS
69
I will start this chapter with a brief description of the RMT results from the open
regime that have been adapted to study the dephasing time in a ballistic quantum
dot with two fully-transmitting QPCs. I will then explain how we determine that the
quantum point contacts are open to N = 1 and present the magnetic field, bias and
temperature dependence of the quantum dot conductance as well as the detection
of the quantum dot charge using the response of the adjacent small quantum dot
conductance.
5.1
Open Quantum Dots
In the case of a disordered conductor, when the mean free path is smaller than the
device size (le < L), the interference of the different electron trajectories through the
system gives rise to random but reproducible conductance fluctuations as a function
of parameters such as magnetic field, gate voltage or electron energy. The statistics
of these fluctuations are universal: they are determined only by the symmetries of
the system and scaling parameters and do not depend on material properties [118].
Similar universal conductance fluctuations same are present in open ballistic quantum dots (le > L) as long as the motion of the electrons inside the quantum dot is
chaotic. The physical origin of these fluctuations is the same as in disordered conductors: the quantum interference of different classical paths that the electron can
take, scattering elastically off the disorder potential in one case, or the confinement
potential in the other. These fluctuations are called universal conductance fluctuations [119] and their statistical moments can be used to extract the dephasing time
in these systems [120].
At zero magnetic field the quantum correction to the probability that an electrons
returns to its original position at a given time leads to weak localization (WL) of the
electrons at the impurities in one case or inside the dot in the other. This constructive
interference of time reversed paths is manifested as a suppression of the conductance
at zero magnetic field compared to the finite magnetic field value.
The number of electrons inside quantum dots are usually large enough that a
statistical description of the system is possible. Several methods whose regimes of
70
CHAPTER 5. QUANTUM DOTS: PERFECTLY TRANSMITTING QPCS
validity depend on whether the quantum dot is open or closed have been very successful in explaining equilibrium transport features [121]. The semiclassical description
and RMT are two of the approaches that describe diffusive as well as chaotic ballistic
quantum dots.
In the semiclassical picture we can think of electrons as moving along classical
trajectories. By summing over the classical trajectories weighted correctly, transport
characteristics of the quantum system can be predicted. Two conditions have to be
fulfilled for the semiclassical results to be valid: a large number of modes that couple
the quantum dot to its leads (N → ∞) is required and the energies have to be larger
than the single particle energy level spacing ∆ of the quantum dot. Semiclassical
results cannot be used if N < 3 [99].
The quantum fluctuations in disordered conductors and chaotic ballistic systems,
in the limit of a large number of open modes in the QPC can also be described
using RMT [122]. This technique was initially introduced to describe the statistics of
eigenfunctions and eigenvalues of many-body quantum systems and was successfully
applied to atomic nuclei. Although quantum dots with single mode QPCs have been
referred to as open, results derived in the limit N → ∞ have to be used with particular
care. As described below, some of the results from RMT and from the semiclassical
description can be modified to describe the N = 1 regime.
In the RMT description, the Hamiltonian of the system is chosen to belong to a
Gaussian ensemble of random Hamiltonians and a similar approach can be applied
for the scattering matrix that describes transport through mesoscopic systems [121].
This picture is valid when the only important features of the quantum system are its
symmetries. Depending on what kind of symmetry describes the system, a different
ensemble of Hamiltonians has to be used to obtain statistics. If the system is invariant
under time reversal the Hamiltonian has to be a real symmetric matrix and the
ensemble is called orthogonal. If time reversal symmetry is broken, the Hamiltonian
has to be a complex Hermitian matrix and the ensemble is called unitary. This is the
case when a magnetic field is present. A third ensemble is the symplectic ensemble and
it describes systems with strong spin-orbit interactions (broken rotational symmetry).
RMT is applicable in the regime E < ET h [99]. Its predictions coincide with
5.1. OPEN QUANTUM DOTS
71
the semiclassical results for energies E > ∆ but unlike the semi-classical results, the
RMT predictions are also valid for E ≈ ∆. A long as τd ≫ τcross they can be used for
diffusive quantum dots as well. The conductance distribution functions and statistical
moments such as average conductance and variance can be calculated for the different
ensembles. The lineshape of the average conductance as a function of magnetic field
is found to be approximated by a Lorentzian:
hG(B)i = hGiB6=0 −
A
1 + (2B/Bc )2
(5.1)
where hGiB6=0 is the average conductance at finite field, and A and Bc are the depth
and width of the weak localization dip. In the semiclassical picture this results is
actually exact. If the quantum dot is not chaotic, the lineshape is expected to be
more “V” shaped.
The conductance fluctuations in a chaotic dot as a function of magnetic fields are
described by a correlator whose semiclassical expression is given by:
cG (∆B) = hG(B + δB)G(B)i − hG(B)i2 ∝
1+
1
∆B
2Bc
(5.2)
2
RMT calculations confirm this result in the limit T → 0 [123].
The effect of dephasing on the average conductance and on the variance of the
conductance of a chaotic quantum dot for the case of one or multiple modes in the
QPCs can be accounted for using a spatially distributed voltage probe model [124].
For single mode leads, the analytic expression that connects the size of the weak
localization correction of the average conductance to the dimensionless dephasing
rate γφ (this is the dephasing rate normalized by the escape rate per lead) can be
approximated by [104]:
δg =
1
e2 /h
2N + 1 + γφ
(where N is the number of open spin degenerate modes in the leads, γφ =
(5.3)
2π~
∆τφ
is the
dimensionless dephasing rate). The average conductance is only implicitly affected
by temperature through the dependence of the dephasing time.
72
CHAPTER 5. QUANTUM DOTS: PERFECTLY TRANSMITTING QPCS
The variance of the conductance is affected by both dephasing and thermal averaging. If the temperature is larger than the total level brodening (kB T > ∆(1 + γφ /2)),
it can be well approximated by the expression [101, 120]:
 q
q

45
( 16 + 13 γφ )−2 for β = 1
∆(1 + γφ /2)
Var(G) =
 (√3 + γφ )−2
6kB T
for β = 2
(5.4)
A third method for extracting the dephasing time is based on the conductance fluctuations at high magnetic field where the cyclotron radius is smaller than the size of the
quantum dot. In this case, the electrons are confined to the edges of the quantum dot
and the correlation in magnetic field can be used to determine the dephasing time.
It is also possible to extract the dephasing time from the weak localization dip width
Bc or from the power spectrum of the fluctuations versus magnetic field.
