S
PIN
, V
IBRATIONS AND
R
ADIATION
S
PIN
, V
IBRATIONS AND
R
ADIATION
IN
S
UPERCONDUCTING
J
UNCTIONS
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus Prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op 18 februari 2013 om 15:00 uur
door
Ciprian P
ADURARIU
Master of Science, Jacobs University Bremen geboren te Roman, Roemenië.
Prof. dr. Yu. V. Nazarov
Samenstelling promotiecommissie: Rector Magnificus, voorzitter
Prof. dr. Yu. V. Nazarov, Delft University of Technology, promotor Prof. dr. L. Y. Gorelik Chalmers University of Technology
and Göteborg University Prof. dr. J. Zaanen, Leiden University
Prof. dr. D. Esteve Universit´e Pierre et Marie Curie Prof. dr. J. Aarts Leiden University
Prof. dr. Y. Blanter, Delft University of Technology Prof. dr. L. P. Kouwenhoven, Delft University of Technology
Prof. dr. H. W. Zandbergen, Delft University of Technology, reservelid
Printed by: Ipskamp Drukkers B.V.
Front cover: Representation of a Cooper pair in a superconducting junc-tion. A particle moving in the disordered potential follows a scattering trajectory as indicated by the dotted line.
Copyright © 2013 by C. Padurariu Casimir PhD Series, Delft-Leiden 2012-25 ISBN 978-94-6191-620-4
An electronic version of this dissertation is available at
To my Teachers.
P
REFACE
This thesis marks the conclusion of my four year PhD training in Delft. Ahead of me now lie great changes.
In smaller ways we always change — we forget some things and learn new oth-ers. It is not just our bodies that are slowly eroded, and slowly rebuilt. It is the personality too.
I would explain my PhD experience in terms of changes. The events of the past four years have not allowed a slow change. On the one hand I went through a fast paced learning experience that contained teachings going well beyond skills of theoretical physics, to the core of my ideas of expression, thought, and values. On the other hand there was another expansion-wave type change going in a rather perpendicular direction: it affected my immediate habits and my long-term goals, relationships with others and motivation. The combined effect of these transfor-mations has been at times rather unsettling, so that I had to use all the power in my grip to hold on and adapt.
Now I can say it has been a very enjoyable journey. Along the way I have found happiness and meaning, and also many challenges for the future. This was possi-ble because I have had great help from wonderful people I have been fortunate to meet in Delft. I am grateful to them and will make an attempt to acknowledge and thank some of them within these few pages.
First of all, I would like to thank Prof. Yuli Nazarov for giving me a chance to learn from him as a PhD student. Yuli is a highly dedicated and talented teacher, a researcher with deep and original insight, a defender of the scientific ethics, a hard-working and reliable colleague and above all a well-meaning and compas-sionate person. And let me not forget, he seems to always try to better himself. Yuli can teach a variety of lessons and does so with generosity: his time manage-ment always seems to favour teaching. I wish I could have stayed on and learned more from him. I feel that I will continue learning from Yuli, also when away from Delft. Alas, Yuli’s qualities for teaching do not speed up the learning process. There is a bottleneck: the process is difficult and demanding on the part of the student as well. I think that anyone who is ready and willing to learn will find themselves lucky to have Yuli as their supervisor.
While learning from Yuli I found it helpful to observe his other students. I want to thank Jeroen, Alina, Frans, and Mihajlo for offering me new perspectives on my
own experience. Mihajlo in particular has left a deep impression that I cannot express. In my opinion he shares more qualities with Yuli than anyone else I’ve met.
I have been privileged to collaborate in Delft with the scientific powerhouse that is the Quantum Transport group. I am very grateful to Prof. Leo Kouwenhoven for his insightful comments during our meetings and for his positive energy. Leo introduced me to Han and Sergey soon after I started in Delft. Once it happened that the three of us, myself, Han and Sergey, took a walk on a wooded path around a lake in Bavaria, heading to a monastery. Han produced the entertainment in the form of short stories about the bravery of animals during war. Sergey was super-vising him, with his smile and healthy humour. We also collaborated on the topic of Josephson electro-mechanical junctions. They were always just as fun, cheerful and imaginative. I want to thank both Han and Sergey for our collaboration. A while later Vincent has joined us bringing a lot of enthusiasm and energy. These have served him well and I want to thank him for his sincere friendliness.
I also want to thank the people interested in nano-mechanics in the group of Prof. Herre van der Zant. I am very grateful to have been able to join their seminar. Everyone there has been friendly, enthusiastic about science and willing to discuss their work. I have particularly learned a great deal from Gary who seems always able to make even the more subtle physical phenomena seem simple. I am grateful to him for his advice and for taking part in my evaluation meeting.
It has been my privilege to be part of the Theoretical Physics section in Delft and meet the members of the faculty. In Delft we are encouraged to refer to them by first names, so I want to thank Gerrit, Yaroslav, Miriam, Jos and once again Yuli. Observing them I have learned that working in scientific research does not have to determine the personality: indeed, each of them could not be more different from the others. This has given me a reason to smile more than once. I want to particularly thank Yaroslav for his gentle presence during most coffee breaks and for his help with recommendation letters. It helped my confidence to notice that as I observed him he was observing me right back.
I want to also thank the two very helpful secretaries of our group: the youthful and joyful Marjolein and the discrete and elegant Erika. I’ve noticed they have one thing in common: whenever they are present I feel that they contribute exactly what I thought was the missing ingredient to the moment.
Now I would like to thank another teacher. This teacher is not one person, but a collection of wonderful people and personalities that I would like to call my friends. However, it seems I am forced to be unfair: not to acknowledge you here would obviously be unfair and to acknowledge any of you is equally unfair due to the limited space.
PREFACE ix
If you have even once offered me your smile and your time, then I want to let you know that you have contributed to the very foundation of my experience in Delft. I would like to call you my friend and thank you from all my heart for being such a wonderful person.
This statement is made true by the fact that the people I have met and talked to during these four years are truly amazing. Unique, sometimes confused, some-times not knowing themselves how wonderful they really are. I try to thank you every day by giving something back and by offering the compliment I feel is most honest: an attempt to copy something from you, something you all have in com-mon.
I hope that as you browse through this text looking for a mention of your name, you will arrive here and find it, as if by magic. Some of you may think that my gratitude could not extend to you, that our encounter was too brief, or that our opinions were too divergent. This is not so.
Finally, I would like to thank all the people who have made it possible for a simple child from a poor family in a poor town to study and complete a PhD the-sis at the prestigious Delft University of Technology. Once again I would like to thank Prof. Yuli Nazarov. I would also like to thank my Bachelor’s and Master’s supervisor at Jacobs University Bremen Prof. Ulrich Kleinekathöfer for his excel-lent classes on Quantum Mechanics. I would like to thank the most dedicated, generous and always young teachers: profesoarei mele de fizic˘a, doamna Rodica Perjoiu, ¸si profesoarei mele de matematic˘a ¸si dirigint˘a, doamna Mihaela Cianga. I would like to thank the eternal Principle of the "Costache Negruzzi" High School Iasi, profesorului meu de matematic˘a, domnul Gheorghe Ilie, for his great impact on my early education. I am indebted to my chess coach, domnul Vasile Manole, for his guidance and lessons beyond the chess board.
I would like to express my warmest gratitude to my family, my dear sister for her
unique love that still has so much to teach me, my ambitious and loving mother,
who has been motivating me to study and praying for me for so many years now, and to my loving father, whose lessons changed my life more than once in the most amazing ways — most notably I would like to thank you, tat˘a, for teaching me the rules of the game of chess. Thank God for that, as it was the start of everything!
