Josephson effects in carbon nanotube mechanical resonators and graphene

(1)

Josephson eff

ec

ts in c

arbon nanotube mechanic

al r

esona

tors and graphene Han K

eijz

ers

Josephson effects in carbon nanotube

mechanical resonators

and graphene

Han Keijzers

Uitnodiging

Josephson effects in carbon nanotube

mechanical resonators

and graphene

voor het bijwonen van de openbare

verdediging van mijn proefschrift

en bijbehorende stellingen

op maandag 15 oktober

om 15:00 uur in de Aula

van de TU Delft

Mekelweg 5 te Delft

Een half uur voor aanvang

(14:30 uur) zal ik het onderwerp van

mijn promotie kort toelichten.

Han Keijzers

Boerhaavelaan 114

2334 ET Leiden

Paranimfen:

Vincent Mourik, 06 3061 0308

Ciprian Padurariu, 06 4511 8617

Josephson effects in carbon nanotube

mechanical resonators

and graphene

Han Keijzers

A carbon nanotube is a unique one dimensional tube in which the quantum nature of electrons can be studied. When cooled

down to temperatures close to absolute zero, a nanotube can become superconducting by connecting it to a superconductor.

Due to the Josephson effect, a dissipationless supercurrent oscillating at several GHz, can flow through the nanotube.

Suspended nanotube mechanical resonators are extremely sensitive force and mass sensors. We have made a unique

resona-tor, in which the suspended nanotube is a Josephson junction. These superconducting nanotube guitar strings can vibrate at

GHz frequencies. We present theoretical and experimental studies on the role of superconductivity on mechanical resonance

in these unique systems.

In a second experiment we have worked on superconductivity in graphene. Graphene is a two dimensional sheet of carbon

atoms. We have studied the effect of magnetic field on supercurrent in graphene, and present our experimental results.

Casimir PhD Series, Delft-Leiden 2012-19

ISBN: 978-90-8593-130-0

Josephson eff

ec

ts in c

arbon nanotube mechanic

al r

esona

tors and graphene Han K

eijz

ers

Josephson effects in carbon nanotube

mechanical resonators

and graphene

Han Keijzers

Uitnodiging

Josephson effects in carbon nanotube

mechanical resonators

and graphene

voor het bijwonen van de openbare

verdediging van mijn proefschrift

en bijbehorende stellingen

op maandag 15 oktober

om 15:00 uur in de Aula

van de TU Delft

Mekelweg 5 te Delft

Een half uur voor aanvang

(14:30 uur) zal ik het onderwerp van

mijn promotie kort toelichten.

Han Keijzers

Boerhaavelaan 114

2334 ET Leiden

Paranimfen:

Vincent Mourik, 06 3061 0308

Ciprian Padurariu, 06 4511 8617

Josephson effects in carbon nanotube

mechanical resonators

and graphene

Han Keijzers

A carbon nanotube is a unique one dimensional tube in which the quantum nature of electrons can be studied. When cooled

down to temperatures close to absolute zero, a nanotube can become superconducting by connecting it to a superconductor.

Due to the Josephson effect, a dissipationless supercurrent oscillating at several GHz, can flow through the nanotube.

Suspended nanotube mechanical resonators are extremely sensitive force and mass sensors. We have made a unique

resona-tor, in which the suspended nanotube is a Josephson junction. These superconducting nanotube guitar strings can vibrate at

GHz frequencies. We present theoretical and experimental studies on the role of superconductivity on mechanical resonance

in these unique systems.

In a second experiment we have worked on superconductivity in graphene. Graphene is a two dimensional sheet of carbon

atoms. We have studied the effect of magnetic field on supercurrent in graphene, and present our experimental results.

Casimir PhD Series, Delft-Leiden 2012-19

ISBN: 978-90-8593-130-0

Josephson eff

ec

ts in c

arbon nanotube mechanic

al r

esona

tors and graphene Han K

eijz

ers

Josephson effects in carbon nanotube

mechanical resonators

and graphene

Han Keijzers

Uitnodiging

Josephson effects in carbon nanotube

mechanical resonators

and graphene

voor het bijwonen van de openbare

verdediging van mijn proefschrift

en bijbehorende stellingen

op maandag 15 oktober

om 15:00 uur in de Aula

van de TU Delft

Mekelweg 5 te Delft

Een half uur voor aanvang

(14:30 uur) zal ik het onderwerp van

mijn promotie kort toelichten.

Han Keijzers

Boerhaavelaan 114

2334 ET Leiden

Paranimfen:

Vincent Mourik, 06 3061 0308

Ciprian Padurariu, 06 4511 8617

Josephson effects in carbon nanotube

mechanical resonators

and graphene

Han Keijzers

A carbon nanotube is a unique one dimensional tube in which the quantum nature of electrons can be studied. When cooled

down to temperatures close to absolute zero, a nanotube can become superconducting by connecting it to a superconductor.

Due to the Josephson effect, a dissipationless supercurrent oscillating at several GHz, can flow through the nanotube.

Suspended nanotube mechanical resonators are extremely sensitive force and mass sensors. We have made a unique

resona-tor, in which the suspended nanotube is a Josephson junction. These superconducting nanotube guitar strings can vibrate at

GHz frequencies. We present theoretical and experimental studies on the role of superconductivity on mechanical resonance

in these unique systems.

In a second experiment we have worked on superconductivity in graphene. Graphene is a two dimensional sheet of carbon

atoms. We have studied the effect of magnetic field on supercurrent in graphene, and present our experimental results.

Casimir PhD Series, Delft-Leiden 2012-19

ISBN: 978-90-8593-130-0

Josephson eff

ec

ts in c

arbon nanotube mechanic

al r

esona

tors and graphene Han K

eijz

ers

Josephson effects in carbon nanotube

mechanical resonators

and graphene

Han Keijzers

Uitnodiging

Josephson effects in carbon nanotube

mechanical resonators

and graphene

voor het bijwonen van de openbare

verdediging van mijn proefschrift

en bijbehorende stellingen

op maandag 15 oktober

om 15:00 uur in de Aula

van de TU Delft

Mekelweg 5 te Delft

Een half uur voor aanvang

(14:30 uur) zal ik het onderwerp van

mijn promotie kort toelichten.