Extracting the dephasing time in quantum dots as a function of temperature
has proven to be rather difficult, despite the many methods available to connect
transport quantities to dephasing. Early experiments on quantum dots [102, 98]
observed a saturation of the dephasing time extracted from the dot conductance at
low temperatures. This saturation was attributed to an extraction method that failed
at low temperatures due to the breakdown of the semiclassical model [98] and to the
discreteness of the dot spectrum at low temperatures [102]
More recent experiments that used RMT calculations to interpret their data [104,
105, 106, 107] still observed a saturation. The experiment in ref. [104] excluded
electron heating effects as a possible cause for the observed saturation. Dephasing
processes that occur in the constrictions connecting the dot to its leads [105, 107]
were proposed as the source of the saturation. As a result, these experiments also
noted a correlation between the temperature at which the saturation occurred and
the temperature at which the coherence time became equal to the dwell time.
The only theoretical prediction for the temperature dependence of the dephasing
time in the 0D regime was given in ref. [93]. For a disordered closed quantum dot the
expected temperature dependence is ∝ T −2 . The same power law has been predicted
for mesoscopic rings [125]. In general, two classes of scattering processes determine the
5.2. THE QUANTUM DOT CONDUCTANCE AT N = 1
73
behavior of the dephasing time: large energy transfer scattering events with E ≥ T
that lead to a square dependence on temperature and small energy exchange processes
with E ≪ T which result in a linear dependence on temperature [14]. Because at low
temperature the Pauli exclusion principle prevents scattering processes with E ≪ T ,
the dominant contribution to dephasing is from scattering processes with E ≈ T .
5.2
The Quantum Dot Conductance at N = 1
The large quantum dot in fig. 2.3 with area 2.6µm2 can be used to study interference
√
and decoherence in the open regime. Because le ≫ A the quantum dot is in the
ballistic regime. The coupling of the electrons in the dot to the electrons in the leads
is controlled via the voltages on gates bw1, bw2 and n. A different set of gates affects
the shape of the quantum dot: c1, c2 and bp are used to gather statistics of the
conductance which offer information about the amount of dephasing present in this
system. In the measured temperature regime (13 mK to 1 K) the dominant processes
that causes dephasing are electron-electron scattering events.
The QPC Plateau
Fig. 5.1 shows how we determine the QPC transmission. In a separate cool-down
(without positive bias) of the device, when voltage is applied only to the gates bw1,
n and bw2, the quantum dot is not formed and clear conductance plateaus quantized
at integer multiples of 2e2 /h are visible at 13mK and at 600 mK. The fact that the
plateaus are not broadened by this increase in temperature indicates that the energy
needed to reach the next QPC subband (and open another transmission mode) is
larger than ≈ 50 µeV
It is more complicated to identify the gate voltage at which the QPCs are open to
one spin-degenerate mode (N = 1 and GQP C1,2 = 2e2 /h) when the dot is formed. In
the absence of interference, the dot conductance is the series conductance of the two
QPCs and as a result a plateau at G = 1e2 /h will appear as the voltages on gates bw1
and bw2 are swept. Interference effects inside the dot can obscure this plateau at low
temperatures and zero magnetic field. Fig. 5.1(b) shows the conductance through
74
CHAPTER 5. QUANTUM DOTS: PERFECTLY TRANSMITTING QPCS
(b)
(a)
8
13 mK
600 mK
G (e 2/h)
6
4
2
0
-800
-600
-400
V bw1(mV)
Figure 5.1: (a) Quantized conductance plateaus of QPC1 at 13 mK and 600 mK (the
600 mK data is shifted horizontally for clarity). (b) The 1 e2 /h conductance plateau
of the quantum dot at T = 960 mK and B = 22 mT. The black circle, square and
triangle mark the three QPC voltage settings used in the measurements.
-3
-3
x10
(a)
dG/d(V
-4
)
bw1
0
x10
4
(b)
-400
dG/d(V
)
bw1
-10
-5
V
(mV)
0
5
10
-400
V
V
2wb
2wb
)Vm(
)Vm(
-500
-500
-400
-300
V
bw1
(mV)
-400
-300
bw1
Figure 5.2: Derivative of the plateau with respect to the gate voltage on QPC1 (a)
and QPC2 (b). The dashed lines indicate the border between high and low values of
the derivative. The (Vbw1, Vbw2) coordinates of the plateau center are marked with a
black circle in (a) and (b), respectively.
5.2. THE QUANTUM DOT CONDUCTANCE AT N = 1
75
the dot with increasing QPC conductance at T = 960 mK and B = 22 mT. A
plateau at 1e2 /h in the dot conductance corresponding to to the 2 e2 /h plateau in
the conductance of each QPC can be seen.
It is difficult to consistently identify the middle of the plateau directly from this 2D
conductance map. Instead we take the derivative (shown in fig 5.2) of the conductance
in fig. 5.1(b) with respect to Vbw1 (fig. 5.2(a)) and Vbw2 (fig. 5.2(b)) and identify the
transition between regions with small and large values of the conductance derivative.
The region with a low value for the conductance derivative corresponds to a low slope
in conductance. The center of this region in both graphs is marked by the black dot
in fig 5.2. The corresponding coordinates are indicated by a black dot in fig 5.1. The
two other markers, (black square and triangle) identify adjacent QPC voltage settings
that should also be be very close to perfect QPC transmission.
From here on, unless otherwise noted, the data shown is taken with the QPCs
tuned to one of the three indicated settings. How closely they correspond to perfect transmission will be determined via a second method that is based on the dot
conductance averaged over different dot shapes.
Universal Conductance Fluctuations and Coulomb Blockade
The conductance through the dot shows large aperiodic variations in G on the scale
of tens of mV in gate voltage around e2 /h, also known as UCFs, as a function of
magnetic field and gate voltage. At zero temperature, the size of the fluctuations
should be 1e2 /h. The magnetoconductance for a given shape of the dot is shown in
fig. 5.3a at 15 mK and at 720 mK. At a fixed magnetic field, changing the shape
of the dot via the gate voltage c2 also leads to fluctuation at low temperatures (80
mK) as seen in fig. 5.3b. In comparison, the fluctuations at higher temperatures are
drastically reduced by decoherence and thermal broadening.