Ciprian Padurariu Delft, ianuarie 2013
C
ONTENTS
1 Introduction 1
1.1 Quantum transport . . . 1
1.2 Superconductivity . . . 3
1.3 The Josephson effect . . . 5
1.4 Quantum information processing . . . 8
1.5 Quantum memory devices . . . 10
1.6 Nano-electro-mechanical systems . . . 12
1.7 Outline of the thesis . . . 13
References . . . 14
2 Theoretical Concepts 19 2.1 Transport in small junctions . . . 19
2.2 Charging effects and Coulomb blockade . . . 23
2.3 Superconducting transport . . . 26
2.4 Dynamics in Josephson junction . . . 30
2.5 Counting statistics . . . 31
2.6 Spin-dependent phenomena . . . 33
References . . . 35
3 Theoretical proposal of superconducting spin qubits 37 3.1 Introduction . . . 38
3.2 Phenomenological Description . . . 39
3.2.1 Effective Hamiltonian . . . 40
3.2.2 Spontaneous Currents . . . 41
3.2.3 Single Qubit Manipulation . . . 42
3.2.4 Design of Qubit-Qubit Interaction . . . 43
3.3 Microscopic Description . . . 45 3.3.1 Hamiltonian . . . 45 3.3.2 Josephson Energy . . . 46 3.4 Estimations . . . 49 3.5 Numerical Analysis . . . 51 3.6 Conclusion . . . 55 References . . . 56 xi
4 Spin-blockade qubit in a superconducting junction 61
4.1 Introduction . . . 62
4.2 Theoretical model . . . 64
4.3 Estimations . . . 67
4.4 Spin-orbit relaxation of the triplet states . . . 69
4.5 Singlet-to-triplet manipulation . . . 69
4.6 Triplet-to-triplet manipulation . . . 71
4.7 Spin qubit . . . 72
4.8 Conclusion . . . 73
References . . . 73
5 The Effect of Mechanical Resonance on Josephson dynamics 75 5.1 Introduction . . . 76
5.2 The setup . . . 79
5.2.1 Electrical Setup . . . 79
5.2.2 Mechanical Setup . . . 81
5.3 Coupling and non-linearities . . . 83
5.3.1 Electrostatic energy . . . 84 5.3.2 Coupling quantities . . . 86 5.3.3 Lorentz force . . . 87 5.3.4 Analysis of non-linearities . . . 88 5.3.5 Workflow . . . 91 5.3.6 Parameters . . . 92 5.4 Phase bias . . . 93 5.4.1 Fano-type response . . . 95 5.5 D.C. voltage bias . . . 96
5.5.1 Excitation by higher harmonics . . . 100
5.5.2 Parametric excitation . . . 100
5.6 Shapiro steps at resonant driving . . . 101
5.6.1 First step . . . 102
5.6.2 Higher steps . . . 108
5.7 Shapiro steps at non-resonant driving . . . 111
5.8 Conclusions . . . 112
References . . . 113
Appendix 117 .1 Lagrangian formalism . . . 119
CONTENTS xiii 6 Statistics of radiation at Josephson parametric resonance 121
6.1 Introduction . . . 122
6.2 Setup . . . 124
6.3 Keldysh action . . . 125
6.4 Full counting Statistics . . . 130
6.4.1 Average intensity and Fano factor . . . 132
6.4.2 Big deviations . . . 133
6.5 interpretation: bursts . . . 135
6.6 Limits . . . 137
6.6.1 Large detuning . . . 137
6.6.2 Vicinity of the threshold . . . 138
6.7 Time-dependent correlations . . . 139 6.8 Frequency-resolved correlations . . . 141 6.9 Conclusions . . . 143 References . . . 144 Appendix 147 .1 Propagator . . . 149 Summary 153 Samenvatting 155 Curriculum Vitæ 157 List of Publications 159
1
I
NTRODUCTION
1.1
Q
UANTUM TRANSPORT
Technology has an increasing impact on our daily lives. Sophisticated elec-tronic devices such as mobile phones and computers are changing every aspect of society by easing our communication. We now rely on some form of modern electronic equipment, if only to remain competitive in the job market. This has motivated a great amount of scientific research into ways to improve electronic devices.
What all electronic devices have in common is their ’workhorse’: the electron. Electric fields generated by voltage differences control the transport of electrons through a combination of structured materials that form the electrical circuit.
When I was a child my father explained the transport in electrical circuits by comparing it to the flow of water through pipes. In this analogy electrical voltage is like water pressure and an electrical resistor is like a constriction in the pipe that slows down the flow of water. The analogy works very well in some conditions, for example if the size of the circuit is similar to the size of typical water pipes. This is of course not the case for modern devices.
The desire for smaller and more functional devices has motivated the miniatur-ization of circuit elements. State-of-the-art fabrication technology permits control of the circuit dimensions at the nano-meter scale (10−9meters scale). The small dimensions reveal the quantum mechanical properties of electrons: in nano-sized circuits electrons may interfere [1], may pass through a constriction one at a time manifesting their discrete charge [2] and the electron spin may play a crucial role
to permit or blockade the transport [3]. No simple analogy can be used to un-derstand the transport at these small scales. The study of quantum transport at the nano-scale has been a topic of intense scientific research. In addition, it turns out that the manifestation of quantum effects in electrical devices does not neces-sarily require small sizes: some metals exhibit superconductivity, an intrinsically quantum mechanical phenomenon that may manifest at any size when the metal is cooled down to low temperatures of the order of a few degrees Kelvin.
Apart from adding functionality to modern electronic equipment, devices ex-hibiting quantum transport find numerous other applications. For instance, in metrology superconducting devices have been used to increase the precision of the voltage standard [4] and of the magnetic flux quantumφ0= πħ/e, 2πħ
be-ing the Plank constant and e bebe-ing the electron charge. A technique based on re-lated properties of superconducting devices aims to improve the current standard [5]. Other experiments using single electron quantum transport devices have been proposed to improve the capacitance standard, termed the electron-counting ca-pacitance standard [6]. A related branch of research is dedicated to finding new ways to exploit quantum properties such as entanglement to yield better measure-ment precision compared to ’classical’ statistical averaging.
Quantum transport devices can also be designed to function as excellent sen-sors. Superconducting quantum interference devices are very sensitive magne-tometers that have already found commercial applications in non-invasive medi-cal imaging of magnetic brain activity and imaging of the minute magnetic fields produced by the human heart [7]. Similar interference devices are being devel-oped for applications as high-frequency amplifiers, biosensors and for realization of magnetic resonance imaging. Superconductivity-based devices are also used for very sensitive radiation detection, for instance in experimental astronomy [8]. Superconducting single photon counters are able to measure a single quantum of electro-magnetic radiation. These are just a few examples of sensing applications; there are of course many more that cannot be covered in one short review.
Recently, an active field of research has been developed that combines the mechanical properties of small-sized conductors with the properties of quantum transport in a single device [9]. Such nano-electro-mechanical systems hold great promise not only for diversifying sensing applications by allowing sensitive detec-tion of small mass and force, but also for answering a fundamental quesdetec-tion of physics: that of the quantum properties of massive objects. At temperatures as low as 10 mK achievable in experimental laboratories, the mechanical motion of a small conductor is frozen close to its quantum mechanical ground state such that electrons passing through it can probe its quantum properties. Such studies demonstrate for the first time the quantum behaviour of objects much larger than the atomic scale [10].