Han Keijzers

Boerhaavelaan 114

2334 ET Leiden

Paranimfen:

Vincent Mourik, 06 3061 0308

Ciprian Padurariu, 06 4511 8617

Josephson effects in carbon nanotube

mechanical resonators

and graphene

Han Keijzers

A carbon nanotube is a unique one dimensional tube in which the quantum nature of electrons can be studied. When cooled

down to temperatures close to absolute zero, a nanotube can become superconducting by connecting it to a superconductor.

Due to the Josephson effect, a dissipationless supercurrent oscillating at several GHz, can flow through the nanotube.

Suspended nanotube mechanical resonators are extremely sensitive force and mass sensors. We have made a unique

resona-tor, in which the suspended nanotube is a Josephson junction. These superconducting nanotube guitar strings can vibrate at

GHz frequencies. We present theoretical and experimental studies on the role of superconductivity on mechanical resonance

in these unique systems.

In a second experiment we have worked on superconductivity in graphene. Graphene is a two dimensional sheet of carbon

atoms. We have studied the effect of magnetic field on supercurrent in graphene, and present our experimental results.

Casimir PhD Series, Delft-Leiden 2012-19

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Propositions

belonging to the thesis

Josephson effects in carbon nanotube mechanical resonators and graphene

C.J.H. Keijzers

1. The presence of the AC Josephson effect enhances transduction of mechan-ical displacement to electrmechan-ical signals by up to two orders of magnitude.

Chapter 5 of this thesis.

2. Even during a complete suppression of the observable DC supercurrent, the amplitude and sign of AC Josephson currents can still be detected.

3. The main reason for the inefficient energy exchange between a carbon nan-otube resonator and the AC Josephson current is the small linewidth of the resonator in comparison to the large linewidth of the Josephson current.

4. The prohibition of the vereniging Martijn∗_{does not protect law and order.}

∗_{Ruling Rechtbank Assen, June 27, 2012.}

5. To prevent argumentation errors within the judicial process, the legitimacy of all arguments of prosecutor and defender must be subject to review by lawyers schooled in empirical research.

6. Creative technical solutions can only arise in complete artistic freedom, while innovation is only possible when it is carefully constrained.

7. The best way to prevent the impending shortage∗ _{of technically schooled}

personnel is doubling the initial salary for this group.

After Ad Lagendijk, NRC, September 17, 2012.

∗_{“Toekomst van de industrie”, May 18, 2012.}

8. The pertinacious use of terms such as “peace mission” and “reconstruction mission” by government and press increases misunderstanding of psycho-logical problems in veterans.

9. A speaker who adds a progress indicator to his PowerPoint presentation will never make his audience so engrossed in his subject that they lose the sensation of time.

10. The strongest propositions deserve a bit of nuance.

These propositions are considered opposable and defendable and as such have been approved by the supervisor, Prof. dr. ir. L. P. Kouwenhoven.

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behorende bij het proefschrift

Josephson effecten in koolstofnanobuis mechanische resonatoren en grafeen

C.J.H. Keijzers

1. De aanwezigheid van het AC Josephson effect verhoogt de transductie van mechanische verplaatsing naar elektrische signalen met tot wel twee ordes van grootte.

Hoofdstuk 5 van dit proefschrift.

2. Zelfs terwijl de waarneembare DC superstroom volledig is onderdrukt, kun-nen de amplitude –en het teken–, van AC Josephson stromen nog steeds gedetecteerd worden.

3. De voornaamste reden voor de ineffici¨ente energie uitwisseling tussen een koolstofnanobuis resonator en de AC Josephson stroom is de kleine lijnbreedte van de resonator ten opzichte van de grote lijnbreedte van de Josephson stroom.

4. Het verbieden van de vereniging Martijn∗ _{is in strijd met de bescherming}

van de openbare orde.

∗_{Uitspraak Rechtbank Assen, 27 juni 2012.}

5. Om argumentatiefouten binnen de rechtsgang te voorkomen, moet de geldigheid van alle argumenten van aanklager en verdediger getoetst worden door ju-risten geschoold in empirisch onderzoek.

6. Creatieve technische oplossingen kunnen alleen ontstaan in complete artistieke vrijheid, terwijl innovatie alleen mogelijk is door de zorgvuldige begrenzing daarvan.

7. De beste manier om het dreigend tekort∗_{aan technisch geschoold personeel}

te voorkomen is het verdubbelen van de aanvangssalarissen voor deze groep.

Naar Ad Lagendijk, NRC, 17 september 2012.

∗_{“Toekomst van de industrie”, 18 mei 2012.}

8. Het hardnekkig gebruik van termen als “vredesmissie” en “opbouwmissie” door de overheid en media vergroot het onbegrip voor psychische problemen bij veteranen.

9. Een spreker die een voortgangsindicator toevoegt aan zijn PowerPoint pre-sentatie zal er nooit in slagen zijn publiek de tijd te laten vergeten. 10. De krachtigste stellingen verdienen een beetje nuancering.

Deze stellingen worden opponeerbaar en verdedigbaar geacht en zijn als zodanig goedgekeurd door de promotor, Prof. dr. ir. L. P. Kouwenhoven.

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Josephson eﬀects in carbon nanotube

mechanical resonators

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Josephson eﬀects in carbon nanotube

mechanical resonators

and graphene

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 15 oktober 2012 om 15:00 uur

door

Christianus Johannes Henricus KEIJZERS

natuurkundig ingenieur geboren te Deurne.

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Prof. dr. ir. L. P. Kouwenhoven

Samenstelling promotiecommissie:

Rector Magniﬁcus, voorzitter

Prof. dr. ir. L.P. Kouwenhoven, Technische Universiteit Delft, promotor Prof. dr. ir. J.E. Mooij, Technische Universiteit Delft

Prof. dr. Y.V. Nazarov, Technische Universiteit Delft Prof. dr. ir. L.M.K. Vandersypen, Technische Universiteit Delft Prof. dr. J. Aarts, Universiteit Leiden

Prof. dr. ir. H. Hilgenkamp, Universiteit Twente

Dr. ir. G.A. Steele, Technische Universiteit Delft

Prof. dr. H.W. Zandbergen, Technische Universiteit Delft, reservelid

Keywords: Josephson eﬀect, quantum dots, graphene, carbon nanotubes,

nanomechanical devices, NEMS, QNEMS, π-junctions.