At the lowest temperatures we see strong periodic oscillations with gate voltage
tuning, superimposed on the UCFs. This is illustrated in fig. 5.4(a). The periodic
oscillation is easily distinguishable from the aperiodic UCFs due to the very different
scale of their variation.Finite bias measurements show a suppression of the conductance around zero bias, reminiscent of Coulomb blockade diamonds, is associated with
76
CHAPTER 5. QUANTUM DOTS: PERFECTLY TRANSMITTING QPCS
(a)
(b)
2.0
720 mK
80 mK
1.2
Gdot (e /h)
2
2
Gdot (e /h)
720 mK
15 mK
1.0
1.0
0.8
0.0
10
20
B (mT)
-700
-600
Vc2 (mV)
Figure 5.3: Dot conductance as a function magnetic field (a) and gate voltage (b) for
GQPC1,2 = 2 e2 /h shows universal conductance fluctuations around 1 e2 /h at low temperatures (blue line). The magnitude of the UCFs is decreased at high temperatures
(red line).
the oscillations in gate voltage (fig.5.4(b)).
To establish that this periodic oscillation is indeed Coulomb blockade, we measure
the Coulomb Blockade peak spacing when the QPCs are in the tunneling regime in
fig 5.5(a). This spacing ∆Vc2 = 1.5 mV coincides with the period of the conductance
oscillations at at N = 1. A zoom-in of the oscillation in the open regime is shown for
comparison in fig 5.5(b).
We also follow the evolution of the Coulomb diamonds from the tunneling regime
to above the middle of the plateau in fig 5.5(c) The voltage on gate n affects the
QPC conductances and changes the number of electrons on the dot. For Vn . −370
mV neither QPC is fully transmitting and we see clear Coulomb diamonds with
a charging energy of ≈ 115 µeV. As Vn is made less negative, the conductance of
the QPCs increases and the effective capacitance of the dot increases. This causes
U =
e2
C
to be renormalized [109, 110]. We observe in fig 5.5 (c) that the vertical
size of the diamonds shrinks with increasing QPC conductance, but does not vanish
even at Vn ≈ −315 mV where the QPCs are fully transmitting. These diamonds are
superimposed on larger UCFs which form a Fabry-Perot pattern in gate voltage and
bias [126].
To determine the size of the renormalized charging energy, the Fabry Perot bias
5.2. THE QUANTUM DOT CONDUCTANCE AT N = 1
77
1.0
2
Gdot (e /h)
(a)
0.5
-700
-650
-600
V c2 (mV)
Vds (μ V)
(b)
d I/d Vds
2
(e / h)
20
0.8
0
0.6
-20
0.4
-625
-620
Vc2 (mV)
Figure 5.4: (a) Dot conductance at 13 mK and B = 0 shows both UCFs and Coulomb
blockade oscillations with gate voltage. (b) Coulomb blockade diamonds with a small
charging energy U ∗ = 16µeV for the QPCs at N = 1 (dashed white lines are guides
to the eye).
CHAPTER 5. QUANTUM DOTS: PERFECTLY TRANSMITTING QPCS
(a)
(b)
0.4
0.2
0.0
1.0
2
2
Gdot (e /h)
0.6
Gdot (e /h)
78
-444
-442
-440
0.8
0.6
-438
-452
-450
-448
-446
Vc2 (mV)
Vc2 (mV)
(c)
d I /dVds
40
2
Vds (μV)
(e / h)
1.0
0
0.5
-40
0.0
-380
-360
-340
-320
-300
Vn (mV)
Figure 5.5: (a) The Coulomb blockade peak spacing in gate voltage when the QPCs
are in the tunneling regime. (b) The period of the conductance oscillation when the
QPCs are at N = 1 as a function of the same gate voltage as in (a). (c) Conductance
as a function of Vds and Vn at B = 0 and T = 13 mK shows the transition between
Coulomb blockade in the closed regime to a Fabry-Perot pattern in the open regime.
5.2. THE QUANTUM DOT CONDUCTANCE AT N = 1
0.5
-40
d I /dVds (e 2/h )
0
0.1
0
0.05
0.00
-40
Δ d I /dVds (e 2/h )
40
1.0
Vds (μ V )
0.15
(b)
40
Vds ( μV )
(a)
79
-0.05
-340
-320
V ( μV)
-300
-340
-320
V ( μV)
-300
Figure 5.6: (a) The Fabry-Perot pattern of the bias and gate voltage dependence
of the UCFs for the QPCs at N = 1 obtained by averaging over the period of the
Coulomb diamonds. (the original data in fig. 5.5) (c) By subtracting the FabryPerot from the original data, the Coulomb diamond component of the finite bias
measurement (dashed white lines are guides to the eye) becomes apparent. The
renormalized charging energy is U ∗ = 16 mV.
dependence has to be subtracted. Fig. 5.6(a) shows the broad underlying Fabry
Perot pattern that is obtained when the conductance is averaged over the period of
the Coulomb diamonds, ≈ 3 mV in gate voltage. When this background is subtracted
from the raw data, the component corresponding to the Coulomb blockade oscillation
is isolated and a renormalized charging energy can be extracted from fig. 5.6b: U ∗ ≈
16 µeV at Vn = −315 mV.
The corresponding periodicity in gate voltage as well as the smooth evolution of
the Coulomb diamonds from the tunneling regime to the open regime suggest that
we observe Coulomb blockade oscillations in this quantum dot at low temperatures
when the QPCs are open to allow one spin-degenerate mode to pass though.
The presence of Coulomb Blockade can be due to two factors: finite reflection of
the QPCs or coherent effects. By investigating the behavior of the average conductance though the quantum dot as a function of temperature we can determine the
amount of coherence that is present as well as whether the QPCs are indeed perfectly
transmitting.
The Ensemble Average and Variance
To determine the conductance average, we have to change the shape of the dot controlled by the voltage on gates c1 and c2. Fig. 5.7 shows the conductance through
80
CHAPTER 5. QUANTUM DOTS: PERFECTLY TRANSMITTING QPCS
1.0
ρG(ΔVc2)
B= 0 mT
B= 30 mT
0.5
0.0
-100
-50
0
50
100
Vc2 (mV)
Figure 5.7: (a) Dot conductance at 720 mK and 22 mT as a function of VS1 and VS2 .