1.2. SUPERCONDUCTIVITY 3 The successes of quantum transport devices have been decisively influenced by the fascinating properties of superconductors. This is why the next two sections contain a brief discussion of the phenomenon of superconductivity and of the be-haviour of small-size superconducting junctions. For a more detailed description of superconductivity in general, and superconducting junctions in particular, the reader is referred to the excellent textbook of Tinkham [11].
1.2
S
UPERCONDUCTIVITY
Superconductivity was first observed by a scientist from the University of Lei-den, Heike Kammerlingh Onnes whom, after being the first to liquefy helium, de-veloped the refrigeration technique to cool down metals to very low temperatures of few degrees Kelvin. In 1911 he reported the following intriguing finding: metals such as mercury, lead, and tin exhibit a large enhancement of the electrical con-ductivity at temperatures below a certain critical value Tcof the order of 1 to 10
Kelvin, unique for each material [12]. His finding, which brought him the Nobel prize in physics in 1913, was verified and confirmed by many, more precise, exper-iments. These showed that below their critical temperature Tcmetals exhibit
per-fect conductivity, that is, no dissipation of energy occurs when an electric current passes through the metal. The novel phenomenon was termed superconductivity and the metal that exhibits it a superconductor. A second striking feature of super-conductors was discovered in 1933 by Meissner and Ochsenfeld [13]. They have found that as a metal is cooled down to a superconducting state it expels any mag-netic field lines that may have threaded the metal while it was in a normal state (above Tc). These experimental observations suggested that the ensemble of
elec-trons in a metal has different properties in the superconducting state and in the normal state, but at that time the microscopic origins of superconductivity were not apparent.
One important clue was revealed by the calorimetric and spectroscopic mea-surements of a metal in the superconducting state [14]. In the normal metallic state electrons occupy quantum levels in a continuous energy spectrum. The su-perconducting state however, exhibits an energy gap of a size∆ of the order of kBTc, kBbeing the Boltzmann constant.
In 1957, forty four years after the observation of Kammerlingh Onnes, a mi-croscopic theory of superconductivity was advanced by John Bardeen, Leon Neil Cooper, and John Robert Schrieffer, theory now known as the BCS theory of super-conductivity [15]. The same theory was advanced independently by Bogoliubov on the other (Eastern) scene of scientific research. A key result leading to our under-standing of superconductivity was presented in 1956 by Cooper [16]. He showed that if an attractive interaction exists between pairs of electrons, regardless of how
small, then electron states close to the Fermi energy of the metal combine in pairs and mix their quantum mechanical wavefunction to form a collective state of lower energy. Thereby an energy gap opens up around the Fermi energy. Electrons with opposite spin and lattice momentum ~k vector are strongly correlated due to the
attractive interaction, these form a Cooper pair.
In the BCS theory, it is shown that Cooper pairing is the mechanism of super-conductivity. However, it is counter-intuitive that electrons, or any particles of the same charge, would have an attractive interaction. How does it take place? Firstly, direct Coulomb repulsion of electrons in a metal is efficiently screened at distances larger than the atomic scale. Secondly, the BCS theory shows that an attractive electron-electron interaction occurs in metals and is mediated by the distortions of the ion core lattice of the metal, termed lattice phonons. The attractive interac-tion is second order in the electron-phonon coupling. When the electron-phonon coupling is sufficiently strong it can overcome the electron repulsion. Letλ be a dimensionless parameter that quantifies the electron-phonon coupling andωD
be the Debye frequency characterizing the energy of lattice distortions. The BCS theory predicts a superconducting energy gap given by
∆ = ħωDexp(−1/λ).
The phenomenon described by Cooper is an example of a phase transition, that of the metallic electron ensemble changes. Phase transitions occur by breaking or gaining a new symmetry of the system. That the superconducting state can be envisioned as a different phase of the metallic electron cloud was already under-stood by Ginzburg and Landau, who have proposed in 1950 a phenomenological theory of superconductivity characterizing the superconducting phase transition by a complex order parameterψ(~r) = (n(~r))1/2exp(iϕ(~r)), depending on the lo-cal coordinate vector~r. The BCS theory provides microscopic reasoning to the conclusions of Ginzburg and Landau. The amplitude of the order parameter n(~r) is identified as the local density of Cooper pairs. The global phase variableϕ(~r), termed the superconducting phase, characterizes the superconducting state. Typ-ically global phases can be removed by a transformation of the global gauge of electro-magnetism. However, the gauge symmetry is precisely the symmetry be-ing broken in the transition to the superconductbe-ing state, therefore the supercon-ducting phase remains well defined.
Excitations from the ground state in normal metals are represented by elec-trons and holes (holes represent the lack of an electron). In the superconducting state the electron-phonon interaction mixes electrons and holes, therefore excita-tions in superconductors can be suitably described by introducing the
1.3. THEJOSEPHSON EFFECT 5 holes. The energy spectrum of superconducting quasiparticles is given by
En=
q ∆2+ ξ2
n,
ξnbeing the excitation energy of the electrons in the normal state. The minimum
excitation energy in a superconductor is given by the energy gap∆. The density of states predicted by the BCS theory is given by
ν(E) = X n δ(E − q ∆2+ ξ2 n) = νN 2E p E2− ∆2θ µ E − ∆ ∆ ¶ , θ(x) = ½ 0, if x < 0 1, if x ≥ 0 ,
νN being the average normal metal density of electron states in a window of
en-ergy 2∆ around the Fermi energy. This describes the energy spectrum of a BCS superconductor: there are no available states with excitations below energy∆, the number of states diverging as (E − ∆)−1/2as the energy approaches∆.
It is fascinating that the BCS theory, a microscopic theory that heavily relies on quantum mechanics, can explain properties of superconductors that manifest at macroscopic length scales, such as the dissipation-free current and the Meiss-ner effect. At such length scales it is possible to control properties of devices that retain the quantum nature of superconductivity. This is why superconducting de-vices have attracted a lot of attention soon after the BCS theory was advanced. The main element of superconducting devices is the superconducting junction, that is discussed below.
1.3
T
HE
J
OSEPHSON EFFECT
The formulation of the BCS theory coincided with a period of intense scien-tific interest into the quantum properties of electron transport, particularly the phenomenon of electron tunnelling through a potential barrier. In the late 1950’s, early 1960’s, state-of-the-art technology allowed fabrication of metal electrodes in-terrupted by a thin region of insulating material, such as a thin layer of oxide, that creates an energy barrier for travelling electrons. Upon connecting the electrodes to a voltage source a small current is measured, despite the fact that the voltage applied is much smaller compared to the energy of the barrier. Classical trans-port through the barrier is forbidden, therefore the observed current must have quantum mechanical origins. These origins were well understood at the time: the wave character of electrons extends into the classically forbidden region creating a finite probability to traverse it. The tunnelling current decreases exponentially as a function of thickness of the barrier, therefore thin barriers are an important requirement to observe the tunnelling phenomenon.
The formulation of the BCS theory of superconductivity naturally attracted in-terest to the properties of tunnelling current in superconductors. The first such ex-periment was reported by Ivar Giaever in 1960, who measured quasiparticle tun-nelling between two superconducting electrodes by applying a sufficiently large voltage, V > ∆/e, in order to excite superconducting quasiparticles [17]. The ex-perimental current-voltage curves showed the signatures of the superconducting density of states of the two electrodes. This experiment has confirmed the pre-dictions of the BCS theory, yet the tunnelling current measured does not have the properties of a superconducting current.