Published by: C.J.H. Keijzers

Cover design: C.J.H. Keijzers

Front: Shapiro steps on mechanical resonance in a suspended CNT

Josephson junction, Ch. 5, Fig. 5.8, of this thesis.

Back: Electron micrograph of a suspended CNT Josephson junction.

Printed by: Ipskamp Drukkers BV, Enschede

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Chapter 1 Introduction

1.1 Quantum nanoscience

The physics of electrons in small electronic structures can be very rich, interest-ing and potentially useful. This is especially the case when measurements are performed in conditions that permit observation of the quantum nature of the electrons in the system, or of the structure itself.

The development of nanotechnology in the last two decades makes it now possible to build structures where quantum mechanical effects become important. It offers a toolbox with which it is possible to make and measure structures in which, for example, the wave nature of electrons dominates the conductance. Typical dimensions that are required to reach this regime are on the order of 1 to 100 nm. Typically experiments take place at temperatures on the order of 100 mK. In the research groups of the Quantum Nanoscience (QN) department the quan-tum nature of nanoscale electronic and mechanical systems is studied (or pursued) by means of electrical or optical interfaces. One of the main goals is the realiza-tion of building blocks for the quantum computer (QC). This is a new type of computer that uses quantum mechanics to perform calculations in a completely different way compared to ordinary computers. When a QC is sufficiently large, it can outperform ordinary computers on specific tasks. Information in quantum computers is stored in quantum bits (or qubits) rather than conventional bits. Small quantum computers are already around in research labs, and much effort is made to investigate existing building blocks, develop new types, and combine them to make larger quantum computers. A review on superconducting qubits can be found in Ref. [1].

On the road towards the quantum computer, many interesting things can be learned about the nature of electrons and the devices in which they reside. We have worked on graphene and carbon nanotubes, two materials that have poten-tial as building blocks for a QC. The experiments in this thesis are both done in new regimes where these materials have not been operated before. The tech-niques we have developed, together with our experimental observations, can be useful in future quantum electronic devices.

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In this thesis we will present the results of two experiments. The first experiment is on graphene Josephson junctions in large magnetic fields. Our goal is to realize a π-junction in graphene, by means of a magnetic field. Such junctions have specific applications and are of fundamental interest.

The second experiment is the main experiment of this thesis. It is a curiosity-driven study where two ﬁelds of physics that are usually separated are brought closely together in a single suspended carbon nanotube (CNT). These ﬁelds are nanomechanics and superconductivity. Our goal is to investigate superconductivity-mediated transduction of the nanotube mechanical vibrations to electrical signals. This work is potentially useful for the study of the quantum nature of a mechan-ical resonator. There are practmechan-ical applications of such resonators, and they are also of fundamental interest.

In this chapter we will introduce the main topics of this thesis: Superconductivity, graphene and carbon nanotubes, nanomechanics and clean carbon nanotubes.

1.2 Superconductivity

In both of our experiments carbon based materials are combined with super-conductivity. When cooled below the critical temperature, superconductors lose their resistance. In the superconducting state, a superconductor can carry a cur-rent without dissipating energy. This is called a supercurcur-rent. Aluminum and niobium are two common superconductors. In these (and many other) metals the interaction of electrons with phonons causes pairing of electrons in Cooper pairs, that form a condensate that can carry a supercurrent. This macroscopic quantum-mechanical eﬀect occurs when the thermal energy of the system is below the binding energy associated to Cooper pairing [2].

To observe the superconducting state a superconductor has to be cooled below its critical temperature. Superconductivity can then be inferred by measuring the electrical resistance. This was ﬁrst done in Leiden, by Heike Kamerlingh Onnes, who discovered superconductivity in mercury in 1911 [3]. The discovery of superconductivity opened up a new ﬁeld of physics that is still growing today.

1.3 Carbon based nano-electronics

1.3.1 Bottom-up nanofabrication

A very successful approach to access the quantum regime with nanotechnology, involves a carbon-based bottom-up fabrication method. In the bottom-up method a (usually small) structure is taken and lithographic techniques are used to in-terface with it. In the top-down method, these techniques are not only used to access, but also to deﬁne the structure. The advantage of the bottom-up method is that structures can be chemically synthesized, or extracted from a larger crys-tal, with almost perfect crystal structure. In the top-down method it is also

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1.3. Carbon based nano-electronics

possible to define structures atom by atom, but this is relatively hard [4]. Molecular configurations (allotropes) of carbon have been the workhorse of bottom-up nanofabrication since 1985. In this year 0D buckyballs were chemically syn-thesized [5]. This was followed by the discovery of multiwall and single-wall nan-otubes in respectively 1991 and 1993 [6,7]. In 2004 graphene, a 2D sheet of carbon atoms was extracted by mechanical exfoliation from graphite [8, 9]. In Fig. 1.1 we give an overview of carbon allotropes. Carbon nanotubes and graphene have exceptional electronic and mechanical properties. This has spurred a great effort by the physics community to study these materials [10, 11].

In the following part of this subsection we will give some examples of the excep-tional properties of CNTs and graphene.

Figure 1.1: Overview of carbon allotropes. Graphene can be formed in (from left

to right) 0D buckyballs, 1D nanotubes and 3D graphite. Figure adapted from Ref. [11].

1.3.2 Carbon nanotube transistors

Carbon nanotubes are usually semiconducting and can have a varying bandgap depending on its chirality (the way graphene is rolled up to form a nanotube). Small bandgap nanotubes are called metallic nanotubes and have gaps on the order of 50 meV. Large bandgap nanotubes have gaps on the order of 500 meV.