The black dots are spaced by 21 mV in gate voltage apart and show the 196 points
used for averaging. (b) The conductance correlation function in gate voltage c2 at 13
mK. The rapid decrease of the correlation indicates that conductance measurements
spaced more than 10 mV apart are independent.
the dot for a 300 mV change in each c1 and c2 voltage at high temperatures and
finite magnetic field where the average conductance is 1e2 /h and small UCFs are visible. To pick a subset of independent shapes from this range, we investigate the low
temperature correlation function. An example of the correlation along c2 is shown in
fig. 5.7(b) for the lowest temperatures. The width of the approximately Lorentzian
correlation indicates that a change of ∆Vc1 , ∆Vc2 > 10mV is needed to modify the
shape of the dot to the extent that a new set of trajectories determines conductance.
At higher temperatures the correlation length will be longer because thermal averaging will mix the trajectories more. Thus, we pick gate voltage that are spaced 21 mV
apart in c1 and c2 as indicated by the black dots superimposed on the conductance
map in Fig. 5.7 (a).
The chosen gate voltages have to form an ensemble of independent and identically
distributed dot shapes. As a consequence, the area of the dot has to remain constant
and the gate voltage has to only affect the shape. Otherwise, it is not a true parametric
fluctuation. A gate voltage that affects the area has a similar effect to energy and acts
as an intrinsic parameter giving rise to slightly different fluctuations. We determine
the area of the dot for the extreme values of c1 and c2 voltage from the magnetic
5.2. THE QUANTUM DOT CONDUCTANCE AT N = 1
81
field dependence of the dot conductance in the regime where the electron trajectories
are skipping around the edge of the dot. The Fourier spectrum of the conductance at
high magnetic field shows that certain oscillation frequencies, those that correspond
to the enclosed area, predominate. They can be used to determine that over the
shape gate voltages used the area of the dot changes by less than ±5% of 2.6 µm2 .
An additional complication is introduced by the capacitive coupling of all the
gate to the QPC conductance. Although the gates c1 and c2 are far from the QPCs,
a 300 mV change in gate voltage corresponds to an effective detuning of the QPC
by 10mV in bw1 or bw2 voltage. This is not enough to move the quantum dot
off the plateau, but it is enough to decrease the average conductance. As such, we
compensate for the effect on the QPCs of changing c1 and c2 by simultaneously
trimming the voltages on bw1 and bw2 to keep the quantum dot in the middle of
the plateau. We identify the middle of the plateau for five voltage combinations on
c1 and c2 ((−500, −500), (−500, −800), (−650, −650), (−800, −500), (−800, −800)
mV) and use a linear interpolation to identify the correct bw1, bw2 voltage values to
remain at N = 1 for the intermediate values in c1 and c2 voltages .
The conductance as a function of magnetic field, averaged over the ensemble of
196 dots is shown in fig. 5.8(a). The dip in the average conductance at zero magnetic
field δg is the signature of coherent backscattering and is due to weak localization.
Its width is the field scale necessary to break time-reversal symmetry, which at zero
temperature corresponds to the magnetic field needed to thread one flux quantum
through the dot: BΦ = Φ0 /ADOT = 1.6 mT. We extract δg and as a function of
temperature by fitting a Lorentzian to the curves.
hG(B)i = hGiB6=0 −
δg
1 + (2B/Bc )2
(5.5)
where the fitting parameters hGiB6=0 ,δg = ∆hGiB6=0 − hGiB=0 and Bc are the average conductance at finite field, the depth and width of the weak localization dip,
respectively.
From the three temperature curves displayed, we see that the dip has a strong
82
CHAPTER 5. QUANTUM DOTS: PERFECTLY TRANSMITTING QPCS
temperature dependence shown in fig. 5.8(b). This temperature dependence is entirely due to the changing coherence time. The size of the dip is expected to reach
1
3
e2 /h at T = 0. The continued increase of the conductance difference at the lowest
temperatures indicate that the coherence in the system also continues to increase. To
determine the coherence time using eq. 5.3 we have to determine the area of the dot
and the reflection coefficient.
We determine the area of the dot from the higher magnetic field oscillations of the
conductance. We expect the reflection coefficient to be r ≈ 0 as the QPCs have been
tuned to N = 1. However, even at the optimal QPC settings there may still be a
small reflection coefficient in the QPCs. The effect of the reflection coefficient on the
dot conductance can be observed in the conductance at finite magnetic field where
weak localization is absent. To ensure that our results are not affected by any small
imperfections in the QPCs we measure the ensemble conductance at three different
QPC settings on the plateau.
To determine the reflection coefficient from the temperature dependence of the
finite magnetic field average ∆hGiβ=2 we use the prediction from ref. [95] for the case
of a phase coherent dot with 4N ≫ 1. We find the difference:
∆hGiβ=2 = hG(T )iB6=0 − hG(T0 )iB6=0
(5.6)
where T0 is the highest measured temperature point and use
∆hGiβ=2 =
r2
T
ln
2
T0
(5.7)
to extract an overall reflection coefficient. In this equation the reflection coefficients
r 2 of the two QPCs are assumed to be equal and are defined by GQPC = 2e2 /h (1−r 2 ).
The solid black line in fig. 5.8(c) shows the results of fitting the data to eq. 5.7. From
the fit we obtain r 2 ≈ 2% for QPC setting A (triangle) and B (circle) and r 2 ≈ 1%
for QPC setting C (square marker).
Fig. 5.8(c) shows that there is a very small temperature dependence of the dot
conductance, for all three datasets corresponding to the three different locations on
5.2. THE QUANTUM DOT CONDUCTANCE AT N = 1
83
(a)
2
<G> (e /h)
1.0
935mK
310mK
36mK
0.8
0.6
0
10
B (mT)
(b)
20
(c)
0
∆<G>β=2 (e /h)
0.2
2
2
∆<G>β=1 (e /h)
0.3
0.1
0.0
-20
point A
point B
point C
-3
2
3
4 5 6
2
100
T(mK)
3
4 5 6
-40x10
2
10
3
4 5 6
2
100
T(mK)
3
4 5 6
1000
Figure 5.8: (a) The dot conductance (QPC setting C) averaged over 196 different
dot shapes as a function of magnetic field at three different temperatures. The solid
lines are the Lorentzian fit to the data. (b) The size of the dip at zero magnetic
field as a function of temperature. (c) The difference between the finite field average
conductance and the highest temperature average conductance value as a function of
temperature. The solid black line is a fit to the data (QPC setting A) used to extract
the QPC reflection coefficient as described in the text. Each marker indicates one of
the three measured QPC settings.