A spectacular prediction was made by Brian David Josephson in 1962, while studying to get his PhD in theoretical physics at Cambridge University. Josephson understood the implications of the BCS theory for the tunnelling of Cooper pairs between two superconductors and predicted that the superconducting current can tunnel through a barrier without need for any voltage bias. As a consequence of the presence of tunnelling current, Josephson predicted that the superconducting phases of the two junction electrodesϕ1andϕ2will differ [18]. In fact, he obtained
the following equation that relates the tunnelling superconducting current to the superconducting phase difference I = Icsin(ϕ), Icbeing a maximum tunnelling
current that the junction can sustain at vanishing bias voltage andϕ ≡ ϕ1− ϕ2.
In his derivation Josephson assumes tunnelling through a barrier, therefore his relation between current and phase applies to superconducting tunnel junctions. However, the implication of this pioneering result is that any junction formed by two superconducting electrodes interrupted by a sufficiently thin region of non-superconducting nature (vacuum, insulator, semiconductor etc.) exhibits tun-nelling of current, now termed Josephson current, at vanishing voltage bias. The si-nusoidal current-phase relation does not hold for a general superconducting junc-tion. However, the current must be a 2π periodic function of the phase, I(ϕ) =
I (ϕ + 2π), and the condition I(ϕ = 0) = 0 must be satisfied. In general, the current
can be expanded as a Fourier sine series as a function ofϕ,
I (ϕ) =
∞
X
n=1
Insin(nϕ).
A simple experimental confirmation of Josephson’s prediction can be realized by passing a dc current through a superconducting junction and measuring the voltage across it. It is observed that no voltage drops across the junction if the cur-rent stays below a critical value. This value corresponds to Icfor superconducting
tunnel junctions and is termed the critical current of the junction. For currents larger than the critical value, a voltage develops across the junction, the dc tun-nelling current being the same as that studied by Giaver [17]. This test is realized
1.3. THEJOSEPHSON EFFECT 7 almost every day in quantum transport laboratories at TU Delft and across the world. It is an important step in characterizing superconducting junctions. The presence of a dc superconducting tunnelling current at zero voltage is usually re-ferred to as the dc Josephson effect, and the equation I = Icsin(ϕ) is termed the first
Josephson relation.
The second prediction of Josephson concerns the tunnelling of superconduct-ing current in the presence of a finite bias voltage. He showed that apart from the quasiparticle current observed in the experiment of Giaever, a second type of cur-rent is present that has superconducting nature. The superconducting tunnelling current at finite voltage bias V cannot be observed in a dc measurement. It is an ac current that oscillates at the Josephson frequencyωJ ≡ 2eV /ħ. Josephson
described the relation between the dc voltage and the time evolution of the su-perconducting phase differenceϕ and found the expression of the ac current for superconducting tunnel junctions.
V = ħ 2e dϕ d t, I = Icsin ¡ ωJt¢ .
The second prediction of Josephson reveals the non-linear nature of the electri-cal response in superconducting junctions, whereby a finite dc voltage gives rise to an ac superconducting current. This effect is usually referred to as the ac
Joseph-son effect, and the equation V = (ħ/2e)(dϕ/d t) is termed the second JosephJoseph-son
relation.
The work of Josephson contributed significantly to a better understanding of superconductivity and superconducting junctions, particularly the role of the su-perconducting phase. It also triggered experiments on the interference of super-conducting currents flowing in junctions embedded in the same supersuper-conducting loop. The large sensitivity of such superconducting quantum interference devices to variations of the magnetic field threading the loops is the basis of a large num-ber of important applications. For his work on the tunnelling of superconducting current Brian David Josephson received the Nobel prize in 1973, sharing half of it with Ivar Giaever and Leo Esaki for their experiments on tunnelling in solid state junctions. This took place only one year after a shared Nobel prize was awarded to Bardeen, Cooper, and Schrieffer, for their discovery of the microscopic theory of superconductivity.
In the past twenty years, developments in the design and functionality of su-perconducting devices have been advancing rapidly. An important application of superconducting devices that motivates a large part of the work presented in this thesis relates to quantum information processing. The section below briefly de-scribes the concept of quantum information processing, while the next section
discusses quantum memory devices. For a more detailed description of these con-cepts the reader is referred to the excellent textbook of Nielsen and Chuang [19].
1.4
Q
UANTUM INFORMATION PROCESSING
Quantum information processing is one of the most fascinating applications of quantum transport devices. It had a very important role in stimulating the rapid development of quantum transport research.
In general, information processing concerns with ways to store facts, operate on them, and retrieve the results in a reliable fashion. The tools of information processing are the language, a set of rules and operations used to represent facts, and the memory, a location where facts can be stored for later use. For exam-ple, computers realize information processing by representing information using a minimal unit: the bit, an object that may hold either value ’0’, or ’1’. The phys-ical realization of a bit is a device that provides the possibility to switch between two stable states in a controllable fashion. These ’classical’ stable states are each interpreted as a different value of the bit.
However, there is no reason why the device representing the bit could not have quantum mechanical properties. Two quantum mechanical states of such a device are relevant for a bit, |0〉 and |1〉. An important difference between ’quantum’ and ’classical’ bits is that the quantum bit does not have to represent either value ’0’ or ’1’, but may assume any mixed stateψ,
ψ = cos(θ)|0〉 + eiϕsin(θ)|1〉 ,
for any value of the anglesθ and ϕ. Repeated measurement of state ψ yields |0〉 with probability cos2(θ) and state |1〉 with probability sin2(θ), while repeated mea-surement of a classical qubit state will always provide the same classical state. It becomes obvious that a quantum bit stores much more information than just a bi-nary number — it stores a real numberθ. To store the same information classically requires a large number of bits. This is one advantage of quantum information
processing compared to the classical method.
Another advantage of quantum bits, in short qubits, consists of the way linear functions that transform stored information are evaluated. To evaluate a linear function f classically one needs to initialize a bit in state ’0’, apply the function
f and read the result f (0). The same procedure must be performed for the result
of f (1). In the case of a qubit, the operation of f on the stateψ leads to the mixed state f (ψ) = cos(θ)f (|0〉)+eiϕsin(θ)f (|1〉). The read-out of f (ψ) provides the result of f at once for both qubit states. Consider a computation using a large number of bits N , the striking conclusion is that, while classical computation power scales linearly with N , quantum computation power increases exponentially with N . It
1.4. QUANTUM INFORMATION PROCESSING 9 is easy to understand the huge motivation to design a quantum computation ma-chine with a large number of qubits. A quantum computer with, for example, one hundred qubits would vastly outperform any modern computer, as well as any fu-turistic computer design one can imagine that relies solely on the laws of classical mechanics.
This section concludes with a brief discussion of the ingredients needed to re-alize a quantum computer that can perform general computational tasks, that is, a
’universal’ quantum computation machine. Such a quantum machine requires the
realization of stable qubits that can retain the information stored in their wave-function for a sufficiently long time to operate transformations on this informa-tion and to read-out the results. Single qubit transformainforma-tions can be represented mathematically as a rotation on the Bloch sphere, the unit sphere parametrized by the anglesθ and ϕ that define any qubit state ψ. An arbitrary rotation of angle η around an axis defined by unit vector~n of the qubit state can be represented by a 2 × 2 matrix in the space spanned by states |0〉 and |1〉.