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A nanotube based FET was ﬁrst demonstrated in 1998 [12]. A lot of interest is in the development of CNT FETs, because of their exceptional properties. Carbon nanotubes can have a very high mobility, exceeding 105_cm2_{/Vs at room}

temperature. This is much higher than the mobility of graphite (∼ 2·104cm2/Vs). Possibly, the origin of this high mobility is the 1D nature of the nanotube. Elec-trons can only go forward or backward and not to the sides, which makes it harder for them to scatter [13]. Because in addition CNTs have very few struc-tural defects, they behave as ballistic conductors. For these reasons, CNTs make exceptionally good FETs. Recently a sub-10 nm transistor made from a CNT was reported that is smaller and performs better than current silicon transistor technology [14, 15].

1.3.3 Carbon nanotube quantum dots

Their small dimensions and low scattering make CNTs ideal materials for con-ﬁning electrons and holes in quantum dots [16]. Single-electron spins in CNT quantum dots can be used as building blocks for a quantum computer. For this reason, much eﬀort is being made to study and control spins in nanotubes [17]. We will discuss CNT quantum dots in Sec. 2.1.

1.3.4 Superconducting carbon nanotubes

When two superconducting leads are coupled by a weak link, a supercurrent can ﬂow from one lead to the other. It is as if the weak link has become supercon-ducting itself, a phenomenon called proximity-induced superconductivity. Such a device is called a Josephson junction. The most common junction is made by employing a natural oxide on a superconductor as a weak link, in a sandwich structure (for example Al/Al2O3/Al). Nanotubes can also be used as weak link,

and have certain advantages over typical Josephson junctions. We point out two advantages of CNT junctions: Gate tunable supercurrents and small dimensions. Because the CNT is a semiconductor, its conductance can be changed with a gate. In this way the coupling between the two superconductors (and with this the maximum supercurrent) can be changed in situ, which is not possible with an oxide junction. The ﬁrst superconducting CNT transistors were made in 2006 [18, 19].

Two parallel CNT junctions form a nano-SQUID (superconducting quantum in-terference device). Because of the small size of a nano-SQUID it can be brought very close to magnetic molecules. This allows in principle for extreme sensitiv-ity to the magnetic ﬂux generated by such molecules. This sensitivsensitiv-ity could be employed to characterize molecules, which is one application of such a device [20]. Josephson junctions have strong non-linear IV characteristics. The current through a Josephson junction is determined by the phase diﬀerence between the contacts rather than the voltage bias. As will be discussed in Ch. 2, the phase is also a function of the voltage bias. When an RF voltage bias is applied to the junc-tion, the time-averaged voltage displays steps as a function of applied current

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1.3. Carbon based nano-electronics

bias. The height and width of these Shapiro steps are a direct consequence of the (time averaged) dynamics in the junction, and depend on the power and fre-quency of the applied RF bias. In nanotube Josephson junctions, Shapiro steps of 2 . . . 20 μV are typically observed by applying an RF drive with a frequency in the 1 . . . 10 GHz range [21]. With Josephson junctions an RF signal can be transduced to a DC signal. We will see later that this is interesting in relation to mechanics, because it allows detection of RF signals within a low-bandwidth measurement setup. Such setups are typically necessary to observe supercurrents in CNTs.

1.3.5 Special properties of graphene

Since its discovery, the unique properties of graphene result in a continuous stream of publications. Here we will give a few examples of the special prop-erties of graphene.

Electrons in graphene are described by the Dirac equation rather than the Schrödinger equation [22]. As a result, electrons behave differently in graphene compared to any other solid state system. Examples of this are the half-integer quantum Hall effect and Klein tunneling through a high potential barrier with approaching 100% transmission probability for certain angles [9, 23–25].

Graphene is a semi-metal (zero-gap semiconductor). Its bandstructure looks like a Dirac cone. Because of the absence of a bandgap it is not possible to create quantum dots in as-is graphene. However, a bandgap can be introduced in bilayer graphene and in graphene nano-ribbons [26–28]. It has been found that disor-der makes it very difficult to define quantum dots in graphene, but it has been achieved by a few groups now [29–31]. Many efforts have been made to reduce the disorder in graphene and improve its mobility. Several approaches have been followed including suspending graphene and placing it on a boron nitride sub-strate [32, 33]. In this way, graphene with a mobility on the order of 105cm2/Vs (at reduced temperature) can be achieved. Large sheets of graphene (∼ 80 cm) can be grown in an industrial setting, and (industrial scale) graphene transistors are expected to outperform silicon based transistors on specific tasks [34, 35].

1.3.6 Superconducting graphene

Early experiments on weak localization have shown that phase coherence and time reversal symmetry (TRS) were absent or strongly suppressed in some graphene samples [36]. Since both phase coherence and TRS are necessary for Andreev reﬂection, it was not expected that graphene could support supercurrents. Fur-thermore, since intrinsic graphene has a vanishing density of states at the Dirac point, carrying supercurrent there was considered to be unfavorable also because of this reason.

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This was disproved in an experiment where graphene was contacted to two super-conducting leads, forming a weak link Josephson junction [37]. The experiment showed that the maximum supercurrent can be gate tuned, and carried by holes as well as electrons. Transport in graphene is phase coherent even at the Dirac point, where the supercurrent was still ﬁnite. Graphene Josephson junctions are unique, because they are the only junctions where the weak link is truly 2D. This property is exploited in the experiment described in Ch. 4 of this thesis.

1.4 Nanomechanics

1.4.1 Quantum mechanics

In condensed matter, quantum mechanical behavior has long been limited to the domain of single electrons and atoms. The advance of nanotechnology has made it possible to test on-chip the limits of quantum mechanics for systems with a large number of atoms. One way of distinguishing quantum behavior from classical behavior is by putting the object of interest in a superposition state. For a single electron this could be a superposition of spin, for a single atom it could be a superposition of position. In the macroscopic, classical world of tables and chairs such exotic phenomena are not observed. Quantum nanomechanics oﬀers a platform to study the emergence of the classical world from quantum mechanics [38].