84
CHAPTER 5. QUANTUM DOTS: PERFECTLY TRANSMITTING QPCS
the QPC plateau. This indicates that indeed a small residual reflection coefficient
is still present in our QPCs. Because the reflection coefficients are on the order of
2% or less, effects that are higher order in r 2 should be suppressed in the average
conductance. The variance of the conductance should be more sensitive to a non-zero
reflection coefficient. The magnetic field dependence of the conductance variance for
three different temperature values is illustrated fig. 5.9(a).
The variance depends on both temperature and dephasing. The effect of temperature can be isolated from the variance according to eq. 5.4 because the effective
level-broadening ∆′ = ∆(1+γφ/2) of our system is smaller than 0.6 kB T for the entire
temperature range. We use the dephasing times determined from the average conductance (as shown in the following section) to calculate the corresponding variance
and compare to the measured variance.
The ratio of the zero magnetic field variance to the finite field value is independent
of temperature and is expected to increase with increasing dephasing. In fig. 5.9(b)
we observe a behavior that is similar to that observed in previous experiments [101]
where the measured ratios can be up to a factor of two larger or smaller than the
ratio expected from the dephasing rate corresponding to the weak localization signal.
Fig. 5.9(c) shows the temperature dependence of the zero and finite magnetic
field conductance variance as well as the calculated variances based on eq. 5.4 for the
dephasing rates extracted from the weak localization correction (red and blue lines).
They follow the same temperature dependence, although a factor of 2 difference is
also observed, most of which is accounted for by the error bars. The error bars have
been determined from the statistical error of the ensemble in the case of the variance.
The calculated variances also have an error stemming from the Lorentzian fit error
as well as from the statistical error.
An important source of error in these kinds of electrical measurements is that the
temperature of the electrons is not measured directly, but using a nearby calibrated
thermometer to indicate the corresponding electron temperature. Even if at higher
temperatures the change in thermometer temperature corresponds to the change in
electron temperature, at low temperatures it becomes difficult for electrons to cool
(less phonons to dissipate energy) and the electron temperature may saturate even
5.2. THE QUANTUM DOT CONDUCTANCE AT N = 1
85
(a)
960 mK
310 mK
121 mK
-2
4
2
VarG (e /h )
2x10
1
0
0
(b)
10
B (mT)
(c)
4
-2
10
3
4
2
VarG (e /h )
VarGβ=1/VarGβ=2
20
-3
VarG (β =1)
VarG β =1 (WL)
VarG ( β =2)
VarG (β =2) (WL)
10
2
-4
10
5 6
2
1
3
4 5 6
γφ
2
10
3
4 5
3
4
5 6 7 8
2
100
T (mK)
3
4
5 6 7 8
1000
Figure 5.9: (a) The variance of the conductance (at QPC setting C) of the ensemble as
a function of magnetic field for three different temperatures. (b) The ratio of the zero
magnetic field and finite magnetic field variance as a function of the dimensionless
dephasing rate. The solid line is the calculated ratio based to the dephasing times
extracted from the average conductance. (c) The measured conductance variance as
a function of temperature. The solid lines are the calculated variance based on the
dephasing time extracted from the average conductance.
though the thermometer temperature continues to decrease. Due to its direct relation
to the electron temperature, the variance and its continued increase with decreasing
temperature shown in fig. 5.9(c) indicates that potential electron heating effects are
small.
86
CHAPTER 5. QUANTUM DOTS: PERFECTLY TRANSMITTING QPCS
5.3
The Dephasing Time in a Quantum Dot at N =
1
The extraction method of τφ based on eq. 5.3 is very sensitive to the exact tuning of
the QPCs [127]. For large dephasing, the presence of a small reflection coefficient does
not significantly affect the accuracy of its extraction from eq. 5.3, but for dephasing
rates of the order of the escape rate, the reflection coefficient has to be taken into
account [127].
To ensure that we have accounted in a consistent manner for the small deviations
from perfect transmission of the QPCs, we measure the ensemble conductance at three
different QPC settings (black circle, square and triangle in Fig. 5.1(b)) and extract
the dephasing time from δg using an analytic equation that includes the effects of a
reflection coefficient (r 2 = 2% for QPC settings A and B and 1% for QPC setting C)
to first order [private comm. P. Brouwer].
By carefully tuning the QPC to be as close as possible to perfect transmission
and by using an extraction method that accounts for the remaining imperfection we
extract a dephasing time that does not saturate as shown in fig. 5.10(a) and thus
differs from previous experimental results [104, 107] in its low temperature behavior.
Th error bars account for the finite size of the ensemble, for the fitting error and for
a 5% error in determining the dwell time, stemming from the error in the area of the
dot and therefore ∆.
The power law we observe has a large contribution from a dephasing mechanism
that is linear in temperature and a small addition from a T −2 power law. This does
not match the theoretical expectation of a T −2 power law for a 0D system [93, 125]:
~
32∆ 9 kT
=
τφ
π 4 ET h
(5.8)
Since there is no exact expression for the dephasing time in a 0D ballistic system
(other than the Fermi liquid prediction) and eq. 5.8 is for a diffusive closed quantum
dot , we compare to the 2D case as well. The 2D behavior in the disordered case is
dominated by a small energy transfer between electrons which results in a dephasing
5.3. THE DEPHASING TIME IN A QUANTUM DOT AT N = 1
(a)
(b)
6
5
3
point A
point B
point C
4
3
2
-9
2
10
-9
9
8
7
6
5
9
8
7
6
4
τφ(s)
τφ (s)
10
87
5
3
4
2
3
-10
2
10
-10
10
9
8
7
6
10
9
8
7
6
point A
point B
point C
WL fit
5
4
2
3
4 5 6
100
T (mK)
2
3
3
4 5 6
1000
10
2
3
4 5 6
2
100
T (mK)
3
4 5 6
1000
Figure 5.10: (a) The dephasing time as a function of electron temperature extracted
from the B=0 conductance dip for the three different QPC settings. The red line is a
fit of the combined datasets discussed in the text. (b) The dephasing time extracted
from the conductance variance for the three different QPC settings as a function of
electron temperature. The red line is a fit to the data for QPC setting B from 1 K
to 60 mK (= TCB ). The dashed black line is the fit from (a).