R(η,~n) = exp£iη(~σ ·~n)/2¤,
where the vector of Pauli matrices~σ has been introduced, its entries being given by σx= µ 0 1 1 0 ¶ ; σy= µ 0 −i i 0 ¶ , andσz= µ 1 0 0 −1 ¶ .
It turns out that for universal quantum computations arbitrary single qubit ro-tations are not sufficient. An additional requirement is the realization of condi-tional logic gates. These are operations performed on the state of multiple coupled qubits that conditionally change the state of one qubit based on the state of the others. An example of a conditional logic gate is the conditional NOT gate, also referred to as the CNOT gate. The CNOT gate acts on the state of two qubits, one of them playing the role of control qubit, while the other plays the role of target qubit. If the control qubit is in state |0〉, the CNOT gate acts as the identity trans-formation on the target qubit, while if the control qubit is in state |1〉, the CNOT gate acts as the NOT transformation on the target qubit. The NOT transformation is equivalent up to a global phase to the rotation R(π,~x) on the Bloch sphere. The CNOT gate can be represented in a compact way by a 4 × 4 matrix in the two-qubit space spanned by the states {|0,0〉,|0,1〉,|1,0〉,|1,1〉}, where the first (second) index labels the state of the control (target) qubit.
GCNOT= 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 .
An example of a sufficient set of operations that enable universal quantum computations is the set of arbitrary single qubit rotations of each qubit state to-gether with a CNOT gate for each pair of qubits.
The realization of universal quantum computations imposes some require-ments, that have been summarized by physicist David DiVincenzo into five cri-teria. These are quoted below [20]:
1. A scalable physical system with well characterized qubits;
2. The ability to initialize the state of the qubits to a simple fiducial state, such
as |000...〉;
3. Long relevant decoherence times, much longer than the gate operation time; 4. A ’universal’ set of quantum gates;
5. A qubit-specific measurement capability.
The first criterion, of scalability, implies that suitable qubits are those for which a tunable qubit-qubit coupling can be engineered such that a large number of qubits can participate in a quantum computation. In general, this requires that the qubit states couple easily to other degrees of freedom. However, the third criterion, of long decoherence time, implies the opposite: that the qubits are well isolated from other degrees of freedom. The decoherence time T2, that is, the time in which
quantum information stored in a qubit is lost, results from unwanted evolution of the qubit driven by the interaction with degrees of freedom of its environment. The stronger the interaction with the environment is, the shorter the decoherence time. Therein lies a paradox that needs to be addressed by qubit designs: the qubits must couple well with each other and with the measuring apparatus, but also re-main well isolated from all unwanted interactions. Modern research has revealed many approaches to alleviate this problem. However, no qubit design is presently sufficient to build a large scale quantum computer.
The following section provides a brief discussion of a variety of realizations of qubit devices in view of the criteria formulated by DiVincenzo.
1.5
Q
UANTUM MEMORY DEVICES
A variety of ways to build solid state qubits have been proposed and imple-mented. A majority of successful proposals exploit the quantum properties of spin, or those of superconducting circuits.
The spin degree of freedom provides the canonical representation of a qubit [21]. The two qubit levels are the ’up’ and ’down’ states of the spin magnetic mo-ment. In order to serve as a qubit, the spin must be confined to a sufficiently small volume such that it can be addressed for manipulation and read-out. Proposals of spin-based qubits include devices that confine a single electron spin in quantum dots (small regions of space where electrons are isolated from the outside by
en-1.5. QUANTUM MEMORY DEVICES 11 ergy barriers) [22], impurities [23] and lattice defects, such as nitrogen defects in diamond [24]. Qubits based on single electron spins are in general very successful in storing and protecting quantum information against decoherence, mainly due to the fact that the environment couples mostly to the electron charge, but also because the magnetic moment of a single electron spin is small. Spin qubits built using nitrogen vacancy diamond show the longest coherence times of any solid state qubit, T2' 1.8 ms at room temperature [25]. In addition, techniques of fast
coherent control of the qubit spin state have been developed [26], allowing a large number of coherent single qubit rotations. However, electron spins do not nat-urally couple well with each other when separated by any reasonable distance in space. Therefore scalability of spin qubit devices is very challenging.
A different approach is to create an ’artificial spin’ by addressing two quantum levels of a system localized in a potential well. This can be achieved if the profile of the potential is sufficiently an-harmonic such that the spacing between quantum levels becomes significantly different at the first few excitations. This requires a strong non-linearity in the system, strong enough to be manifested in the quan-tum regime. It turns out that the highly non-linear electro-magnetic properties of superconducting junctions are particularly suitable for this task. The main advan-tage of this approach is that superconducting devices have many features that can be tuned by employing existing technology.
The first demonstration of a superconducting qubit was reported in 1999, show-ing that in a superconductshow-ing junction states of different charge can be controlled and measured similarly to the state of a spin [27]. The decoherence time of this first superconducting qubit device was short, about ' 1ns. In the meantime, over the span of about ten years, superconducting qubit designs have flourished and the decoherence time has been increased by over a factor of one thousand, to val-ues of ' 1..10 µs [29]. Many of the later supercoducting qubits do not rely on the charge degree of freedom, but rather on the superconducting phase of junctions embedded in a superconducting loop [28]. The latter can be controlled by the magnetic flux threading the loop or by the bias current. Although superconduct-ing qubits do not boast the very long coherence times measured in state-of-the-art spin qubits, when it comes to scalability superconducting qubits have the advan-tage. Qubits included in the same superconducting circuit can be engineered to interact capacitively [30], or inductively [31]. It is also possible to create a long-range interaction by using a superconducting cavity mode (a mode of a low dis-sipation superconducting LC -type resonator) to mediate the interaction [32]. Im-portantly, the resulting qubit-qubit interaction can be tuned, thereby allowing op-eration of logical gates. Using state-of-the-art technology, a small superconduct-ing quantum computer was realized, capable of factorsuperconduct-ing the number fifteen [33]. This accomplishment matched in a solid state superconducting device a previous
achievement obtained using liquid state nuclear magnetic resonance [34]. How-ever, further scalability of the superconducting quantum computer proves to be a technological challenge at the moment.
The advantages of spin qubits and superconducting qubits complement each other. This provides a strong motivation to combine the features of both designs in such a way as to retain all advantages. A hybrid superconducting spin qubit design can be realized in general by trapping a single electron spin inside a super-conducting junction. An early proposal of such a device [35] exploits the spin of a superconducting quasiparticle localized in a junction. Chapters 3 and 4 of this thesis propose and investigate superconducting spin qubits, exploiting the spin of an unpaired electron in a quantum dot superconducting junction (Chapter 3) and the phenomenon of spin blockade of quasiparticles localized in the superconduct-ing junction (Chapter 4). However, such devices have not yet been realized. It is my hope that the work in this thesis can promote the development of hybrid spin and superconducting qubits.
1.6
N
ANO
-
ELECTRO
-
MECHANICAL SYSTEMS
In nano-electro-mechanical systems (NEMS) mechanical motion is coupled to the electrical degrees of freedom to produce a measurable current. NEMS have ap-plications as sensitive detectors of mass, force and electrical charge. Recently, im-provement of fabrication techniques have encouraged the idea of building and de-tecting the quantum mechanical motion of a massive mechanical resonator, giving a dimension of fundamental science to NEMS research. Previously, observation of quantum mechanical behaviour was restricted to objects of atomic scale dimen-sions. State-of-the-art NEMS with macroscopic dimensions (about 1012 atoms) have been cooled down to the quantum mechanical ground state [36, 37]. Fur-thermore, in systems where NEMS couple to a superconducting qubit, the quan-tum coherence of the coupled dynamics was studied [36].