Since 2010 the ﬁrst experiments have been reported where a mechanical resonator has been brought into a superposition of its state of motion [39–42]. These exper-iments show that a macroscopic resonator can be in two places at the same time, just like a single atom can. It shows that quantum mechanics is not limited to systems made of only a few atoms, but holds for macroscopic systems containing ∼ 1012 _{of atoms as well. Theory predicts that by study of the decoherence rate}

of macroscopic superposition states, we can learn about the fundamental origin of decoherence [43]. Experiments are proposed in which this is used to directly test the limits of quantum theory [44].

A mechanical resonator becomes a quantum resonator when its average thermal occupation (¯n = [exp(ωr/kBT )− 1]−1) drops below one. In practice, this means

that the temperature of the resonator has to be below its quantization energy: kBT ωr. A direct approach to reach this regime is to cool a high frequency

(fr > 1 GHz) resonator in a dilution fridge (T < 50 mK). However, such high

frequency resonators will have small zero point motion and hence their motion will be hard to detect. As will be discussed in the next subsection, CNTs are very promising nanomechanical systems (NEMS), because they combine high frequency modes with large zero point motion and low damping. Much eﬀort is been made now to make the ﬁrst quantum resonator in a suspended carbon nanotube.

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1.5. Clean carbon nanotubes

1.4.2 Carbon nanotube mechanics

From an engineering viewpoint, CNT NEMS are interesting systems because they are the most sensitive force and mass sensors. Mass detection with a resolution on the order of 1.7 yoctogram (1 yg = 10−24g) has recently been achieved using a CNT detector [45]. This corresponds to weighing the mass of a single proton. The large sensitivity is mainly due to the large Young’s modulus (E = 1.25 TPa) and low mass density (ρ = 1350 kg/m3) of a CNT [46, 47]. In general, a large E implies a large spring constant and a high stiﬀness. In combination with a small mass (on the order of m = 2× 10−18g = 2 ag), this results in high resonance frequencies. In CNTs bending modes have now been detected with frequencies in the 100 MHz to 40 GHz range [48–51]. In this thesis we will report on CNTs with fr∼ 1 GHz.

Ultimately the force sensitivity of a mechanical resonator is limited by its zero point motion (zpm). From the engineering viewpoint of building a quantum resonator with a CNT, it is convenient when the zero point motion is large, because then it is easier to detect. Because of their low mass, CNTs can combine a (relatively) large zpm (∼ 1 pm) with large eigenfrequencies (∼ 1 GHz) and for this reason they are very suitable for embedding a quantum resonator [38]. An additional advantage of bottom-up CNT resonators compared to top-down fabricated NEMS, is their lack of structural and surface defects. It is likely that the origin of damping of mechanical vibrations lies in such defects in top-down fabricated NEMS [52]. Small damping implies long ring down times. In a quan-tum nanotube resonator, this could be used to manipulate and store quanquan-tum states.

1.5 Clean carbon nanotubes

The outstanding intrinsic electronic and mechanical properties of CNT devices were for a long time masked by environmental eﬀects introduced by the nanofab-rication steps necessary to make a device. In other words, the CNT got dirty during fabrication. The development of a new fabrication method, where the CNT is kept clean by suspending it on the chip at the last fabrication step, enabled the start of a new generation of CNT experiments [50, 53–55].

Clean nanotubes show very low disorder, which is reﬂected in their electronic and mechanical properties. Quantum dots made in clean nanotubes can very easily be tuned in the few electron/hole regime and are very stable. The mechanical properties are also outstanding, because they combine fundamental resonance frequencies on the order of f ∼ 1 GHz with very large quality factors, exceed-ing Q = 105. This discovery boosted the potential of CNTs in the ﬁeld of nanomechanics, because such high Q· f product has not been achieved in other nanomechanical systems [51].

At this moment two clean fabrication methods are being in use in Delft. We call the ﬁrst (oldest) fabrication method conventional, and the second (newer) new.

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In both methods, the device contacts and gates are made before the deposition of the nanotube.

In the conventional method, all the gates and contacts have to be compatible with the CNT growth condition (∼ 900 °C in a ﬂow of methane). Our major achievement is that we have found a superconductor that is compatible with the conventional clean fabrication method. This opens up a new playground for experiments with CNT resonators and superconducting devices. In Ch. 5 and 6 we report on the simplest example of such a system, a single CNT Josephson junction made of a suspended CNT.

In the new type of clean fabrication, the nanotube is grown on a mother chip and then stamped onto a receiver chip on which all the gates and contacts have been pre-fabricated [56, 57]. This new type of fabrication offers advantages above the existing one. The first advantage is that the structures on the receiver chip do not have to be compatible with the CNT growth. The second advantage is that a single CNT can (in principle, this has not been done yet because it is hard to see exactly where the tube is on the mother chip) be picked and placed on a predefined position. The conventional clean fabrication method which we have used relies heavily on statistics and has a very low yield on the order of 1%.

1.6 Outline of this thesis

In Ch. 2 we will discuss theoretical concepts that are relevant in our experiments. Especially our CNT devices display very rich physics. They are: Quantum dots, Josephson junctions and mechanical resonators all at the same time. We will not discuss many theoretical concepts in the experimental chapters, but refer to the theory chapter when necessary. Chapter 3 is on the nanofabrication of graphene and suspended CNT devices.

In the main part of this thesis we will discuss two experiments. The ﬁrst experi-ment is on graphene Josephson junctions and the second (and main) experiexperi-ment on suspended nanotube Josephson junctions. In the experiment on graphene, the dependence of supercurrent on an in-plane magnetic ﬁeld is studied with the goal of creating a Zeeman π-junction. This work is presented in Ch. 4. In Ch. 5 we report on our study of suspended CNT Josephson junctions with GHz res-onance frequencies. We report on a novel mixing signal that we contribute to AC Josephson dynamics. In Ch. 6 we discuss an experiment in which we probe the superconducting CNT mechanical vibrations by driving the system at a sub-harmonic of the mechanical mode. In Ch. 7 we discuss the current status and challenges regarding CNT resonators, and suggest a broad range of experiments with suspended CNT Josephson junctions. I conclude Ch. 7 with a personal consideration of future directions.