88
CHAPTER 5. QUANTUM DOTS: PERFECTLY TRANSMITTING QPCS
time inversely proportional to temperature:
−1
τphi
=
πlelastic
k B T λF
ln
2π~ lelastic
λF
(5.9)
where λF is the Fermi wavelength and lelastic is the mean free path. For ballistic
transport in 2D, Fermi liquid physics [128] is expected to determine the behavior of
the dephasing time which is then determined by the Fermi energy of the electrons EF
and temperature [129]:
−1
τphi
=
π (kB T )2
EF
ln
4 ~EF
kB T
(5.10)
We fit the data to a combination of these two laws τφ−1 = aT + bT 2 . For the
parameters of our 2DEG we expect a transition between these two regimes around 100
mK, which is the temperature at which τthermal = τelastic . The black line in fig. 5.10(c)
is the best fit to the three combined data sets. The only free parameter in the fit
is the mean free path that together with the 2DEG density determines coefficient
a = 1.2 × 1010
1
.
s K
Coefficient b is fixed by the 2DEG density to be 5 × 109
1
.
s K2
This
power law covers the entire measured temperature range and it is consistent with the
behavior observed in the high temperature region of previous experiments [103, 101],
but predicts parameters that are different from the 2DEG parameters. The best fit
value for the mean free path in this device is 268 ± 8 nm, which is far smaller than
the 2DEG mean free path of 15 µm but in agreement with the value extracted from
measurements of quantum dots of different sizes (0.4 − 4 µm2 ) in the experiment in
ref. [103].
We can also extract the dephasing time from the variance of the conductance. We
use the finite temperature variance, averaged over seven magnetic fields in the 15 mT
to 25 mT. From eq. 5.4 we find the dephasing time values shown in fig. 5.10(b). The
temperature dependence from 1 K to 80 mK is similar to the temperature dependence
of the dephasing times extracted using the average conductance method (dashed black
line in fig. 5.10(c)) and the fit to the dephasing time extracted from the variance (red
line in fig. 5.10(b)) yields a mean free path of lelastic = 500 ± 70 nm. The equivalence
of the dephasing time extracted from the weak localization signal and the dephsing
time extracted form the UCF variance has been experimentally observed in 1D and
5.3. THE DEPHASING TIME IN A QUANTUM DOT AT N = 1
89
2D AuPd wires [130]. Fig. 5.10 indicates that the two methods yield similar dephasing
times in 0D systems as well, but only at high temperatures.
As mentioned in the previous section, at low temperature the conductance through
the dot shows Coulomb Blockade oscillations that are superimposed on the UCFs.
This complicates the extraction of the dephasing time from the UCFs. One reason
why this method for determining τφ is less reliable than the previous method is because
it assumes that the UCF variance that we extract from our data is equivalent to the
ensemble variance. This is a reasonable assumption at high temperatures but at low
temperatures, in the absence of a theoretical analysis of this regime where Coulomb
Blockade oscillations of the conductance appear in addition to the UCFs, we are
unable to distinguish between the Coulomb Blockade and the UCF contribution to
the ensemble variance. In addition, it is not clear that eq. 5.4 would still be valid
if a small but finite reflection coefficient in the QPCs is present. As a consequence
we cannot use the extraction of the dephasing time from the variance using eq.5.4 to
reach any conclusions about the behavior below 60 mK.
We can compare this extracted temperature dependence to the relevant time and
energy scales of our quantum dot as well to the theoretical predictions. Fig. 5.11(a)
indicates where the relevant time and energy scales are with respect to our measured
depashing times. The Thouless energy of the quantum dot is larger than all but one
temperature and as such it is appropriate to use RMT results. There are no features
in the data at the temperature at which we see Coulomb Blockade emerging and thus
we conclude that, as expected, the average conductance and this extraction method
for the dephasing time are not affected by Coulomb blockade. We reach temperatures
that are smaller than the single particle level spacing and even low enough that the
dephsing time is 5 times larger than the dwell time but do not observe a saturation
as previous experiments have [102, 107].
The thermal time is larger than the elastic collision time at temperatures below
100mK which would correspond to the diffusive regime in a 2D system. However, the
thermal time is larger than the time to cross the sample for the entire temperature
range of the experiment which indicates that the quantum dot is 0D even though the
temperature is smaller then the level spacing. The time the electron spends on the
90
CHAPTER 5. QUANTUM DOTS: PERFECTLY TRANSMITTING QPCS
(a)
6
5
4
(b)
∆=34 mK
thermal time
2
ETh=930 mK
-7
10
τdwell=0.8 ns
-9
10
6
5
4
-8
10
τφ(s)
3
τ φ(s)
2D (ballistic)
2D (disordered)
1D (disordered)
0D (closed)
TCB=60 mK
3
2
-9
τe=0.07 ns
-10
10
10
6
5
4
3
τEhrenfest=0.02 ns
-10
10
2
τcross=0.008 ns
-11
10
-11
2
10
3
4 5 6
2
100
T (mK)
3
10
4 5 6
1000
2
10
3
4 5 6
2
100
3
4 5 6
1000
T (mK)
Figure 5.11: (a) The calculated values of the dwell time, elastic scattering time,
Ehrenfest time and the time to cross the dot are indicated with respect to the measured dephasing time (black dots). The red line is the thermal time. The energy
level spacing of the quantum dot, the temperature below which Coulomb blockade
appears and the Thouless energy are compared to temperature. (b) Calculated dephasing times for a 2D ballistic system (dark blue), 2D disordered system (black line),
1D disordered system (light blue line) and 0D diffusive system (red line) for the same
density, mobility and area as the measured quantum dot.
dot is longer than the Ehrenfest time, that is the necessary time for an interference
pattern to build up. Because the time between elastic collisions is larger than the
time an electrons take to cross the quantum dot, we expect the motion to be ballistic.
Fig. 5.11(b) shows how the measured dephasing time (black dots) compare to the
theoretical predictions using the 2DEG and quantum dot parameters. The dephasing
times in a 2DEG with the same density and mobility as our 2DEG are expected to
follow the ballistic T −2 law (eq. 5.10) at high temperatures and the disordered T −1
law at low temperatures, but the amount of dephasing is about an order of magnitude
5.4. MESOSCOPIC COULOMB BLOCKADE IN A QUANTUM DOT AT N = 191
smaller than the values we measure in the quantum dot. Similarly, dephasing that
would take place in the 1D parts of the 2DEG, the leads, shown as the blue line
would also not account for the measured dephasing. The 0D law from eq. 5.8 predicts
a magnitude of the dephasing time that matches the measured values although the
exponent 2 does not describe our temperature dependence.