A typical implementation of NEMS involves coupling mechanical motion ca-pacitively to the current in a junction. This is enabled by charging effects. In the regime where charging effects are significant the conductance of the junction G(q) is sensitive to the gate-induced charge q = CgVg. Mechanical motion of either part
of the junction, or of part of the gate electrode, directly modifies the capacitance
Cg(y), y being the amplitude of the motion. This gives rise to a contribution to the
current that allows characterization of the amplitude of motion. A modulation of the gate voltage at the frequency of the mechanical resonance can be used to excite the mechanical motion, by modulating the gate force Fg = (dCg(y)/d y)Vg(t )2/2.
The design has the added benefit that the static gate voltage modifies the mechani-cal tension in the system, allowing tuning of the mechanimechani-cal resonance in the same
1.7. OUTLINE OF THE THESIS 13 way guitar strings are tuned.
Recently, NEMS have been built using a superconducting quantum interfer-ence device (SQUID) consisting of a superconducting loop connecting two or more superconducting junctions. The superconducting current in a SQUID depends on the magnetic flux flowing through the loop, SQUIDs in fact being the most sensi-tive detectors of magnetic fields. The sensitivity can be exploited to study mechan-ics by embedding the mechanical resonator into the loop such that the area of the loop is modified by the mechanical motion. If a constant magnetic field threads the loop, the modulation of the area gives rise to a modulation of the flux, induc-tively coupling the electrical and mechanical degrees of freedom.
In terms of the realization of the mechanical resonator, these have been ei-ther one dimensional structures built by suspending a thin wire that also provides electrical conductivity, or two dimensional suspended "drum"-type structures. In-ductive coupling in SQUIDs has been realized using mostly one-dimensional res-onators. However, capacitive coupling favours large area drum-type resonators that allow large mechanical modulations of the capacitance. The trade-off is that drum-type resonators typically have lower resonance frequencies compared to one-dimensional structures, assuming materials with similar elastic constants are used. In resonator wires resonance frequencies of the order of 10..100 MHz have been routinely realised and measured. Using exceptional light and high elasticity mate-rials, the state-of-the-art being carbon nanotubes, resonance frequencies of sev-eral GHz have been realised, reaching the regime of microwave frequencies. This offers further possibilities to combine the physics of (superconducting) microwave electrodynamical resonators and that of mechanical resonators into one device.
1.7
O
UTLINE OF THE THESIS
This thesis presents the theoretical study of superconducting transport in sev-eral devices based on superconducting junctions. The important feature of these devices is that the transport properties are modified by the interaction with an-other physical system integrated in the superconducting circuit. In the case of the device studied in Chapter 3 the additional physical system is a single local-ized spin-1/2 electron. In Chapter 4 the device properties are modified by the spin state of two spin-1/2 superconducting quasiparticles trapped in a supercon-ducting junction. In Chapter 5 we study the effect of mechanical vibrations on the transport in a superconducting junction. The fourth device presented in Chapter 6 consists of a superconducting junction coupled to an electro-magnetic resonance. In this device we have investigated the electro-magnetic radiation emitted in the conditions of parametric driving below the classical parametric instability thresh-old.
The outline of the thesis is as follows. In chapter 2 I have briefly presented sev-eral theoretical concepts that are relevant for the topics discussed in the following chapters. Chapters 3-6 present the main work of this thesis.
In Chapter 3 we present our proposal of the spin superconducting qubit device. The spin superconducting qubit consists of a single spin confined in a Josephson junction. We describe single qubit operations and more complicated quantum gates based on the tunable interaction of spins in different junctions. We discuss the qubit properties at the phenomenological level and also present a microscopic theory that enables us to make accurate estimations of the qubit parameters.
In Chapter 4 we present another proposal of a novel qubit design that exploits the spin state of two quasiparticles trapped in a superconducting junction. We detail the single qubit operations and provide extensive microscopic estimations of the parameters of our model.
In Chapter 5 we study dynamics in a Josephson junction coupled to a ical resonator. We show that charging effects in the junction give rise to a mechan-ical force that depends on the superconducting phase difference. We develop a model that encompasses the coupling of electrical and mechanical dynamics and compute the mechanical response (the effect of mechanical motion) in a variety of bias regimes. Our results are applicable to recent experiments on suspended ultra-clean carbon nanotube superconducting junctions. We augment theoretical estimations with the values of setup parameters measured in the samples fabri-cated.
Finally, in Chapter 6 we study the properties of radiation emitted below the threshold of parametric resonance in a superconducting circuit with an electro-magnetic resonance. The correlations of the radiation are described by deriving a closed expression for the full counting statistics in the limit of long measurement times. We provide an interpretation of our results in terms of uncorrelated bursts of photons.
R
EFERENCES
[1] R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, Phys. Rev. Lett. 54, 2696 (1985).
[2] Y. Nagamune, H. Sakaki, L.P. Kouwenhoven, L.C. Mur, C.J.P.M. Harmans, J. Motohisa, and H. Noge, Appl. Phys. Lett. 64, 2379 (1994).
[3] A.T. Johnson, L.P. Kouwenhoven, W. de Jong, N.C. van der Vaart, C.J.P.M. Har-mans, and C.T. Foxon, Phys. Rev. Lett. 69, 1592 (1992); J. Weis, R.J. Haug, K. v. Klitzing, and K. Ploog, Phys. Rev. Lett. 71, 4019 (1993).
REFERENCES 15 [4] Yonuk Chong, P. D. Dresselhaus, and S. P. Benz, Appl. Phys. Lett. 86, 232505 (2005); Yonuk Chong, N. Hadacek, P. D. Dresselhaus, C. J. Burroughs, B. Baek, and S. P. Benz, Appl. Phys. Lett. 87, 222511 (2005).
[5] O. V. Astafiev, L. B. Ioffe, S. Kafanov, Yu. A. Pashkin, K. Y. Arutyunov, D. Shahar, O. Cohen, and J. S. Tsai, Nature 484, 355 (2012).
[6] E. R. Williams, R. N. Ghosh, and J. N. Martinis, J. Res. Natl Inst. Stand. Tech-nol. 97, 299 (1992); M. W. Keller, A. L. Eichenberger, J. M. Martinis, and N. M. Zimmerman, Science 285, 1706 (1999).
[7] SQUID Sensors: Fundamentals, Fabrication and Applications, edited by H. Weinstock, (Kluwer, Dordrecht, 1996).
[8] A. Endo, J.J.A. Baselmans, P.P. van der Werf, B. Knoors, S.M.H. Javadzadeh, S.J.C. Yates, D.J. Thoen, L. Ferrari, A.M. Baryshev,f, Y.J.Y. Lankwarden, P.J. de Visser, R.M.J. Janssen, and T.M. Klapwijk, submitted for publication in the SPIE Astronomical Telescopes + Instrumentation 2012.
[9] K. L. Ekinci and M. L. Roukes, Rev. Sci. Instrum. 76, 061101 (2005).
[10] A. D. O’Connell, M. Hofheinz, M. Ansmann, Radoslaw C. Bialczak, M. Lenan-der, Erik Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, John M. Martinis, and A. N. Cleland, Nature 464, 697 (2010).
[11] M. Tinkham, Introduction to Superconductivity, (McGraw-Hill, New York, 1975).