In App. A we present additional data on CNT and graphene experiments. In App. B we present a general analysis of the eﬀect of RF signals on the conductance of a suspended CNT quantum dot. Appendix C contains our theory preprint (see next paragraph). In App. D we present our results on the characterization of rhenium ﬁlms. Appendix E contains the fabrication recipes of graphene and

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1.6. Outline of this thesis

nanotube devices, and in App. F we provide the recipe for a superconducting coil such as we have used in our graphene experiment.

We present an analysis of resonant interaction between AC Josephson eﬀect and CNT mechanics in our preprint that we have included as-is in App. C. Unfor-tunately most of its contents is not directly relevant to our experiments. In our paper we focus on a regime of resonant interaction in which energy is eﬃciently transferred between resonator and supercurrent. Our experiment is probably not in this regime, due to a large mismatch between the linewidth of AC Josephson dynamics and the linewidth of the resonator. This will be discussed more in detail at the end of Sec. 5.10.

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Chapter 2 Theoretical concepts

In this chapter we will present the three main ingredients of our experiments: Quantum dots in carbon nanotubes (Sec. 2.1), superconductivity and the Joseph-son effect (Sec. 2.2-2.3), and carbon nanotube mechanical reJoseph-sonators (Sec. 2.4). In a separate section we will consider interaction between Josephson effect and nanomechanics (Sec. 2.5). In the final section of this chapter (Sec. 2.6) we discuss the magnetic field dependence of the Josephson current. Properties of graphene will be discussed in the section on nanotubes.

2.1 Carbon nanotube quantum dots

Quantum dots are small structures in which the wave-nature of electrons or holes dominates their behavior. Nanotubes are narrow hollow cylinders made entirely out of carbon, in which electrons or holes are naturally conﬁned. In this section we will discuss the nature of electrons and holes in graphene and CNTs, and discuss how quantum dots can be made in CNTs.

2.1.1 The band structure of carbon nanotubes and graphene

In carbon nanotubes and graphene the carbon atoms are arranged in a hexagonal lattice that gives graphene and CNTs a unique electronic structure: An electron-hole symmetric linear dispersion relation near two points, that are referred to as

Kand K. We will see that this has important consequences for the behavior of electrons and holes.

The organization of this subsection is as follows: First we will discuss the band-structure of graphene, and how it is affected by quantization conditions in CNTs. After that we will discuss the effect of symmetry breaking by magnetic field and spin-orbit interaction, and we will conclude with a short discussion on the bandgap of CNTs and the effect of longitudinal quantization.

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Graphene bandstructure

A CNT can be modeled as a rolled up sheet of graphene. The bandstructure of CNTs is very similar to that of graphene, but is modified by quantization conditions that are due to the spatial confinement of electrons. To understand the bandstructure of CNTs it is necessary to first understand the bandstructure of graphene.

Graphene consists out of a two-dimensional honeycomb lattice of carbon atoms (Fig. 2.1). There are two inequivalent sites in the graphene lattice, labeled A and B. Each orbital can be occupied by an electron with spin up or spin down, a four-fold basis (A↑,A↓,B↑,B↓) is suﬃcient to describe the system.

Each carbon atom is covalently bound to three neighboring atoms, with which it shares one electron forming sp2σ-bonds. The fourth valence electron occupies a pzorbital. The allowed range of energies of this electron deﬁnes the

bandstruc-ture. All pz orbitals can mix together, forming delocalized electron states that

determine the conductivity. The bandstructure can be found with a tight-binding approximation, in which only the nearest neighbor coupling of pz orbitals has to

be considered [1]. Instead of reproducing the mathematical derivation we will give the outcome and comment on some important consequences. We will use a graphical approach to discuss how the CNT bandstructure appears from the graphene bandstructure.

In Fig. 2.1b the bandstructure of graphene is given. The valence and conduction bands meet at six points at the Fermi energy. These points coincide with the corners of the hexagonal Brillouin zone, the Fermi surface is reduced to these six points. These points are called the charge neutrality points. We ﬁnd that graphene is a semimetal, a zero band gap semiconductor.

Electrons are deﬁned as excitations above the Fermi energy, holes are deﬁned as excitations below the Fermi energy. The dispersion relation near the charge degeneracy points is conical, as is shown in Fig. 2.1c. The Brillouin zone has two inequivalent points that are called K and K =−K (Fig. 2.1d). These points are sometimes referred to as valleys, or as isospin [3] and should not be confused with pseudospin [4].

The pseudospin describes the amplitude of the electronic wavefunction on the sublattice atoms A and B. It can be shown that a pseudospin can be deﬁned in such a way that it points parallel to the direction of propagation k for electrons near the K point, and antiparallel near the K point [4]. The pseudospin gives electrons in graphene a chirality, or “handedness”. States close to K correspond to right-handed electrons with pseudospin parallel to k and states close to K are left-handed, with pseudospin antiparallel to k. The pseudospin assignment is reversed for holes. One important implication of pseudospin is the suppression of backscattering. In 1D metallic CNTs backscattering corresponds to scattering from kx to −kx. It can be shown that this is forbidden, because the overlap

integral of two antiparallel pseudospins in the same valley is zero [4]. This is one reason for ballistic transport in metallic CNTs.

In graphene there is an extra dimension, and scattering can happen in 2D. This makes it harder to achieve ballistic transport in graphene. Intervalley scattering processes (from K to K) are however suppressed in graphene, because they

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2.1. Carbon nanotube quantum dots E

-

EF k_x k_y E k_|| k valence conduction B A x y a kx ky K’ K K’ K Δk=μμμμB/ƫννννF BII E cw ccw ccw A B C D E F kŏ k_II kŏ BII=0 B_II>0 kŏ μ μ μ μ II B cw EG

Figure 2.1: Electronic band structure of carbon nanotubes. (a) The unit

cell of graphene contains two carbon atoms (A and B), separated by a≈ 1.42 ˚A. (b) Bandstructure of graphene. Conduction and valence band meet at the K and Kpoints where the dispersion can be approximated by Dirac cones. (c) Wavevectors of electrons traveling around the CNT (k⊥) have periodic boundary conditions which results in quantization. This set of discrete states is depicted with red lines. Each line is a 1D subband. (d) In a small bandgap nanotube, the quantization lines almost pass through a K point. The oﬀset determines the bandgap Eg at B = 0 T. (e) When a magnetic

field is applied parallel to the CNT, an AB phase modifies the quantization conditions by a shift Δk. This decreases the bandgap associated with K electrons and increases the bandgap for Kelectrons. (f ) When contributions due to Zeeman splitting are ignored, each electronic state shifts in energy according to its orbital magnetic moment. For example, the level marked with a blue dot corresponds to a clockwise moving electron, its orbital magnetic moment is aligned parallel to the magnetic field. Figure and caption are adapted from Ref. [2].

require a change of momentum on the order of the reciprocal lattice vector. This can only be provided by short-range scatterers (defects) that act on the order of the spacing between sites A and B.