We find that the temperature dependence includes contributions from both a T −2
and a T −1 law, similar to the results of Huibers et al. However, unlike in [103, 104,
107], we observe that the dephasing time increases monotonically down to the lowest
experimentally accessible temperature of 13 mK. This temperature is well below the
value for onset temperature of the saturation Tsat observed in the experiments in ref
[104] as well as the value of 118 mK given by the empirical formula for Tsat determined
by Hackens et al. [107].
5.4
Mesoscopic Coulomb Blockade in a Quantum
Dot at N = 1
The small value for the QPC reflection coefficients determined from the temperature dependence of the average conductance at finite magnetic field suggest that the
strong Coulomb blockade oscillations we observe as a function of gate voltage have
to be mesoscopic in nature and cannot be due to the intrinsic reflection of the QPCs
alone. This connection to the electron interference caused by long coherence times is
confirmed by the magnetic field and temperature dependence of the oscillation.
Fig. 5.12(a) shows the evolution of the Coulomb blockade oscillation with increasing magnetic field. We observe that large regions of periodic oscillations appear at
small magnetic field while smaller regions appear at finite magnetic field. This behavior is easier to see in the map in fig. 5.12 where the background was eliminated and
the local oscillation maxima are denoted in blue while oscillation minima are denoted
in red. This evolution is consistent with MCB being sensitive to an applied magnetic
field that disrupts the constructive interference between time-reversed paths.
Fig. 5.13a shows a zoom in of line cuts taken from fig. 5.12(a) at several fields
CHAPTER 5. QUANTUM DOTS: PERFECTLY TRANSMITTING QPCS
(a)
(b)
Vc2(mV)
-520
-520
2
G(e /h)
1.6
-540
1.2
Vc2(mV)
92
-540
0.8
-560
-560
0.4
20
5
15
30
B(mT)
25
40
35
50
5
20
15
30
B(mT)
25
40
35
50
Figure 5.12: (a) The dot conductance as a function of gate voltage and magnetic field.
The fine oscillations in gate voltage are Coulomb blockade oscillations and they are
superimposed on the larger UCF background. (b) Positions of maxima and minima
of the oscillations in (a) are indicated in blue and red.
which show that the oscillation is suppressed at finite field in comparison to zero field.
To determine how large the magnetic field is that suppresses the MCB, we Fourier
transform the data and integrate the power spectral density around the frequency of
the oscillation to find the power PM CB [113]. The results are shown as the solid line
in fig. 5.13b. The dotted line shows the weak localization dip at 13 mK for QPC
setting A. The fact that the amplitude of the oscillation decreases over the same field
scale is strong evidence that the oscillation is MCB.
For B > 5 mT the power of the oscillation is small but non-zero because some
oscillations are still present at some gate voltages and magnetic fields, as can be
seen in fig. 5.12. Without the constructive interference of time-reversed paths, the
oscillations are weaker and less frequent. They can be due to the fact that some
magnetic fields, certain electron paths can still constructively interfere to cause coherent backscattering and Coulomb blockade oscillations and/or because of the < 2%
remaining QPC reflection coefficient.
The field scale over which PM CB decreases in fig. 5.13b is that necessary to introduce 1 − 2 flux quanta in the dot. The size of a QPC is much smaller than this, so
we would expect the amplitude of Coulomb blockade oscillations caused by a finite
r 2 without any contribution from coherent backscattering in the dot, to change over
5.4. MESOSCOPIC COULOMB BLOCKADE IN A QUANTUM DOT AT N = 193
(b)
4
2
2
4
-3
2
1.0
2.2 mT
10 mT
1
-560
-550
-540
Vc2 (mV)
2
2
Gdot (e /h)
1.4 mT
3
<Gdot > (e /h)
4
0.4 mT
PMCB ( x10 e / h )
(a)
0.8
0
0
10
20
30
B (mT)
Figure 5.13: (a) Conductance trace vs Vc2 at several magnetic fields. Traces are offset
by 1 e2 /h. (b) PM CB (solid line, left axis) is obtained by Fourier transforming the
data in fig. 5.12(a). The dotted line shows the average conductance vs magnetic field.
All data are taken at 13 mK.
a field scale that is much larger than that observed. As a consequence, this residual
small reflection cannot be used to account for all the observed oscillations.
Fig. 5.14(a) shows that increasing temperature also causes the oscillation to become weaker: MCB is suppressed for T & 54 mK. This temperature can be qualitatively understood by comparing the dephasing time to the time the electron spends
on the dot. Once τφ becomes smaller than the dwell time τd ≈ 0.8 ns, the electron
spends enough time on the dot to lose the phase coherence, the interference decreases
and MCB should indeed be weaker. From fig. 5.10 we see that τφ = τd at T ≈ 80
mK, which agrees with our observations.
Fig. 5.14b show the results of extracting PM CB for different temperatures at B = 0
(filled circles) and B = 30 mT (open squares). The oscillation decreases quickly with
increasing T (the saturation at PM CB = 2 × 10−5 e4 /h2 is from the noise floor). Both
the Coulomb blockade peaks as well as the UCFs are affected by dephasing and
by thermal broadening. There is no apriori reason to expect the same temperature
dependence for the two, but we can compare the MCB power to the variance dependence on temperature (blue crosses) We see that PM CB is at least as sensitive to T as
Var(G), supporting the conclusion that the oscillations depend on phase coherence.
CHAPTER 5. QUANTUM DOTS: PERFECTLY TRANSMITTING QPCS
2
22 mK
3
10
10
10
31 mK
2
10
1
-720
-710
Vc2 (mV)
10
-1
-3
10
-2
-4
10
-3
2
54 mK
-2
4
Gdot (e /h)
13 mK
4
(b)
Var(Gdot) (e /h )
(a)
PMCB (e4/h2)
94
-5
10
100
T (mK)
Figure 5.14: (a) Conductance vs Vc2 at B = 0 and several temperatures. Traces
are offset by 1 e2 /h. (d) PM CB obtained from data taken from Vc2 = −500 mV to
Vc2 = −800 mV at several values of Vc1 , for B = 0 (filled triangles, left axis) and
B = 30 mT (open squares, left axis). For comparison we show measurements of
Var(G) (crosses, right axis) at B = 0.