[12] H. Kamerlingh Onnes, Comm. Phys. Lab. Univ. Leiden 120b (1911); H. Kamer-lingh Onnes, Comm. Phys. Lab. Univ. Leiden 122b (1911); H. KamerKamer-lingh Onnes, Comm. Phys. Lab. Univ. Leiden 124c (1911).
[13] W. Meissner and R. Ochsenfeld, Naturwissenschaften 21, 787 (1933).
[14] W. S. Corak, B. B. Goodman, C. B. Satterthwaite, and A. Wexler, Phys. Rev. 96, 1442 (1954) and Phys. Rev. 102, 656 (1956); R. E. Glover and M. Tinkham, Phys. Rev. 104, 844 (1956) and Phys. Rev. 108, 243 (1957).
[15] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957). [16] L. N. Cooper, Phys. Rev. 104, 1189 (1956).
[17] I. Giaever, Phys. Rev. Lett. 5, 147, 464 (1960). [18] B. D. Josephson, Phys. Letters 1, 251 (1962).
[19] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum
Infor-mation, (Cambridge University Press, Cambridge, 2000).
[20] D. P. DiVincenzo, Fortschr. Phys. 48, 771 (2000).
[21] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vander-sypen, Rev. Mod. Phys. 79, 1217 (2007).
[22] D. Press, T. D. Ladd, B. Zhang, and Y. Yamamoto, Nature 456, 218-221 (2008). [23] R. Vrijen, E. Yablonovitch, K. Wang, H. W. Jiang, A. Balandin, V. Roychowdhury,
T. Mor, and D. DiVincenzo, Phys. Rev. A 62, 012306 (2000).
[24] F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, Phys. Rev. Lett. 92, 076401 (2004).
[25] G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, Nat. Mater. 8, 383 (2009).
[26] L. Childress, M. V. Gurudev Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin, Science 314, 281 (2006).; W. F. Koehl, B. B. Buckley, F. J. Heremans, G. Calusine, and D. D. Awschalom, Na-ture 479, 84 (2011).
[27] Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, Nature 398, 786 (1999).
[28] J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian, C. H. van der Wal, and S. Lloyd, Science 285, 1036 (1999).
[29] H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, Phys. Rev. Lett. 107, 240501 (2011).
[30] Yu. A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D. V. Averin, and J. S. Tsai, Nature 421, 823 (2003).
[31] M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, A. D. O’Connell, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, Nature 467, 570 (2010).
[32] L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schus-ter, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Nature 460, 240 (2009).
REFERENCES 17 [33] E. Lucero, R. Barends, Y. Chen, J. Kelly, M. Mariantoni, A. Megrant, P. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, Y. Yin, A. N. Cleland, and J. M. Martinis, Nat. Phys. 8, 719 (2012).
[34] L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, Nature 414, 883 (2001).
[35] N. M. Chtchelkatchev and Yu. V. Nazarov, Phys. Rev. Lett. 90, 226806 (2003). [36] A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E.
Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, Nature 464, 679 (2010).
[37] J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, Nature 475, 359 (2011).
2
T
HEORETICAL
C
ONCEPTS
2.1
T
RANSPORT IN SMALL JUNCTIONS
Among the quantum properties of electron transport in small junctions, per-haps the most spectacular is the quantization of conductance. It was first observed in 1988 [1, 2], by measuring the conductance of a narrow constriction between two wide electrically conducting regions. Such a constriction was termed a quantum
point contact (QPC). In [1], the QPC was built using the two-dimensional layer of
mobile electrons that forms at the surface of a GaAlAs-GaAs semiconductor het-erostructure. A tunable constriction is defined by placing gate electrodes as de-picted in the inset of Fig. 2.1 and applying a negative voltage that electrostatically repels electrons. For sufficiently negative gate voltages the constriction becomes so narrow that no electron can pass and the conductance vanishes. As the gate voltage is increased, the width of the constriction increases and as a result, the conductance of the junction increases as one would expect.
However, it was surprising to find that the increase occurs in quantised steps. As presented in Fig. 2.1, the conductance shows well separated plateaus located at multiples of the quantity GQ= e2/πħ, the quantum of conductance. An important
note is that the experiments were performed at very low temperatures, below one Kelvin. The effect of temperature is to smear out the quantization plateaus. In order to observe pronounced quantization, as seen in Fig. 2.1, low temperature is required.
The quantum of conductance is a universal constant, not depending on the details of the junction. This was an important hint that quantum transport is gov-erned by universal laws valid for any concrete realization of small junctions,
FIGURE2.1: Observation of conductance quantization. The conductance increases in plateaus located at multiples of the conductance quantum GQ= e2/πħ, as the gate voltage is increased. Inset: the
ge-ometry of the gate electrodes that define the constriction (from [1]).
sitive to the chemistry of the materials or to the imperfections of the fabrication techniques.
The explanation of the observed quantization of conductance was already pro-posed in [1]. It is based on the idea that the transverse momentum of electrons in the QPC, the momentum along the y-axis in Fig. 2.1, is quantized.
Quantum mechanics predicts a discrete energy spectrum for any particle con-fined to a region of space. A well known example is the problem of a quantum particle in a box [3]. For a particle in a rectangular box of size Lx, Lyand Lz, the
levels are labelled by three integers nx, nyand nzand their energy is given by:
E (nx, ny, nz) =π 2 ħ2 2m à n2x L2 x + n2y L2 y +n 2 z L2 z ! , (2.1)
m being the mass of the particle. The smaller the dimensions of the box, the larger
the energy of the particle excitations. This statement is valid for a general confine-ment profile. The energy spacing of discrete levels can be modified by tuning the size of the confinement.
The situation in small junctions is similar. In the region of the junction, elec-trons are confined in the transversal direction to the transport, while in the direc-tion of transport there may be no confinement. Transversal confinement gives rise to discrete levels that can be labelled by the corresponding quantum number. The corresponding wavefunctions are delocalized along the transport direction, these
2.1. TRANSPORT IN SMALL JUNCTIONS 21 are transport channels each corresponding to a discrete energy. The discrete en-ergy spectrum implies that the number of transport channels that are occupied at any finite Fermi energy is also finite, assuming the effects of temperature can be neglected. As suggested by the experiment [1], junctions may have a small num-ber of occupied channels that can be modified by tuning the dimensions of the transversal confinement. In this case, the effect of opening a single transport chan-nel can be significant for the transport properties.
One can easily show that the conductance contributed by each open transport channel corresponds to the quantum of conductance (for a comprehensive deriva-tion, the reader is referred to [4]). The typical length scale of the confinement that allows experimental observation of conductance quantization is the Fermi wave-lengthπkF−1, estimated to a few tens of nanometers. In [1] a number of eleven conductance steps have been measured by varying the width of the constriction from W = 0 to a maximum of W = 250 nm. The picture of transport channels requires that the profile of the constriction W (x) changes adiabatically along the transport direction at the scale of the Fermi wavelength. This forces the electron wavefunction to locally satisfy the constriction boundaries and ensures the dis-crete spectrum of transport channels.
Up to this point we have neglected the barriers along the transport direction. If this is the case, the conductance can be determined by simply counting the num-ber of open transport channels. A more general picture takes into account poten-tial barriers along the transport direction, to model, for instance, junctions with an insulating layer in electrical contact with conducting leads. Let us assume a barrier of width d and general profile U (x) and discuss the general implication for transport.