The states of electrons close to the charge neutrality points and with wavenumber

k= K +κ are described by the Dirac Hamiltonian:

H =vFσ · κ , (2.1)

here vF ≈ 106m/s is the Fermi velocity andσ are the Pauli matrices acting on

the pseudospin. The eigenvalues of the Dirac Hamiltonian are E = ±vF|κ|.

The eigenvectors of the Dirac Hamiltonian are the pseudospinors. At the charge neutrality points graphene has a linear dispersion relation, which means that the eﬀective mass of electrons/holes is zero. A consequence of this is that electrons in graphene always move at the same velocity. In this sense electrons in graphene

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behave as charged photons in free space, but with a velocity that is roughly 300 times smaller than the velocity of light.

Electron-wave quantization in a CNT

Due to its cylindrical shape electron waves propagate in spirals on the surface of a CNT. This results in two quantization conditions. The ﬁrst quantization condition is due to the phase accumulation along its circumference. This leads to a quantization of k_⊥, the wave number perpendicular to the CNT axial direction. The second quantization condition is due to the phase accumulation along the longitudinal direction and leads to a quantization of the wavenumber k. The quantized values of k_⊥are determined by:

πDk_⊥+ 2πΦ

Φ0

= 2πn . (2.2)

Here n is an integer, D is the diameter of the CNT, Φ0= h/e is the ﬂux quantum,

and Φ = BπD2/4 is the magnetic flux threading the CNT. The first term in this equation is the dynamical phase that the electron wave picks up by traveling a distance πD (moving around the CNT once). The second term is the Aharonov-Bohm (AB) phase that the electron wave picks up by encircling the magnetic flux [5].

Because the diameter of a CNT is very small (∼ 1 nm) compared to its length

(∼ 1 μm), electrons are free to move over much larger distances along the

longi-tudinal direction. This results in a strong quantization along the perpendicular direction and a weak quantization along the longitudinal direction. The electron wavenumber k is eﬀectively continuous on the scale of k_⊥. The continuum of these k states in each k_⊥ mode are called 1D subbands. In Fig. 2.1c,d the 1D subbands are indicated by the red lines. Of course subbands only appear if the quantization energy is above kBT .

There are always two quantization lines that are the nearest to a charge neutrality point, one at K and one at K. These lines correspond to k_⊥ and−k_⊥, indicat-ing that (in the absence of symmetry breakindicat-ing) electrons can move clockwise or anticlockwise at the same energy. This is referred to as valley-degeneracy. Intu-itively one can say that electrons in the K and Kvalleys encircle the CNT with opposing handedness. This is similar to the chirality of electrons in graphene, but here the picture is more intuitive.

Since each channel can be occupied with one electron with spin-up and one with spin-down, CNTs have four transport channels. Its maximum conductance Gm

is two times the conductance quantum, Gm= 4e2/h, Rm= h/4e2= 6.45 kΩ [6].

Symmetry breaking by magnetic ﬁeld

One way to break the symmetry is by applying a magnetic field. As is shown in Eq. (2.2) a magnetic field parallel to the CNT axis will change the quantization of k_⊥. Graphically this results in different spacing of the red lines in Fig. 2.1d. This can have different consequences for electrons close to the K and K points, as is shown in Fig. 2.1e. In this case the shift of k_⊥is positive at Kand negative at K. Since the dispersion relation is linear, we can write ΔE =vFΔk, which

yields the shift of energy that is due to the magnetic ﬁeld: ΔE = evFD

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2.1. Carbon nanotube quantum dots

Here we have deﬁned μorb≈ D [nm]0.2 meV/T as the orbital magnetic moment

that is associated with an electron encircling a CNT of diameter D. This shows that energy levels of states in a CNT shift when a magnetic ﬁeld is applied parallel to the CNT axis. The convention is that states that move up in energy are labeled counterclockwise (ccw) and states that move down in energy are labeled clockwise (cw). This is shown in Fig. 2.1f. Note that for typical CNTs μorbis much larger

than the Bohr magneton (μB≈ 0.058 meV/T).

Symmetry breaking by spin-orbit interaction

The four-fold degeneracy of quantized states in a CNT can be lifted by spin-orbit interaction. At zero ﬁeld spin-orbit interaction splits each orbital state into two sets of twofold degenerate states. Without going into details we will present the most important points and refer to Ref. [2] for a detailed discussion.

The atomic spin-orbit coupling of a free carbon atom is Δ ∼ 12 meV. This is relatively low, because carbon has a small mass. In graphene the intrinsic spin-orbit splitting close to the charge neutrality points is even smaller (∼ 1 μeV) due to symmetry reasons. In CNTs there is an enhanced spin-orbit interaction close to the charge neutrality points, due to their curvature. On a curved surface pz

orbitals are not aligned parallel with respect to each other, but are under an angle. An electron with spin-up in a tilted pz orbital on atom A can couple directly to

a px-up orbital on atom B (these are perpendicular to pzorbitals) and stay here

for a while before it is ﬂipped by the (weak) atomic spin-orbit interaction, to the pz-down orbital on atom B. This process depends on the curvature of the CNT

and results in a diameter depended spin-orbit interaction.