To quantify the amount of residual charge quantization that is associated with this
oscillation, we perform capacitive measurements: we use the conductance through the
adjacent small quantum dot to detect the voltage change cause by the addition of
one electron to the large dot. The charge on the large dot is varied by changing the
voltage on gate bp. In response, the conductance GCS of the adjacent charge sensor
changes by an amount ∆GCS . This amount is determined by the change of electric
potential due to the accumulation of charge on gate bp as well as due to the addition
of charge to the large dot. We can convert ∆GCS into an effective voltage change
Veff , which would produce the same ∆GCS if applied to the small dot gate sp. This
correspondence is shown in Fig. 5.15a.
When the large dot is in the tunneling regime, the strong charge quantization is
easily detectable by the small dot conductance. When an electron is added to the
large dot, a clear Coulomb blockade peak is seen in the large dot conductance (black
trace, right axis) and a decrease of Veff is seen in the small dot (blue trace, left axis) as
illustrated by the data in fig. 5.15b. Veff initially increases as Vn is made less negative
because of the capacitive coupling between the gate and the small dot. However at
5.4. MESOSCOPIC COULOMB BLOCKADE IN A QUANTUM DOT AT N = 195
2
ΔGCS
0.2
0.0
-246
-245
Vsp (mV)
0.04
0.10
0.02
0.05
-390
-389
2
Veff
Veff (mV)
(b)
Gdot (e /h)
GCS (e /h)
(a) 0.4
0.00
-388
Vn (mV)
Figure 5.15: (a) Coulomb blockade peak of the small dot used for charge sensing. We
convert the change in the conductance ∆GCS of the charge sensor into an effective
voltage change Veff . (b) Simultaneous measurement of charge sensing signal Veff (left
axis) and conductance (right axis) of the large dot with both QPCs in the tunneling
regime.
the value of Vn where an electron is added to the dot, there is a sharp decrease in Veff .
The correspondence between the decrease in Veff and the Coulomb blockade peaks in
the large dot is most easily seen in the derivative D = dVeff /dVn . We refer to this
derivative as the charge sensing signal. A dip in charge sensing signal corresponds to
a step-like increase/decrease (after correcting for the capacitive gate-sensor coupling)
of Veff associated with the addition of one electron to the large dot.
When the large dot is in the fully transmitting regime, the quantization of charge
is suppressed and the effect it has on the small dot conductance becomes increasingly small. The sensitivity of our quantum dot charge sensor is 1.7 × 10−4 e/Hz1/2
referenced to the detector. To increase our charge sensitivity we average together
300 individual charge sensing measurements taken over the same range of Vn which
corresponds to an factor of 17 improvement in sensitivity.
Background charge fluctuations in the donor layer can cause small shifts of the
Coulomb blockade peaks in Vn and in the charge sensing signal. The resulting misalignment has to be corrected for before averaging over the individual traces. When
the charge sensing signal is large (tunneling regime), the shifts are easily identified
and corrected. However, when the charge sensing signal is small (open regime) detecting the shifts becomes increasingly difficult. To overcome this difficulty, we use
96
CHAPTER 5. QUANTUM DOTS: PERFECTLY TRANSMITTING QPCS
(a)
(b)
1.0
1.0
2
-360
Gdot (e /h)
-380
2
0.5
-0.2
dVeff / dVn
0.010
Gdot (e /h)
dVeff / dVn
0.0
0.005
0.0
-340
Vn (mV)
0.5
-315
-310
-305
Vn (mV)
Figure 5.16: (a) Simultaneous measurement of the charge sensing signal in the small
dot (dVeff /dVn ) (blue dots, left axis) and and the large dot conductance (black dots,
right axis). The solid lines show fits discussed in the text. (b) Charge sensing data
(left axis) and large dot conductance (right axis) at the values of Vn for which the
two QPCs are open to 2 e2 /h conductance. The solid red line is a fit described in the
text.
the Coulomb oscillations that are visible in transport though the large dot over the
entire range of QPC transmission to identify the shifts, align the charge sensing data
and average it correctly.
We follow the evolution of the Coulomb oscillations and of the charge sensing
signal with increasing QPC transmission in fig. 5.16(a). For Vn < −375 mV the
conductances of both QPCs are less than 2 e2 /h and we have well-defined CB on
the large dot. The charge sensing signal shows a large dips that corresponds to the
Coulomb blockade peaks. As the QPCs are opened by increasing Vn , the dips remain
aligned to peaks in Gdot .
The measurements of charge quantization for QPCs open to exactly one fully
transmitting spin-degenerate mode ( GQP C1,2 = 2e2 /h) are shown in fig. 5.16b. We
see a periodic variation of the charge sensing signal with the dips corresponding to
peaks inthe large dot conductance which confirms that the conductance oscillation
corresponds to a residual quantization of charge on the dot.
To estimate the quantized charge to the size of the charge sensing signal, we use
a model similar to ref. [111] where D is determined by the rate of change of electron
number on the dot with gate voltage dNd /dVn and by the capacitances of the dot
d and of the charge sensor CS. For Vn < −385 mV the theoretical predictions for
5.4. MESOSCOPIC COULOMB BLOCKADE IN A QUANTUM DOT AT N = 197
dNd /dVn from ref. [131] and the capacitance ratios discussed in ref. [132] fit the large
dot conductance and the charge sensing signal lineshapes very well (solid red lines in
fig 5.16).
For −370 < Vn < −340 mV the data is well fitted by the prediction for a one-
leaded dot without phase coherence and with GQP C ≈ 2 e2 /h [94] (solid line in
fig. 5.16a)). However, this theory predicts there should not be a periodic variation
in the charge sensing signal when GQP C = 2 e2 /h which does not correspond to our
observations and can thus not be used in the N = 1 gate voltage region.
For Vn ≈ −310 the theoretical model for MCB in a one-leaded dot [108] predicts
e dNd /dVn = Cn,d (1 + (A/e) cos(2πCn,d Vn /e)) where A gives the residual charge quantization. This one-leaded model is not a valid description of our system, but it fits the
data very well (solid red line in fig. 5.16(b)) and indicates that a significant amount
of charge A/e = 0.27+0.21
−0.08 is still quantized.
In conclusion, we observe Coulomb blockade oscillations in the conductance of a
quantum dot where each QPC has a fully-transmitting spin-degenerate mode that is
due to electron interference effects. This type of dot was thought to be described by
neglecting explicit Coulomb interactions. Our results demonstrate that the behavior
of quantum dots with two leads and few transmitting modes is more similar to the
behavior of quantum dots with one closed off lead than to the behavior of quantum
dots with many transmitting modes when the phase coherence time of the system is
long.
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