U (x) =
½ U
0(x), for |x| ≤ d/2,
0, for |x| > d/2. (2.2)
An electron coming from one of the conducting leads, for instance the left lead
x ≤ −d/2, scatters at the barrier and is either reflected with a probability R, the
re-flection coefficient, or transmitted with probability T , the transmission coefficient,
T = 1 − R. To quantify transmission and reflection, let us analyse electron
scatter-ing in one of the transport channels. Let the initial electron wavefunction have a plane wave-type dependence on the x coordinate,ψ0= ei kx, k being the x
compo-nent of the wave-vector, the y and z dependence being fixed by the labels (ny, nz)
of the transport channel. The electron energy is E = (ħk)2/2m + Ey z(ny, nz). After
the x coordinate.
ψ(x) =
½
exp(i k x) + r exp(−i kx), for x < −d/2,
t exp(i kx), for x > d/2, (2.3)
r (t ) being complex reflection (transmission) amplitudes. The states with
wave-vector −k correspond to propagation in the opposite direction, away from the bar-rier.
The wavefunction in the region of the potential barrier is determined by the barrier shape and the energy E of the electron. The same factors determine the properties of transport, the reflection (transmission) coefficient R = |r |2(T = |t2|). For instance, for a constant barrier profile U0(x) = U0the transmission coefficient
is given by T (E ) = 4k 2κ2 (k2− κ2)2sin2(κd) + 4k2κ2, κ = (ħ2k2− 2mU0)1/2 ħ . (2.4)
The result suggests that an electron has a finite probability to pass the barrier de-spite its energy being smaller than the barrier height, E < U0; the electron tunnels
through. In this case, the parameterκ is imaginary and the transmission amplitude decays exponentially with the barrier width, T (E < U0) = exp(−2d|κ(E < U0)|) ¿ 1.
The scattering formulation allows for a general description of quantum trans-port in any small junction using the concepts of electron reservoir, scattering region and transport channels [4]. In a typical junction the scattering region is connected to two electron reservoirs, corresponding to the two macroscopically-sized con-ducting leads, referred to as the left and the right lead. Since the leads are macro-scopic, the number of transport channels is generally infinite, however most of these are closed, or have little transmission coefficients, such that the conductance of the junction remains finite. The occupation of states in the left and right reser-voirs is determined by the Fermi distribution function
fL,R(E ) =
1
1 + exp[(E − µL,R)/kBT ]
, µL,R= EF+ eVLR, (2.5)
T being the temperature,µLand VL(µRand VR) being the chemical potential and
voltage respectively, in the left (right) reservoir.
A general expression for the conductance in a small junction at low tempera-ture kBT ¿ eV , V ≡ VL− VR being the bias voltage, is given by the Landauer for-mula [5]. G = GQ X n ZµL µR d E eVTn(E )( fL(E ) − fR(E )), (2.6)
2.2. CHARGING EFFECTS ANDCOULOMB BLOCKADE 23
n being the index of transport channels. If the energy dependence of the
transmis-sion coefficients is negligible in the energy interval (µL..µL), typically the case, the
formula simplifies to a better known expression
G = GQ
X
n
Tn. (2.7)
To recover the expression of the conductance in a QPC, one sets T = 1 for the open transport channels, and T = 0 for the closed ones. The Landauer formula also accounts for junctions with a conductance smaller than GQ. For instance,
this is the case of tunnelling junctions, where even though the number N of open channels may be large, all the transmission coefficients are small T ¿ 1.
2.2
C
HARGING EFFECTS AND
C
OULOMB BLOCKADE
The discussion of electron scattering in nanostructures provided above disregards interactions between electrons. However, electrons have discrete charge and can in principle interact with each other electrostatically, by the Coulomb interaction. While this interaction can safely be neglected in many cases, in junctions with an isolated region of space where electrons can accumulate charging effects may become important [6]. When charging effects are dominant the quantization of charge plays a crucial role and the physics of transport becomes sensitive to single electron transport events [7]. This is the ultimate limit of sensitivity to electrical currents and opens up exciting applications, such as the single electron transis-tor realized for the first time in [8]. Other important applications are based on the manifestations of charging effects in superconducting Josephson junctions. In this quantum system states of different charge are mixed and can be used to encode quantum information [9].
Following the approach of [4], let us consider an isolated metallic island of roughly cubic shape Lx' Ly' Lz' L, with charge q consisting of N discrete
elec-tron charges q = Ne. The electrostatic energy of the island is a classical quantity that can be expressed in terms of the island capacitance C , E (N ) =2Cq2 = N2 e
2
2C. The
charge on the metallic island is distributed at the surface, therefore the capacitance can be estimated simply as C ' 4π²0L,²0being the electric constant.
One can tune the electrostatic energy of the island by placing it in an external electrical field. The practical realization involves placing a gate electrode near the island, but not in electrical contact with it, and tuning the voltage on the electrode. For a gate voltage Vg, the electrostatic energy becomes E (N ) = N2EC− NeVg, EC≡ e2/2C . However, since no electron reservoir is connected to the island, the number of discrete charges on remains constant.
Let us bring the metallic island in electrical contact with an electron reservoir. The setup of reservoir, metal island, and gate electrode is known as the
single-electron box. For simplicity, we assume for now that the capacitance CLformed
between the metallic island and the reservoir is much larger than the capacitance
Cg formed between the island and the gate electrode. The energy cost of adding
an electron to the island is given by the difference of electrostatic energies, that is, the charging energy, E+. Similarly, the energy cost of removing an electron is E−.
E+=E(N + 1) − E(N ) = EC(2N + 1) − eVg, (2.8) E−=E(N − 1) − E(N ) = −EC(2N − 1) + eVg. (2.9)
If Vgis tuned such that either E+or E−becomes negative, it follows that the
charg-ing state with N electrons on the island is no longer the lowest energy state. In the case E+< 0 (E−< 0), an electron will be transferred to (from) the island. This
requires a charging timeτC' C /G, G being the conductance of the electrical
con-tact to the reservoir. In the stationary case, one electron transfer is guaranteed if the voltage on the gate Vg changes by an amount EC/e. Thereby, the number of
electrons in the ’box’ can be tuned one by one.
An important note is that so far we have been talking about charge states. For the number of electrons to be a good index of a quantum state, we must require that the energy of quantum fluctuations ħ/τCis much smaller than the energy
re-quired to change the charging state EC, so that ħ ¿ ECτC. Remarkably, in this
relation the capacitance of the island simplifies and we find a condition for the conductance of the electrical contact G ¿ GQ. The conclusion is universal for
any such junction: the charging effects are important if the conductance is much smaller than the quantum of conductance and may be neglected in the opposite case [10]. An intermediary case is found if G ' GQ, when the effective charging
energy is renormalized to the value ˜ EC= EC µG GQ ¶ exp µ −αG GQ ¶ , (2.10)
α being a dimensionless parameter of order one depending on the details of the
junction [13]. Therefore, at large conductances charging effects diminish expo-nentially.
The implications of charging effects on transport can be understood by adding an additional reservoir in electrical contact with the island, such that one reservoir can act as a source of electrodes, while the other one can act as a drain. This setup describes a single electron transistor (SET). Let us denote the capacitances to the left (right) reservoirs by CL(CR). Here, we will consider the capacitances finite. The
!["A Descriptive Catalogue of the Greek Papyri in the Collection of Wilfred Merton F. S. A.", vol. I, H. Idris Bell, C. H. Roberts, London 1948 : [recenzja]](data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==)