From a tight binding model it can be found that the curvature induced spin-orbit coupling is:

Δcurv∼ 1.6 meV/D [nm] . (2.4)

The spin-orbit interaction modiﬁes the circumferential quantization conditions and can be studied by tracking the magnetic ﬁeld dependence of states in a quantum dot.

Carbon nanotube bandgap

At zero ﬁeld the k_⊥ spacing is determined by the diameter of the CNT. At a particular spacing the quantization lines cut the Dirac cones exactly at the charge neutrality points. In this case the dispersion relation is linear and the CNT has no bandgap (similar to graphene). Such CNTs are called metallic. In practice curvature, strain or twist can induce a small bandgap on the order of 10 . . . 50 meV in metallic CNTs [3, 7–9].

When the quantization lines cut the Dirac cones not exactly at the charge neu-trality points the CNT is a semiconductor. It has a parabolic dispersion relation with a large bandgap Egthat depends on the diameter in the following way [10]:

Eg≈

0.8

D [nm]eV . (2.5)

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Electron-wave quantization by longitudinal conﬁnement

The 1D subbands are quantized by the longitudinal conﬁnement. This leads to the following boundary condition:

LkF

E

EF

= 2πn . (2.6)

From which a level spacing of ΔE =vFπ/L = hvF/2L is found. Using vF ≈

8.1· 105_{m/s we ﬁnd:}

ΔE≈ 1.67

L [μm]meV , (2.7)

for metallic nanotubes of length L. This approximation holds for metallic nan-otubes, in which the dispersion relation is (close to) linear. In semiconducting nanotubes the dispersion relation is parabolic, which results in a smaller level spacing [11]. Due to its large Fermi velocity, a typical CNT with a length on the order of a few 100 nm, can already have a level spacing that can be observed at temperatures of a few Kelvin. Numerous experimental studies of quantum dots in CNTs have been reported. As we already mentioned in Sec. 1.3.3, one of the main motivations for this research is to use the spin degree of freedom of isolated electrons as a resource for quantum computation [12].

Shell ﬁlling in CNT quantum dots

Shell ﬁlling is a characteristic feature of CNT quantum dots [13]. We will use it to experimentally identify the number of CNTs across a trench, and to determine the transport regime. In Fig. 2.2 we show shell ﬁlling measured by the Hongjie-Dai group at Stanford on devices that are very similar to ours [11].

In panel (a) the energy dispersion of a small bandgap CNT is shown. Due to the longitudinal conﬁnement, only discrete wavenumbers are allowed, that are indicated by the vertical dashed lines. Electrons and holes can occupy states that reside in small bands at the points where the vertical lines cross the parabolic dispersion. These bands are indicated by the short horizontal lines. The states in these bands are split by the charging energy Ueﬀ and the level spacing ΔE.

States in CNTs are fourfold degenerate due to the spin and valley degrees of freedom. The charging energy is the electrostatic energy that is needed to add a single electron to the CNT. A single orbital level, or shell, can contain four electrons. After the shell is filled (this requires ΔE + 4Ueff), an energy ΔE + Ueff

is needed to add the ﬁfth electron, this is shown in panel (b).

Shell filling of a quantum dot becomes apparent by measuring the conductance at small bias, as a function of gate voltage. By increase/decrease of gate voltage, electrons can be added/removed from the quantum dot. When an empty state becomes available (in the energy window that is set by the applied bias), electrons or holes can tunnel on/off the quantum dot from/to the leads. In panel (c) shell filling is shown, measured in four different CNTs, with different bandgap Eg. The

top curve is for a small bandgap device, the bottom curve for a large bandgap device.

The eﬀects of tunnel coupling and bandgap on transport are clearly visible. In the top curve the dot is in the strong coupling regime where Ueﬀ≈ 0, and transport

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2.1. Carbon nanotube quantum dots

the lower curve the dot is in the weak coupling regime, where Ueﬀ  0. In this

regime the charge on the dot is quantized and transport is dominated by Coulomb blockade. In this experiment the electrons in small bandgap nanotubes are weakly confined (small tunnel barrier) and in large bandgap nanotubes they are strongly confined (large tunnel barrier). In Sec. 2.1.2 we will discuss (in a simplified model) how the CNT bandgap determines tunnel coupling of a quantum dot to the leads. 50 40 30 -40 0 40 k(106 _m-1₎ E(mV) G(e 2/h) 0 2 4 CB FP Ueff Eg increasing 0 0.5 1 Vg(a.u.) Ueff+ǻn Ueff K’Ļ K’Ĺ KĻ KĹ (a) (b) (c)

Figure 2.2: Shell ﬁlling in CNT quantum dots. (a) Energy dispersion E(k)

of electrons/holes in a CNT quantum dot. The quantization of wavevectors in the longitudinal direction (kn = nπ/L) is indicated by the evenly spaced vertical lines.

Each kngives rise to a shell consisting of four states corresponding to K, K, spin-up

and spin-down. The shells are indicated by the horizontal lines. (b) Each shell can be ﬁlled with four electrons. For the addition of one electron a charging energy of Ueﬀ

is needed. When a shell is full, Ueﬀ+ ΔE is required to put the next electron on the

dot. (c) Shell filling is apparent in conductance vs. gate voltage traces. Here are traces shown from four different CNTs. This figure has been adapted from Ref. [11].

2.1.2 Tunnel barriers at the metal-nanotube interface

To achieve longitudinal quantization the wavefunction of electron-waves in the CNT has to be confined, or reflected, at barriers defining a nanotube segment. These barriers can be defined by local doping of the CNT with small gates or by the naturally occurring barrier at the interface of the nanotube with the contact metal [14]. In this section we will discuss the nature of the metal-nanotube interface, because this will be important in choosing the contact metal for our devices.

The size of the barriers determine the coupling of the quantum dot to the leads and the transport through it. The barrier width and height can be estimated by the Schottky barrier theory [15]. Important are: The diﬀerence in electron work function of nanotube and contact, the CNT bandgap, nanotube doping, Fermi level pinning and parasitic charge close to the interface. In the simple approximation Fermi level pinning and parasitic charge are neglected, which of course does not necessarily mean they cannot be important.